ECOLOGY, ECO LOGY, GENETICS, G EN ETICS, AND EVOLUTION AND EVOLUTION OF OF METAPOPULATIONS M ETAPOPULATIO N S
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COLOGY, CS, E ECO LOGY, GENETI G EN ETICS, AND EVOLUTION AND EVOLUTION OF OF METAPOPULATIONS M ETAPO PU LATIO N S
Edited Edited by by
IIkka Hanski Ilkka Hanski
Metapopulation Research Group Department of Ecology Ecology and Systematics University of Helsinki, Finland
Oscar Oscar E. E. Gaggiotti Gaggiotti
Metapopulation Research Group Department of Ecology Ecology and Systematics University of Helsinki, Finland Finland
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C ONTENTS CONTENTS
CONTRIBUTORS PREFACE PREFACE
ix
xiii
ACKNOWLEDGMENTS xv
PERSPECTIVES PERSPECTIVES ON ON SPATIAL SPATIAL DYNAMICS DYNAMICS 1 1
11
METAPOPULATION METAPOPULATION BIOLOGY: BIOLOGY PAST, PRESENT, PRESENT, AND FUTURE 3 3 ANDFUTURE by Ilkka Hanski and Oscar E.Gaggiotti E.Gaggiotti
2 2
METAPOPULATION METAPOPULATION DYNAMICS: DYNAMICS: PERSPECTIVES PERSPECTIVES FROM FROM LANDSCAPE LANDSCAPE ECOLOGY ECOLOGY
23 23
by Kimberly A. With With
3 3
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS FOR FOR POPULATION POPULATION DYNAMICS DYNAMICS
45 45
by by Benjamin M. M . Balker Bolker
II METAPOPULATION METAPOPULATION ECOLOGY ECOLOGY 4 4
71 71
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
73 73
by by Otso Otso Ovaskainen and Ilkka Ilkka Hanski v V
vi
CONTENTS CONTENTS
5 5
APPLICATION OF STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS TO REAL METAPOPULATIONS METAPOPULATIONS 105 105 by Rampal S. Etienne, Cajo].F. Caj0J.F. ter Braak, and Claire C. Vos by
6 6
FROM METAPOPULATIONS METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES FROM
133 133
by Mathew A. Leibold and Thomas E. Miller by
III METAPOPULATION METAPOPULATION GENETICS GENETICS
1151 51
7 SELECTION SELECTION AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS 153 153 7 by Michael C. C. Whitlock by 8 8
METAPOPULATIONS METAPOPULATIONS AND COALESCENT COALESCENT THEORY
175 175
by John Wakeley by John Wakeley 9 9
METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS: GENETICS: THE METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS OF POPULATION POPULATION
DIFFERENTIATION DIFFERENTIATION
199 199
by Charles Charles J. Goodnight by
Part IV
EVOLUTIONARY DYNAMICS DYNAMICS EVOLUTIONARY IN METAPOPULATIONS METAPOPULATIONS 225 225 IN 10 10
LIFE HISTORY HISTORY EVOLUTION EVOLUTION IN IN METAPOPULATIONS METAPOPULATIONS 227 227 LIFE by Ophelie Ophe'lie Ronce Rome and Isabelle Olivieri by
11 11
IN METAPOPULATIONS: METAPOPULATIONS: THE COEVOLUTION COEVOLUTION SELECTION IN SELECTION OF PHENOTYPE PHENOTYPE AND CONTEXT CONTEXT 259 259 by Michael JJ.. Wade Wade by
12 12
SPECIATION SPECIATION IN IN METAPOPULATIONS METAPOPULATIONS 275 275 by by Sergey Sergey Gavrilets Gavrilets
Partt V
INTEGRATION AND APPLICATIONS INTEGRATION APPLICATIONS 13 13
305 305
CAUSES, MECHANISMS MECHANISMS AND CONSEQUENCES CONSEQUENCES DISPERSAL 307 307 OF DISPERSAL by Jean Clobert, Clobert, Rolf Rolf Anker Ims, and Franfois FranGois Rousset by
CONTENTS CONTENTS
14 14
vii vII MECHANISMS MECHANISMS OF POPULATION POPULATION EXTINCTION EXTINCTION
by Oscar E. Gaggiotti and Ilkka Hanski
15 15
337 337
MULTILOCUS MULTILOCUS GENOTYPE GENOTYPE METHODS METHODS FOR FOR THE STUDY STUDY OF OF METAPOPULATION METAPOPULATION PROCESSES PROCESSES 367 367
by Oscar E. Gaggiotti
16 16
ECOLOGICAL AND EVOLUTIONARY OF ECOLOGICAL AND EVOLUTIONARY CONSEQUENCES CONSEQUENCES OF SOURCE-SINK SOURCE-SINK POPULATION POPULATION DYNAMICS DYNAMICS
by Tadeusz J. Kawecki
17 17
387 387
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS OF INFECTIOUS INFECTIOUS DISEASES DISEASES 415 415
by Matt J. Keeling, Bjornstad, and Bryan T. Keeling, Ottar N. Bj~rnstad, T. Grenfell
18 18
TOWARD A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR FOR PLANTS PLANTS 447 447 by N.J. Ouborg and O. Eriksson
19 19
LONG-TERM STUDY OF OF A A PLANT-PATHOGEN LONG-TERM STUDY PLANT-PATHOGEN METAPOPULATION METAPOPULATION
by Janis Antonovics
20 20
471 471
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS: ENVIRONMENTS" BUTTERFLY BUTTERFLYRESPONSES RESPONSES TO HABITAT HABITAT AND CLIMATE AND CLIMATE CHANGE CHANGE 489 489 by Chris Chris D. Thomas and Ilkka Ilkka Hanski
21 21
INFERRING AND PROCESS INFERRING PATTERN PATTERN AND PROCESS IN IN SMALL SMALL MAMMAL MAMMAL METAPOPULATIONS: METAPOPULATIONS" INSIGHTS INSIGHTS FROM FROM ECOLOGICAL ECOLOGICAL AND GENETIC AND GENETIC DATA DATA 515 515
by Xavier Lambin, Jon Aars, Aars, Stuart B. Piertney, Piertney, and Sandra Sandra Telfer
22 22
METAPOPULATION AND RESERVE METAPOPULATION DYNAMICS DYNAMICS AND RESERVE NETWORK NETWORK DESIGN DESIGN 541 541
by Mar Cabeza, Cabeza, Atte Moilanen, and Hugh P. P. Possingham
23 23
VIABILITY ANALYSIS ANALYSIS FOR VIABILITY FOR ENDANGERED ENDANGERED METAPOPULATIONS: METAPOPULATIONS" APPROXIMATION APPROACH APPROACH 565 A DIFFUSION DIFFUSION APPROXIMATION 565 by E.E. Holmes and B.X. B.X. Semmens REFERENCES REFERENCES 599 599
INDEX
683 683
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C ONTRIBUTORS CONTRIBUTORS
Numbers Numbers in parentheses parentheses indicate indicate the pages on which which the authors' authors" contributions contributions begin. Jon Jon Aars Aars (515) (515) NERC Molecular Genetics Genetics in Ecology Ecology Initiative, Aberdeen Aberdeen
Population Ecology Ecology Research Unit, School School of Biological Biological Sciences, Sciences, University University of Aberdeen, Aberdeen, Aberdeen Aberdeen AB24 AB24 2TZ, Scotland; Scotland; current address: address: Norwegian Polar Institute, Troms0, Norway Institute, Polarmilj0senteret, Polarmiljosenteret, N-9296 Tromso, Janis Antonovics (471) Biology Department, University of Virginia, Charlottesville, Virginia Virginia 22904 Ottar N. Bjornstad Bj0rnstad (415) Departments of Entomology and Biology, Biology, Penn State University, University Park, Pennsylvania Pennsylvania 16802 University, University Benjamin Zoology Department, University of Florida, Benjamin M. Bolker (45) (45) Zoology Florida, Gainesville, Florida 32611 J.F. ter Braak (105) Biometrics, Cajo J.F. Biometrics, Wageningen Wageningen University University and Research Centre, Box 100, NL-6700 AC Wageningen, The Netherlands Mar Mar Cabeza (541) Metapopulation Research Group, Department of Ecology Ecology and Systematics, Systematics, University University of Helsinki, FIN-00014 Helsinki, Finland Jean Clobert (307) Laboratoire d'Ecologie, Universite Universit~ Pierre Pierre et Marie Curie, Batiment B~timent A, 75252 75252 Parix cedex OS, 05, France O. Eriksson (447) Department of Botany, Botany, Stockholm University, University, SE-I06 SE-106 91 Stockholm, Sweden Rampal S. Etienne Etienne (105) (105) Community and Conservation Ecology Ecology Group, University University of Groningen, Box 14, NL-9750 AA Haren, The Netherlands Oscar E. Gaggiotti Gaggiotti (3, 337, 367) Genomique G~nomique de Populations Populations et Biodiversite, Biodiversit~, LECA-CNRS LECA-CNRS UMR 5553 Universite Universit~ Joseph Fourier, Fourier, F-38041 F-38041 Grenoble Cedex 9, France Sergey Sergey Gavrilets (275) (275) Department of Ecology Ecology and Evolutionary Biology, Biology, University of Tennessee, Department of Mathematics, University Tennessee, Knoxville, Knoxville, Tennessee Tennessee 37996 37996 Ix ix
CONTRIBUTORS CONTRIBUTORS
xx
Charles J. Goodnight (201) Department of Biology, Biology, University University of Vermont,
Marsh Life Life Sciences Sciences Building, Building, Burlington, Vermont 05405 05405
Bryan T. Grenfell Grenfell (415) Department of Zoology, Zoology, University University of Cambridge,
Downing Street, Street, Cambridge CB2 3EJ, 3EJ, England
I1kka 337, 489) Metapopulation Research Group, Department Ilkka Hanski (3, 73, 73,337,
of Ecology University of Helsinki, FIN-00014 Ecology and Systematics, Systematics, University FIN-00014 Helsinki, Finland E.E. Holmes Holmes (565) (565) National Marine Fisheries Fisheries Service, Service, Northwest Fisheries Fisheries Science Science Center, Center, Seattle, Seattle, Washington 98112 Rolf Anker Anker Ims (307) (307) Institute of Biology, Biology, University University of Troms0, Tromso, N-9037 Troms0, Tromso, Norway Tadeusz J. Kawecki Kawecki (387) Unit Unit for Ecology Ecology and Evolution, Department of Biology, University Biology, University of Fribourg, CH-1700 Fribourg, Switzerland Matt Matt J. Keeling (415) Maths Institute and Department of Biological Biological Sciences, Sciences, University AL, England University of Warwick, Coventry, Coventry, CV4 77AL, Xavier Lambin Aberdeen Population Ecology Lambin (515) Aberdeen Ecology Research Unit, Unit, School School of Biological Sciences, Sciences, University University of Aberdeen, Aberdeen, Aberdeen Aberdeen AB24 2TZ, 2TZ, Scotland Mathew Mathew A. Leibold Leibold (133) Department of Ecology Ecology and Evolution, University University of Chicago, Chicago, Illinois Illinois 60637 (133) Department of Biological Science, Florida State Thomas E. Miller (133) Biological Science, State University, University, Tallahassee, Florida 32306 Atte Atte Moilanen Moilanen (541) Metapopulation Research Research Group, Department of Ecology Ecology and Systematics, Helsinki, FIN-00014 Helsinki, Helsinki, Finland Systematics, University University of Helsinki, Isabelle (229) Institut des Sciences IsabeUe Olivieri (229) Sciences de l'Evolution UMR5554, Universite Universit~ Montpellier II, Place Place Eugene Eugene Bataillon, Bataillon, 34095 Montpellier cedex 5, France N.J. Ouborg (447) Department of Molecular Ecology, Ecology, University University of Nijmegen, Nijmegen, The Netherlands Toernooiveld 1, 6525 Ed Nijmegen, Otso Otso Ovaskainen (73) (73) Metapopulation Research Research Group, Department of Ecology Helsinki, Ecology and Systematics, Systematics, University University of Helsinki, Helsinki, FIN-00014 Helsinki, Finland Stuart B. Piertney Piertney (515) (515) NERC Molecular Genetics Genetics in Ecology Ecology Initiative, Aberdeen Population Population Ecology Ecology Research Unit, Unit, School School of Biological Biological Sciences, University University of Aberdeen, Aberdeen Aberdeen AB24 AB24 2TZ, Scotland Hugh Hugh P. P. Possingham (541) Departments of Zoology Zoology and Mathematics, The University University of Queensland, Queensland, St Lucia Lucia QLD 4072, 4072, Australia Ophelie (229) Institut des Sciences Oph~lie Ronce (229) Sciences de l'Evolution UMR5554, Universite Universit~ Montpellier II, Place Place Eugene Eugene Bataillon, Bataillon, 34095 34095 Montpellier cedex 5, France Fran�ois Francois Rousset Rousset (307) (307) Institut des Sciences Sciences de l'Evolution, l'Evolution, UMR5554 UMR5554 Universite Universit~ Montpellier, 34095 Montpellier cedex cedex 5, France B. Semmens Semmens (565) (565) Zoology Department, University University of Washington, Seattle, Washington 98195 98195 Sandra Animal Infectious Disease Sandra Telfer Telfer (515) (515) Small Small Animal Disease Group, Leahurst, University University of Liverpool, Neston CH64 7TE, 7TE, England Chris Chris D. Thomas Thomas (489) Department of Biology, Biology, University University of Leeds, Leeds, Leeds Leeds LS2 9JT, England Claire C. Vos (105) (105) Alterra Green Green World Research, Research, Box 47, NL-6700 NL-6700 AA Wageningen, The Netherlands
CONTRIBUTORS CONTRIBUTORS
Michael J.
xl xi
Wade (259) Department of Biology, Biology, Indiana University, Bloomington, Indiana 47405 47405 John John Wakeley Wakeley (175) Biological Biological Laboratories, Harvard University, University, Cambridge, Massachusetts Massachusetts 02138 02138 Michael Michael C. Whitlock (153) (153) Department of Zoology, University University of British Columbia, Vancouver, Vancouver, BC V6T 1 1 Z4, Z4, Canada Kimberly A. With With (23) (23) Division of Biology, Biology, Kansas State University, Manhattan, Kansas Kansas 66506 66506
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CE PREFA PREFACE
Over past 15 years, metapopulation metapopulation biology has Over the the past has developed from from a set of ideas, simple models, models, and and a limited number number of case studies studies to to an an essential part of population population biology. Some areas population biology continue to part areas of meta metapopulation continue to flourish flourish with bold new new visions and and attempts attempts to to clarify them with models, models, but but other areas areas have already become consolidated consolidated into into a solid body of theory and and have have been thoroughly investigated empirically. Progress Progress has has been so great great that that contents of this this volume bear the contents bear only superficial resemblance resemblance to to the contents contents Metapopulation Biology of the predecessor, predecessor, Metapopulation Biology (Hanski (Hanski and and Gilpin, Gilpin, 1997), 1997), to to say Metapopulation Dynamics nothing nothing about about the the first edited edited volume in this series, Metapopulation Dynamics (Gilpin and Hanski, 1991). and Hanski, 1991 ). In this volume volume we have achieved, for the first time, an equal equal coverage of metapopulation metapopulation ecology, meta population genetics, metapopulation genetics, and and evolutionary meta population biology. There is no parity, however. metapopulation biology. There no complete parity, however. Metapopulation which was at the stage of conceptual Metapopulation ecology, which conceptual development and turned to and budding budding empirical empirical studies studies 15 years ago, has has by now now turned to a well wellestablished established discipline with with substantial substantial impact on on practical practical conservation. conservation. In contrast, metapopulation meta population genetic and contrast, and evolutionary studies studies are at an an earlier stage, stage, with with less well-developed integration integration of theoretical and and empirical work. work. But such integration integration is undoubtedly undoubtedly coming, coming, and and it is hoped hoped that that this volume will stimulate further further development in this direction. chapters in this volume are entirely new, All the chapters new, nothing nothing has has been copied copied from from Metapopulation Metapopulation Biology Biology.. The The previous previous volume includes contributions contributions that are well worth that worth reading reading even today, today, but but we we did not not include them them here in the the interest of giving space space to to a new set of authors authors and and chapters, chapters, and and also because because the previous volume is still available. available. One One important important similarity remains. This is an remains. This an edited edited volume volume in which which we have not not forced forced the same app approach roach in treatment of the subject matter matter in all the the chapters. Some chapters are primarily or or even entirely theoretical, theoretical, whereas whereas others others are based based on on empirical research. present an metapopulation research. Some chapters chapters present an overview of one slice of metapopulation xlii xiii
xxlv lv
PREFACE PREFACE
biology, whereas others are focused more narrowly on new developments. There are up-to-date reviews of all areas of metapopulation biology. We are confident that there is not a single population biologist in this world world who would find nothing new in this volume, nor are there many who would find all the chapters easy bed-time reading. reading. But we trust that that most of our readers will appreciate the diversity and the challenge, and will be inspired by at least some of the visions, comprehensive empirical studies, and modeling efforts described in this volume. volume. Our Our aim was to produce a volume that serves serves both both as a reference for researchers and as a text for advanced students in ecology, genetics, evolutionary biology, and conservation biology. The emphasis is on integration across disciplines. disciplines. Several chapters are relevant for conservation conservationists in setting the stage for new applications. It is hoped that graduate students will find material in this volume for innovative Ph.D. Ph.D. projects. We are grateful to a large number of colleagues who provided truly helpful Araujo, Frederic Austerlitz, Hans reviews of particular particular chapters: Miguel Arafijo, Baveco, Peter Beerli, Beerli, Thomas Thomas Berendonck, Ben Bolker, Bolker, Cajo ter Braak, Mark Mark Burgman, Burgman, Jeremy Burdon, Burdon, Dennis Couvet, Michael Michael Doebeli, Stephen Ellner, Rampal Rampal Etienne, Patrick Foley, Foley, Robert Freckleton, Sylvain Gandon, Gandon, Gisela Garcia, Nicholas Gotelli, Mikko Mikko Heino, Jessica Hellmann, Eric Imbert, Imbert, Rolf Ims, Par P~r K. K. Ingvarsson, Kevin Kevin Laland, Xavier Lambin, Russ Lande, Martin Martin Lascoux, Richard Law, Michel Lareau, Loreau, Michael McCarthy, Juha Merila, Meril~i, Atte Moilanen, Moilanen, Allen J. Moore, Moore, Isabelle Olivieri, Olivieri, Otso Otso Ovaskainen, Ovaskainen, John Pannell, Craig Primmer, Jonathan Pritchard, Pritchard, Chris Ray, Ray, Steven Riley, Riley, Ilik Saccheri, Saccheri, Mikko Jonathan Silvertown, Peter Smouse, Mikko J. Sillanpaa, Sillanp~i~i, Jonathan Smouse, Per Sj6gren-Gulve, Chris Thomas, Thomas, Xavier Vekemans, Jana Jana Verboom, and Franjo Weissing. We thank thank Marjo Marjo Saastamoinen and Tapio Gustafsson for indispensable secretarial help. Chuck Crumly from Academic Press had trust in this volume from our very first correspondence, correspondence, and Kelly Kelly Sonnack, Angela Dooley, Michael Sugarman and and Eric DeCicco at Academic Press made our task as editors as easy as possible. Finally, our thanks to all the authors authors for showing great enthu enthusiasm and keeping deadlines. deadlines. Ilkka Hanski Oscar Oscar Gaggiotti April 2003, Helsinki
A CKNOWLEDG MENTS ACKNOWLEDGMENTS
CHAPTER CHAPTER 1 1
We thank thank Rolf Ims and Chris Thomas for comments on the chapter. Supported by the Academy of Finland (Centre of Excellence Programme 2000-2005). 2000-2005).
CHAPTER CHAPTER 2 2
I thank thank Hans Baveco, Ilkka Hanski, Greg Schrott, Per Sjogren-Gulve, Sj6gren-Gulve, and Jana Verboom for their comments on the chapter. chapter. My research on the effects of landscape structure and and dynamics on extinction risk for spatially structured populations populations has been supported by past grants from the National Science Science Foundation and, most recently, by the u.S. U.S. Environmental Protection Agency (R829090).
CHAPTER CHAPTER 3 3
I thank thank the Isaac Newton Newton Institute for supporting a workshop on scaling in biological systems where some of these ideas were developed and Toshinori Okuyama and Graeme Cumming for useful discussions.
CHAPTER CHAPTER 44
We thank thank Ben Bolker, Bolker, Cajo van ter Braak, Rampal Etienne, and Karin Karin Frank for comments on the chapter. chapter. Supported by the Academy of Finland (Centre of Excellence Programme Programme 2000-2005). 2000-2005). XV xv
Part I Perspectives on Spatial Dynamics
sdfsdf
1
META PO PULATION M ETAPO PU LATIO N BIOLOGY: BIOLOGY: PAST, PAST! PRESENT, AND FUTURE PRESENT, AND FUTURE Ilkka Ilkka Hanski Hanski and and Oscar Oscar E. Gaggiotti
11.1 .1
INTRODUCTION INTRODUCTION The The term term metapopulation metapopulation stems stems from from the the general general notion notion of of the the hierarchical hierarchical structure structure of of nature. nature. Just Just like like the the term term population population is is needed needed to to describe describe an an assemblage assemblage of of interacting interacting individuals, individuals, it it seems seems apt apt to to have have aa term term for for an an assem assemblage blage of of spatially spatially delimited delimited local local populations populations that that are are coupled coupled by by some some degree degree of 970) . It of migration m i g r a t i o n- the the metapopulation metapopulation (Levins, (Levins, 11970). It is is conceptually conceptually attract attractive, and explicitly consider ive, and helpful helpful for for the the study study of of population population biology, biology, to to explicitly consider the the sequence of entities from populations. sequence of entities from individuals individuals to to local local populations populations to to meta metapopulations. Theoretical Theoretical studies studies are are greatly greatly facilitated facilitated by by the the view view of of landscapes landscapes as as networks networks of inhabited by local populations. just theory: of habitat habitat patches patches inhabited by local populations. And And it it is is not not just theory: there there are are innumerable innumerable species species that that definitely definitely have have such such aa spatial spatial population population structure structure in in some some landscapes, landscapes, and and continuing continuing habitat habitat loss loss and and fragmentation fragmentation force species to population structure. force ever ever greater greater numbers numbers of of species to conform conform to to aa meta metapopulation structure. Other continuous spatial spatial distributions distributions in Other species species have have more more continuous in less less distinctly distinctly patchy some purposes patchy environments, environments, but but even even for for these these species species and and for for some purposes the the meta population view metapopulation view of of nature nature can can be be helpful. helpful. A metapopulation metapopulation approach approach refers refers to to research research or or management management that, that, in in one one form form or or another, another, adopts adopts the the view view that that local local populations, populations, which which the the metapopula metapopulations (or relatively tions consist consist of, of, are are discrete discrete (or relatively discrete) discrete) entities entities in in space space and and that that these these local populations population local populations interact interact via via migration migration and and gene gene flow. flow. Classic Classic meta metapopulation
Ecology, Ecology, Genetics, Genetics, and Evolution of Metapopulations
3 3
Copyright 2004, Elsevier, Elsevier,Inc. 0-12-323448-4
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ILKKA ILKKA HANSKI HANSKIAND AND OSCAR OSCAR E. E. GAGGIOTII GAGGIOTTI
dynamics 1969, 11970) 970) focus dynamics in in the the sense sense pioneered pioneered by by Levins Levins ((1969, focus on on the the processes processes of local local extinction extinction and and recolonization recolonization in in the the same same manner as as population population dynam dynamof ics are are concerned concerned with with births births and and deaths deaths of of individuals. individuals. However, However, such such popula populaics tion turnover turnover is is not not aa necessary necessary condition condition for for the the meta metapopulation approach to to tion population approach be be useful, useful, nor nor aa characteristic characteristic feature feature of of all all species species that that are are structured, structured, in in some some landscapes, landscapes, into into spatially spatially discontinuous discontinuous local local populations. populations. Important Important questions questions need to be asked asked about the the interaction interaction of permanent local populations, for context of source-sink source-sink dynamics (Chapter (Chapter 116). instance in the context 6). Metapopulation biology represents one one way of of explicitly explicitly putting population biology biology into into aa spatial spatial context. context. The The basic basic tenet tenet of of spatial spatialecology, ecology,which which includes includes metapopulation ecology as well as other approximations approximations (see (see alter) alter),, is that the individuals and populations matter, in the sense of influ influspatial positions of individuals encing the growth populations and growth rate and dynamics of populations and meta metapopulations their competitive, predator-prey and other interactions. Likewise, we may use the term spatial spatial population population biology biology to emphasize the influence of the spatial positions of individuals and populations on their genetic and evolutionary dynamics as well as their ecological dynamics. That That spatial spatial positions positions matter matter is is aa trivial trivial observation observation for for biologists biologists working working on plants and other sessile organisms. Thus Harper ((1977) 1 977) entitled one of the Population Biology of Plants as "The effects of neigh neighfive main sections of his Population bours. bours."" It has been less obvious that spatial positions of individuals matter in the case of mobile animals, which may form more or less less random-mating (panmictic) populations. However, from the point of view of of ecological ecological interactions, spatial positions often do matter even in mobile animals. One example example is is the the large large number number of of insect insect species species with with mobile mobile adults adults but but immobile immobile larval stages. Larvae do most of the interactions and so the spatial distribution of larvae matters to single-species single-species (de (de long, 979), competitive of larvae matters greatly greatly to Jong, 11979), competitive ((Hanski, Hanski, 11981, 9 8 1 , 11990a), 990a), and and predator-prey predator-prey dynamics (Hassell, 11978, 978, 2000). 2000). dynamics (Hassell, Indeed, from the 1970s 1 970s onward, indi onward, the spatial aggregation of interacting individuals has been one of the most important important themes in population dynamics. within-population spatial structures also have evolutionary evolutionary These types of within-population ( 1970), Boorman Boorman and and consequences, which which have been investigated by Levins (1970), ( 1 973 ), Cohen and Eshel (1976), ( 1 976) , Wilson Wilson (Wilson, 1980; 1 980; Wilson aI., Levitt (1973), Wilson et al., 1 992; Mitteldorf Mitteldorf and population 1992; and Wilson, 2000), and others. Interestingly, the the population genetic modeling populations initiated by Wright modeling of continuously continuously distributed distributed populations ( 1 940, 1943, 1 943, 1946) 1 946) and and Malecot Malecot (1948) ( 1 94 8 ) faced faced difficulties difficulties precisely precisely because because of of (1940, the spatial spatial aggregation of of individuals (Felsenstein, (Felsenstein, 1975). 1 975). Much Much progress progress the has been made made in this area in the last decade decade using Monte Monte Carlo Carlo simulations, simulations, has autocorrelation methods, methods, and and lattice lattice models (Eppenson and and Allard, Allard, spatial autocorrelation 1989, 1 989, 1993a,b, 1 993a,b, 1995; 1 995; Rousset, Rousset, 2000). 2000). Taking the population population structure structure in in which reproduction reproduction is panmictic panmictic but but Taking ecological interactions interactions are are localized, localized, as described earlier, as the the starting starting point, point, ecological there are are two two ways ways of of moving moving to to the the domain domain of of metapopulation metapopulation dynamics. dynamics. First, First, there widespread dispersal may may not not occur occur in in every every generation, generation, in which which case patches patches widespread of microhabitat microhabitat harbor harbor not not just just single-generation assemblages assemblages of of interacting interacting of individuals, but but multigeneration multigeneration local local populations. populations. Insects Insects living in decaying individuals, wood provide provide good good examples, examples, ranging ranging from from those those that that disperse disperse completely completely in in wood each each generation generation to to species species that that form form local local populations populations in in particular particular (large) (large) trunks for for tens tens or or even even hundreds hundreds of of generations generations (Fig. 1.1). 1 . 1 ) . The The decisive factor factor is trunks
11.. METAPOPULATION METAPOPULATIONBIOLOGY BIOLOGY
5 5
.1 Oak woodland in Sweden population Fig. 11.1 Sweden where a long-term long,term study has examined the meta metapopulation biology of the beetle Osmoderma Osmodermoeremita, eremita, with long-lasting local local populations populations inhabiting individual oak trees trees (Ranius, (Ranius, 2000; Ranius Ranius and Jansson, Jansson, 2000). Photograph by Jonas Jonas Hedin.
simply simply the the longevity longevity of of the the microhabitat microhabitat in in relation relation to to the the life life span span of of individu individuals, underscoring the more generally valid point that that metapopulation dynamics are are typically typically determined determined as as much, much, or or more, more, by by the the structure structure and and dynamics dynamics of of the the physical environment as by the properties of the species. In the population population genet genetics literature, the sort of situation represented by insect populations inhabiting long-lasting microhabitats microhabitats has been examined under under the rubric of the haystack model (Maynard Smith, 11964; 964; Bulmer and Taylor, 11981). 981). The population domain The second second way way of of moving moving ttoo the the meta metapopulation domain from from panmictic panmictic local populations is simply by expanding the spatial scale: most organisms have limited dispersal powers, hence there is a spatial scale at which most inter interactions, including mating, occur "within populations," populations," whereas at larger spatial scales, scales, these these local local populations populations are are connected connected by by migration migration and and gene gene flow. flow. It It is is especially population approach especially natural natural to to turn turn to to the the meta metapopulation approach if if the the environment environment is is physically fragmented into pieces of habitat habitat that that may support support local populations. Metapopulation Metapopulation biology recognizes that that many, if not not most, ecological, genetic, and evolutionary processes occur at spatial scales that that are greater greater than than the scale within within which which most individuals disperse. Hence there is spatial structure at the meta population scale to still metapopulation scale that that should should not not be be ignored. ignored. Moving Moving to still larger larger spatial spatial scales, to the geographical ranges of species, brings in other processes that that are beyond population concept beyond the the meta metapopulation concept and and domain. domain. We population processes We emphasize emphasize the the significance significance of of meta metapopulation processes rather rather than than spatial structures. It is tempting to attempt attempt to classify different kinds of spatial population 994), and some terminology is needed population structures (Harrison, (Harrison, 1991, 11994), for communication, communication, but the danger is that that we impose an order order to nature nature that that is not not there. Landscapes are all different, hence there must be a huge diversity of "metapopulation - migration, gene "metapopulation structures." structures." Focusing Focusing on on the the processes processes--migration, gene
6 6
ILKKA HANSKI OSCAR E. ILKKA HANSKI AND AND OSCAR E. GAGGIOTTI GAGGIOTTI
flow, flow, spatially spatially correlated correlated dynamics, dynamics, local local extinction, extinction, genetic genetic drift, drift, local local adapta adaptation, tion, and and so so forth forth ~ circumvents circumvents the the need need to to infer infer processes processes from from patterns patterns where necessary ((in in many cases there where this this is is not not necessary many cases there is, is, however, however, valuable valuable informa information in patterns that should not ignored; see tion in patterns that should not be be ignored; see Wiegand Wiegand et et aI., al., 2003). 2003). By By emphasizing emphasizing the the metapopulation metapopulation approach, approach, we we also also underscore underscore the the point point that that this .3). this is is only only one one approach approach and and not not always always the the most most appropriate appropriate one one (Section (Section 11.3). There There is is little little doubt doubt that that spatially spatially localized localized interactions interactions and and movements movements influ influence genetic, and majority of ence the the ecological, ecological, genetic, and evolutionary evolutionary dynamics dynamics of of the the vast vast majority of species. particular approach species. It It is is another another question question which which particular approach is is the the most most effective effective in in uncovering the interactions and uncovering the biological biological consequences consequences of of spatially spatially localized localized interactions and movements for research and movements for both both research and management. management.
11.2 .2
METAPOPULATION IN THE METAPOPULATION BIOLOGY: BIOLOGY: PAST PAST TRENDS TRENDS IN THE LITERATURE LITERATURE The population biology The history history of of research research in in meta metapopulation biology has has been been narrated narrated by by Hanski 1 997) and 1999b). Rather Hanski and and Simberloff Simberloff ((1997) and Hanski Hanski ((1999b). Rather than than repeating repeating it it here, examine that here, we we will will examine that history history in in light light of of the the number number of of citations citations to to relevant relevant key key words. words. Such Such aa systematically systematically "documented "documented history" history" of of metapopulation metapopulation biol biology 970s. We ogy goes goes back back to to the the 11970s. We used used the the BIOSIS BIOSIS database, database, which which yielded yielded 1087 1087 citations citations to to the the key key word word metapopulation metapopulationin in the the title title of of aa paper paper or or in in its its abstract abstract (years 970-2001 ) . To (years 11970-2001). To get get aa fair fair idea idea of of the the temporal temporal patterns patterns in in the the number number of of citations, pooled number citations, we we divided divided the the yearly yearly totals totals by by the the pooled number of of citations citations in in the the database total volume volume of database in in that that year, year, aa measure measure of of the the total of the the literature. literature. Thus measured, the number of citations to Thus measured, the number of citations to metapopulation metapopulation has has increased increased more less linearly linearly since 990 (Fig. (Fig. 11.2), .2), with more or or less since 11990 with only only aa few few earlier earlier citations, citations, even even if already in 970 (Levins, if the the metapopulation metapopulation concept concept itself itself was was introduced introduced already in 11970 (Levins, 11970). 970). Some Some inaccuracy inaccuracy is is due due to to less less thorough thorough coverage coverage of of the the literature literature in in the the database database in in the the 1970s 1970s than than later later on, on, but but this this does does not not change change the the broad broad picture. picture. One 20-yr time One can can think think about about several several reasons reasons for for the the 20-yr time lag lag in in the the wider wider use use of of the the metapopulation metapopulation concept, concept, which which is is in in sharp sharp contrast contrast to to the the early early success success of of the island biogeographic 1963, 1967), the island biogeographic theory theory of of MacArthur MacArthur and and Wilson Wilson ((1963, 1967), pub published 1 969, 11970) 970) metapopulation lished only only aa few few years years prior prior to to Levins's Levins's ((1969, metapopulation idea idea and and model 996). First, model (Hanski, (Hanski, 11996). First, MacArthur MacArthur and and Wilson Wilson published published their their theory theory in in leading journal for population population biology and as a high-profile monograph, a leading whereas papers were published in illustrious journals. whereas Levins's Levins's papers were published in less less illustrious journals. Second, Second, MacArthur MacArthur and and Wilson Wilson were were purposely purposely in in the the business business of of turning turning aa page page in in the the history 1 969) immediate immediate goal history of of biogeography, biogeography, whereas whereas Levins's Levins's ((1969) goal was was more more modest, to model to examine alternative alternative strategies eradica modest, to construct construct aa model to examine strategies of of pest pest eradication. Third, Third, MacArthur MacArthur and Wilson were widely respected scientists, whereas Levins was was aa hero hero for for aa more more limited limited number number of of people. people. Fourth, Fourth, and and what what may may Levins be be really really important, important, the the island island theory theory became became associated associated with with the the species-area species-area relationship, enhancing the ecologists could relationship, enhancing the theory'S theory's popularity popularity because because ecologists could use use it it in research (whether made aa lasting in their their research (whether this this application application of of the the theory theory made lasting contri contribution similar opportunity bution is is another another matter). matter). There There was was no no similar opportunity to to do do empirical empirical work linked with models work that that would would be be similarly similarly linked with Levins's Levins's models ~ aa situation situation that that was 990s with was to to change change only only in in the the 11990s with further further development development of of the the theory theory ((Section Section 1.3). awareness of biological conse 1.3). Finally, Finally, the the heightened heightened awareness of the the dire dire biological consequences 980s onward quences of of habitat habitat loss loss and and fragmentation fragmentation from from the the late late 11980s onward has has
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Fig. .2 Number Number of of citations citations in in the the BIOSIS database to the the key words words indicated indicated in in the the panels panels Fig. 11.2 BIOSIS database divided of citations citations in a particular particular year year (to (to control control for divided by by the the total total number number of for the the increasing increasing total total volume of of literature literature over the years). Note Note that that the the scale on the the vertical vertical axis is different different in different different volume rows of panels. See text for discussion. rows of text for
practically forced interest in metapopulation practically forced an an interest metapopulation biology, making making the the rediscovrediscov ery of of Levins's early work work inevitable. inevitable. The top top row row in Fig. 1.2 1 .2 gives the the number of citations to the the key words land The number of citations to words landand island island biogeograph* hiogeograph* as well as to scape ecology ecology and scape to the the key word word metapopulametapopula as tion ( biogeograph" includes all words words starting starting with with "biogeograph," "biogeograph," such such as tion (biogeograph* "biogeography" "biogeography" and and "biogeographic"). "biogeographic"). The The temporal temporal patterns patterns show show intriguing intriguing differences. Landscape ecology ecology was was established established in the the literature literature in the the beginning beginning differences. Landscape of of the the period period considered, considered, in in 1970, 1 970, but but for for the the next next 15 1 5 years years the the frequency frequency of of citations citations remained remained at at a constantly constantly low low level. A distinct distinct growth growth phase phase began began around around 1985, 1 985, and and definitely definitely earlier earlier than than in the the case case of of metapopulation. metapopulation. At At present, metapopulation is cited cited somewhat somewhat more more frequently frequently than than landscape landscape ecolecol present, metapopulation ogy. Island Island biogeograph* hiogeograph * has has appeared appeared in in the the literature literature since since the the mid-1970s mid-1 970s and and ogy. the the frequency frequency has has remained remained high high until until the the present, present, with with ups ups and and downs. downs. Perhaps Perhaps
8 8
ILKKA AND OSCAR OSCAR E. ILKKA HANSKI HANSK! AND E. GAGGIOTII GAGGIOTTI
surprisingly, surprisingly, the the standardized standardized number number of of citations citations to to island island biogeograph'� biogeograph* was was higher in in 2001 2001 than than ever ever before before m 34 34 years years since since the the classic classic monograph monograph by by higher MacArthur 1967) established MacArthur and and Wilson Wilson ((1967) established the the modern modern era era in in ecological ecological biogeography. biogeography. It It is is noteworthy noteworthy that that the the peaks peaks in in the the time time series series for for landscape landscape 980s, ecology ecology and and island island biogeograph': biogeograph*' agree agree rather rather closely closely since since the the late late 11980s, suggesting suggesting that that many many papers papers refer refer to to both both key key words. words. Next examined combinations including metapopulation Next we we examined combinations of of key key words, words, including metapopulation and something else. second row Fig. 11.2 .2 compares and something else. The The second row in in Fig. compares the the three three subdiscip subdiscip* , and * , genetic * . In cases, the lines lines ecolog ecolog*, genetic*, and evolution evolution*. In all all cases, the first first papers papers were were published 978 and and 1985. Most of these papers listed in in all published in in 11978 1985. Most of these papers were were in in fact fact listed all three Gill's ((1978a,b) 1 978a,b) papers papers on three searches searches and and include include Gill's on the the metapopulation metapopulation ecol ecology ogy of of the the red-spotted red-spotted newt, newt, its its migration migration rate, rate, and and effective effective population population size; size; Couvet 1 985) study Couvet et et al.'s al.'s ((1985) study on on the the population population genetics genetics in in spatially spatially structured structured populations; Fix's ((1985) 1985) theoretical populations; and and Fix's theoretical study study of of the the evolution evolution of of altruism. altruism. Since 990, ecolog has accumulated Since 11990, ecolog** has accumulated many many more more citations citations than than genetic genetic** or or * . The evolution appear to evolution*. The temporal temporal patterns patterns appear to indicate indicate that that while while ecolog ecolog** has has not been growing 994, genetic* not been growing systematically systematically since since 11994, genetic* has has been been growing growing until until the 990s and the late late 11990s and the the number number of of citations citations to to evolution" evolution': appears appears still still to to be be growing. These These trends trends are growing. are consistent consistent with with our our general general perception perception of of shifting shifting research research interests, interests, as as well well as as with with the the change change in in the the contents contents of of the the three three volumes on biology (Gilpin volumes on metapopulation metapopulation biology (Gilpin and and Hanski, Hanski, 1991; 1991; Hanski Hanski and and Gilpin, 11997; 997; present Gilpin, present volume) volume).. A A somewhat somewhat different different interpretation interpretation of of the the fig figures for 997 ures for ecolog* ecolog* associates associates the the peak peak in in the the number number of of citations citations in in 1996-1 1996-1997 to population volume, to the the publication publication of of the the previous previous meta metapopulation volume, which which appeared appeared in in the 996 (Hanski 997). In the year year 11996 (Hanski and and Gilpin, Gilpin, 11997). In any any case, case, it it is is apparent apparent that that the the has increased again since 996. number number of of citations citations to to ecolog': ecolog*' has increased again since 11996. The .2 gives The next next row row in in Fig. Fig. 11.2 gives some some further further comparisons. comparisons. Theory Theory has has main maintained tained its its position position well well over over the the years years (key (key word word model), model), although although evidence evidence also also indicates indicates that that empirical empirical work work has has been been catching catching up up to to theoretical theoretical studies studies in in recent recent years. years. This This is is shown shown by by aa significant significant declining declining trend trend in in the the ratio ratio of of cita cita990 tions tions to to metapopulation metapopulation + model model over over metapopulation metapopulation (yearly (yearly counts counts for for 11990 until 990 count until 2001, 2001, the the 11990 count also also including including all all the the previous previous papers; papers; linear linear regres regression, F course, many sion, F= = 7.76, 7.76, P P = 0.02). 0.02). Of Of course, many of of the the papers papers referring referring to to model model might might not not be be theoretical theoretical papers, papers, and and part part of of the the continuing continuing increase increase in in model model papers population stud papers is is due due to to an an increase increase in in genetic genetic and and evolutionary evolutionary meta metapopulation studies. Conservation ies. Conservation combined combined with with metapopulation metapopulation has has increased increased steadily steadily for for the the past 995-1996, paralleling past decade, decade, with with the the exception exception of of aa striking striking peak peak in in 11995-1996, paralleling (although peak for (although not not exactly exactly matching) matching) the the corresponding corresponding peak for ecolog ecolog*. The very very " . The low frequency frequency of of citations citations to to metapopulation metapopulation + landscape landscape ecology ecology is is not not sur surlow prising disciplines that prising in in the the light light of of the the continuing continuing separation separation of of these these two two disciplines that seemingly (more about seemingly have have so so much much in in common common (more about this this in in the the next next section). section). Let Let us number of citations to us hope hope that that the the relatively relatively large large number of citations to metapopulation metapopulation + + land landscape scape ecology ecology scored scored for for 2001 2001 represents represents the the beginning beginning of of aa new new era! era! Finally, .2 examines Finally, the the last last row row in in Fig. Fig. 11.2 examines three three taxa, taxa, plants, plants, fishes, fishes, and and but butterflies, terflies, all all of of which which show show the the same same increasing increasing trend trend as as metapopulation metapopulation itself. itself. The number of citations to The pooled pooled number of citations to metapopulation metapopulation + "taxon" "taxon" for for the the years years 1996 22; mammal, 1996 to to 2000 2000 is is as as follows follows for for the the following following taxa: taxa: bird, bird, 22; mammal, 85; 85; fish, fish, 38; butterfly, 38; butterfly, 49; 49; and and plant, plant, 94. 94. These These overall overall figures figures are are somewhat somewhat misleading, misleading, however. however. For For instance, instance, there there are are many many more more "hard "hard core" core" metapopulation metapopulation
11.. METAPOPULATION METAPOPULATION BIOLOGY BIOLOGY
9 9
papers on papers on butterflies butterflies than than on on birds birds and and mammals, mammals, undoubtedly undoubtedly because because the the metapopulation many butterflies metapopulation approach approach is particularly particularly applicable applicable to many butterflies (Chapter 20; Hanski, Hanski, 11999, 999, Ehrlich Hanski, 2004). (Chapter 20; Ehrlich and and Hanski, 2004). This This is is also also reflected reflected in in the database for .2. the type type of of the the very very first first papers papers in in the the database for the the taxa taxa shown shown in in Fig. Fig. 11.2. For 1 988) on For butterflies butterflies the the pioneering pioneering study study is is Harrison Harrison et et al. al. ((1988) on the the mainland mainlandisland population structure island meta metapopulation structure in in the the Bay Bay checkerspot checkerspot butterfly butterfly (Euphydryas (Euphydryas editha) in California, fishes and and plants editha) in California, whereas whereas for for fishes plants the the first first papers papers are, are, respect respectively, Hanzelova and 1 992) essentially ively, Hanzelova and Spakulova's Spakulova's ((1992) essentially biometric biometric study study and and Ellstrand et al.'s 1 984) notion Ellstrand et al.'s ((1984) notion of of an an inflorescence inflorescence as as aa metapopulation. metapopulation.
11.3 .3
AN OVERVIEW OVERVIEW OF CURRENT RESEARCH RESEARCH AN OF CURRENT This This section section outlines outlines some some noteworthy noteworthy recent recent developments developments in in metapopula metapopulation tion ecology, genetics, and and evolutionary evolutionary studies studies as well as their their integration. integration. This This section section refers refers extensively extensively to to the the remaining remaining chapters chapters in in this this volume. volume. Although Although the motivation motivation for for research typically stems from past past scientific dis discoveries coveries and and perceived perceived opportunities opportunities for for further further discoveries, discoveries, the the ongoing ongoing loss, loss, alteration, alteration, and and fragmentation fragmentation of of natural natural habitats habitats are are widely widely viewed viewed as as other other important reasons for population biology. important reasons for conducting conducting research research in in meta metapopulation biology.
Ecology Ecology The population approach The meta metapopulation approach is is conceptually conceptually closely closely related related to to the the dynamic dynamic theory island biogeography 1967). Most theory of of island biogeography of of MacArthur MacArthur and and Wilson Wilson ((1967). Most import importantly, islands antly, both both theories theories advocate advocate the the same same "island "island perspective," perspective," whether whether the the islands habitat islands, and both both theories are concerned with local are true islands or habitat extinctions extinctions and and recolonizations, recolonizations, although although this this is is not not an an exclusive exclusive interest interest in in meta population biology, metapopulation biology, as as pointed pointed out out earlier. earlier. The The apparent apparent difference difference in in the the focus population theories focus of of the the island island theory theory on on communities communities and and of of meta metapopulation theories on on single assumes independindepend single species species is is not not aa fundamental fundamental difference, difference, as as long long as as one one assumes ent (as the ent dynamics dynamics in in the the species species that that comprise comprise the the community community (as the basic basic island island model model does). does). The The similarity similarity between between the the island island biogeographic biogeographic model model and and the the classic population model pop classic meta metapopulation model is is underscored underscored by by the the spatially spatially realistic realistic meta metapopulation a; Hanski ulation theory theory (Hanski, (Hanski, 2001 2001a; Hanski and and Ovaskainen, Ovaskainen, 2003; 2003; Chapter Chapter 4; 4; see see later), later), which which adds adds the the effects effects of of habitat habitat patch patch area area and and isolation isolation on on extinctions extinctions and classic meta population theory. and colonizations colonizations into into the the classic metapopulation theory. In In fact, fact, we we can can now now see population model see that that Levins's Levins's meta metapopulation model and and MacArthur MacArthur and and Wilson's Wilson's island island two special cases of a more comprehensive comprehensive model (Hanski, 2001a). model are two One One advantage advantage of of the the metapopulation metapopulation theory theory over over the the island island theory theory is is that that the the former former but not the latter allows each species to have its own patch network network in the the same same landscape, landscape, reflecting reflecting differences differences in in the the habitat habitat selection selection of of the the species. species. In any case, it is intriguing that population theory that the island theory theory and meta metapopulation have been widely have been widely considered considered as as representing representing two two different different paradigms paradigms in in conser conservation (see discussion 997). vation biology biology (see discussion in in Hanski Hanski and and Simberloff, Simberloff, 11997). The island theory and and metapopulation metapopulation theory are are not not the only approaches approaches to spatial ecology. . 3 gives simple classification spatial ecology. Figure Figure 11.3 gives aa simple classification of of three three main main approaches. approaches. The The key key issue issue is is what what is is assumed assumed about about the the structure structure of of the the environment. environment. In In one one extreme, extreme, labeled labeled as as the the theoretical theoretical ecology ecology approach, approach, the the
1100
ILKKA OSCAR E. ILKKA HANSKI HANSKI AND OSCAR E. GAGGIOTTI GAGGIOTTI Metapopulation ecology
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common assumption assumption is that the environment is completely homogeneous. Here the primary aim of research is to elucidate the consequences consequences of spatially restricted interactions and/or and/or migration of individuals to the dynamics and spatial structures of populations. Chapter 3 describes at length this approach approach to spatial ecology. The mathematical tools commonly employed include lattice latticebased models, such as interacting particle systems, cellular automata automata and coupled-map coupled-map lattices, spatial moment equations, and partial differential equa equations, as well as simulations. Recent work on "neutral" "neutral" theories of community structure ((Bell, Bell, 2000; Hubbell, 200 1 ) also fit in this category, although these 2001) Chapter 6). The models deal with evolutionary as well as ecological dynamics ((Chapter assumption assumption of homogeneous space facilitates the study of population population processes as opposed to the heterogeneous heterogeneous landscape in creating and maintaining spatial variation in population densities, but this assumption assumption also practically Chapter 3, eliminates the possibility of testing model predictions. As suggested in Chapter the models studied by theoretical ecologists are strategic models designed to investigate general principles rather than tactical models designed to answer specific questions about about specific populations. populations. Nonetheless, even the general theory has to be related to the real world. It is hence important important that that recent modeling studies in this framework framework have attempted to relax the assumption of homogeneous homogeneous space. For instance, Murrell and Law (2000) have used the method method of moments to model the dynamics of carabid beetles in heterogeneous
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landscapes with three different classes of land type, woodland, woodland, agricultural land, and urban areas, and Keeling (2000b) has applied the method of moments to single-species and predator-prey predator-prey dynamics in coupled local populations in a meta population (for further metapopulation further discussion, see Chapter 33).) . IInn the other extreme depicted iinn Fig. 11.3, .3, which iiss represented bbyy much of landscape ecology, the starting point is just the opposite, a detailed description of of the the often often complex structure structure of of real real landscapes. landscapes. Chapter Chapter 22 presents presents an an overview of landscape ecology as far as it is concerned with population processes. processes. Given Given the the complex complex description description of of the the landscape landscape structure structure and and the the emphasis on individual movements in much of landscape ecology (Schippers et aI., 996; Pither and Taylor, 11998; 998; Haddad, 11999a; 999a; Bunn et aI., al., 11996; al., 2000; Jop.sen Jopsen and Taylor, 2000; Byers, 200 1 ), it is not surprising that 2001), that the prevalent modeling tool has been individual-based simulation (With and Crist, 11995; 995; With With and King, 11999b; 999b; Hill and Caswell, 11999; 999; Fahrig, 2002; Chapter .3, Chapter 2). As seen from Fig. 11.3, we we view view the the metapopulation metapopulation approach approach as as occupying occupying the the middle middle ground ground in in this this classification: the environment is assumed to consist of discrete patches of suit suitable habitat for the focal species, usually ignoring the shape of these patches, sur surrounded by the landscape matrix that that is not suitable for reproduction reproduction but through through which individuals may migrate. These assumptions can be somewhat somewhat relaxed population the relaxed without without compromising compromising the the possibility possibility of of developing developing meta metapopulation theory. For example, one may allow for matrix heterogeneity by calculating effec effective patch connectivities, and one may replace real patch areas by effective areas allowing for spatial variation in habitat patch quality (Moilanen and Hanski, 11998; 998; Hanski, 11999b). 999b). What What still remains intact is the core assumption of dis discrete local populations inhabiting discrete patches of habitat. metapopulation perIn terms of theory in meta population ecology, our admittedly partial per spective spective inclines inclines us us to to emphasize emphasize the the significance significance of of the the spatially spatially realistic realistic metapopulation metapopulation theory theory (SMT). The core mathematical mathematical models in this theory are are stochastic stochastic patch patch occupancy occupancy models models (SPOM). (SPOM). SPOMs assume assume aa network network of of habitat habitat patches, which which have only two two possible states, occupied by the focal species or empty. If there are n n patches in the network, the metapopulation metapopulation has 2 2 nn possible states, which is such a large number for large n n that that a rigorous mathematical analysis is not not possible and some simplification is called for. One simplification is to assume a homogeneous homogeneous SPOM, SPOM, with identical habitat habitat patches, which allows a rigorous analysis of even the stochastic stochastic model (this is the the familiar familiar "island "island model"). model"). Another Another simplification simplification is is to to resort resort to to determin deterministic models that that ignore spatial correlations in the pattern pattern of patch patch occupancy and variability due to a finite number number of patches in the network. network. The Levins model model makes makes both both simplifying simplifying assumptions assumptions at at the the same same time time ~ it it is is aa deter deterministic approximation approximation of a homogeneous homogeneous SPOM. What What we now now know know is that that rigorous just one rigorous theory theory can can be be constructed constructed by by making making just one of of the the simplifying simplifying assumptions. SMT is obtained by combining a heterogeneous SPOM, in which heterogeneous SPOM, patches patches have have different different extinction extinction and and colonization colonization probabilities, probabilities, with with assump assumptions tions as as to to how how the the structure structure of of the the landscape landscape influences influences these these probabilities probabilities (or (or rates in the case of continuous-time continuous-time models). Chapter 4 describes SPOMs and the theory in the spatially spatially realistic realistic metapopulation metapopulation theory in detail. detail. The population theory The spatially spatially realistic realistic meta metapopulation theory makes makes aa contribution contribution toward toward aa unification of research in population population biology in several fronts (Hanski and Ovaskainen, 2003). First, as already pointed out, the island theory and the
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classic metapopulation a). metapopulation theory are two two special cases of SMT (Hanski, 2001 2001a). Second, SMT contributes to to the unification of metapopulation metapopulation ecology and landscape landscape ecology with with its explicit focus on the influence of the structural fea features tures of of the the landscape on on population processes. processes. As As Chapter Chapter 2 2 shows, shows, some some land landscape scape ecologists ecologists have have worked worked toward toward the the same same goal goal from from their their own own tradition. tradition. These developments suggest that the merging of the two fields of metapopulametapopula Third, SMT tion ecology and landscape ecology is finally starting to take place. Third, is mathematically closely related to matrix population models (Caswell, 1) (Caswell, 200 2001) for age-structured and size-structured populations, which have also been employed in the study of source-sink metapopulations Chapter 16). Fourth, metapopulations ((Chapter SMT shares common theoretical underpinnings with with epidemiological theory (Grenfell and Harwood, 997; Ovaskainen and Grenfell, 2003). Fifth, a great Harwood, 11997; advantage of the models stemming from from SMT is that that they can be parameterized rigorously with data on the dynamics and pattern pattern of habitat patch occupancy. Chapter 5 presents a review of the methods of parameter estimation and how the models can be applied to real metapopulations. metapopulations. Chapter 22 employs SMT to combine spatial dynamics with reserve site selection algorithms to incorporate the concept of population persistence into reserve selection procedures. The close linking of theory to empirical research that SMT facilitates is somewhat analogous to the link between the dynamic theory of island biogeography and empirical research on the species-area relationship in the 1970s. The difference, however, is that population models can be parameterized rigorously with that meta metapopulation with empirical data, whereas just documenting the species-area relationship is not sufficient to parameterize, nor to test, the island biogeographic model. The rea reason for the success of the meta population models in this respect is that metapopulation that they are typically applied to meta populations with metapopulations with many and often small local popula populations with with a measurable rate of population turnover. Data available hence relate to spatial dynamics as well as to the consequent spatial patterns patterns of habitat habitat occu occupancy. This is in contrast with with past research on the island theory and species-area relationship, which was largely restricted, due to a low rate of popu population turnover on large islands, to analyses of spatial patterns rather than of processes. Spatially realistic meta population theory is focused on the actual spatial metapopulation structure of metapopulations, metapopulations, in the sense of specifying the probabilities with which particular habitat patches in a fragmented landscape are occupied. Another class of structured meta population models considers the distribution metapopulation of local population sizes but ignores the actual spatial structure by assuming that all local populations are equally equaUy connected (Hanski, 11985; 985; Hastings and Wolin, 11989; 989; Hastings, 1991; GyUenberg 992; GyUenberg Gyllenberg and Hanski, 11992; Gyllenberg et aI., al., 11997). 997). These models are particularly concerned with the the influence of emigra emigration and immigration on local dynamics in the meta population context and are, metapopulation in this respect, akin to source-sink models ((Chapter Chapter 116). 6) . The aforementioned modeling studies assume an infinite number of local populations with deter deterministic local dynamics. Lande et al. ((1998)developed 1 998) developed another class of models structured by local population populations with stochastic population size for finite meta metapopulations local dynamics. The most interesting new phenomenon phenomenon predicted by population size-structured meta population models is the possibility of alternative stable metapopulation equilibria in meta population size, one of which corresponds to meta population metapopulation metapopulation extinction, the other one to a positive and possibly large meta population size metapopulation
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(Hanski, 985; Gyllenberg 992; Hanski 993). (Hanski, 11985; Gyllenberg and and Hanski, Hanski, 11992; Hanski and and Gyllenberg, Gyllenberg, 11993). The The processes processes that that lead lead to to alternative alternative stable stable equilibria equilibria are are the the rescue rescue effect, effect, extinction due to immigration, immigration, and the Allee effect, which reduced rate of local extinction increases per capita increases the the rate rate of of successful successful colonization colonization per capita with with increasing increasing immi immigration gration rate. rate. These These processes processes can can be be added added to to SPOMs SPOMs only only in in aa nonmechanis nonmechanistic manner Hanski, 2001 manner (Ovaskainen and Hanski, 2001),) , as SPOMs are concerned concerned with habitat habitat patch occupancy, not not with numbers numbers of individuals. Another great advantage population models models structured advantage of the meta metapopulation structured by the actual size size of local populations populations is the possibility to extend extend the analysis to evolutionary evolutionary issues. For instance, 1 ), and instance, Ronce Ronce et et al. al. (2000b), (2000b), Metz Metz and and Gyllenberg Gyllenberg (200 (2001), and Gyllenberg Gyllenberg et al. (2002) population size et al. (2002) studied studied the the evolution evolution of of migration migration rate rate with with population sizestructured structured metapopulation metapopulation models models [see [see also Heino Heino and Hanski Hanski (2001 (2001)) for for a spatially model and spatially realistic realistic model and Chapter Chapter 1100 for for comprehensive comprehensive discussion]. discussion]. Ecological population dynamics Ecological models models of of meta metapopulation dynamics tend tend to to make make simple simple assumptions assumptions about about migration. migration. Emigration Emigration is is typically typically assumed assumed to to be be density density independent, and individuals are independent, and migrating migrating individuals are assumed assumed to to follow follow aa correlated correlated random about the random walk walk or or some some less less mechanistic mechanistic simple simple assumption assumption is is made made about the behavior behavior of migrants. Chapter Chapter 1133 presents presents a thorough thorough review of what what is known known about about migration migration at the level of individual individual behavior. Not Not surprisingly, there no strong the simple in most most models. there is is no strong support support for for the simple assumptions assumptions made made in models. contrast, migration migration is seen as a complex complex behavior behavior involving a series of In contrast, decisions decisions that that often often depend depend on on the the state state (condition) (condition) of of individuals individuals and and their their interactions interactions with with other other individuals. individuals. In In particular, particular, migration migration is is often often density density dependent, dependence is dependent, although although both both positive positive and and negative negative density density dependence is commonly reported Chapter 113). 3 ) . Positively commonly reported ((Chapter Positively density-dependent density-dependent emigration emigration and negatively density-dependent density-dependent immigration immigration are expected to enhance enhance the and growth population, increasing growth rate rate of of the the meta metapopulation, increasing the the range range of of conditions conditions under under which Saether et aI., 999). These effects occur which the metapopulation metapopulation is viable ((Saether al., 11999). occur because because the the pattern pattern of of migration migration will will influence influence the the strength strength of of the the rescue rescue effect effect and the probability brief, it is clear that and probability of successful colonization. colonization. In brief, that migrants migrants in behavior than assumed by most in most most species species have have more more sophisticated sophisticated behavior than assumed by most models. would it models. What What is is not not clear, clear, however, however, is is when when would it be be necessary necessary to to (greatly) (greatly) complicate complicate the the models models by by including including many many behavioral behavioral details, details, and and indeed indeed to to what what extent extent should should the the models models be be modified. modified. Turning Turning from from rigorous rigorous mathe mathematical just for matical models models to to simulations simulations just for the the sake sake of of adding adding some some "realism" "realism" is is not not necessarily necessarily warranted. warranted. What What is is needed needed is is aa family family of of models models incorporating incorporating different different amounts amounts of of detail. detail. No No systematic systematic study study of of this this type type has has yet yet been been conducted conducted on migration migration and metapopulation metapopulation dynamics. dynamics. Ecologists Ecologists working working with with population population viability viability analysis analysis tend tend to to prefer prefer individ individual-based (Possingham and Noble, 1991; Ak�akaya Ferson, 11992; 992; Lacy, Akqakaya and Ferson, 11993, 993, 2000; 2000; Ak�akaya, Akqakaya, 2000a) 2000a) or or population-based population-based (Sjogren-Gulve (Sj6gren-Gulve and and Ray, Ray, 11996) 996) simulation these models that any simulation models. models. The The advantage advantage of of these models is is that any processes processes and and mechanisms mechanisms that that the the researcher researcher may may wish wish to to add add to to the the model model can can be be added added readily. readily. The The disadvantage disadvantage is is that that general general insights insights are are difficult difficult to to extract extract from from complex simulations. Furthermore, it is practically impossible to estimate rigor rigorously ously the the often often large large number number of of parameters parameters and and to to test test the the structural structural model model assumptions; assumptions; the the modeling modeling results results are are thus thus of of questionable questionable value value for for manage management. ment. The The best best use use of of these these models, models, as as perhaps perhaps of of any any population population models, models, for for conservation that differ conservation and and management management is is to to contrast contrast alternative alternative scenarios scenarios that differ in in
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only a small number of factors (Hanski, 11997a; 997a; Ralls and Taylor, 997; Beissinger Taylor, 11997; and Westphal, 11998; 998; Ak�akaya Akqakaya and Sjogren-Gulve, Sj6gren-Gulve, 2000). One would hope that the result of such comparisons is relatively insensitive to the many uncertainties uncertainties in parameter values and even even in the structure of the model itself. itself. The interested interested reader is referred to many chapters in two edited volumes (Sjogren-Gulve (Sj6gren-Gulve and Ebenhard, 2000; Beissinger 1 ) . Our emphasis in this vol Beissinger and McCullough, 200 2001). volume is on SPOMs for the reasons that much progress has been made in recent years in developing both the theory ((Chapter Chapter 4) and applications to real metapopulations ((Chapters Chapters 5, 20, and 22; see also Dreschler aI., 2003). Dreschler et al., To To present present aa balanced balanced view view about about the the standing standing of of the the metapopulation metapopulation approach in ecology, it is appropriate to acknowledge the critical opinions that have been voiced about its general significance. Harrison ((1991, 1 99 1 , 11994; 994; Harrison, 11994; Hastings and Harrison, 994; Harrison and Taylor, 11997; 997; Harrison and Bruna, 11999) 999) has suggested repeatedly that that the occurrence of species "in the balance between the extinction and recolonization of populations is an improbable 994, p. 15 ) . To improbable condition" condition" (Harrison, (Harrison, 11994, p. 1115). To some some extent, extent, Harrison's Harrison's population theory, which concerns are answered by the spatially realistic meta metapopulation relaxes many of the simplifying assumptions of the nonspatial homogeneous patch occupancy models, such as the Levins model, and which shows how realistic variation in habitat patch areas and connectivities can be incorporated into models. Another line of response is provided by the scores of empirical studies that that demonstrate the operation, in practice, of metapopulation dynam dynamics with a frequent turnover that lack large and turnover of local populations in systems that permanent "mainland" "mainland" populations. Chapter 20 assesses the performance of the metapopulations metapopulations approach approach in dynamic (nonequilibrium) landscapes, where Harrison's criticisms initially seem most relevant. In fact, the models perform well in the situations examined and can be used to gain valuable insights about the long-term behavior of metapopulations. metapopulations. Research on European butterflies, in particular, has produced much empir empirical evidence for metapopulation processes in shaping not only the ecologiecologi cal dynamics (Thomas, 1994b; Thomas and Hanski, 11997; 997; Hanski, 11999b; 999b; CD. Saccheri et aI., 998; Nieminen C.D. Thomas et aI., al., 2002), but but also genetic ((Saccheri al., 11998; et al., aI., 2001; Scmitt and Seitz, 2002) and evolutionary dynamics (Kuussaari et aI., al., 1 ; Heino and Hanski, 2001 2000; Hanski and Singer, Singer, 200 2001; 2001;; Thomas et aI., al., 2001; al., 2002) of butterflies. Chapter 20 in this volume and a volume 200 1 ; Hill et aI., checkerspot on the biology of checkers pot butterflies (Ehrlich and Hanski, 2004) present two overviews covering much of this research. Butterflies possess several traits that make them a convenient model group of species for metapopulation research: specific host plant and habitat requirements, meaning that that many landscapes are highly fragmented for butterflies; small body size size allowing the presence of local breeding populations in relatively small habitat patches; and high population growth rate but also great sensitivity to environmental condi conditions, leading to high population turnover 990). Additional turnover (Murphy et aI., al., 11990). advantages that that butterflies offer include the facility of estimating population sizes and migration rates with mark-release-recapture mark-release-recapture methods methods and the often great between the and the matrix. It may great distinction distinction between the suitable suitable habitat habitat and the landscape landscape matrix. It may remain a matter of opinion as to how representative, and representative of what, the many butterfly studies are, but minimally we expect that butterflies fairly represent a large number of specialized insect species.
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Chapters 8 , 119, 9, and 21 discuss plant, plant-pathogen, Chapters 118, plant-pathogen, and small mammal metapopulation metapopulation dynamics, respectively. The metapopulation metapopulation dynamics of many plants are influenced by the seed bank and very long-lived adult indi individuals, which complicate empirical empirical studies studies greatly but also mean that that certain phenomena, phenomena, such as long transients in the dynamics of species in changing environments, Chapter 118). 8 ) . The basic issue environments, are especially important important in plants ((Chapter habitat is often often very difficult in the case of of delimiting suitable but occupied habitat plants because plant species typically compete for space and hence a single singlespecies approach approach is likely to be inadequate. Chapter Chapter 1199 on plant-pathogen plant-pathogen metapopulation metapopulation dynamics is focused on a two-species interaction, interaction, which add additionally involves a coupling of ecological and evolutionary dynamics that that may be responsible population sizes. Metapopulation responsible for long-term trends trends in meta metapopulation Metapopulation dynamics in small mammals ((Chapter Chapter 2 1 ) may also often 21) often involve more than than one species, for instance, a specialist predator predator driving some of the population population turnover turnover in the prey species, potentially potentially leading to spatially correlated correlated patterns patterns of habitat occupancy. More More generally, there is a clear need for more studies on meta communities metacommunities m assemblages of interacting metapopulations. metapopulations. Chapter 6 reviews the conceptual conceptual framework framework and current research on metacommunities. metacommunities. Not Not surprisingly, webs of direct direct and indirect indirect interactions interactions in communities, com combined with with webs of spatially connected connected populations, populations, complicate complicate matters greatly, and we may need several different theoretical frameworks frameworks to cover the full range of possibilities that that arise in metacommunity metacommunity dynamics. Returning Returning to the criticism against the general significance of the metapopu metapopuapproach, Fahrig ((1997, 2001,, 2002) has suggested repeatedly lation approach, 1 997, 11998, 998, 2001 that the persistence of species in (increasingly) fragmented landscapes is little affected by habitat fragmentation as such, but rather what what matters is the total area of the (remaining) habitat. In other words, in Fahrig's opinion, opinion, the spatial configuration of the habitat makes little difference. If this were generally the case, much of the contents contents of this volume would would be superfluous. superfluous. We, however, consider that that Fahrig's conclusions are too far-fetched. Considering the plane depicted .4, defined depicted in in Fig. Fig. 11.4, defined by by the the proportion proportion of of the the suitable suitable habitat habitat in in the the landscape landscape and the migration range of the focal species, habitat fragmentation may indeed be of little significance in most parts of this plane. However, a huge number of species/landscape combinations combinations crowd the lower-left corner of Fig. 11.4: .4: highly fragmented landscapes, landscapes, in which which only a small fraction of the total total area is covered covered by the suitable habitat; habitat; and relatively poorly dispersing species at the scale of interest. Furthermore, Furthermore, as we all know, know, human-caused human-caused habitat loss and fragmentation continuously push further combinations combinations of species and landscapes .4, where landscapes to this corner corner in Fig. 11.4, where the spatial configura configuration of the remaining habitat should not be ignored. The metapopulation metapopulation theory ((Chapter Chapter 4) is helpful in delineating the parts of the shaded square in Fig. 11.4 .4 that allow long-term metapopulation metapopulation persistence from those parts that that lead to meta population extinction. metapopulation
Genetics Genetics Metapopulation Metapopulation genetic studies have their roots in Sewall Wright's island model of population 93 1 ), which assumes distinct local population structure structure (Wright, 11931), populations (colonies, demes) connected by migration and gene flow. In this
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0/
Total amount of habitat in the landscape
Fig. 11.4 .4 Habitat fragmentation (the spatial configuration Fig. fragmentation (the configuration of the remaining remaining habitat) habitat) matters on the the abundance in landscapes where the suitable habitat on abundance and persistence persistence of species in landscapes where habitat covers only a small fraction fraction of the total landscape area and and the migration the focal species species is migration range of the limited.
classic model, model, all all local populations populations are identical identical (same (same size) classic size) and equally connected (constant migration rate), which are also the assumptions assumptions of the ecological Levins model (a (a deterministic approximation approximation of a homogeneous homogeneous SPOM, see SPOM, see earlier earlier discussion) discussion).. However, However, while while the the latter latter was was focused focused on on popu population turnover, Wright's island model assumed permanent permanent local populations. populations. The The first first formal formal application application of of the the classic classic ecological ecological metapopulation metapopulation concept concept in the domain domain of population population genetics was a generalization of the island model to to cover cover the the case case where where local local populations populations would would go go extinct extinct and and new new ones ones were were established Slatkin, 11977). 977). The ideas to established ((Slatkin, The extension extension of of these these ideas to stepping-stone stepping-stone models followed shortly afterward (Maruyama and Kimura, 1980). The pion pioneering 1 977) was eering work work of of Slatkin Slatkin ((1977) was followed followed by by studies studies by by Wade Wade and and McCauley 1 9 8 8 ) and McCauley ((1990). 1 990). The McCauley ((1988) and Whitlock Whitlock and and McCauley The aim aim of of all all these these investigations was extinctions and investigations was to to clarify clarify the the effects effects that that extinctions and recolonizations recolonizations have populations, that have on on the the genetic genetic structure structure of of meta metapopulations, that is, is, the the partitioning partitioning of of genetic variability between local like in genetic variability within within and and between local populations. populations. Just Just like in classic classic ecological ecological metapopulation metapopulation models, models, the the effect effect of of local local dynamics dynamics on on genetic genetic structure structure was was ignored ignored to to facilitate facilitate the the study study of of factors factors such such as as the the extinction extinction rate individuals that rate and and the the genetic genetic composition composition of of the the groups groups of of individuals that establish establish new populations. populations. The population genetic The effect effect of of local local dynamics dynamics on on meta metapopulation genetic structure structure has has been been addressed 1992a), Gaggiotti addressed in in aa series series of of papers papers published published by by Whitlock Whitlock ((1992a), Gaggiotti and and Smouse 1 996), Gaggiotti 1 996), and 1 997). These Smouse ((1996), Gaggiotti ((1996), and Ingvarsson Ingvarsson ((1997). These studies studies demonstrate interaction between local dynamics demonstrate that that the the interaction between local dynamics and and migration migration pat patterns consequences for terns can can have have important important consequences for the the genetic genetic structure structure of of metapopu metapopulations. In metapopulations of populations lations. In metapopulations of the the Levins Levins type, type, with with all all local local populations having population size having the the same same carrying carrying capacity, capacity, fluctuations fluctuations in in local local population size and/or and/or migration migration rate rate increase increase genetic genetic differentiation differentiation among among populations populations (Whitlock, (Whitlock, 11992a). 992a). Slow population growth similar effect Slow population growth following following colonization colonization has has aa similar effect when migration rate 997). In when the the migration rate is is constant constant (Ingvarsson, (Ingvarsson, 11997). In the the case case of of
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source-sink populations, the source-sink meta metapopulations, the degree degree of of genetic genetic differentiation differentiation among among sources sources and and sinks, sinks, among among sinks, sinks, and and the the level level of of genetic genetic variability variability maintained maintained by by sink sink populations populations is is largely largely determined determined by by the the variance variance in in propagule propagule size. size. The lower the variance, the higher higher the degree of genetic differentiation differentiation and the lower Gaggiotti lower the the level level of of genetic genetic variability variability maintained maintained by by sink sink populations populations ((Gaggiotti and 996; Gaggiotti, 996). and Smouse, Smouse, 11996; Gaggiotti, 11996). All All the the theoretical theoretical studies studies mentioned mentioned so so far far have have been been concerned concerned with with the the genetic structure neutral genes, most thoroughly genetic structure of of selectively selectively neutral genes, which which is is the the most thoroughly studied population genetics. studied subject subject in in meta metapopulation genetics. Following Following the the publication publication of of comprehensive 1 997) and Rousset ((1999a,b), 1999a,b), comprehensive analyses by Whitlock Whitlock and Barton ((1997) which which extended extended the the results results of of the the previous previous studies studies to to models models that that cover cover aa wide wide variety population scenarios, scenarios, theoretical theoretical research variety of of meta metapopulation research into into the the genetic genetic structure diminished. Presently, structure of of metapopulations metapopulations has has diminished. Presently, the the most most active active area area in in meta population genetics is concerned concerned with selected genes and metapopulation and quantitative quantitative genetic Chapters 77 and genetic variation variation ((Chapters and 9). 9). This This recent recent work work has has added added important important new processes such heterosis, and new processes such as as inbreeding, inbreeding, heterosis, and mutation mutation accumulation accumulation into population approach into the the meta metapopulation approach and and its its application application to to conservation conservation and and management of endangered endangered species (see Chapter Chapter 7). To some extent, extent, further further management advance area is lack of advance in in this this area is hampered hampered by by the the substantial substantial lack of knowledge knowledge that that exists spontaneous mutations exists about about the the rates rates and and effects effects of of spontaneous mutations (discussed (discussed in in Chapter point where Chapter 14). 14). Indeed, Indeed, the the fact fact that that we we have have reached reached aa point where further further progress area of meta population biology progress in in aa specific specific area of metapopulation biology requires requires the the resolution resolution of a fundamental fundamental issue in such an established established discipline as genetics is an indication indication of of how how fast fast the the field field has has progressed. progressed. Studies classic quantitative Studies reviewed reviewed in in Chapter Chapter 99 have have extended extended classic quantitative genetics genetics theory populations. The theory to to meta metapopulations. The classic classic theory theory was was concerned concerned with with measur measuring response to selection and ing the the response to selection and largely largely ignored ignored epistatic epistatic interactions, interactions, whereas population quantitative whereas the the more more recent recent meta metapopulation quantitative genetics genetics theory theory is is concerned concerned with with measuring measuring differentiation differentiation among among populations populations and and empha emphasizes Chapter 99).) . This sizes the the importance importance of of epistatic epistatic interactions interactions ((Chapter This shift shift in in emphasis emphasis has has uncovered uncovered new new mechanisms mechanisms for for speciation speciation and and is is aa good good example populations can example of of how how aa focus focus on on meta metapopulations can shed shed new new light light onto onto key key evolutionary problems. evolutionary problems. Another population biology Another important important recent recent development development in in meta metapopulation biology is is the the extension of the coalescent approach 982a; reviewed by Fu and Li, approach (Kingman, 11982a; 11997) 997) to to cover cover metapopulation metapopulation scenarios scenarios (Wakeley (Wakeley and and Aliacar, Aliacar, 2001 2001).). The The coalescent coalescent approach approach represented represented aa big big leap leap forward forward for for population population genetics genetics because about past because it it provides provides aa theoretical theoretical framework framework to to make make inferences inferences about past events sample representing events based based on on aa genetic genetic sample representing the the present present population. population. The The essence theory is move back essence of of the the coalescent coalescent theory is to to start start with with aa sample sample and and to to move backward ward in in time time to to identify identify events events that that occurred occurred in in the the past past since since the the most most recent recent common common ancestor ancestor of of the the sample. sample. Chapter Chapter 88 provides provides an an overview overview of of the the coa coalescent population context lescent process process in in the the meta metapopulation context and and describes describes ways ways in in which which it it can can be be used used to to make make statistical statistical inferences. inferences. Although Although current current work work in in this this area area is lead to is highly highly theoretical, theoretical, it it will will lead to useful useful applications applications such such as as the the development development of of statistical statistical approaches approaches for for the the analysis analysis of of molecular molecular data data aimed aimed at at making making inferences inferences about about metapopulation metapopulation processes. processes. This This in in turn turn will will facilitate facilitate the the inte integration gration of of theoretical theoretical and and empirical empirical work work as as well well as as the the demographic demographic and and genetic population biology. genetic approaches approaches to to meta metapopulation biology.
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Evolution Evolution Application of the meta population approach metapopulation approach in the domain of evolutionary biology broad issues: issues: the biology has has been been motivated motivated mainly mainly by by three three broad the shifting shifting balance balance theory SBT) (Wright, 9 3 1 , 11940), 940), the theory ((SBT) (Wright, 11931, the evolution evolution of of migration migration rate, rate, and and the the evolution of species' ranges. An important controversy in evolutionary biology e.g., Coyne et aI., 997, 2000; deals with two opposing views of adaptation ((e.g., al., 11997, Wade and Goodnight, 11998; 998; Goodnight and Wade, 2000) 2000).. One view, view, called the Fisherian view, view, advocates that the bulk of adaptive evolution results from Darwinian mass selection. The other view maintains that adaptation cannot be explained and that such as as genetic explained by by selection selection alone alone and that stochastic stochastic processes processes such genetic drift drift often has been been the often play play an an important important role. role. Sewall SewaU Wright Wright has the main main advocate advocate of of this this latter view view and and he shifting balance latter he formalized formalized it it in in his his shifting balance theory. theory. The shifting balance theory is based on the idea that species are subdivided into many local populations (demes) that are weakly connected by migration. The small size allow genetic The small size of of the the local local populations populations would would allow genetic drift drift to to overwhelm overwhelm the effects of natural selection and take the populations to the domain of attraction of new adaptive peaks (phase I). Individual selection could then move the population toward the new peak itself (phase 11), II), at which point selection among the local populations would act to pull the entire species (meta population) toward peak (phase (phase III). (metapopulation) toward the the new new adaptive adaptive peak III). At At the the time time of of the the publication volume (Hanski 997), the publication of of the the predecessor predecessor to to this this volume (Hanski and and Gilpin, Gilpin, 11997), the SBT (Barton and SBT was was imperfectly imperfectly understood understood and and largely largely untested untested (Barton and Whitlock, Whitlock, 11997). 997). However, a large number of theoretical studies have provided new insight into the feasibility of the genetic mechanisms underlying the SBT. SBT. These studies have also uncovered many alternative forms of evolution in "adaptive landscapes" that are theoretically and empirically better supported than the SBT (Whitlock and Phillips, Phillips, 2000). Much of this work was influenced by or even population paradigm. even based based on on the the meta metapopulation paradigm. This This body body of of literature literature and and its its connection 9, 111, 1, and connection to to some some recent recent theories theories are are discussed discussed in in Chapters Chapters 9, and 12. 12. A populations differs A particularly particularly brilliant brilliant example example of of how how evolution evolution in in meta metapopulations differs from large panmictic provided by recent stud from evolution evolution in in large panmictic populations populations is is provided by the the recent studies Chapter 111). 1 ) . IGEs based ies of of indirect indirect genetic genetic effects effects (IGEs, (IGEs, Chapter IGEs are are genetically genetically based environmental influences that are generated whenever the phenotype of one 997). IGEs cre individual acts as an environment for another (Moore et aI., al., 11997). create causal pathways phenotypes of ate causal pathways between between the the genes genes on on individuals individuals and and the the phenotypes of other permitting the other related related or or unrelated unrelated individuals individuals permitting the coevolution coevolution of of phenotype phenotype and Chapter 111). 1 ). Another and context context that that is is unique unique to to metapopulations metapopulations ((Chapter Another import important advance in the evolutionary studies of meta populations is the recently metapopulations developed theory Chapter 12). theory developed theory of of "holey "holey adaptive adaptive landscapes" landscapes" ((Chapter 12). This This theory provides genetically explicit explicit approach the dynamics provides aa genetically approach for for the the study study of of the dynamics of of speciation and diversification in spatially explicit systems. Evolution Evolution of of the the migration migration rate rate is is aa well-studied well-studied topic in in evolutionary evolutionary ecology, metapopulation paradigm paradigm has ecology, but but use use of of the the metapopulation has shed shed new new light light onto onto the the selective pressures created by population turnover (Olivieri and Gouyon, 11997). 997). For 1 0 have shown For example, several several studies studies reviewed reviewed in in Chapter Chapter 10 shown that that under some circumstances, under some circumstances, migration migration is is aa nonmonotonic nonmonotonic function function of of the the extinction extinction rate, rate, with with high high extinction extinction rates rates leading leading to to reduced reduced migration migration propensity, contrary to the prevailing view. now is to find out view. The challenge now
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what 01) what actually actually happens happens iinn real real metapopulations. metapopulations. Heino Heino and and Hanski's Hanski's (20 (2001) modeling study ooff evolution ooff the migration rate iinn checkerspot butterflies demonstrated the possibility of a reduced migration rate with an increasing extinction rate, but they concluded that that this would not occur under conditions met in natural metapopulations of the butterflies. Other studies have made a start in developing a more general framework of life life history evolution in metapopulations, metapopulations, including traits other than migration rate and interactions among different traits. This research is reviewed in Chapter 10. Another Another evolutionary evolutionary problem problem that that has has benefited benefited from from application application of of the the metapopulation approach is the evolution of species' ranges. The basic ques question being populations at tion being asked asked here here is: is: Why Why do do populations at the the range range margin margin not not adapt adapt to their local conditions and then spread outward outward (Kirkpatrick and Barton, 11997)? 997) ? One populations receive One answer answer to to this this question question is is that that peripheral peripheral populations receive migrants migrants from from the the center center of of the the species' species' range. range. These These immigrants immigrants will will be be well well adapted adapted to to the the conditions conditions at at the the range range center center but but not not to to conditions conditions at at the the periphery and, therefore, the genes that that they bring hinder adaptation at the periphery (Mayr, 11963). 963). Thus, peripheral populations are forced into the role of demographic sinks, preventing the range from expanding outward outward ((Kirkpatrick Kirkpatrick and Barton, 11997). 997). An appropriate conceptual framework used to study the interplay between migration and selection in peripheral popula populations is the source-sink metapopulation framework framework (e.g., Holt and Gaines, 11992). 992). The usual approach approach in this context has been to consider the conditions that fit that would allow the increase of a rare allele with antagonistic effects on fitness and Gomulkiewicz, 1 997; Gomulkiewicz ness in in two two habitats habitats (Holt (Holt and Gomulkiewicz, 1997; Gomulkiewicz et et aI., al., 11999; 999; Kawecki, Kawecki, 2000; 2000; Kawecki Kawecki and and Holt, Holt, 2002 2002).) . Use Use of of the the source-sink source-sink meta metapopulation approach has led to an important important general conclusion about sink populations: the parameter that that governs the rate of spread of the beneficial mutation not its relative fitness, as is the mutation is the absolute fitness fitness of the mutant, mutant, not case in Gomulkiewicz, 11997). 997). Use of case in populations populations of of constant constant size size (Holt (Holt and and Gomulkiewicz, Use of the source-sink metapopulation concept has also shed new light on the evolu evolutionary consequences of asymmetric migration in heterogeneous heterogeneous landscapes ((Ronce Ronce and Kirkpatrick, Kirkpatrick, 2001 2001;; Kawecki and Holt, 2002). These studies are described in detail in Chapter 116. 6.
Integration Integration across across Disciplines Disciplines and and Applications Applications A population biology A clear clear indication indication of of the the maturity maturity that that the the field field of of meta metapopulation biology has has reached reached is is the the appearance appearance of of increasing increasing numbers numbers of of studies studies that that attempt attempt to to integrate many or even all of the main subdisciplines covered by the broader field of population biology. The integration of ecology and genetics has been in 93 1 , Sewall in the the minds minds of of population population biologists biologists for for aa long long time. time. As As early early as as 11931, Sewall Wright attempted the integration of ecological and population genetic processes processes through through his his shifting shifting balance balance theory, theory, as as described described earlier, earlier, with with the the aim aim of demonstrating that that evolution could proceed rapidly in spatially structured populations. years that most of both populations. In In the the years that followed, followed, most of the the work work that that included included both ecological and genetic considerations was empirical and did not explicitly attempt such integration. However, the importance importance of such integration was widely paper published published in in 11960 960 by widely accepted accepted as as attested attested by by the the conceptual conceptual paper by
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L. C. Birch. Although using slightly different terms, Birch ((1960) 1 960) referred to many of the problems that are the current focus of meta population biology, metapopulation such as feedback between population dynamics and genetic variability, the importance of sink populations, and so forth, and provided numerous refer references to the empirical work available at that time. As an aside, it is worth not notcontribution to early development of ing that L. C. Birch also made a lasting contribution the ecological metapopulation ideas by his textbook with H. G. Andrewartha (Andrewartha and Birch, 11954). 954). The first step toward toward a more formal integra integraMacArthur ((1962), anation of the two disciplines can be traced back to MacArthur 1 962), who ana lyzed a selection model in which population population regulation plays a central role. Subsequent studies continued continued to explore the way in which which population dynam dynamics affects natural selection (e.g., Anderson, 1971; Asmussen, 11979, 979, 11983a,b), 983a,b), but left most other questions unexplored. but New New impetus for the integration of the two two disciplines came with the realization that human impact is the primary cause of species extinctions in many landscapes and that that extinctions are taking place at an alarming rate. Just Just known about about the interaction of demographic, over a decade ago, little was known 1 988) urged population ecologic, and genetic factors factors in extinction, and Lande ((1988) population biologists to address this fundamental fundamental but difficult problem. Much Much progress contribution, and despite its short history, has occurred since Lande's key contribution, metapopulation metapopulation biology has facilitated substantial progress in this area. Chapters population Chapters 13 to 16 cover many of the key contributions contributions of meta metapopulation biology toward toward the integration integration of population population biology. For this integration to be truly successful, we need to extend it also to the domain domain of empirical research. Current Current developments in the field of statistical genetics provide new tools that Of particular develthat will help accomplish accomplish this goal. Of particular importance importance is the devel opment multilocus genotype to make inferences about about opment of of powerful powerful multilocus genotype methods methods to the origin (natal populations) migrating individuals populations) of migrating individuals (e.g., Smouse et al., aI., 1 990a; Rannala Rannala and Mountain, 1997; 1 997; Pritchard Pritchard et al., aI., 2000; Dawson and and 1990a; and Mountain, 2000; Dawson 200 1 ) . These methods, methods, when when implemented implemented under under the hierarchical Belkhir, 2001). framework, can be used to combine genetic, demographic, and Bayesian framework, demographic, and environmental data model (e.g., (e.g., Gaggiotti aI., 2002). 2002). environmental data in a single statistical model Gaggiotti et al., turn, provides a way way of testing hypotheses about This approach, approach, in turn, about the demo demographic and and environmental factors that control graphic factors that control metapopulation metapopulation processes. These very recent recent developments developments are covered in Chapter Chapter 15. There There are already good examples examples of studies studies that that have have employed employed the the metapopulation metapopulation approach approach good to integrate integrate ecology, ecology, genetics, and/or and/or evolution, evolution, including including studies studies on on to host-pathogen interaction interaction (Chapter ( Chapter 19), 1 9 ), butterflies butterflies (Chapter ( Chapter 20), 20), and and small small host-pathogen mammals (Chapter mammals ( Chapter 21). 21). mentioned earlier, the renewed renewed interest interest in the the metapopulation metapopulation concept concept As mentioned was was fostered fostered by its potential potential application application to to the the field of of conservation conservation biology, and it it is now now clear clear that that the the initial expectations expectations were well founded. founded. The The design of of reserve networks networks (Chapter ( Chapter 22) is a good good example of of a problem problem that that needs to be addressed addressed using the the metapopulation metapopulation approach. approach. Another Another important important to example is the the extension extension of of population population viability analysis (PVA) to to fragmented fragmented populations. In In the the past, past, most most PVA PYA methodologies methodologies either either took took no no account account of of populations. spatial structure structure or or did did so in in ways ways that that have have unrealistic unrealistic data data requirements. requirements. spatial Chapter 23 23 presents presents a practical practical approach approach that that considers considers spatial spatial population population Chapter structure and and can can be be parameterized parameterized using using available available data. data. This This chapter chapter structure
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describes how such a model can be used in the management management of endangered species Columbia basin species using using the the contentious contentious Columbia basin salmon salmon stocks stocks as as an an example. example. Practical applications of the meta population concept metapopulation concept have gone beyond the domain domain of of conservation conservation biology biology and and now now include include epidemiological epidemiological studies studies of of infectious infectious diseases diseases in in humans humans and and domestic domestic animals. animals. Chapter Chapter 1177 explores explores how how meta population theory metapopulation theory at a variety of scales can help understand understand epidemio epidemiological dynamics and how how this newly gained insight can be used in the design of efficient vaccination vaccination programs. programs.
11.4 .4
CHALLENGES FOR THE CHALLENGES FOR THE NEAR NEAR FUTURE FUTURE It is generally difficult ((and and often unnecessary) to try try to predict the the course that research in a particular field will take even over a short short period of a few that years. A truly novel discovery may radically change the way we think about about a particular particular issue; new modeling tools are introduced, introduced, allowing researchers to tackle previously could cumbersome simu tackle questions questions that that previously could be be studied studied only only via via cumbersome simulations; and new methods of field study may open up possibilities that that we could only dream about about in the past. One good example is the study of migra migration and gene flow, which which has benefited greatly from from new statistical models of both both demographic and genetic data data and the combination combination of the two, two, as well as of the the high-resolution genetic markers that that have become recently available. Chapters 15 and 21 illustrate Chapters 15 and 21 illustrate the the power power of of these these new new tools. tools. We anticipate that that the integration of ecological, genetic, and evolutionary studies studies will will continue continue in in the the near near future. future. Metapopulation Metapopulation biology biology is is well well placed placed to make ground-breaking contributions contributions here. Theoretical challenges start from the need need to to combine combine currently currently distinct distinct ecological ecological modeling modeling approaches, approaches, such as as stochastic patch occupancy models, spatial moment moment equations, and metapopu metapopulation models structured by local population population size. Adding realistic description of landscape structure structure into genetic and evolutionary models is another another chal challenge. The new statistical methods that that integrate genetic, demographic, and environmental data ((Chapter Chapter 115) 5 ) offer a route to merging ecology and genetics but also the possibility of linking theory ever more closely with empirical research. Few of these methods are currently widely available, but we expect that many will be developed further in the near feature. feature. Somewhat Somewhat more specific specific research research tasks tasks include include the the need need to to better better understand understand the the interactive interactive effects spatial structure effects of of populations' populations' age/stage-structure age/stage-structure and and their their spatial structure on on the the maintenance clines, inbreeding maintenance of of genetic genetic variability, variability, genetic genetic clines, inbreeding depression, depression, and and so so forth (Mills and Smouse, 11994; 994; Gaggiotti et al., aI., 11997; 997; Gaggiotti and Vetter, 11999). 999). To what extent can the meta population approach metapopulation approach be developed to address such large-scale issues as determination of species' range boundaries and their responses to global changes ((Chapter Chapter 20), and indeed the global extinction risk of species? We have already commented on the relative lack of studies on metacommunities. metacommunities. population context is needed Combining ecology and genetics in the meta metapopulation for conservation and epidemiology. Chapter Chapter 22 takes an important important step forward forward in adding spatial dynamics to existing reserve site selection procedures. procedures. We imagine that that including genetics in the same package package would would be worth population dynamics worth the effort. Research on plant-pathogen plant-pathogen meta metapopulation
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((Chapter Chapter 119) 9 ) shows the way forward for epidemiology ((Chapter Chapter 117) 7) in includ includframework. Finally, ing genetic and evolutionary issues into the demographic framework. ever since Lande's ((1988) 19 8 8 ) key contribution, ecology and genetics have been integral parts of conservation biology. Opinions have shifted over the years on their relative importance ((Chapter Chapter 115). 5 ) . The coming years may demonstrate that asking about the "relative importance" has been a somewhat misleading (although necessary) question, as often the real question is about interactions. That That being being said, said, we we should should not not lose lose perspective perspective on on the the kinds kinds of of threats threats that that operate at present, of which habitat loss and fragmentation are the most important ones. The immediate adverse effects of habitat loss and fragmenta fragmentametapopulation are largely ecological, and it remains a major challenge for metapopula tion biologists to develop predictive models and robust understanding of this key issue to be able to provide solid scientific advice to the society.
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META PO PULATION M ETAPO PU LATIO N DYNAMICS DYNAMICS:: PERS PECTIVES FROM PERSPECTIVES FROM LANDS CA PE ECOLOGY LANDSCAPE ECOLOGY Kimberly A. With
2.1 2.1
INTRODUCTION INTRODUCTION It It is is no no coincidence coincidence that that the the current current biodiversity biodiversity crisis crisis occurs occurs at at aa time time when when landscapes are landscapes are being being transformed transformed faster faster than than ever ever before before in in human human history history (With, 2004). Many conservation issues are ultimately human human land-use issues (Wiens, become (Wiens, 2002), 2002), which which is is why why the the discipline discipline of of landscape landscape ecology ecology has has become increasingly biological diver increasingly relevant relevant for for the the management management and and conservation conservation of of biological diversity (e.g., Gutzwiller, 2002 2002).) . Processes that that operate at broader broader spatial scales likely influence the occurrence and persistence of an organism at a local scale, and ultimately required assessing species' and thus thus aa landscape landscape perspective perspective is is ultimately required for for assessing species' extinction extinction risk. risk. Such Such acknowledgment acknowledgment of of the the importance importance of of landscape landscape ecology ecology for for conser conservation reinforces the common common misconception misconception that that landscape ecology is con concerned cerned solely solely with with broad broad spatial spatial scales, scales, however. however. In In the the present present context, context, this this would entail understanding meta population dynamics at a "landscape metapopulation "landscape scale" (e.g., Rushton et al., 11997). 997). Apart Apart from the usual broad-scale anthropocentric anthropocentric definition landscape, aa landscape landscape is definition of of landscape, is defined defined more more appropriately appropriately as as aa "spa "spatially heterogeneous area" 99 1 ) that area" (Turner and Gardner, 11991) that is scaled relative
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to the process or organism of interest (Wiens, 1989). By this rendering, meta population dynamics can then be studied in fragmented landscapes that that metapopulation range in scale from that encompassing encompassing bacteria and protozoan communities ((Burkey, Burkey, 11997) 997) to spotted owls (Strix (Strix occidentalis; occidentalis; Gutierrez Guti~rrez and Harrison, Harrison, 11996). 996). The landscape thus provides a spatial context context for understanding understanding population dynamics and persistence in frag processes contributing contributing to meta metapopulation fragmented landscapes. Although meta population theory metapopulation theory is the current current paradigm paradigm for the conserva conservation of spatially structured structured populations in fragmented fragmented landscapes (Hanski and Simberloff, 11997), 997), landscape ecology provides an additional perspective and suite of approaches that that can complement complement metapopulation metapopulation theory, particularly in applications that that are not not handled well by existing theory, such as those involving continuous habitat habitat distributions distributions or recently fragmented fragmented landscapes. Metapopulation Metapopulation theory is not applicable to species in landscapes landscapes in which the habitat habitat is not not distinctly patchy or already fragmented extensively (Moilanen and Hanski, 200 1 ) . Nor 2001). Nor is the application application of metapopulation metapopulation theory theory necessar necessarily appropriate appropriate for species in recently fragmented systems, given the assump assumption of equilibrium colonization-extinction colonization-extinction dynamics that that underlies much of the theory [but see Ovaskainen and Hanski (2002) and Chapter Chapter 4 for advances in metapopulation metapopulation theory involving transient transient dynamics] dynamics].. In particu particular, landscapes fragmented by human human land-use activities may represent transient population transient nonequilibrium nonequilibrium dynamics in which a formerly continuous population has become subdivided into smaller, more isolated populations. populations. Dispersal among populations populations is disrupted disrupted such that that a functional metapopulation metapopulation is not not created; created; local extinctions extinctions are not not balanced by recolonization, recolonization, and conse consequently, all populations populations slowly decline to extinction extinction (Hanski and Simberloff, 11997). 997). Thus, a declining population population may superficially resemble a metapopula metapopulation in structure, but not not function function like one. Spatial subdivision is a necessary, necessary, but not not sufficient, condition condition for metapopulation metapopulation dynamics. Metapopulation Metapopulation theory has nevertheless drawn drawn attention attention to the importance importance of landscape landscape structure and dispersal for maintaining population population persistence 996). Indeed, the effect of patch structure on dispersal and coloniza (Wiens, 11996). colonization success is a unifying theme in both metapopulation metapopulation theory and landscape ecology (Wiens, 11997). 997). Colonization success is not simply a function function of the distance distance between between patches, but also depends depends on the nature nature of the intervening habitat or land-use matrix matrix through which organisms disperse, which deter deterIncorporation of mines the "effective isolation" isolation" of patches (Ricketts, 2001 2001).). Incorporation the more complex mosaic structure population structure of real landscapes landscapes into meta metapopulation models has been viewed as the main promise of landscape landscape ecology for metapopulation e.g., Hanski and Simberloff, 11997; 997; Wiens, 11997). 997). metapopulation theory ((e.g., In 1 996, 11997) 997) identified In aa couple couple of of earlier earlier reviews, reviews, Wiens Wiens ((1996, identified several several land landscape ecological concepts metapopulation ecology and concepts that that are relevant to metapopulation which emphasize the dual importance of dispersal dispersal and heterogeneous heterogeneous land landscape mosaics for understanding 1 ) landscape understanding metapopulation metapopulation dynamics: ((1) connectivity, which emerges as the interaction interaction of individual movement with with landscape pattern, pattern, is important important for metapopulation metapopulation persistence; persistence; (2) the land landscape matrix matters for meta population dynamics because it affects dispersal metapopulation and thus colonization success; ((3) 3 ) landscapes landscapes are heterogeneous mosaics of habitats and land uses, such that that habitat quality varies across the landscape,
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setting the stage for source-sink population dynamics ((Chapter Chapter 16); and (4) landscape dynamics may affect, or even generate, meta population dynamics. metapopulation The latter represents landscape heterogeneity in time as well as space. In In addition addition to to these these potential potential contributions contributions of of landscape landscape ecology ecology to to meta population ecology, a more fundamental principle emerges from the metapopulation definition of landscape ecology itself. Landscape ecology is the study of the effect of spatial pattern on ecological process (Turner, 11989). 989). From this it fol follows that that adopting a landscape ecological perspective to metapopulation dynamics entails understanding understanding how spatial pattern, pattern, such as habitat fragmen fragmentation or heterogeneity, affects processes that contribute to the dynamics of spatially structured populations. This might involve, for example, under understanding the relative effects of habitat fragmentation on dispersal (coloniza (colonization) and demography demography on metapopulation metapopulation persistence. This expanded expanded perspective of landscape ecology is adopted adopted in this chapter. The 1 ) demonstrate The objectives objectives in in this chapter are are thus thus to to ((1) demonstrate what what aa landscape landscape ecological perspective can contribute population contribute toward toward understanding understanding meta metapopulation dynamics, beyond the usual suggestions that that landscape ecology offers a broader scale perspective or more spatially complex rendering of landscape structure; (2) discuss how landscape structure is expected, or has been demon demonstrated, to affect various processes (dispersal, demography) that affect meta population persistence and thus extinction risk; ((3) 3 ) assess the implica metapopulation implications of adopting a landscape ecological perspective for management and con conservation; and (4) identify theoretical and empirical research needs that that would help contribute contribute to the further development of this "exciting scientific synthe synthesis" between meta population biology and landscape ecology (see metapopulation (see Hanski and 991). Gilpin, 11991).
2.2 2.2
CONTRIBUTIONS CONTRIBUTIONS OF OF LANDSCAPE LANDSCAPE ECOLOGY ECOLOGY TO TO UNDERSTANDING ETAPOPULATION DYNAMICS UNDERSTANDING M METAPOPULATION DYNAMICS This section addresses how how landscape structure affects, or is expected to affect, the dynamics of meta populations. This includes a discussion on issues metapopulations. pertaining to landscape connectivity, landscape connectivity and dispersal thresholds, the relative importance importance of dispersal for metapopulation persist persistence, landscape effects on demography and extinction risk, the source-sink potential of landscapes, extinction risk in dynamic landscapes, and the relative effects of habitat loss and fragmentation on meta population persistence. metapopulation
Landscape Landscape Connectivity Connectivity Issues: Issues: Patch-Based Patch-Based vs vs Landscape-Based Landscape-Based Measures Measures Habitat Habitat connectivity is a central theme in both landscape ecology and metapopulation 999a; Tischendorf and Fahrig, 2000a). metapopulation ecology (Hanski, 11999a; Connectivity refers to the ability of organisms to to access habitat, which affects colonization rates and thus metapopulation persistence on the landscape (e.g., Gonzalez et aI., 998; Kindvall, 11999). 999). The emphasis in meta population ecol al., 11998; metapopulation ecology, however, has been on deriving patch-based patch-based measures measures related to the prox proximity and area of neighboring patches, which quantify the accessibility of
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999a; Moilanen habitat patches to an individual on the landscape (Hanski, 11999a; and Hanski, 200 1 ) . An and Hanski, 2001). An overall overall measure measure of of patch patch connectivity connectivity for for the the landscape landscape can be obtained as a weighted average of patch patch isolation, which then gives the amount amount of habitat accessible to a random random individual on the landscape (Hanski, 11999a). 999a). Overall patch connectivity may give an indication indication of landscape con connectivity, nectivity, but but the the latter latter is is not not formally formally derived derived mathematically mathematically from from such such patch-based Fahrig, 2001 connect patch-based measures measures (Tischendorf (Tischendorf and and Fahrig, 2001).). Patch-based Patch-based connectivity measures are best applied to extensively fragmented or distinctly patchy landscapes and are less applicable to more continuous continuous habitat habitat distributions (Moilanen 1 ). (Moilanen and and Hanski, Hanski, 200 2001). In contrast, landscape ecologists have focused on deriving measures of overall landscape landscape connectivity. Habitat Habitat connectivity is thus being assessed at different different scales scales ~ patch patch based based vs vs landscape landscape based based ~ in in these these two two disciplines. disciplines. Landscape connectivity is defined as the degree to which various habitat habitat types 993; With facilitate movement across the landscape (Taylor et aI., al., 11993; With et aI., al., 11997; 997; Tischendorf Tischendorf and and Fahrig, 2000a) 2000a) and and can can thus be assessed for continu continuous habitat distributions heterogeneous landscapes Schippers et ous habitat distributions and and heterogeneous landscapes ((Schippers et aI., al., 11996; 996; With et aI., al., 1997). Landscape connectivity can be quantified in a number of ways, such as by the use of percolation theory and its neutral neutral landscape derivatives ((Gardner Gardner et aI., 987; With, 997), al., 11987; With, 1997, 1997, 2002; With With and and King, 11997), graph theory 1 ) , and theory (Urban and and Keitt, 200 2001), and various other other approaches approaches (e.g., Schumaker, 996; Tischendorf Schumaker, 11996; Tischendorf and and Fahrig, Fahrig, 2000a,b). 2000a,b). Although Although aa full full render rendering lies beyond beyond the ing of of how how landscape landscape connectivity connectivity can can be be quantified quantified lies the scope scope of of this common theme this chapter, chapter, the the common theme underlying underlying all all of of these these approaches approaches is is how how the the movement movement behavior behavior of of organisms organisms interacts interacts with with the the patch patch structure structure of of land landscapes. Landscape connectivity thus emerges as a species-specific response to landscape habitat affinities, landscape structure structure based based on on factors factors such such as as the the species' species' habitat affinities, gap-crossing abilities, movement movement rates, response to patch boundaries, boundaries, and differential mortality through 993; through elements of the landscape (Wiens et a!., al., 11993; Dale et aI., 994; With, 11997; 997; With et aI., 997; Tischendorf al., 11994; al., 11997; Tischendorf and Fahrig, 200 1 ; Vos et aI., 1). 2001; al., 200 2001). Landscape connectivity is important important for understanding understanding the emergence of spatial structure in populations, populations, which in turn turn is expected to have implications for the persistence and dynamics of meta populations. As an example of how metapopulations. how species-specific species-specific responses responses to to heterogeneity heterogeneity affect affect landscape landscape connectivity connectivity and and population 1 995) used habitat-specific rates of population distributions, distributions, With and Crist ((1995) movement in an individual-based simulation model inspired by percolation theory to predict the distributional patterns of two two acridid grasshopper species in a heterogeneous landscape within the shortgrass steppe of the North North American Great Plains. The largest species (Xanthippus (Xanthippus corallipes) corallipes) moved rapidly through through the grass matrix matrix (65% of the landscape), suggesting that that the overall of this overall landscape landscape was was highly highly connected connected from from the the standpoint standpoint of this species. species. Its Its reduced rate of movement in the remaining third of the landscape resulted in the observed patchy population population distribution, distribution, consistent with model expecta expectations that that good dispersers should exhibit patchy distributions distributions when the landscape contained :::; 3 5 % preferred habitat because their high mobility -<35% allows individuals to locate and allows individuals to locate and aggregate aggregate within within the the preferred preferred habitat habitat (assum (assuming that that individuals reduce their rates of movement and exhibit greater resi residence dence times times in in preferred preferred habitats). habitats). In In contrast, contrast, the the lower lower mobility mobility of of the the
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smaller smaller species species (Psoloessa (Psoloessa delicatula) delicatula) prevented prevented large large numbers numbers of of individuals individuals from locating and aggregating within its preferred preferred habitat, habitat, which constituted aa minor ((8%) 8 % ) component component of of the the landscape. landscape. Because Because of of its its relatively relatively greater rates of movement through species was rates of movement through other other grassland grassland habitats, habitats, this this species was expected expected to landscape, which to be be distributed distributed randomly randomly across across the the landscape, which was was consistent consistent with with its its observed distribution distribution in the field.
Landscape Connectivity Issues: Issues: Data Requirements Patch-based population Patch-based connectivity connectivity measures measures that that form form the the basis basis of of meta metapopulation theory Hanski, theory have have the the distinct distinct advantage advantage of of ease ease of of model model parameterization parameterization ((Hanski, 11999b; 999b; Hanski Hanski et aI., al., 2000; Chapter Chapter 5). Consistent Consistent with with metapopulation metapopulation the theory's abstraction abstraction of landscape structure as discrete patches embedded in an ecologically neutral matrix, matrix, patchwise patchwise measures of connectivity have often been been based based on on simplistic simplistic measures measures such such as as nearest-neighbor nearest-neighbor distances distances (Moilanen and Nieminen, Nieminen, 2002 2002).) . More More sophisticated measures of patch patch connectivity connectivity have have been been developed, developed, however, however, which which incorporate incorporate patch-area patch-area effects effects on on emigration emigration and and immigration immigration rates rates and and species-specific species-specific dispersal dispersal distances 1 ; Chapter distances (Moilanen and Hanski, Hanski, 2001 2001;; Vos et aI., al., 200 2001; Chapter 4). As the aforementioned aforementioned grasshopper grasshopper example example illustrates, illustrates, however, however, the the connectivity connectivity of of habitat habitat patches is not just a simple function function of the distance between between patches. Because the intervening matrix matrix may determine determine the effective isolation of patches (Ricketts, 2001 aI., 2001),) , and thus thus overall overall landscape connectivity (With et al., 11997), 997), explicit consideration consideration of how how the complex complex mosaic mosaic structure of hetero heterogeneous landscapes affects population extinction geneous landscapes affects colonization colonization success success and and population extinction risk risk has typically been viewed as one of the most most important important contributions contributions that that landscape has to population ecology ecology ((Hanski Hanski and and landscape ecology ecology has to offer offer meta metapopulation Simberloff, 11997; 997; Wiens, 11997). 997). The The connectivity of heterogeneous landscapes is not easily captured captured by a simple index or landscape metric unfortunately, unfortunately, but but is commonly tackled with an data on an individual-based individual-based simulation simulation modeling modeling approach. approach. Empirical Empirical data on habitat habitatspecific specific movement movement parameters parameters or or residence residence times times within within different different elements elements of of the the landscape landscape are are used used to to parameterize parameterize aa rulerule- or or vector-based vector-based movement movement model to simulate dispersal across a heterogeneous landscape map. Landscape connectivity is then inferred by extrapolating extrapolating habitat-specific rates of move movement, and perhaps perhaps other information information (if available) about about behavior at habitat edges (Lidicker and Koenig, 11996) 996) or mortality mortality risk while dispersing through through the different elements of the landscape, to determine determine whether whether individuals are able to able to colonize colonize aa suitable suitable habitat habitat successfully. successfully. Some Some presumed presumed correlate correlate of of aa connected landscape, such as dispersal success, degree of population population aggrega aggregation 995; With 997), or tion (With (With and and Crist, Crist, 11995; With et et aI., al., 11997), or population population connectivity connectivity (Schippers et aI., 996), is then used as an indirect measure of landscape al., 11996), connectivity. Admittedly, quantifying the resistance of different different habitat habitat types to move movement is a challenge in practice. practice. Direct Direct observation observation of individual movement movement responses to landscape structure structure is time intensive and is necessarily limited in temporal and therefore bound to change temporal ((and therefore spatial) extent, extent, although although this is bound with with the the increasing increasing availability availability of of satellite-tracking satellite-tracking devices devices that that permit permit the the near-continuous near-continuous monitoring monitoring of individuals. As an alternative, alternative, investigators
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typically rely upon typically rely upon mark-recapture mark-recapture techniques techniques to to derive derive estimates estimates of of inter interpatch patch movements movements in in different different matrix matrix types types (e.g., (e.g., Pither Pither and and Taylor, Taylor, 1998; 1998; Ricketts, 1 ) or Ricketts, 200 2001) or make make inferences inferences about about matrix matrix resistance resistance based based on on observed e.g., observed patterns patterns of of patch patch occupancy occupancy in in different different landscape landscape contexts contexts ((e.g., Moilanen 99 8 ) . For Moilanen and and Hanski, Hanski, 11998). For example, example, mark-recapture mark-recapture data data were were used used to to quantify quantify the the resistance resistance of of different different matrix matrix types types to to butterfly butterfly movement movement within naturally heterogeneous heterogeneous landscape within a naturally landscape located located in an alpine alpine valley of the Colorado 1 ). Movement Colorado Rocky Rocky Mountains Mountains (Ricketts, (Ricketts, 200 2001). Movement through through coniferous coniferous forests likely than forests was was 3-12 3-12 times times less less likely than movement movement through through willow willow thickets thickets for for the these meadow meadow butterfly the majority majority of of these butterfly species. species. Meadows Meadows separated separated by by coniferous coniferous forest forest are thus thus effectively more more isolated isolated than than meadows meadows separated separated by by aa similar similar distance, distance, but but embedded embedded in in aa willow willow matrix. matrix. Because Because of of the the diffi difficulty and and cost cost of culty of attempting attempting to to reconnect reconnect habitat habitat fragments fragments with with corridors corridors or stepping stepping stones stones to to enhance enhance dispersal dispersal and and thus thus colonization colonization success, it might practical to isolation of of patches patches by alter might be be more more practical to reduce reduce the the effective effective isolation by altering management management practices practices in the surrounding surrounding matrix Ricketts, 200 1). matrix ((Ricketts, 2001). Corridors Corridors need need not not bbee linear linear features features ooff the the landscape, landscape, but but can can occur occur or or be be created created through through the the juxtaposition juxtaposition of of certain certain matrix matrix types, types, such such as as different different habitats funnel individuals habitats or land land uses, which which serve to funnel individuals among among habitat habitat patches Gustafson and 996; Vos et al., aI., 2002) patches ((Gustafson and Gardner, Gardner, 11996; 2002)..
landscape Landscape Connectivity Connectivity Issues: Issues: Asymmetrical Asymmetrical Connectivity Connectivity Among Among Populations Populations Connectivity Connectivity in in heterogeneous heterogeneous landscapes landscapes may may thus thus be be difficult difficult to to identify identify from a simple analysis of landscape structure. Nor landscape structure. Nor is the transfer transfer of individu individuals among patches als among patches necessarily necessarily symmetrical symmetrical in in aa heterogeneous heterogeneous landscape landscape matrix. 1 996) developed an individual-based matrix. Gustafson Gustafson and Gardner Gardner ((1996) individual-based simula simulation tion model model to to explore explore how how altering altering landscape landscape heterogeneity heterogeneity affected affected dispersal dispersal success success for for aa generic generic organism organism among among fragments fragments of of deciduous deciduous forest forest in in several several agricultural agricultural landscapes landscapes located located in in the the midwestern midwestern United United States. States. Emigration Emigration and among forest and immigration immigration rates rates among forest fragments fragments often often were were not not symmetrical, symmetrical, leading Gustafson 1 996) to Gustafson and and Gardner Gardner ((1996) to speculate that that such asymmetrical asymmetrical transfers rule rather transfers may may be be the the rule rather than than the the exception exception in in heterogeneous heterogeneous land landscapes. If true, true, this presents presents a problem problem for for patchwise patchwise connectivity connectivity measures measures in which which colonization colonization probabilities probabilities are are based based on on interpatch interpatch distances distances that that ignore ignore matrix problem, distances would matrix effects. effects. To To overcome overcome this this problem, distances between between patches patches would need inter need to to be be specified specified by by direction direction and and weighted weighted by by the the resistance resistance of of the the interUrban and 1 ). vening habitat to vening matrix matrix habitat to movement movement (i.e., (i.e., dij dii *' 4: dji; dii; see see Urban and Keitt, Keitt, 200 2001). The potential The potential for for asymmetrical asymmetrical connectivity connectivity among among populations populations of of European European badgers (Meles (Meles meles) meles) was indicated indicated by a GIS-based random random walk walk model model applied applied to to aa landscape landscape in in the the central central part part of of The The Netherlands Netherlands (Schippers (Schippers et 996). Urban areas, canals, et aI., al., 11996). Urban areas, canals, and and motorways motorways either either created created barriers barriers to to movement movement or or increased increased mortality mortality such such that that asymmetries asymmetries in in the the connectivity connectivity of of populations populations emerged. emerged. Asymmetrical Asymmetrical connectivity connectivity has has also also been been found found among among populations populations of of the the critically critically endangered endangered Iberian Iberian lynx lynx (Lynx (Lynx pardinus) pardinus) in in aa human-dominated region in southwest human-dominated and and therefore therefore extensively extensively fragmented fragmented region in southwestern Spain (Ferreras, 200 1 ). Although emi 2001). Although two two source populations populations had had similar emigration rates, they differed in their connectivity connectivity to outlying populations populations
2. 2.
M ETAPOPULATION DYNAMICS DYNAMICS METAPOPULATION
29
because of of differences differences in in the the matrix type surrounding surrounding each each source. source. One One of of the the because matrix type sources was embedded in an an agricultural agricultural matrix, matrix, which which lynx lynx avoided, avoided, and and sources was embedded in were instead instead funneled funneled along along aa narrow narrow corridor corridor of of aa more more suitable suitable habitat habitat to to were populations in in the the south south and and west. west. The The other other source source population population was was located located in in populations the southwestern southwestern region, which consisted consisted mainly mainly of of Mediterranean Mediterranean scrubland scrubland the region, which and tree tree plantations, plantations, and and most most dispersing dispersing individuals individuals from from this this population population and settled in this this region region instead instead of of dispersing dispersing toward toward the the northeast. northeast. Thus, Thus, there there is is settled in an asymmetrical asymmetrical transfer transfer of of individuals individuals that occurs among among populations: indi an that occurs populations: individuals to disperse disperse from from the the northeast northeast to to the the southwest, southwest, but but not not in in the the viduals tend tend to opposite direction. direction. Such Such asymmetries asymmetries could could lead lead to to aa reduction reduction in in the the effective effective opposite connectivity, and and thus thus metapopulation metapopulation capacity capacity (see (see Hanski Hanski and and Ovaskainen, Ovaskainen, connectivity, 2000), of of the the landscape landscape for for the the species. species. 2000),
Landscape Connectivity Connectivity Issues: Issues: Thresholds Thresholds in in Connectivity Connectivity Landscape Because of landscape connectivity the Because of the the importance importance of of landscape connectivity for for evaluating evaluating the structure metapopulations, it it would advantageous to to structure and and dynamics dynamics of of metapopulations, would be be advantageous identify when when landscapes become disconnected disconnected and and thus thus when when metapopulametapopula identify landscapes become tion likely to to be be disrupted. disrupted. Both Both patch tion processes processes such such as as colonization colonization rates rates are are likely patch connectivity (Hanski, 1999a) 1 999a) and and measures measures of of landscape landscape connectivconnectiv connectivity measures measures (Hanski, ity predict critical critical thresholds thresholds in habitat connectivity, where the the ity (With, (With, 2002) 2002) predict in habitat connectivity, where habitat network becomes disconnected at at a critical level of remaining habitat network becomes abruptly abruptly disconnected of remaining habitat. percolation-based approaches, approaches, for landscape connectivity habitat. In In percolation-based for example, example, landscape connectivity cluster) spans spans whether a single habitat habitat cluster cluster (the percolating cluster) is assessed by whether the landscape. when the the critical critical habitat habitat the landscape. Landscape Landscape connectivity connectivity is is disrupted disrupted when nodes nodes forming forming the the "backbone" "backbone" of of the the percolating percolating cluster cluster are are destroyed, destroyed, which which abruptly abruptly breaks breaks the the percolating percolating cluster cluster into into two two or or more more fragments. fragments. The The critical critical level level of of habitat habitat at at which which landscape landscape connectivity connectivity becomes becomes disrupted disrupted (percolation threshold) depends depends on on aa number number of of assumptions assumptions regarding regarding species-specific (gap-crossing abilities, species-specific movement movement attributes attributes (gap-crossing abilities, movement movement rates rates through habitat types, mortality) and through different different habitat types, matrix matrix mortality) and the the representation representation and landscape itself habitat and configuration configuration of of the the landscape itself (grid (grid geometry, geometry, degree degree of of habitat fragmentation) issue, however, fragmentation) (for (for aa review, review, see see With, With, 2002). 2002). The The issue, however, is is whether whether thresholds thresholds in in landscape landscape connectivity, connectivity, or or measures measures of of landscape landscape connectivity connectivity more generally, success and more generally, relate relate to to processes processes such such as as colonization colonization success and local local extinction extinction rates, rates, which which are are important important for for predicting predicting metapopulation metapopulation persistence persistence on on landscapes. landscapes. In In other other words, words, is is landscape landscape connectivity connectivity both both aa necessary population persistence e.g., necessary and and aa sufficient sufficient condition condition for for meta metapopulation persistence ((e.g., With, With, 1999)? 1999) ?
Landscape Landscape Connectivity Connectivity Thresholds Thresholds and and Dispersal Dispersal Success Success Dispersal populations together Dispersal is is the the "glue" "glue" that that keeps keeps meta metapopulations together (Hansson, (Hansson, 11991), 99 1 ) , and and thus thus colonization colonization success success is is deemed deemed crucial crucial to to metapopulation metapopulation persistence. persistence. Clearly Clearly there there should should be be some some relationship relationship between between landscape landscape connectivity connectivity and and dispersal dispersal (colonization) (colonization) success: success: dispersal dispersal success success is is expected expected to to be be higher higher in in landscapes landscapes with with aa high high degree degree of of connectivity. connectivity. What What is is less less clear, clear, however, however, is is whether whether thresholds thresholds in in landscape landscape connectivity connectivity should should neces necessarily sarily coincide coincide with with thresholds thresholds in in dispersal dispersal or or colonization colonization success. success.
WITH KIMBERLY A. WITH
30 30
To this, With With and and King King (1999a) ( l 999a) quantified quantified dispersal dispersal success success on on a To address this, of landscapes landscapes with with complex complex (fractal) (fractal) habitat habitat distributions distributions that that reprerepre series of sented a gradient gradient of of fragmentation fragmentation severity severity (Fig. 2.1a). 2.la). Dispersal Dispersal success success was was sented defined as the the proportion proportion of of independent independent dispersers dispersers that that successfully successfully located located a defined suitable suitable habitat habitat patch patch (cell). (cell) . Consider Consider that that if if dispersal dispersal is truly random, random, such such that dispersal dispersal occurs occurs to to a random random point point on on the the landscape, landscape, then then the the underlying underlying that spatial spatial pattern pattern of of the the landscape landscape is unimportant unimportant for for predicting predicting dispersal success success
a
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120 100 .S 80 2!.� 60 40 j 20 x
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1 .0 0.9 s: 0.8 or. 0.7 ti: 0.6 Q; 0.5 'E - Lacunarity 0.4 C Fragmented 0.3 � 0.2 :E Clumped 0.1 e , 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0
(f', (f', a: c..:
(f', c (f',
Cl: .c a..
C
g.
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� 0.9
� 0.8 -al 0.7 13 0.6 Clumped g? 0.5 Fragmented 8 0.4 2!- 0.3 15 0.2 0.1 0:: O.O-t-�-T--'-----;r--.--�--r---.,..---, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 Proportion suitab le habitat c
I'
Fig. Fig. 2.1 (a) Examples of fractal landscape patterns illustrating extremes in fragmentation fragmentation severity severity ((H, success declines precipitously below H, spatial autocorrelation of habitat). (b) Dispersal success 20% habitat (dispersal threshold), coinciding with the lacunarity threshold of landscape struc structure, which isis a landscape-wide measure of interpatch distances. The lacunarity index isis not affected by the landscape pattern for analyses conducted at the finest scale scale (1 (1 x x 11 grid cell), shown here for clarity of of presentation. The lacunarity curve at other scales scales isis qualitatively simi similar, but lacunarity indices tend to be higher in clumped fractal landscapes than in fragmented lar, ones due to the greater variability in gap sizes. sizes. (c) Percolation thresholds, aa patch-based assess assessment 2-cell ment of of landscape landscape connectivity, connectivity, do not coincide coincide with dispersal thresholds (assuming aa 112-cell dispersal neighborhood). Modified 999a). Modified from from With With and King (1 (1999a).
31 31
2. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS 2.
and only only the the fraction fraction of of habitat habitat (h) (h) and and number number of of dispersal dispersal steps steps (m; (m; equivaequiva and lently the the "dispersal "dispersal neighborhood" neighborhood" or or "dispersal "dispersal ability" ability" of of the the species) species) affect affect lently dispersal success success as as dispersal Pr( success ) = Pr(success) = 1 1 -- (1 ( 1 -- h)m. h)m.
(2.1) (2. 1 )
Equation (2.1) (2. 1 ) represents represents the the mean-field mean-field approximation. approximation. If If dispersal dispersal is is instead instead Equation constrained to to movement movement through through adjacent adjacent cells cells (but ( but still still random random in in direction) direction) constrained to force force individuals individuals to to interact interact with with the patch structure structure of of the landscape, then then to the patch the landscape, dispersal on fractal fractal landscapes landscapes can can no no longer longer be be derived first dispersal success success on derived from from first principles and and may may lack lack aa closed-form closed-form solution. solution. Thus, Thus, dispersal dispersal success success on on principles fractal landscapes landscapes had to be be obtained obtained through through numerical numerical simulations simulations (With (With fractal had to and King, King, 1999a). 1 999a). and As expected, dispersal dispersal success success declined declined with with decreasing decreasing habitat habitat and and As expected, increasing fragmentation fragmentation of of the the landscape, landscape, but but the the rate rate of of this this decline decline accelaccel increasing erated 1 0-20% (Fig. 2 . 1 b ) . In In other erated once once the the amount amount of of habitat habitat fell fell below below 10-20% (Fig. 2.1b). other words, response to habitat amount. amount. words, dispersal dispersal success success exhibited exhibited aa threshold threshold response to habitat This with percolation to This dispersal dispersal threshold threshold did did not not coincide coincide with percolation thresholds thresholds used used to quantify landscape connectivity, allowing for dispersal quantify landscape connectivity, even even after after allowing for aa larger larger dispersal neighborhood in which neighborhood to to define define habitat habitat connectivity, connectivity, in which individuals individuals could could move move through cells nonhabitat in their search habitat site site ((as as in in the the through cells of of nonhabitat in their search for for aa suitable suitable habitat simulation; 2.1c). Intuitively, Intuitively, dispersal to decline as simulation; Fig. Fig. 2.1c). dispersal success success is is expected expected to decline as patches become more isolated because the up patches become smaller smaller and and more isolated because the disperser disperser ends ends up spending of its its time in the the matrix be greater. spending much much of time in matrix where where mortality mortality may may be greater. Lacunarity analysis the "gap landscapes and is related related Lacunarity analysis quantifies quantifies the "gap structure" structure" of of landscapes and is to variance-to-mean ratio to the the variance-to-mean ratio of of the the distances distances among among patches patches on on the the land landscape 993). The scape (Plotnick (Plotnick et et aI., al., 11993). The higher higher the the lacunarity lacunarity index, index, the the greater greater the the variability variability in in distances distances among among patches. patches. Lacunarity Lacunarity is is not not merely merely the the inverse inverse of of some some measure measure of of patch patch structure, structure, such such as as the the fractal fractal dimension dimension of of the the land landscape, scape, however; however; it it can can resolve resolve differences differences in in landscape landscape pattern pattern that that may may be be obscured obscured by by patch-based patch-based measures measures (Plotnick (Plotnick et et aI., al., 1993 1993).) . Landscape Landscape lacu lacunarity narity exhibited exhibited aa strong strong threshold threshold effect effect around around 20% 20% habitat; habitat; interpatch interpatch dis distances tances became became greater greater and and more more variable variable when when habitat habitat fell fell below below this this critical critical level 999a; Fig. level (With (With and and King, King, 11999a; Fig. 2.1b). 2.1b). Thresholds Thresholds in in dispersal dispersal success success thus thus coincide percolation thresholds coincide with with lacunarity lacunarity thresholds thresholds rather rather than than percolation thresholds of of landscape landscape connectivity. connectivity. Empirical Empirical tests tests of of percolation percolation theory theory have have been been performed performed in in the the field field with with insects insects moving moving across across experimental experimental "microlandscapes" "microlandscapes" in in which which habitat habitat (grass (grass sod) 997; sod) was was arrayed arrayed as as either either aa random random or or aa fractal fractal distribution distribution (Wiens (Wiens et et aI., al., 11997; 999; With 999). Although McIntyre Mclntyre and and Wiens, Wiens, 11999; With et et aI., al., 11999). Although it it is is not not clear clear whether whether parameters parameters that that describe describe movement movement pathways pathways should should exhibit exhibit threshold threshold behav behavior, ior, let let alone alone coincide coincide with with percolation percolation thresholds thresholds in in landscape landscape connectivity connectivity (With (With et et aI., al., 1999), 1999), tenebrionid tenebrionid beetles beetles (Eleodes (Eleodes obsoleta) obsoleta) nevertheless nevertheless exhibited exhibited threshold threshold behavior behavior in in several several movement movement parameters parameters when when grass grass cover cover fell fell below below 20% 997). This 20% (Wiens (Wiens et et aI., al., 11997). This is is in in the the domain domain of of lacunarity lacunarity thresholds thresholds (With (With and 999), suggesting and King, King, 11999), suggesting that that landscape landscape measures measures of of gap gap structure structure may may be better better predictors predictors of of dispersal dispersal success success than than landscape landscape measures measures of of ultimately be patch population theory, patch structure. structure. This This reiterates reiterates one one of of the the main main tenets tenets of of meta metapopulation theory, that that patch patch isolation isolation measures measures (and (and therefore therefore patch-based patch-based connectivity connectivity measures measures
32 32
KIMBERLY KIMBERLYA. A. WITH WITH
that that incorporate incorporate interpatch interpatch distances) distances) are are aa strong strong correlate correlate of of colonization colonization suc success, population cess, at at least least in in extensively extensively fragmented fragmented landscapes landscapes that that meet meet the the meta metapopulation ideal ideal of of habitat habitat patches patches embedded embedded in in an an ecologically ecologically neutral neutral matrix. matrix. As As dis discussed cussed previously, previously, the the relationship relationship between between landscape landscape structure structure and and colonization colonization success landscapes, where success is is more more complicated complicated in in heterogeneous heterogeneous landscapes, where patch patch isolation isolation may may be be less less important important than than the the quality quality of of the the matrix matrix habitat habitat through through which which the the organism organism disperses. disperses. For For example, example, large-scale large-scale forestry forestry in in Sweden Sweden resulted resulted in in extensive extensive ditching ditching to to drain drain clear-cut clear-cut areas, areas, which which created created an an inhospitable inhospitable matrix matrix colonizing breeding that that prevented prevented pool pool frogs frogs (Rana (Rana lessonae) lessonae) from from colonizing breeding ponds, ponds, irre irrespective 996). spective of of their their proximity proximity to to an an occupied occupied pond pond (Sjogren-Gulve (Sj6gren-Gulve and and Ray, Ray, 11996). The The lack lack of of concordance concordance between between percolation percolation thresholds thresholds and and dispersal dispersal suc success has led investigators to invent other, seemingly more-relevant more-relevant measures of landscape landscape structure structure for for predicting predicting dispersal dispersal or or colonization colonization success success (e.g., (e.g., Schumaker, 996; Tischendorf Schumaker, 11996; Tischendorf and and Fahrig, Fahrig, 2000a) 2000a).. The The problem problem is is not not the the measure used to measure used to quantify quantify landscape landscape connectivity, connectivity, however, however, but but with with the the scale scale at at which which habitat habitat connectivity connectivity is is assessed assessed relative relative to to the the scale scale of of dispersal. dispersal. Landscape Landscape connectivity connectivity relates relates to to the the potential potential of of organisms organisms to to traverse traverse the the entire entire landscape, landscape, whereas whereas dispersal dispersal or or colonization colonization success success pertains pertains only only to to the the likelihood organism will will successfully suitable habitat likelihood that that aa dispersing dispersing organism successfully find find aa suitable habitat patch patch (or (or cell cell in in aa grid-based grid-based landscape). landscape). Although Although the the two two are are related, related, assess assessments ments of of landscape landscape connectivity connectivity and and dispersal dispersal success success are are ultimately ultimately per performed at different scales. Individual movement movement is constrained constrained in the latter formed assessment individuals that locate suitable suitable habitat assessment (success (success is is scored scored for for individuals that locate habitat within within aa dispersal dispersal neighborhood, point colonization neighborhood, at at which which point colonization occurs occurs and and individuals individuals are are assumed assumed to to stop stop moving), moving), but but not not in in the the former former where where the the emphasis emphasis is is on individuals to landscape (whether on the the ability ability of of individuals to move move across across the the entire entire landscape (whether the the organism organism actually actually does does or or not). not). Thus, Thus, the the grain grain of of movement movement may may be be the the same individuals move same ~ how how individuals move within within or or between between habitat habitat types types or or cells cells ~ but but the the spatial spatial extent extent of of movement movement is is different. different. Habitat Habitat connectivity connectivity is is obviously obviously important important for for colonization colonization success success at at some some scale. scale. The The challenge challenge is is to to identify identify what what scale scale is is appropriate appropriate for for predicting predicting colo colonization nization success success in in aa given given species, species, however. however. This This involves involves adopting adopting aa species' species' perspective perspective of of habitat habitat connectivity connectivity (Wiens (Wiens and and Milne, Milne, 1989; 1989; With, With, 1994; 1994; Pearson Pearson et 996; Vos 1 ) . Although et aI., al., 11996; Vos et et aI., al., 200 2001). Although this this has has been been done done using using percolation percolationbased neutral related approach based neutral landscape landscape models models (see (see With, With, 2002), 2002), aa related approach involves involves the the use use of of graph graph theory. theory. In In graph graph theory, theory, the the grid grid structure structure of of the the landscape landscape is is represented represented as as aa graph graph in in which which habitat habitat patches patches (vertices (vertices or or nodes) nodes) are are con con(lines or nected nected across across varying varying distances distances (lines or edges) edges) (Urban (Urban and and Keitt, Keitt, 2001 2001).) . The The graph representation permits measure of graph representation permits aa process-based process-based measure of connectivity connectivity for for indi individual patches Overall connectivity vidual patches as as well well as as the the entire entire landscape. landscape. Overall connectivity of of the the graph graph (i.e., assessed in (i.e., landscape) landscape) is is simply simply assessed in terms terms of of whether whether each each node node is is connected connected to node. Although several ways to some some other other node. Although there there might might be be several ways to to connect connect the the various various nodes nodes of of the the graph graph to to form form aa spanning spanning tree, tree, the the one one with with the the shortest shortest minimum spanning length length is is termed termed the the minimum spanning tree. tree. There There is is aa critical critical threshold threshold distance disconnected, reminiscent reminiscent of distance at at which which the the graph graph becomes becomes disconnected, of the the percola percolation tion threshold threshold of of landscape landscape connectivity connectivity for for grid-based grid-based landscapes landscapes (Urban (Urban and and Keitt, Keitt, 2001 2001).) . Using Using aa graph-theoretic graph-theoretic approach, approach, van van Langevelde Langevelde (2000) (2000) identi identified fied different different scales scales of os connectivity connectivity and and related related this this to to colonization colonization patterns patterns of os the the European European nuthatch nuthatch (Sitta (Sirra europaea) europaea) occupying occupying woodlots woodlots within within fragmented fragmented
METAPOPULATIONDYNAMICS 2. METAPOPULATION
33 33
Netherlands (Fig. 2.2a). Patch Patch occupancy patterns of landscapes of The Netherlands nuthatches were correlated with with a critical threshold distance distance of 2.4-3 2.4-3 km nuthatches that woodlots woodlots located located >3 >3 km from a neighboring neighboring forest patch (Fig. 2.2b), such that were unlikely to be colonized by dispersing nuthatches (van Langevelde, 2000). nuthatch populations populations is related to to both both the connectivity The extinction of local nuthatch size of forested patches (Verboom et aI., al., 11991) and the size 99 1 ) and underscores again habitat connectivity ~ at some scale ~ for population population per perthe importance of habitat Merriam, 11985). metapopulation dynamics dynamics of sistence (e.g., Fahrig and Merriam, 985). The metapopulation nuthatches within this fragmented landscape have also been assessed using an incidence function model (Ter Braak et aI., al., 11998). incidence 998).
a
Above threshold (4 km)
Below threshold (2 km)
b � "5
U Q)
c c 0 u
'0
� Q)
"0
Ol Q)
�
Q) >
0
1 2,000 1 0,000 8,000 6,000 4,000 2,000 0 0
2
3
4
5
6
7
8
9
10
Threshold distance (km) Fig. 2.2 2.2 (a) A A graph graph theoretic of habitat habitat connectivity connectivity for for European nuthatches nuthatches (Sitta (Sitta Fig. theoretic analysis of europaea) in an agricultural agricultural landscape based based on on an analysis assuming assuming a dispersal distance distance of of 2 km km (just Gust below below the the landscape connectivity connectivity threshold; threshold; b) b) and and a dispersal distance distance of of 44 km km (above (above the the connectivity threshold; threshold; b). Gray areas are habitat habitat fragments, fragments, and and black black lines indicate indicate landscape connectivity connections connections among among patches based on on the the indicated dispersal distance. distance. (b) (b) Landscape Landscape connectivity connectivity exhibits a threshold threshold in in this landscape landscape at at about about 2.5 km. Patches farther farther than than 3 km km apart apart were were unlikely to to be colonized colonized by nuthatches. nuthatches. Modified Modified from from van van Langevelde Langevelde (2000). (2000).
KIMBERLY A. A. WITH WITH KIMBERLY
334 4
Relative Importance Importance of of Dispersal Dispersal for for Metapopulation Metapopulation Persistence Persistence Relative Studies on how how landscape landscape structure structure and and matrix matrix heterogeneity heterogeneity affect affect dispersal dispersal Studies on (colonization) assume assume that that the the fine-scale fine-scale movements movements of of individuals individuals translate translate (colonization) into broader broader patterns patterns of of population population distribution distribution (e.g., (e.g., Turchin, Turchin, 1991; 1 99 1 ; With With into and Crist, Crist, 1996; 1 996; With With et et al., al., 1997), 1 997), which which in in turn turn may may have have consequences consequences for for and metapopulation persistence persistence on on the the landscape. landscape. How How important important is is dispersal dispersal for for metapopulation predicting metapopulation metapopulation persistence? persistence? Dispersal Dispersal is is aa key key component component of of most most predicting spatially explicit explicit population population models, models, which which serve serve as as the the main main arsenal arsenal in in the the spatially landscape ecological ecological approach approach to to predicting predicting metapopulation metapopulation persistence persistence in in landscape fragmented landscapes, landscapes, particularly particularly in in evaluating evaluating the the consequences consequences of of differdiffer fragmented ent scenarios scenarios of of land-use land-use change change on on extinction extinction risk risk for for species species of of conservation conservation ent concern (Dunning (Dunning et et al., al., 1995). 1 995). As As mentioned mentioned previously, previously, it it is is difficult difficult to to concern obtain speciesspecies- and and habitat-specific habitat-specific information information on on dispersal, dispersal, which which may may result result obtain in errors errors in in the the estimation estimation of of dispersal dispersal success. success. Such Such estimation estimation errors may in errors may propagate in spatially affect estimates of species' species' extincextinc propagate in spatially explicit explicit models models and and affect estimates of tion risk (Ruckleshaus (Ruckleshaus et et al., al., 1997), 1 997), although although the magnitude of of these these errors tion risk the magnitude errors may have been been overestimated overestimated (Mooij (Mooij and and DeAngeles, DeAngeles, 1999). 1 999). may initially initially have Landscape structure not always always important important for predicting dispersal dispersal sucsuc Landscape structure is is not for predicting cess, however. Using an an individual-based individual-based model model of dispersal on on neutral neutral land cess, however. Using of dispersal landscape King and and With that the the mean-field mean-field approximation approximation scape models, models, King With (2002) (2002) found found that [Eq. (2. 1 )] was was sufficient sufficient for for predicting predicting dispersal dispersal success when >40% >40% of of the the [Eq. (2.1)] success when landscape below this this level, level, specifics related to to dispersal dispersal behavior behavior landscape was was suitable; suitable; below specifics related and landscape pattern became more important. Given and landscape pattern became more important. Given that that many many species species of of conservation in landscapes <40% suitable conservation concern concern occur occur in landscapes with with substantially substantially <40% suitable habitat, however, however, it is likely that landscape structure the configuration and habitat, it is likely that landscape structure m the configuration and heterogeneity generally be heterogeneity of of land-cover land-cover types types - - will will generally be important important for for predicting predicting dispersal dispersal success. success. Although Although dispersal dispersal (colonization) (colonization) success success is is considered considered an an important important process process necessary population persistence, necessary for for meta metapopulation persistence, demographic demographic factors factors that that affect affect extinction may actually more important some species 999; extinction risk risk may actually be be more important for for some species (South, (South, 11999; With 999b). This With and and King, King, 11999b). This is is especially especially true true for for good good dispersers, dispersers, such such as as birds, birds, where landscape landscape structure has a greater effect effect on reproductive reproductive output output through through edge 995b; Dooley Dooley and edge effects effects than than on on immigration immigration rates rates (e.g., (e.g., Donovan Donovan et et al., al., 11995b; and Bowers, 998). The landscape structure Bowers, 11998). The effect effect of of landscape structure on on demography demography and and extinction extinction risk risk on on metapopulation metapopulation persistence persistence is is explored explored in in the the next next section. section.
Landscape Landscape Effects Effects on on Demography Demography and and Extinction Extinction Risk Risk Habitat Habitat loss and fragmentation fragmentation pose the greatest threats to biodiversity (Wilcove 998) and (Wilcove et et al., al., 11998) and are are the the inevitable inevitable consequence consequence of of the the transforma transformation tion of of landscapes landscapes by by humans humans (With, (With, 2004). 2004). Beyond Beyond the the sheer sheer magnitude magnitude and and rate problem: the rate of of this this transformation transformation lies lies aa more more insidious insidious problem: the effects effects of of habi habitat tat loss loss and and fragmentation fragmentation on on population population viability viability are are not not linear. linear. Habitat Habitat loss loss may may precipitate precipitate aa sudden sudden and and rapid rapid decline decline in in the the probability probability of of metapopula metapopulation tion persistence persistence (i.e., (i.e., aa threshold). threshold). Using Using aa demographic demographic model model founded founded on on Levins' 1 969) classic 1987) first Levins' ((1969) classic metapopulation metapopulation model, model, Lande Lande ((1987) first defined defined extinction extinction thresholds thresholds for for territorial territorial vertebrates vertebrates as as aa function function of of their their demo demopotentials (k, (k, a composite composite parameter parameter derived algebraically from graphic potentials
2. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS 2.
335 5
life-history parameters parameters such such as as net net lifetime lifetime reproductive reproductive output, output, R'o, R' 0' and and disdis life-history persal ability, ability, m). m). The The extinction extinction threshold threshold is is the the critical critical level level of of habitat habitat (h~) (hcl persal at which which the the population population no no longer longer occurs occurs on on the the landscape landscape (patch (patch occupancy, occupancy, at p'~ p * == 0) 0) and and is is defined defined mathematically mathematically as as hhec == 11 -- k.k. The The decline decline in in patch patch occupancy accelerates accelerates past past aa certain certain reduction reduction in in habitat, habitat, such such that that the the occupancy approach to to the the extinction extinction threshold threshold (h,) (he) is is usually usually nonlinear. nonlinear. approach Lande's (1987) ( 1 987) model model was was in in the the tradition tradition of of the the classical classical metapopulation metapopulation Lande's model, which which only only assumes assumes the the existence existence of of habitat habitat patch patch structure structure (i.e., (i.e., itit is is model, spatially implicit). implicit). Habitat Habitat is is assumed assumed to to be be distributed distributed randomly randomly across across the the spatially landscape or, or, alternatively, alternatively, is is randomly randomly accessible accessible by by dispersing dispersing individuals. individuals. landscape Bascompte and and Sold Sole (1996) ( 1 996) developed developed aa spatially spatially explicit explicit realization realization of of this this Bascompte model using using grid-based grid-based landscapes landscapes with with random random habitat habitat distributions distributions and and model found that that extinction thresholds generally generally occurred occurred at at about about the the same same level level found extinction thresholds (he) as as in in Lande's Lande's (1987) ( 1 987) spatially spatially implicit model, but but the the decline decline in in patch patch (h,) implicit model, occupancy occurred occurred faster, faster, resulting resulting in in steeper steeper thresholds. thresholds. occupancy The effect effect of of habitat habitat fragmentation fragmentation on on extinction thresholds has has been been The extinction thresholds explored using using fractal fractal landscapes, landscapes, which which generate generate complex complex landscape landscape patterns patterns explored across aa gradient gradient of of fragmentation fragmentation severity severity (e.g., (e.g., Fig. Fig. 2.1a; 2. 1a; With With and and King, King, across 1 999b; Hill and and Caswell, Caswell, 1999). 1999). Because of of the the complexities complexities of of how how species 1999b; interact with with fractal fractal landscape landscape patterns, patterns, itit was was necessary necessary to to parse parse the the demodemo interact (R' 0 and m) to to evaluate evaluate the the graphic potential (k) graphic potential (k) into into its its constituent constituent parameters parameters (R'o and m) these life-history life-history parameters on extinction. Different combin relative effects effects of of these parameters on extinction. Different combinrelative R' 0 and and rn m may may give give rise rise to to the the same potential (k), (k), but but ations ations of of R'o same demographic demographic potential have population persistence persistence on on the the landland have very very different different consequences consequences for for meta metapopulation scape extinction thresholds. thresholds. On On fractal fractal landscapes, reproduc scape in in terms terms of of their their extinction landscapes, reproductive 0 ) had population persistence tive output output (R' (R'o) had aa much much greater greater effect effect on on population persistence (he) (h~) than than dispersal ability (m), (m), which which is the opposite opposite of what what was found found in Lande's ((1987) 1 987) model, (in essence) essence) aa random model, which which assumes assumes (in random landscape landscape (i.e., (i.e., compare compare the the rate rate at at which which he hc declines declines as as aa function function of of increasing increasing R' R'o0 as as opposed opposed to to increasing increasing m m in in fractal fractal landscapes, landscapes, relative relative to to the the rate rate at at which which those those same same parameters parameters decline decline in in the the random random landscape, landscape, Fig. Fig. 2.3). 2.3). Enhancing Enhancing reproductive reproductive output, such as through through the conservation conservation of high-quality habitats habitats or supple supplementation mentation of of nesting nesting habitat, habitat, may may thus thus have have aa greater greater effect effect than than enhancing enhancing dispersal dispersal success, success, by by the the maintenance maintenance or or restoration restoration of of habitat habitat connectivity, connectivity, on on mitigating mitigating extinction extinction risk. risk. This This is is not not to to say say that that landscape landscape structure structure had had no no effect effect on on population population per persistence, .0) sistence, however. however. Populations Populations in in landscapes landscapes that that were were not not fragmented fragmented (H = = 11.0) were were generally generally able able to to persist persist throughout throughout almost almost the the entire entire range range of of habitat habitat avail availability .0, Fig. ability (i.e., (i.e., he h~ :::; --- 0.1; 0.1; H H = = 11.0, Fig. 2.3). 2.3). Reducing Reducing fragmentation fragmentation and and maintain maintaining habitat connectivity thus mitigate extinction risk, as expected. In fact, fact, species with with low low demographic demographic potentials, potentials, due due to to aa combination combination of of low low reproductive reproductive out output 0 ) and put (R' (R'o) and poor poor dispersal dispersal ability ability (m), (m), generally generally went went extinct extinct sooner sooner on on frag frag1987) model. mented mented landscapes landscapes (H = = 0.0) 0.0) than than predicted predicted by by Lande's Lande's ((1987) model. Because Because many many species species of of conservation conservation concern concern have have these these combined combined traits traits of of low low fecund fecundity ity and and poor poor dispersal dispersal ability, ability, such such species species may may be at at a greater risk of extinction from from habitat habitat loss loss and and fragmentation fragmentation than than previously previously suspected. suspected. The The problem problem of of how how habitat habitat fragmentation fragmentation affects affects extinction extinction thresholds thresholds has has also also been been tackled tackled by by evaluating evaluating the the metapopulation m e t a p o p u l a t i o n capacity capacity of of the the landscape. landscape. The population capacity The meta metapopulation capacity is is basically basically the the sum sum of of the the relative relative contribution contribution of of
KIMBERLY KIMBERLY A. WITH
36 36 Fractal H = O.O
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Fig. Fig. 2.3 2.3 Fragmentation effects effects on extinction thresholds for species species with with different demo demographic potentials (the combined effects of reproductive output, output, R'o, Ro, and dispersal ability, m). Random landscapes are the most extensively fragmented; .0 are the fragmented; fractal landscapes, H H= = 11.0 least fragmented (i.e., (i.e., clumped; Fig. Fig. 2.1 a). Data from and King (1 999b). 2.1a). from With and (1999b).
individual individual patches patches (their (their "value" "value")to ) to metapopulation metapopulation persistence persistence based based on their size and degree of connectivity connectivity to other other habitat habitat fragments fragments in the landscape landscape (Hanski and Ovaskainen, 2000; (Hanski 2000; Chapter Chapter 4). This is a spatially realistic exten extension 1 969) model landscape structure sion of of Levin's Levin's ((1969) model in in which which landscape structure (patch (patch area area and and iso isolation) is allowed to affect the metapopulation metapopulation processes processes of colonization colonization and and extinction. extinction. For For example, example, the the probability probability of of extinction extinction is is calculated calculated as as aa function function of of the the inverse inverse of of patch patch area area because because extinction extinction is is more more likely likely in in small small patches patches than (see Chapter than in in large large ones ones (see Chapter 4 4 for for the the mathematical mathematical details details of of this this model). model). For population capacity For aa given given landscape, landscape, the the meta metapopulation capacity increases increases with with the the disper dispersal enhances connectivity sal range range of of the the species species because because dispersal dispersal enhances connectivity and and thus thus patch patch colonization colonization rates. rates. For For aa given given species, species, landscapes landscapes can can be be ranked ranked according according to to their capacity populations. A their capacity to to support support viable viable meta metapopulations. A landscape landscape is is capable capable of of supporting supporting a viable metapopulation metapopulation if its metapopulation metapopulation capacity (analogous to model) exceeds by to the the fraction fraction of of habitat habitat in in Lande's Lande's model) exceeds aa threshold threshold determined determined by the potential" of the species' the "metapopulation "metapopulation potential" of the the species species (the (the ratio ratio of of the species' extinc extinction rates, analogous analogous to potential in tion to to colonization colonization rates, to the the demographic demographic potential in Lande's Lande's model). Landscape fragmentation, fragmentation, created model). Landscape created by by the the random random destruction destruction of of habi habitat, tat, resulted resulted in in aa decline decline in in the the metapopulation metapopulation capacity capacity of of the the landscape landscape that that was habitat was roughly roughly proportional proportional to to the the amount amount of of habitat habitat lost. lost. Destruction Destruction of of habitat in population capacity in large large blocks blocks caused caused the the meta metapopulation capacity to to decline decline slower slower than than the the loss loss of of habitat, habitat, and and therefore therefore was was less less detrimental detrimental to to metapopulation metapopulation persist persistence. ence. Landscape Landscape structure structure thus thus affects affects extinction extinction thresholds thresholds and and metapopula metapopulation tion persistence, persistence, which which is is consistent consistent with with the the findings findings of of other other spatially spatially realistic realistic metapopulation models (Hill and 999; With 999b). metapopulation models (Hill and Caswell, Caswell, 11999; With and and King, King, 11999b). Although Although these these theoretical theoretical investigations investigations demonstrate demonstrate that that some some critical critical level level of of habitat habitat is is required required for for metapopulation metapopulation persistence, persistence, what what empirical empirical evidence evidence is is there support the extinction thresholds? there to to support the existence existence of of extinction thresholds? Although Although extinction extinction thresholds thresholds have have been been quantified quantified mathematically mathematically for for various various species species based based on on available available demographic demographic information information and and estimates estimates of of the the fraction fraction of of suitable suitable habi habitat tat in in the the landscape landscape (e.g., (e.g., Lande, Lande, 1988; 1988; Carlson, Carlson, 2000), 2000), extinction extinction thresholds thresholds have have been been identified identified empirically empirically as as an an abrupt abrupt decline decline in in the the occupancy occupancy of of habitat habitat patches patches across across aa series series of of landscapes landscapes that that vary vary in in the the amount amount of of habitat. habitat. For For example, (Melitaea cinxia) cinxia) exhibited exhibited example, the the endangered endangered Glanville Glanville fritillary fritillary butterfly butterfly (Melitaea
2. METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICS 2.
37 37
metapopulation capacity of patch net neta threshold response to declines in the metapopulation works distributed among the A land Islands in southwest Finland (Hanski and ]kland Ovaskainen, Ovaskainen, 2000; 2000; Chapter Chapter 4). 4). Many Many birds birds may may not not exhibit exhibit aa threshold threshold response response to the amount of habitat, however, particularly if they are migratory and exist regionally due to coupled source and sink landscape dynamics (With and King, 200 1 ). Such species may be able to occupy all remaining habitat fragments even 2001). found for most neotropical migra migrain extensively fragmented landscapes, as was found tory songbirds across a landscape gradient of increasing agricultural dominance al., 1999). Threshold Threshold responses to a in southern Ontario, Canada (Villard et aI., reduction reduction in in forest forest cover cover were were generally generally absent absent for for most most species, species, except except for for two two species (ovenbird, Seiurus Seiurus aurocapillus; aurocapillus; black-and-white warbler, Mniotilta Mniotilta varia) varia) <10% that were not found in landscapes with < 1 0% mature forest. For For many many species, species, such such as as neotropical neotropical migrants, migrants, the the efficiency efficiency of of patch patch occupancy occupancy does does not not decline decline with with habitat habitat loss loss and and fragmentation. fragmentation. Many Many plant plant species may also maintain maintain constant constant patch occupancy occupancy despite a reduction reduction in i.e., suitable habitat, habitat, due to life-history strategies such as seed dormancy dormancy ((i.e., species escape in time as well as space; Chapter 8 ) . Although Chapter 118). Although the extinction extinction threshold threshold is defined as the fraction fraction of all sites that that are suitable but but not not occu occulandscape, Eriksson and and Kiviniemi ((1999) performed a modified modified cal calpied on a landscape, 1999) performed culation in which which the "quasi-equilibrium" "quasi-equilibrium" threshold threshold was obtained obtained as only the culation fraction of suitable sites that that were not not occupied occupied (hi (h'c). 44% fraction c ) . By this measure, 44% ((8/18) 8/1 8 ) of plants they Sweden were were of the the grassland grassland plants they evaluated evaluated in in southeastern southeastern Sweden extinction threshold, threshold, which which means that that although such existing below the extinction species were currently found found in these landscapes, landscapes, the amount amount of suitable habi habitat tat was was not not sufficient sufficient to to permit permit the the long-term long-term persistence persistence of of these these species; species; species response to habitat loss. loss. Thus, Thus, the the identification identification of of species exhibited exhibited aa lagged lagged response to habitat extinction and availability availability of suitable suitable extinction thresholds, thresholds, based based on site occupancy occupancy and habitat, may not always be sufficient population persistence and habitat, may not always be sufficient for for evaluating evaluating population persistence and extinction an analysis of population responses responses to to extinction risk. In such cases, an of lagged lagged population Extinction Risk in Dynamic Dynamic Landscapes). Landscapes). landscape landscape change change is required required (see Extinction Risk in
The Source-Sink Source-Sink Potential Potential of of Landscapes The Landscapes Spatial resulting from from differences in in the size, shape, and quality quality Spatial heterogeneity, resulting the size, shape, and of habitats habitats comprising the landscape, affects species' demographic demographic rates. of Reproductive success may be be maximized, maximized, or or survivorship survivorship may may be in Reproductive success may be minimized, minimized, in habitat. Habitat-specific Habitat-specific survivorship survivorship and and reproductive reproductive success set aa particular particular habitat. success set the stage source-sink dynamics dynamics (Pulliam, (Pulliam, 1988) 1988) in in which population growth growth the stage for for source-sink which population rates are are positive positive (birth ( birth rates rates exceed exceed death death rates) rates) in in some some patches patches (sources) (sources) but but rates are negative negative in in others others (sinks). (sinks). The The relative relative amount amount of of source source and and sink sink habitat habitat on on are landscape may may thus thus affect affect persistence persistence of of the the metapopulation metapopulation at at the the landscape landscape a landscape scale (Pulliam (Pulliam and and Danielson, Danielson, 1991; 1 99 1 ; Donovan Donovan et et al., aI., 1995b; 1 995b; Chapter Chapter 16). 1 6) . landscape perspective perspective is ultimately ultimately required required to to assess assess source-sink source-sink dynamdynam A landscape and to to evaluate evaluate how how changes changes in landscape landscape structure, structure, such such as from habitat ics and from habitat fragmentation or fragmentation or land-use land-use change, change, may may affect affect these these dynamics dynamics and and thus thus meta population persistence. Unfortunately, Unfortunately, most most of of the the previous previous efforts efforts to to metapopulation model source-sink source-sink dynamics dynamics have have been been spatially spatially implicit implicit (e.g., (e.g., Pulliam Pulliam model and and Danielson, Danielson, 1991), 1 99 1 ), including including those those that that have have attempted attempted to to determine determine the the effects of of habitat habitat fragmentation fragmentation on on the the source-sink source-sink status status of of populations populations effects
KIMBERLY WITH
KIMBERLY A. A. WITH
38 38
( e.g., Donovan Donovan et et al., aI., 1995a; 1 995a; but but see see Ritchie, Ritchie, 1997). 1 997). Although Although demographic demographic (e.g., rates vary vary spatially spatially in in such such models, models, they they are are usually usually fixed fixed input input parameters parameters rates that are are independent independent of of landscape landscape structure. structure. In In birds, birds, for for example, example, landscape landscape that structure is is known known to to affect affect reproductive reproductive output output in in many many species species due due to to higher higher structure edge effects effects in in fragmented fragmented landscapes landscapes in in which which nesting nesting success success is is lower lower in in habihabi edge tat fragments fragments because because of of greater greater nest nest predation predation or or brood brood parasitism parasitism (such (such as as tat Moluthrus in North America ) along frag by the the brown-headed by brown-headed cowbird, cowbird, M o l u t h r u s ater, ater, in North America) along fragedges (Donovan (Donovan et et al., aI., 1995b, 1 995b, 1997). 1 997). Thus, Thus, reproductive reproductive output output (a (a ment edges ment demographic rate) rate) is spatially dependent dependent and and varies varies as as aa function function of patch size size demographic is spatially of patch and shape, shape, being being reduced reduced in in fragments fragments dominated dominated by by edge edge and and maximized maximized in in and large patches patches of of contiguous contiguous habitat. habitat. large With and King King (2001) (20 0 1 ) devised devised aa functional functional relationship patch With and relationship between between patch structure and and reproductive reproductive success success for for neotropical neotropical migratory migratory songbirds songbirds as as part part structure of aa spatially spatially structured structured demographic demographic model model developed developed to to assess assess the the of source-sink potential potential of of fragmented landscapes. Reproductive Reproductive success success declines declines source-sink fragmented landscapes. as effects) . For For example, example, some some as aa function function of of increasing increasing edge edge (i.e., (i.e., negative negative edge edge effects). species were "edge sensitive" and decline in in reproductive reproductive out species were "edge sensitive" and exhibited exhibited aa steep steep decline output in small irregularly shaped shaped patches were dominated put in small or or irregularly patches that that were dominated by by edge edge (edge (edge index � .0; Fig. Fig. 2.4a). 2Aa). The The demographic of landscape landscape structure structure index --~ 11.0; demographic consequences consequences of were assessed the expected expected number of female female offspring offspring produced produced per per were assessed as as the number of female, per per patch, patch, for for all across the the entire entire landscape landscape (bL). ( bLl. A A simple simple female, all individuals individuals across and two-stage table combining combining fecundity fecundity (bL) (bLl and survivorship (juvenile, (juvenile, So two-stage life life table and survivorship So and adult, s) was for each 2Aa) in in aa given given landscape adult, s) was then then constructed constructed for each species species (Fig. (Fig. 2.4a) landscape (e.g., Fig. Fig. 2.1a). 2.1a). From From the the life table, we we calculated calculated the finite rate rate of of increase increase for for (e.g., life table, the finite the entire entire landscape landscape population population (A.Ll as the the solution solution to to the the characteristic characteristic equa the (XL) as equation 988): tion (Lande, (Lande, 11988): kL,~ _ skL,~-1
-
-
bLl~ = 0
(2.2)
for for A.k � - 11 and and 00 < < ss < < 1, 1, where where lex Is is is survivorship survivorship at at the the age age of of first first breeding, breeding, ss isis the (> 11 yr), yr), and the annual annual probability probability of of survivorship survivorship for for breeding breeding adults adults (> and bL bE is is
derived derived from from the the population population across across the the entire entire landscape. landscape. The The landscape landscape popu population was .0, declining lation was stable stable when when A.L XL = 11.0, declining when when A.L XL < <11 and and increasing increasing when when A.L 1 . The annual rate change in %/yr) is XL > >1. The annual rate of of change in the the size size of of the the metapopulation metapopulation ((%/yr) is (A.L .0) >,* 1100. 00. Thus, (XL -- 11.0) Thus, this this modeling modeling approach approach treats treats aa demographic demographic rate rate (b) (b) as as aa spatially spatially dependent dependent variable; variable; it it is is aa model model output output rather rather than than aa fixed fixed model model parameter parameter as as in in traditional traditional demographic demographic models. models. Furthermore, Furthermore, this this approach approach extends extends the the concept concept of of source-sink source-sink populations populations from from the the scale scale of of patches to patches to the the entire entire landscape, landscape, such such that that the the potential potential of of aa given given landscape landscape to to function function as as aa population population source source or or sink sink is is ultimately ultimately assessed. assessed. For For species species with with low low edge-sensitivity edge-sensitivity (Fig. (Fig. 2Aa), 2.4a), landscapes landscapes supported supported viable viable metapopulations metapopulations and and had had the the potential potential to to function function as as sources sources across across aa wide 1 ) . Fragmented wide range range of of available available habitat habitat (Fig. (Fig. 2Ab; 2.4b; With With and and King, King, 200 2001). Fragmented landscapes populations of landscapes (random) (random) could could not not support support viable viable meta metapopulations of this this species, species, however, .0) when however, and and functioned functioned as as sinks sinks (A.L (XL < < 11.0) when habitat habitat fell fell below below 30% 30% (Fig. (Fig. 2Ab). 2.4b). The The situation situation was was bleaker bleaker for for species species with with high high edge-sensitivity, edge-sensitivity, which which had had aa difficult difficult time time persisting persisting in in landscapes landscapes with with <50% <50% habitat habitat even even when when the the landscape landscape was was managed managed to to preserve preserve large large tracts tracts of of contiguous contiguous habi habi.0, Fig. tat tat (H (H = = 11.0, Fig. 2Ac). 2.4c).
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Fig. decline in in reproductive Fig. 2.4 2..4 (a) (a) Degree Degree of of edge edge sensitivity s e n s i t i v i t y- the the decline reproductive success success as as aa function function of increasing edge songbirds. (b) of increasing edge m for for aa couple couple of of generic generic migratory migratory songbirds. (b) Effect Effect of of fragmentation fragmentation (random (random = = maximum maximum fragmentation) fragmentation) on on aa species species with with low low edge-sensitivity. edge-sensitivity. (c) (c) Effect Effect of of frag fragmentation high edge-sensitivity. King (2001 mentation on on aa species species with with high edge-sensitivity. Modified Modified from from With With and and King (2001).).
This This spatially spatially structured structured avian avian demographic demographic model model was was parameterized parameterized for for the the Henslow's Henslow's sparrow sparrow (Ammodramus (Ammodramus henslowii) henslowii) in in aa heavily heavily managed managed land landscape of north-central aI., 2000). north-central Kentucky in the eastern eastern United United States (King et al., Henslow's Henslow's sparrow sparrow is is an an area-sensitive area-sensitive species species that that requires requires large large tracts tracts of of dense dense 994) and tallgrass tallgrass prairie prairie for for nesting nesting (patch (patch sizes sizes �30 ->30 ha; ha; Herkert, Herkert, 11994) and is is thus thus aa species of concern given 1 % of species of conservation conservation concern given that that < <1% of the the historical historical tallgrass tallgrass prairie Plains of (Knopf and prairie remains remains throughout throughout the the Great Great Plains of North North America America (Knopf and Sampson, 11994). 994). Only Only 0.6% suitable breeding breeding Sampson, 0.6% of of the the managed managed landscape landscape was was suitable habitat, which barely habitat, which was was fragmented fragmented and and consisted consisted of of many many small small patches patches barely large area requirements large enough enough to to support support aa breeding breeding pair, pair, let let alone alone meet meet the the area requirements of this species (the largest patch was 5511 ha ha).l . The finite rate of increase for for the metapopulation in the Henslow's Henslow's sparrow sparrow metapopulation in this this landscape landscape was was "-L kL = - 0.86. 0.86. Thus, Thus, the Henslow's sparrow sparrow was declining at an annual annual rate of 14%/yr 14 %/yr such that that the
KIMBERLY KIMBERLY A. A. WITH WITH
40 40
landscape output, b, be increased landscape was was aa sink sink for for this this species. species. Reproductive Reproductive output, b, must must be increased by about 1.5 by about 1.5 times times its its current current level level to to restore restore the the Henslow's Henslow's sparrow sparrow to to aa stable stable or population, which or increasing increasing meta metapopulation, which would would require require an an increase increase in in the the land landscape-wide 39 to 8 % . To scape-wide nesting nesting success success from from 39 to 558%. To reverse reverse the the current current decline decline in in this population, aa land minimize disturbances this meta metapopulation, land manager manager should should thus thus minimize disturbances that that contribute habitat fragmentation contribute to to habitat fragmentation of of grassland grassland habitat, habitat, thereby thereby increasing increasing reproductive output. output. patch sizes and minimizing minimizing edge effects that decrease reproductive Although Although this this spatially spatially structured structured demographic demographic model model focuses focuses on on how how patch patch geometry geometry for for aa single single habitat habitat type type affects affects reproductive reproductive success, success, this this approach approach can easily to landscapes in can be be extended extended easily to heterogeneous heterogeneous landscapes in which which reproductive reproductive success or varies as success or survivorship survivorship additionally additionally varies as aa function function of of habitat habitat quality. quality.
Extinction Extinction Risk Risk in in Dynamic Dynamic Landscapes Landscapes The population processes The effects effects of of landscape landscape structure structure on on the the meta metapopulation processes discussed discussed thus assumed aa static landscape in thus far far have have assumed static landscape in which which the the amount, amount, suitability, suitability, and and configuration of patches remain remain unchanged configuration of habitat habitat patches unchanged on on the the landscape. landscape. Even Even studies that habitat loss studies that have have explored explored the the effects effects of of habitat loss and and fragmentation fragmentation on on meta population persistence persistence have been conducted metapopulation have been conducted on on aa series series of of static static land landscapes habitat availability scapes representing representing aa gradient gradient of of habitat availability and and fragmentation fragmentation sever severity 9 96; Hill Caswell, 11999; 999; With ity (e.g., (e.g., Bascompte Bascompte and and Sole, Sol~, 11996; Hill and and Caswell, With and and King, King, 11999b, 999b, 2001 2001),) , which which assumes assumes that that these these landscapes landscapes all all lie lie on on aa particular particular tra trajjectory ectory of of landscape landscape change. change. Real Real landscapes landscapes are are not not static, static, however, however, especially especially given given the the current current rate rate at at which which most most landscapes landscapes are are being being transformed transformed by by human human land-use land-use activities. activities. Different Different trajectories trajectories of of land-use land-use change change could could gen generate consequences for erate similar similar landscape landscape patterns, patterns, but but have have very very different different consequences for the the dynamics persistence of dynamics and and persistence of metapopulations metapopulations on on these these landscapes. landscapes. The The rate rate of of landscape landscape change change is is an an important important component component of of landscape landscape structure 9 92; Keymer structure that that affects affects extinction extinction risk risk (e.g., (e.g., Fahrig, Fahrig, 11992; Keymer et et aI., al., 2000) 2000) and and which which may may be be responsible responsible for for generating generating metapopulation metapopulation dynamics, dynamics, particularly successional habitat Hanski, 11999a; 999a; Johnson, particularly in in ephemeral ephemeral or or successional habitat ((Hanski, Johnson, 2000). 2000). Patch Patch demographics, demographics, such such as as the the life life span span of of aa patch, patch, drive drive the the dynamics dynamics of population in critical of the the meta metapopulation in these these systems. systems. For For aa given given species, species, there there is is aa critical rate changes too relative to rate of of patch patch turnover turnover in in which which the the landscape landscape changes too fast fast relative to the the scale population persist scale of of the the extinction-colonization extinction-colonization process process to to permit permit meta metapopulation persistence ence (Keymer (Keymer et et aI., al., 2000). 2000). Metapopulation Metapopulation extinction extinction is is thus thus predicted predicted to to occur more frequently than in static landscape occur more frequently in in dynamic dynamic than in static landscape scenarios. scenarios. These These recent recent theoretical theoretical treatments treatments of of metapopulations metapopulations on on dynamic dynamic land landscapes scapes have have been been concerned concerned primarily primarily with with systems systems in in which which there there is is aa constant constant rate habitat turnover (but see Hanski and rate of of habitat turnover (but see Hanski and Ovaskainen, Ovaskainen, 2002; 2002; Ovaskainen Ovaskainen and and Hanski, Hanski, 2002; 2002; Chapter Chapter 4). 4). Many Many landscapes landscapes are are subjected subjected to to chronic chronic habi habitat tat loss loss and and fragmentation, fragmentation, however, however, in in which which habitat habitat that that has has been been destroyed destroyed is restored. To this latter is not not restored. To address address this latter scenario, scenario, Schrott, Schrott, With With and and King King (unpub (unpublished) extended the spatially structured structured avian demographic model of With and King (200 1 ) to dynamic landscape King (2001) to aa dynamic landscape context context in in which which habitat habitat was was destroyed destroyed at at various %/yr) until denuded. The various rates rates (0.5, (0.5, 11 and and 55%/yr) until the the landscape landscape was was entirely entirely denuded. The most surprising result result of population appeared appeared to most surprising of this this analysis analysis was was that that the the meta metapopulation to persist across aa greater range of lost rap persist across greater range of habitat habitat destruction destruction when when habitat habitat was was lost rapthan when it was destroyed slowly (0.5%/yr); in words, idly ((5%/yr) 5%/yr) than in other words,
2.
METAPOPULATION DYNAMICS DYNAMICS METAPOPULATION
41 41
extinction extinction appeared appeared to to occur occur sooner sooner in in landscapes landscapes subjected subjected to to lower lower rates rates of of disturbance disturbance (Fig. (Fig. 2.5a) 2.5a).. This This paradox paradox is is resolved resolved by by considering considering the the decline decline in in metapopulation metapopulation growth growth rates rates as as aa function function of of time time (Fig. (Fig. 2.5b), 2.5b), which which demon demonstrates strates that that populations populations on on landscapes landscapes subjected subjected to to rapid rapid rates rates of of habitat habitat loss loss ((5%/yr) 5%/yr) will will go go extinct extinct within within 20 20 yr yr (the (the time time to to total total landscape landscape denudation), denudation), whereas whereas populations populations in in landscapes landscapes subjected subjected to to lower lower rates rates of of habitat habitat loss loss can can apparent prolonged prolonged persistence of persist for up to three times as long. The apparent meta populations in metapopulations in landscapes landscapes undergoing undergoing rapid rapid change change results results from from aa lagged lagged response response by by the the species. species. The The generic generic migratory migratory songbird songbird being being modeled modeled in in this this study %/yr, total study had had aa life life span span of of 88 yr, yr, such such that that at at aa habitat habitat loss loss rate rate of of 55%/yr, total denudation denudation of of the the landscape landscape would would occur occur in in aa little little over over two two generations. generations. The The landscape landscape is is changing changing more more rapidly rapidly than than the the demographic demographic potential potential of of the the species, species, and and thus thus declines declines in in the the metapopulation metapopulation growth growth rate rate (ALJ (kL) lag lag behind behind the rate of habitat loss. This "extinction debt" has been demonstrated to be
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Years Years % habitat habitat destroyed/yr) destroyed/yr) on on metapopumetapopu Fig. 2.5 2.5 (a) Effect of the the rate of of habitat habitat destruction destruction (r, Fig. Effect of (r, % lation persistence for aa species species with with intermediate intermediate edge edge sensitivity sensitivity in in moderately fragmented lation persistence for moderately fragmented 0.5) dynamic dynamic landscapes. landscapes. Because Because of of the the initial initial conditions conditions of of the the model, model, in in which which the the (H == 0.5) (H (AL == 1.0), 1 .0), the the landscape landscape population population can can only only decline decline metapopulation growth rates are are stabilized stabilized (~kL metapopulation growth rates as habitat is lost. lost. (b) (b) Time TIme to to extinction extinction for for the the same same species species in in dynamic dynamic landscapes landscapes undergoing undergoing as habitat is moderate fragmentation fragmentation (H (H -= 0.5) 0.5) at at different different rates. From From Schrott, Schrott, With With and and King King (unpublished). (unpublished). moderate
KIMBERLY WITH KIMBERLY A. A. WITH
42 42
especially great for meta populations that especially great for metapopulations that are are close close to to their their extinction extinction thresholds thresholds following habitat loss following habitat loss and and fragmentation fragmentation (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2002). 2002). These These results results emphasize emphasize the the importance importance of of understanding understanding the the historical historical landscape patterns, habiforces shaping landscape patterns, such as the rate of land-use change or habi tat tat destruction, destruction, for for assessing assessing species' species' extinction extinction risk. risk. Two Two landscapes landscapes with with iden identical tical metrics metrics could could have have achieved achieved their their present present state state via via vastly vastly different different trajectories change, with trajectories of of landscape landscape change, with different different consequences consequences for for metapopula metapopulation (Fig. 2.5a). landscape metrics tion persistence persistence (Fig. 2.5a). Conventional Conventional landscape metrics thus thus cannot cannot be be used populations in used to to evaluate evaluate extinction extinction risk risk for for meta metapopulations in dynamic dynamic landscapes. landscapes. Time Time lags lags in in species' species' responses responses to to landscape landscape change change may may also also create create relict relict dis distributions species' occurrence explained by tributions in in which which the the species' occurrence is is better better explained by aa historical historical landscape configuration "ghosts of landscape past"; landscape configuration than than the the current current one one (("ghosts of landscape past"; Nagelkerke Nagelkerke et et aI., al., 2002). 2002). For For example, example, the the distribution distribution of of carabid carabid beetles beetles (Abax (Abax parallelepipedus) within an parallelepipedus) in in aa hedgerow hedgerow network network within an agriculturally agriculturally dominated dominated landscape France was landscape of of France was better better explained explained by by the the well-connected well-connected hedgerow hedgerow distribution from Burel, 1998). distribution from 40 40 yr yr ago ago than than the the current current network network (Petit (Petit and and Burel, 1998). From From aa conservation conservation and and land-management land-management standpoint, standpoint, the the potential potential for for lagged lagged population responses is disquieting because population responses is especially especially disquieting because the the effects effects of of landscape landscape change change may may go go unnoticed unnoticed for for long long periods periods of of time, time, such such that that the the window window of of opportunity opportunity for for affecting affecting aa recovery recovery may may close close before before the the problem problem is is realized realized and action is taken. Alternatively, such lagged effects may buy the necessary time measures before time in in which which to to implement implement conservation conservation and and restoration restoration measures before the the species goes extinct. This assumes that that the problem problem can be recognized in time, which argues for the importance importance of performing performing theoretical theoretical and empirical analy analyses landscape change ses of of the the effect effect of of dynamic dynamic landscape change on on extinction extinction risk. risk.
Relative Relative Effects Effects of of Habitat Habitat Loss Loss and and Fragmentation Fragmentation on on Metapopulation Metapopulation Persistence Persistence A A number number of of empirical empirical and and theoretical theoretical investigations investigations have have attempted attempted to to the relative relative importance importance of of the the amount amount of of habitat habitat versus versus the the degree degree of of assess the habitat habitat fragmentation fragmentation on species occurrence occurrence and and extinction extinction thresholds thresholds (e.g., McGarigal 995; Trzcinski et aI., 999; Villard et aI., 999; McGarigal and and McComb, McComb, 11995; al., 11999; al., 11999; Fahrig, 997, 2002; Fahrig, 11997, 2002; Flather Flather and and Bevers, Bevers, 2002). 2002). Part Part of of the the difficulty difficulty in in evalu evaluating the relative relative effects of these two two components components of landscape landscape structure, structure, however, thresholds are however, has has been been differences differences in in how how extinction extinction thresholds are defined, defined, which which is measure of is aa consequence consequence of of the the specific specific modeling modeling approach approach or or measure of population population viability viability used used to to assess assess extinction extinction risk. risk. Depending Depending on on the the modeling modeling construct, construct, extinction thresholds thresholds have been defined variously as extinction have been defined variously as the the critical critical level level of of habi habitat 1 ) is tat at at which which the the population population ((1) is unlikely unlikely to to persist persist for for (or (or after) after) some some speci specified amount individual-based simulation amount of time (e.g., duration duration of run in individual-based simulation models; Fahrig, 997; Flather 1 ), (2) models; Fahrig, 11997; Flather and and Bevers, Bevers, 200 2001), (2) no no longer longer occupies occupies any any of the available habitat habitat (metapopulation (metapopulation models, where p'; p* = 0; Lande, 1987; Bascompte 996; With 999b), or 3 ) is Bascompte and and Sole, Sold, 11996; With and and King, King, 11999b), or ((3) is no no longer longer stable, stable, as as assessed assessed by by some some demographic demographic index, index, such such as as net net reproductive reproductive rate rate (Ra) (Ro) or or the population growth the finite finite rate rate of of population growth (A) (k) (spatially (spatially structured structured demographic demographic models, .0; With measures of pop models, where where Ra Ro or or Ak < < 11.0; With and and King, King, 2001 2001).). Some Some measures of population viability may be more more sensitive to fragmentation fragmentation effects, however however ((Flather Flather and Bevers, Bevers, 2002); measures of population 2002); for example, example, patch-based patch-based measures population
2.
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viability (patch occupancy, p" p*,, of metapopulation metapopulation models) may be more sen sensitive to fragmentation fragmentation effects than than probabilities probabilities of population population persistence derived from individual-based simulation models. Thus, the debate over the relative effects of habitat population persist habitat loss and fragmentation fragmentation on meta metapopulation persistence is muddled muddled by the specific modeling approach approach employed in theoretical studies (Flather and Bevers, 2002), 2002), the corresponding corresponding measure of population population viability used, and the taxonomic taxonomic group group being assessed assessed in the case of empirical studies. For example, birds may not be the best test of the relative importance importance of fragmentation effects, effects, as many species are efficient at occupying available habitat 999). habitat even when habitat habitat is rare (e.g., Villard Villard et ai., al., 11999). Nevertheless, population Nevertheless, habitat amount is generally the best predictor of meta metapopulation persistence, although fragmentation m the explicit arrangement of patches in space m becomes increasingly important below the extinction threshold (Fahrig, 11997; 997; Flather and Bevers, 2002). As pointed out in the previous section, however, the discussion of the relative effects of habitat loss and fragmentation on metapop metapopinformation on landscape dynamics. ulation persistence is moot in the absence of information The rate of patch turnover turnover or habitat destruction destruction has been shown to have a more profound profound effect on extinction thresholds than either the amount amount or the fragmen fragmentation of habitat (Keymer et ai., al., 2000; Schrott, With and King (unpublished). (unpublished).
2.3 2.3
SUMMARY SUMMARY Although landscape population ecology share a common landscape ecology and meta metapopulation common goal in predicting predicting the persistence of spatially structured structured populations populations in frag fragmented landscapes, they differ in scope and academic tradition, tradition, which is reflected in the different approaches approaches typically employed by ecologists from the two two disciplines. Landscape Landscape ecology is an interdisciplinary field that that arose from European European traditions traditions of regional geography and vegetation science, science, and which combined combined the spatial approach of the geographer with the functional functional approach approach of the ecologist (Turner et ai., 1 ) . As a consequence, there has been greater al., 200 2001). emphasis placed on remotely sensed data, geographical information information systems, and spatial statistics to generate, display, and analyze complex landscape landscape pat patterns within within landscape landscape ecology, as opposed opposed to the more abstract abstract representation representation of landscapes inspired by the patch-based ecological theory adopted adopted by meta population ecology. In North metapopulation North America, the systems ecology background background of many "first-generation" "first-generation" landscape ecologists contributed contributed to the widespread widespread use of computer computer simulation models to tackle problems related to the effect of land-use change on resource resource management, management, an application application that that additionally could take advantage advantage of GIS. GIS. Subsequently, this led to the development development of spa spatially explicit population population models, the main tool of the landscape landscape ecologist for assessing population population viability in fragmented landscapes. In contrast, contrast, metapopu metapopulation ecologists hail from a background population ecology that background in population that is rich in mathematical mathematical theory and thus tend to approach approach the problem of population population persistence persistence in fragmented landscapes analytically rather rather than than numerically. Despite Despite its its diverse diverse disciplinary disciplinary breadth, breadth, landscape landscape ecology is is fundamentally fundamentally concerned with the effects effects of spatial pattern on ecological processes (Turner, 11989), 989), at whatever scale spatial heterogeneity emerges. In the context context metapopulation dynamics, any study that that incorporates incorporates the effect of spatial of metapopulation
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KIMBERLY KIMBERLYA. WITH
pattern, such as habitat fragmentation or matrix heterogeneity, on processes con contributing to metapopulation persistence is, by definition, adopting a landscape ecological perspective. Thus, it could be argued that the "exciting scientific synthesis" between landscape ecology and metapopulation theory that was envienvi sioned by Hanski and Gilpin ((1991) 1 99 1 ) a decade ago is well underway. This is evident in current metapopulation metapopulation theory that that utilizes spatially explicit (or realis realistic) models (e.g., Hanski, 11999b) 999b) and by the use of analytical approaches from metapopulation 999b). metapopulation theory in landscape ecology (e.g., With and King, 11999b). The goal of this chapter has has been to demonstrate demonstrate that landscape ecology has more to offer meta population ecology than metapopulation than just the incorporation incorporation of a broader spatial scale or landscape heterogeneity to existing meta population theory by metapopulation providing a spatial context for understanding processes contributing to the populations. Nevertheless, there are still sev dynamics and persistence of meta metapopulations. several areas at this nexus of landscape ecology and meta population biology that metapopulation that are in need of further research: First, spatially structured demographic models require further development and testing. Demographic factors, such as repro reproductive output output and survivorship, may be spatially dependent. Estimates of meta population viability that ignore this spatial dependency may give erroneous metapopulation and overly optimistic assessments of a species' status on a landscape (With and King, 200 1 ) . Second, matrix effects on dispersal success and extinction risk need 2001). importance of to be evaluated further. Research has demonstrated the potential importance managing the matrix for enhancing colonization success and reducing extinction risk, and thus we need to move beyond assessment of mere patch size size and iso isolation effects on metapopulation persistence (e.g., Fleishman et aI., al., 2002), par particularly in understanding understanding asymmetrical flows among populations populations and evaluating the source-sink potential of landscapes for species of conservation concern. Third, the effect of landscape dynamics on extinction thresholds deserves greater attention (e.g., Keymer et aI., al., 2000; Schrott, With and King (unpublished)). Chronic habitat loss and fragmentation increase the potential for lagged responses to landscape change, which may produce an extinction debt (Hanski and Ovaskainen, 2002), such that the status and future viability of meta populations may not be well predicted by current landscape patterns. metapopulations Fourth, meta population viability analysis needs to be extended to a broader, metapopulation approach" is particularly important important regional scale. A "metalandscape modeling approach" for assessing dynamics among among source-sink source-sink landscape populations populations (e.g., Donovan 995a). Finally, empirical work to address these issues, analo Donovan et aI., al., 11995a). analogous to how population theory has been applied to real meta populations in how meta metapopulation metapopulations metapopulation metapopulation ecology (Chapters 4 and 5), should be a research priority in landscape ecology. Otherwise, Otherwise, continued continued progress progress toward toward this developing syn synthesis between landscape ecology and metapopulation metapopulation ecology will slow.
3
CONTINUOUS-S PA CE CONTINUOUS-SPACE MODELS MODELS FOR PO PULATION POPULATION DYNAMICS DYNAMICS Benjamin M. Bolker
3.1 3.1
INTRODUCTION INTRODUCTION Metapopulation Metapopulation ecologists have explored the dynamics of biological popu populations that because they live in patchy that are discontinuous discontinuous both spatially ((because patchy habi habitats) and temporally ((because because local populations .The populations frequently go extinct) extinct).The success of meta population theory comes both from metapopulation from the ecological importance of such populations (e.g., populations populations in fragmented fragmented landscapes) landscapes) and and from the simplicity of meta population theory. Dealing with population dynamics in metapopulation more general continuous continuous landscapes m those that that are neither neither completely patchy nor completely hhomogeneous o m o g e n e o u s- requires some compromise between generality and tractability, which has in turn turn divided ecologists into two two camps. Landscape Landscape ecologists ecologists have focused on large-scale patterns patterns of popula population distributions distributions in heterogeneous heterogeneous environments, environments, generally considering the effects of exogenous environ exogenous heterogeneity imposed by the abiotic or biotic environment with less less concern for the endogenous endogenous heterogeneity caused by interac interactions within the populations. populations. In contrast, contrast, spatial spatial ecologists ecologists have focused on smaller scale patterns patterns of population population distribution or expansion expansion and have been more interested in endogenous endogenous than in exogenous heterogeneity. Landscape Landscape
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BENJAMIN BENJAMIN M. BOlKER BOLKER
ecologists 982), designed ecologists favor favor tactical tactical models models (Nisbet (Nisbet and and Gurney, Gurney, 11982), designed to to answer questions about populations (e.g., answer specific specific questions about specific specific populations (e.g., distributions distributions of of extinction extinction times times under under different different management management scenarios), scenarios), whereas whereas spatial spatial ecolo ecologists "toy" models, gists generally generally prefer prefer strategic strategic or or "toy" models, which which are are more more abstract abstract and and are designed to explore general principles of spatial dynamics (see Chapter ). dynamics (see Chapter 11). Landscape Landscape ecologists ecologists use use computational computational tools tools such such aass geographic geographic information information systems GIS) and models, which flexible enough systems ((GIS) and individual-based individual-based models, which are are flexible enough to to incorporate realistic patterns incorporate realistic patterns of environmental environmental heterogeneity heterogeneity and individual individual behavior, behavior, whereas whereas spatial spatial ecologists ecologists use use simpler simpler models, models, such such as as partial partial differ differential equations equations (PDEs) and interacting interacting particle systems (IPSs), which may lead to to more more general general results results and and are are easier easier to to analyze. analyze. (As (As discussed discussed in in Chapters Chapters 11 and population models models are used in entire range tactical to and 4, 4, meta metapopulation are used in the the entire range from from tactical to strategic modeling.) This chapter brief! ) overview of models for ecological dynamics chapter gives a ((brief!) dynamics in continuous continuous space, space, focusing focusing on on spatial spatial rather rather than than landscape landscape ecology. ecology. In In partic particular, ular, it it explores explores spatial spatial moment moment eequations, relatively new new framework framework for for ana anaquations, aa relatively lyzing population densities lyzing spatial spatial dynamics dynamics in in terms terms of of mean mean population densities and and spatial spatial covariances. covariances. The The chapter chapter gives gives aa sample sample derivation derivation of of aa set set of of spatial spatial moment moment equations; equations; summarizes summarizes various various applications applications of of moment moment equations equations for for single singlespecies species and and community community dynamics; dynamics; contrasts contrasts the the strengths strengths and and weaknesses weaknesses of of the the IPSs; and discusses future approach with other frameworks, frameworks, such as PDEs and IPSs; directions moment equations. directions and and potential potential of of spatial spatial moment equations. Although Although this this chapter chapter concentrates endogenous rather concentrates on on the the effects effects of of endogenous rather than than exogenous exogenous variability, variability, it it also also describes describes some some strategies strategies for for incorporating incorporating both both kinds kinds of of heterogeneity heterogeneity and and bridging the gap between spatial and landscape ecology. Spatial moment moment equa equations are a powerful tool for this task; other other advantages advantages (not to leave the reader reader in suspense for too long) include preservation preservation of the spatial and and stochastic character analytical tractability; character of of ecological ecological systems; systems; analytical tractability; and and simple simple connections connections to individual dispersal to field field data data on on individual dispersal and and performance performance and and to to well-established well-established spatial spatial statistical statistical measures. measures.
3.2 3.2
OVERVIEW OVERVIEW OF OF CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS The The full full range range of of continuous-space continuous-space models models for for ecological ecological systems systems is is far far too too large to review properly properly in a short short book book chapter, but but this section gives abbrevi abbreviated ated descriptions descriptions that that serve serve at at least least to to put put spatial spatial moment moment equations equations in in perspective. For each category discussed earlier, the the text text describes the the basic approach; approach; gives gives some some examples examples of of how how it it has has been been used used to to study study ecological ecological dynamics; dynamics; discusses discusses how how models models of of this this type type have have been been used used to to explore explore the the combination of endogenous combination endogenous and exogenous exogenous variability; and and gives some starting points in in the literature literature for further points further exploration. exploration. Table Table 3.1 3.1 categorizes categorizes mathematical mathematical models models for for population population dynamics dynamics in in con continuous tinuous space. space. Most Most of of these these models models are are discussed discussed in in the the following following sections. sections. Any Any I) can model model with with discrete discrete individuals individuals (type (type = = I) can be be considered considered an an individual-based individual-based model model (although (although the the following following section section focuses focuses on on relatively relatively complex, complex, flexible flexible IBMs), IBMs), whereas whereas models models with with continuous continuous individuals individuals are are covered covered in in the the section section on on continuum continuum models. models. All All of of the the discrete-space discrete-space models models in in Table Table 3.1 3.1 have have regularly regularly L) arranged arranged sites sites or or patches patches and and so so fall fall under under the the rubric rubric of of lattice lattice models models (type (type = = L)
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TABLE TABLE 3.1 3.1
i
Partial Enumeration of Continuou s-Space Models Continuous-Space Models
Space
Time
Population
Random Random
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Continuous Discrete Continuous Continuous Continuous Discrete Discrete
Continuous Continuous Continuous Continuous
Discrete Discrete Either Either
Continuous Continuous
Continuous Continuous Continuous Continuous
Continuous Continuous
Continuous Continuous Continuous Continuous
Deterministic Deterministic Cellular Cellular automaton (CA) (CA) Stochastic Stochastic Probabilistic Probabilistic CA CA = = stochastic stochastic CA CA Either Either Coupled-map Coupled-map lattice Stochastic Stochastic Interacting Interacting particle particle system system =pair ~pair approximations Either Either Integrodifference equation equation Stochastic Spatial point Stochastic Spatial point process process =spatial ~spatial moment moment equations Deterministic Integrodifferential Deterministic Integrodifferential equation (IDE), (IDE), equation (PDE) partial differential differential equation (PDE) = = reaction-diffusion reaction-diffusion equation Stochastic Stochastic IDE, PDE Stochastic StochasticIDE, PDE
Continuous Continuous Discrete Discrete
Model
Type Type"" I,L I,L L I,L I,L C I C
C C
aa "Type" model; L, lattice model; C, continuum continuum model) "Type" (I, (I, individual-based individual-based model; lattice model; model) gives gives a reference reference to the section(s} section(s) ill
i
i
i
i
that describe describe different different classes classes of models models -- some some models models fall fall in more than one class. class. As As discussed discussed in the text, "continuum" refers refers to continuous continuous population densities densities rather than continuous space. space.
(as opposed (as opposed to to the the "continuous-space" "continuous-space" models, models, where where continuity continuity is is used used in in the the narrow narrow mathematical mathematical sense sense rather rather than than the the broader broader sense sense used used elsewhere elsewhere in in the the chapter). equations, discussed chapter). Pair Pair approximations approximations and and spatial spatial moment moment equations, discussed in in some some detail approximations to models; both detail later, later, are are two two classes classes of of approximations to stochastic stochastic spatial spatial models; both eliminate the eliminate the explicitly explicitly stochastic, stochastic, discrete-individual discrete-individual nature nature of of their their parent parent mod models els but but preserve preserve some some of of the the important important properties properties of of discrete discrete stochastic stochastic spatial spatial dynamics 994). Other lead dynamics (Durrett (Durrett and and Levin, Levin, 11994). Other approximations approximations and and limits limits also also lead to models: for to connections connections between between different different classes classes of of models: for example, example, interacting interacting par particle ticle systems systems converge converge to to PDEs PDEs in in the the "hydrodynamic" "hydrodynamic" limit limit where where individual individual movement movement is is on on aa rapid rapid timescale timescale relative relative to to intraintra- and and interspecific interspecific interactions interactions (Durrett 994). (Durrett and and Neuhauser, Neuhauser, 11994).
Individual-Based I n d i v i d u a l - B a s e d Models Models By By definition, definition, an an individual-based individual-based model model tracks tracks the the fates fates of of all all of of the the indi individual vidual organisms organisms within within an an ecological ecological community. community. IBMs IBMs need need not not be be spatial, spatial, but but their their flexibility flexibility appeals appeals to to spatial spatial and and landscape landscape ecologists. ecologists. In In addition, addition, keeping keeping track track of of unique unique individuals individuals is is often often the the simplest simplest way way to to model model spatial spatial pattern pattern in in populations. populations. Spatial Spatial IBMs IBMs are are one one kind kind of of spatially spatially explicit explicit popula populamodels that individuals as tion tion models models (SEPMs): (SEPMs): models that take take individuals as discrete discrete individuals individuals occupying occupying aa continuous, continuous, usually usually two-dimensional two-dimensional landscape. landscape. If If individual individual locations locations are are described described as as discrete, discrete, infinitesimal infinitesimal points points (rather (rather than than disks disks with with aa finite finite radius, radius, for for example) example),, then then the the model model is is aa spatial spatial point point process process model, model, which which will will form form the the basis basis for for the the spatial spatial moment moment approximations approximations later later in in this this chapter. chapter. Spatial Spatial IBMs IBMs allow allow dynamics dynamics to to be be stochastic stochastic and and are are usually usually compu computational tational rather rather than than analytical. analytical. IBMs IBMs are are widely widely used used in in wildlife wildlife and and conserva conservation about organisms tion ecology, ecology, where where researchers researchers have have questions questions about organisms moving moving and and interacting aI., 1998). interacting on on very very specific specific landscapes landscapes (Wiegand (Wiegand et etal., 1998). They They are are convenient convenient because because they they can can incorporate incorporate aa wide wide range range of of individual individual behaviors behaviors
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and and landscape landscape structures. structures. The The downside downside of of these these models models is is that that they they may may be be computationally computationally intensive, data data hungry, and relatively hard hard to generalize (Wennergren ai., 11995; 995; Ruckelshaus ai., 11997; 997; Mooij (Wennergren et et al., Ruckelshaus et et al., Mooij and and Deangelis, Deangelis, 11999; 999; South, 999). South, 11999). In In addition addition to to aa large large variety variety of of tactical tactical simulation simulation models models (Turner (Turner et et ai., al., 11994; 994; Deangelis 998; Johnson 998; Werner Deangelis et et ai., al., 11998; Johnson et et ai., al., 11998; Werner et et ai., al., 2001; 2001; Mullon Mullon et et ai., al., 2002; 2002; Woodbury Woodbury et et ai., al., 2002), 2002), spatial spatial IBMs IBMs have have been been used used to to study study the the endogenous endogenous spatial spatial dynamics dynamics of of predator-prey predator-prey and and larger larger trophic communities and trophic communities and to to compare compare their their results results with with those those from from other other model model formulations (McCauley et 993; Wilson ai., 11993; 993; Keitt, formulations (McCauley et ai., al., 11993; Wilson et et al., Keitt, 1997; 1997; Schmitz Schmitz and 997; Wilson, 998; Donalson 999). Forest and Booth, Booth, 11997; Wilson, 11998; Donalson and and Nisbet, Nisbet, 11999). Forest models models are are aa well-developed well-developed subset subset of of spatial spatial IBMs IBMs that that are are individually individually based based but but only indi only sometimes sometimes fully fully spatially spatially explicit; explicit; they they usually usually track track the the locations locations of of individual 00 m vidual trees trees only only to to within within the the nearest nearest "gap," "gap," an an area area of of 1100 m 22 or or so so that that rep represents resents the the crown crown size size of of an an adult adult canopy canopy tree, tree, although although more more recent recent models models have localized localized individuals 984; Urban individuals to points points in the plane plane (Shugart, 11984; Urban et ai., al., 1991; 993; Chave, 999; Phillips 2003). Spatial 1991; Pacala Pacala et et ai., al., 11993; Chave, 11999; Phillips et et ai., al., 2003). Spatial forest forest models are tactically and models are used used both both tactically and strategically. strategically. The flexibility the effects effects on on popu The flexibility of of spatial spatial IBMs IBMs lends lends itself itself to to modeling modeling the population lation movement movement and and dynamics dynamics of of different different configurations configurations of of barriers barriers and and habitat habitat types types in in the the landscape landscape (Jonsen (Jonsen and and Taylor, Taylor, 2000; 2000; Cumming, Cumming, 2002). 2002). While While strategic strategic spatial spatial IBMs IBMs focus focus on on endogenous endogenous heterogeneity, heterogeneity, tactical tactical spatial spatial IBMs IBMs often often incorporate incorporate exogenous exogenous heterogeneity heterogeneity ~ frequently frequently derived derived from from aerial or satellite images images of a particular Caspersen et ai., 999; particular landscape landscape ((Caspersen al., 11999; Murrell Murrell and and Law, Law, 2000; 2000; Mooij Mooij et et ai., al., 2002). 2002). Further 1 992) gives Further reading. The The classic classic book book by by DeAngelis DeAngelis and and Gross Gross ((1992) gives aa good good introduction introduction to to IBMs IBMs in in general general (although (although most most of of the the material material does does not not empha emphasize spatial models), whereas Grimm and co-workers co-workers give more recent reviews and major directions directions in research (Grimm, 999; and highlight highlight some some of of the the major in IBM IBM research (Grimm, 11999; Grimm Grimm et et ai., al., 1999). 1999).
Lattice Lattice Models Models Lattice models models represent represent a continuous continuous landscape landscape as a regular (usually) square square lattice. lattice. Each Each lattice lattice cell cell may may contain contain aa single single individual; individual; aa population; population; or individuals or populations of or individuals or populations of multiple multiple species, species, depending depending on on the the model. model. Lattice individuals are Lattice cells cells change change their their state state ((individuals are born, born, grow, grow, or or die, die, or or are are eaten eaten by populations of by predators, predators, or or sites sites are are taken taken over over by by populations of other other species) species) deter deterministically or ministically or stochastically stochastically according according to to rules rules based based on on the the local local density density of of prey, prey, predators, predators, or or competitors. competitors. Interacting Interacting particle particle systems systems are are stochastic stochastic lat lattice tice models models that that run run in in continuous continuous time. time. Stochastic Stochastic or or probabilistic probabilistic cellular cellular automata model stochastic site occupancy automata model stochastic changes changes in in individual individual site occupancy in in discrete discrete model continuous continuous (deterministic time, whereas whereas coupled-map time, coupled-map lattices lattices model (deterministic or or stochastic) populations stochastic) populations in in discrete discrete time. time. Lattice Lattice models models are are another another form form of of SEPM spatial IBMs. IBMs. A individu SEPM that that overlaps overlaps with with spatial A lattice lattice model model where where single single individuals equivalent to spatial IBM within-species variation als occupy occupy cells cells is is equivalent to aa spatial IBM with with no no within-species variation (so that (so that the the only only unique unique property property of of an an individual individual is is its its spatial spatial location) location).. However, and IPSs However, lattice lattice models models ((and IPSs iinn particular) particular) have have their their own own literature, literature, which general theoretical theoretical questions questions such which focuses focuses on on general such as as competitive competitive coexistence coexistence
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
49 49
and and the connections connections between IPS and partial differential questions. In contrast contrast to the situation for complex spatial IBMs, there is a well-developed mathe mathematical theory that that can be used to construct construct formal proofs of persistence or extinction in IPS ((Durrett, Durrett, 1988; Bramson ai., 11991; 9 9 1 ; Durrett, 992; Durrett Bramson et al., Durrett, 11992; Durrett and Neuhauser, 11994). 994). IPS and stochastic stochastic cellular automata automata have been used to Sole, 11997); 997); explore persistence of single-species populations populations (Bascompte and So16, competitive interactions (Harada 994; Schwinning and (Harada and and Iwasa, 11994; and Parsons, 11996; 996; Takenaka et ai., 997); predator-prey al., 11997); predator-prey and parasitoid-host parasitoid-host interactions interactions ((Hassell Hassell et ai., 9 9 1 ; Comins et ai., 992; Wilson et ai., 993; Cuddington al., 11991; al., 11992; al., 11993; Cuddington of epidemics and Yodzis, 2000; 2000; Hosseini, 2003); 2003); and dynamics and evolution evolution of ((Sat6 Sat6 et ai., 994; Rand 995; Boots and Sasaki, 2000, al., 11994; Rand et ai., al., 11995; 2000, 2002). 2002). Some researchers researchers have used lattice models to to explore the effects of envir environmental heterogeneity, particularly the viability of single-species populations populations degraded or fragmented habitats. habitats. Bascompte and Sole Sol~ ((1997) in degraded 1 997) and Hiebeler (2000) 1 ) understand (2000) have have both both used used IPS IPS to to ((1) understand how how population population densities densities vary vary in in response to the amount amount and pattern pattern of habitat habitat destruction destruction and (2) compare the results to the predictions of patch occupancy models under under similar scenarios al., 11994; (Gotelli, 11991; 99 1 ; Tilman et ai., 994; Lavorel and Chesson, 11995). 995). Further reading. Durrett and Levin's classic paper paper on on "The Importance Importance of Being Discrete (and Spatial) 1 994) lucidly compares Spatial)"" ((1994) compares the dynamics and coexis coexistence properties of nonspatial models, continuum (see below), and IPS. IPS. continuum models (see
Pair Approximation
Pair Pair approximation, approximation, an approximate approximate method method for analyzing IPS that that focuses on the joint occupancy probability probability of neighboring pairs of cells, has been an important important complement complement to more rigorous analytical techniques techniques (Tainaka, 1988, 11994; 9 94; Harada 994; Harada 995; Nakamaru 997; Harada and Iwasa, 11994; Harada et ai., al., 11995; Nakamaru et ai., al., 11997; Takenaka ai., 11997; 997; Iwasa, 2000; Sat6 Takenaka et al., Sat~ and Iwasa, Iwasa, 2000; 2000; Boots and Sasaki, 2000 2000).) . Pair approximation approximation makes the assumption assumption of conditional conditional independ independence. Pairs of neighbors are assumed to be independent independent (the conditioning is on the presence of a focal individual) so that that the probability of a particular particular con configuration of a focal individual (e.g., a predator) predator) and and two two neighbors neighbors (e.g., two different prey individuals) is the product product of the probabilities probabilities of each pair of neighbors. neighbors. Although a great deal of foundational foundational work in IPSs IPSs has been done on simple models, such as the contact contact process and the biased voter voter model (the IPS analogues of logistic growth growth and and simple competition, respectively), pair approximations approximations allow rapid rapid construction construction and analysis of models dealing with larger communities or more complex complex biological rules. In addition addition to basic pair pair approximation approximation on the lattice lattice with a single, fixed interaction interaction scale, variants of pair approximation approximation have been developed to handle handle more general spaces, such as networks 997; van Baalen and Rand, 998; Rand, 1 999; networks (Keeling et ai., al., 11997; Rand, 11998; Rand, 1999; Keeling, 11999a,b; 999a,b; van van Baalen, 2000); to to approximate approximate the spatial dynamics of expanding 99 8 ); and expanding populations populations (Ellner et ai., al., 11998); and to model systems with multi multiple scales of movement and interaction interaction (Ellner, (Ellner, 2001 2001).) . Further reading. Chapters 1 999) give the Chapters bbyy van Baalen (2000) and Rand ((1999) clearest general explanation of pair approximation approximation derivations (they both dis discuss network network models, which are more general than than lattices); chapters by Sat6 Sat~ and and Iwasa (2000) and Iwasa (2000) are also usefui. useful.
BENJAMIN BENJAMINM. M. BOlKER BOLKER
50 50
C o n t i n u u m Models Models Continuum
The The "continuum" "continuum" in in the the definition definition of of continuum continuum models models refers refers primarily primarily to to the the representation representation of of local local populations populations as as continuous continuous densities densities rather rather than than dis discrete crete numbers, although although the the most most common continuum continuum models models (partial (partial differ differential equation and and integrodifferential equations) also operate operate in continuous continuous space and time. Continuum Continuum models are are more common in strategic strategic than in in tactical tactical applications. applications. The The simplest simplest reaction-diffusion reaction-diffusion models models allow allow only only completely completely local local reaction reaction terms terms (predator-prey (predator-prey interactions, interactions, competition, competition, etc.) etc.) that are are formulated similarly to to their nonspatial analogues analogues coupled with move movement by local diffusion; more detailed integrodifferential models allow interacindividuals to move or interact nonlocally according to movement or interac tion kernels. Continuum models can also be defined in discrete time, in which case they become integrodifference equations. The continuous-population assumption eliminates demographic demographic stochas stochasticity from continuum models, although the effects of demographic stochas stochasticity can be reintroduced either by assuming Poisson or multinomial statistical variation in the local population ((Durrett Durrett and Levin, 11994; 994; Griinbaum, 11994) 994) or by adding a stochastic term to create a stochastic POE PDE or integral equation (Lande et aI., 999; Engen et aI., al., 11999; al., 2002) 2002).. Continuum models models have have aa proud proud history history in in ecology; ecology; before before computational computational models models were were practical, they were the only way to model dynamics in continuous space, foundation. Continuum mod modand they still benefit from a strong theoretical foundation. wavefront dynamics of els are particularly powerful for modeling spread and wavefront genes, diseases, and populations (Metz 995), but genes, diseases, and populations (Metz and and van van den den Bosch, Bosch, 11995), but they they are also useful for studying equilibrium states and complex spatiotemporal dynamics (Holmes et ai., 1 994; Klausmeier, 1999; 1 999; Briscoe et al., ai., 2002). al., 1994; Although many ecologists perceive them them as oversimplified, PDEs equations Although incorporate fairly sophisticated models of behavior (Kareiva and Odell, can incorporate 1 9 8 7; Griinbaum, 1 9 94; Turchin, 1 998; Griinbaum, 1 998; Moorcroft et al., aI., 1987; Griinbaum, 1994; Turchin, 1998; Griinbaum, 1998; Moorcroft et 1999). 1 99 9 ) . Historically, most most continuum continuum models models have have explored explored the the effects effects ooff endogenendogen Historically, of work work on the distribudistribu ous variability only. One exception is the large body of tion of animals following different different foraging and and dispersal strategies tion strategies in heterogeneous environments (Griinbaum, (Griinbaum, 1994); 1 994); classical work on population population heterogeneous work on and competition in the presence of of environmental environmental features, persistence and features, such as resource patches patches and and gradients gradients (Kierstead and and Slobodkin, Slobodkin, 1953; 1 953; Pacala Pacala and and resource Roughgarden, 1982); 1 982); and and a series series of of papers papers by Roughgarden Roughgarden on on the the shape shape and and Roughgarden, of spatial distributions distributions of of organisms organisms in the presence of of particular particular envirenvir scale of (Roughgarden, 1974, 1974, 1977, 1 9 77, 1978; 1 978; Sasaki, 1997). 1 997). onmental heterogeneities (Roughgarden, [Klausmeier (1999) ( 1 999) shows shows how how the the combination combination of of endogenous endogenous dynamics with with [Klausmeier in smooth slope slope can generate generate the the heterogeneous heterogeneous pattern pattern of of "tiger "tiger bush" bush" in a smooth semi-arid landscapes.] landscapes.] More More recently, recently, Lande Lande et et al., ai., (1999) ( 1 999) and and Engen Engen et et al., ai., semi-arid (2002) (2002) have have developed developed a series of of approximate approximate and and exact exact stochastic stochastic PDE PDE modmod els for for the the spatial spatial synchrony synchrony of of population population fluctuation fluctuation that that extend extend els Roughgarden's work. work. Results Results from from this this approach approach of of modeling modeling noise-driven noise-driven Roughgarden's correlations on on an an exogenously exogenously variable variable landscape landscape (also (also used used by by Snyder Snyder and and correlations Chesson, 2002) 2002) have have begun begun to to converge converge on on some some of of the the results results of of spatial spatial Chesson, moment moment equations equations described described later. later.
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
51 51
Further reading. Continuum Continuum models models are are presented presented in in Okubo's Okubo's book book on on dif diffusion Okubo, 11980) 980) and, and, more more recently, recently, by fusion models models ((Okubo, by Shigesada Shigesada and and Kawasaki Kawasaki ((1997) 1 997) [the [the edited edited volume volume by by Okubo Okubo and and Levin Levin (2002) (2002) is is more more up up to to date date than than Okubo ((1980), 1 980), but Okubo but gives gives more more of of an an overview overview of of developments developments in in ecological ecological diffusion 1 998) does diffusion models models than than an an introduction]; introduction]; Turchin Turchin ((1998) does aa particularly particularly good job connecting good job connecting continuum continuum models models with with data data on on animal animal movement. movement. Books Books by 1 990) and 1 988) cover by Murray Murray ((1990) and Edelstein-Keshet Edelstein-Keshet ((1988) cover the the use use of of PDEs PDEs in in math mathematical ematical biology biology more more generally, generally, but but do do have have useful useful ecological ecological material. material.
3.3 3.3
SPATIAL MOMENT MOMENT EQUATIONS EQUATIONS SPATIAL How SMEs) fit How do do spatial spatial moment moment equations equations ((SMEs) fit into into this this assemblage assemblage of of mod models tools? The els and and tools? The basic basic structure structure behind behind spatial spatial moment moment equations equations is is that that of of spatial (possibly heterogeneous) spatial point point processes processes in in aa (possibly heterogeneous) two-dimensional two-dimensional space. space. As As pair pair approximations approximations do do for for lattice lattice models, models, SMEs SMEs attempt attempt to to reduce reduce aa spatial den spatial point point process process to to aa set set of of equations equations for for the the dynamics dynamics of of population population densities and and spatial spatial pattern pattern in in the the system. SMEs capture the spatial spatial properties properties of of sities system. SMEs capture the an tracking the mean dens an ecological ecological system system by by tracking the first first two two spatial spatial moments, moments, the the mean densdifferent species and and the spatial correlations among among individuals of all ities of different species, species, which which are are analogous analogous to to the the mean mean and and variance variance of of aa nonspatial nonspatial distri distribution. Like models, the representations of spatial pattern bution. Like continuum continuum models, the actual actual representations of spatial pattern are continuous, differentiable are continuous, differentiable functions. functions. While While moment moment equations equations are are not not technically technically stochastic stochastic m they they represent represent average average behavior behavior over over space space or or across across ensembles similar ecological ensembles of of similar ecological arenas arenas - - they they do do preserve preserve some some of of the the import important ant effects effects of of stochasticity stochasticity in in finite finite populations populations that that are are lost lost from from typical typical con continuum tinuum models. models. Many Many nonspatial nonspatial models models use use similar similar methods methods to to track track the the properties properties of of population 964) can population distributions. distributions. Moment Moment generating generating functions functions (Bailey, (Bailey, 11964) can be be used used to to analyze analyze the the entire entire infinite infinite series series of of moments moments of of aa distribution, distribution, but but are are difficult nonlinear processes, difficult to to apply apply to to nonlinear processes, let let alone alone nonlinear nonlinear spatial spatial processes. processes. Nonspatial Nonspatial moment moment equations equations that that use use some some rule rule to to approximate approximate higher higher moments (including constant, or common in moments (including setting setting them them constant, or to to zero) zero) are are common in ecology, ecology, epidemiology, 992; Turelli epidemiology, and and genetics genetics (Dobson (Dobson and and Hudson, Hudson, 11992; Turelli and and Barton, Barton, 11994; 994; Grenfell 995; Dushoff, 999; Cornell Grenfell et et aI., al., 11995; Dushoff, 11999; Cornell et et aI., al., 2000). 2000). For For example, example, Anderson 1 985) modeled Anderson and and May May ((1985) modeled macroparasite macroparasite dynamics dynamics by by assuming assuming that that the binomial distribution the number number of of parasites parasites per per host host has has aa negative negative binomial distribution with with aa fixed fixed overdispersion overdispersion parameter parameter so so that that the the variance variance can can be be calculated calculated for for any any given value of given value of the the mean. mean. The The next next section section derives derives the the spatial spatial moment moment equations equations for for the the simplest simplest par parasitoid-prey asitoid-prey system, system, equivalent equivalent to to Lotka-Volterra Lotka-Volterra predator-prey predator-prey equations equations with with aa predator predator efficiency efficiency of of one one (predator (predator bith bith rate rate equal equal to to predation predation rate). rate). The The parasitoid parasitoid moves moves by by random random jumps, jumps, while while the the prey prey is is sessile sessile and and disperses disperses only close enough only as as aa seed seed or or propagule. propagule. When When the the parasitoid parasitoid is is close enough to to the the prey, prey, it prey into into aa single This system, rather than than the it turns turns the the prey single new new parasitoid. parasitoid. This system, rather the analogous been chosen chosen for analogous equations equations for for competition competition or or epidemics, epidemics, has has been for novelty; novelty; derivations derivations of of competitive competitive and and epidemic epidemic systems systems appear appear elsewhere elsewhere (Bolker (Bolker and 999; Bolker, 999). Moment and Pacala, Pacala, 11999; Bolker, 11999). Moment equations equations have have been been used used to to model model
52 52
M. BOLKER BOlKER BENJAMIN M.
predator-prey metapopulations metapopulations (Keeling, (Keeling, 2000; 2000; Keeling Keeling et et al., aI., 2002) 2002) and and spaspa predator-prey tial epidemics epidemics without replenishment (Bolker, (Bolker, 1999; 1 999; Brown, Brown, 2001), 200 1 ), and and pair pair tial without replenishment approximations approximations have have been been used used for for more more general general epidemics epidemics (Sato (Saw et et al., aI., 1994), 1 994), spatial moment moment equations equations for for predator-prey predator-prey systems systems have have not not been been published published spatial (M. Desai, Desai, unpublished unpublished manuscript). manuscript). The The derivation derivation is is sketched, leaving out out the the (M. sketched, leaving straightforward but but scary-looking algebra; this this approach approach makes makes things slightly straightforward scary-looking algebra; things slightly the serious serious reader reader who who wants wants to to follow follow the the algebra algebra line line by by line, line, but but harder for the harder for it it allows allows the the discussion discussion to to focus focus on on the the important important steps steps in in the the derivation, derivation, pointing out out where where alternative alternative assumptions assumptions or or procedures procedures are are possible. possible. [Similar [Similar pointing derivations can can be be found found in in papers papers by by Bolker Bolker and and Pacala Pacala (1999) ( 1 999) and and by by derivations Dieckmann and and Law Law (2000).] (2000).] Dieckmann Further reading. reading. Other Other introductions introductions to to SMEs SMEs can be found found in in chapters chapters by by Further can be Bolker et et al. al. and and Dieckmann Dieckmann et et al. ai. in in Dieckmann Dieckmann et et al. ai. (2000) (2000) and and in in other other papers papers Bolker by Law, Law, Dieckmann, Dieckmann, and and Murrell Murrell (Law (Law and and Dieckmann, Dieckmann, 2000; 2000; Law Law et et al., aI., 2001, 2001, by 2003). 2003).
Write Rules W r i t e down down Stochastic Stochastic Rules We by writing writing down change in in the of aa small We start start by down the the expected expected change the occupancy occupancy of small patch of w located in aa landscape landscape of patch of size size 00 located at at position position x x over over aa small small time time M, At, in of O. In In this ( V) reproduce reproduce at at aa constant constant per capita rate patches D.. patches this system, system, prey prey (V) per capita rate t fvv (leading to aa Poisson-distributed number of of offspring per unit unit time) time) and and disperse disperse (leading to Poisson-distributed number offspring per them in aa random with aa distance given by by the kernel 73v. Vv. them in random direction direction with distance given the dispersal dispersal kernel [Kernels are that determine the rate rate or probability with [Kernels are continuous continuous functions functions that determine the or probability with which will interact interact with with another another site site or or individual individual at given disdis which an an individual individual will at aa given tance. kernels represent probability distributions tance. Dispersal Dispersal kernels represent probability distributions and and so so must must integrate integrate to all possible also scale scale interacinterac to l1 over over all possible locations; locations; for for analytical analytical convenience, convenience, we we also tion kernels Kernels are tion kernels so so they they integrate integrate to to 11.. Kernels are typically typically symmetric, symmetric, and and to to avoid avoid interactions at infinite distances, distances, kernels distance becomes becomes interactions at infinite kernels must must approach approach zero zero as as distance large. We typically typically use use simple simple decreasing decreasing functions functions such such as as the the Laplacian Laplacian (back (backlarge. We to-back exponential) Gaussian distribution, distribution, but to-back exponential) or or Gaussian but other other shapes shapes are are feasible.] feasible.] Prey Prey die die at at aa density-independent density-independent per per capita capita rate rate !.Lv, ~v, and and intraspecific intraspecific competition leads leads to additional mortality neigh competition to additional mortality at at aa rate rate proportional proportional to to the the neigh- xl) all conspecific borhood borhood density, density, measured measured as as the the sum sum of of aUYY{ly ~Uvv(lYxl) for for all conspecific neighbors, overall strength neighbors, where where a c~ is is the the overall strength of of competition competition and and Uw b/w (p) (P) is is the the rel relative ative strength strength of of competition competition at at aa distance distance p. p. [In [In the the standard standard nonspatial nonspatial Lotka-Volterra Lotka-Volterra model, model, specifying specifying birth birth and and density-independent density-independent death death rates rates separately separately would would be be redundant, redundant, but but in in stochastic stochastic and and spatial spatial models models the the ratio ratio of of birth birth to to death death rates rates as as well well as as the the difference difference between between birth birth and and death death rates rates is 999).] Self-limitation is important important (Bolker (Bolker and and Pacala Pacala 11999).] Self-limitation of of the the prey prey may may seem seem to to be be an an unnecessary unnecessary complication, complication, but but it it is is actually actually required required to to keep keep the the spa spatial Desai, unpublished tial moment moment equations equations from from blowing blowing up up (M. (M. Desai, unpublished manuscript). manuscript). Parasitoids Parasitoids (P) (P) die die at at aa density-independent density-independent per per capita capita rate rate !.Lp, ~p, and and at at rate rate mp ump from mp they they jjump from their their current current location location in in aa random random direction direction with with aa dis dispersal persal kernel kernel Vp• De. Parasitism Parasitism occurs occurs as as aa function function of of parasitoid-prey parasitoid-prey distance, distance, with is the with aa rate rate proportional proportional to t o 'YUpY{ly ~/b/PV(ly - xl) xl) [again, [again, 'Y ~/is the overall overall rate rate and and Upv btpv(p) (p) predation rate at different distances]. When When predation governs the relative predation occurs, occurs, the the prey prey turn turn into into newborn newborn parasitoids. parasitoids. The The processes processes described described here here can can be be written written formally formally as as stochastic stochastic rates: rates:
3. 3.
CONTINUOUS-SPACE MODELS MODELS CONTINUOUS-SPACE
53 53
Event Event
Change Change
Rate Rate
Prey birth birth Prey
V(x)-; V(x) ++ 11 V(x)--->V(x)
( ly -- xl)~176 ~L yEf/fvV(y)Dv yE n fv V(y )D v (lY xl )w
Prey death death Prey (density independent) independent) (density
V(x)-;V(x) V(x) -- 11 V(x)--->
ILVV(X) ~vV(x)
Prey death death Prey (density dependent) dependent) (density
V(x)->V(x) -- 11 V(x)-+V(x)
V(x) ~ aV(y) Uvv( l y -- xxI) V(x) ] y~a L Y E n oLV(y)Uw(ly l)
Parasitoid movement Parasitoid movement
P(x)->P(x) -- 11 P(x)-+P(x) P(y)->P(y) + + 11 P(y)-->P(y)
mpp P(x) P(x) D/, xl )w ( ly -- xl)~ m Dp (lY
Parasitoid death death Parasitoid
P(x)-;P(x) -- 11 P(x)-+P(x)
f.LpP(x) ~/,P(x)
V(x)-; V(x) -- 11 V(x)-+V(x)
xl ) V(x) ~ yP(y) UepvY( (l yly -- xl) V ( x ) ~L yyEn ),P (y) U
Parasitism Parasitism
P(x) -;P(x) + + 11 P(x)-+P(x)
The limits represent represent the locations of patches in the landland The summation summation limits the set set of of locations of all all patches in the scape. All All events involve the the probability probability of of aa propagule or an an individual scape. events that that involve propagule or individual landing in in aa particular particular patch prey birth birth rate rate and movement landing patch (the (the prey and parasitoid parasitoid movement rates in this this case) patch size size (00) (w) term term because because the the probability probability of an rates in case) contain contain aa patch of an individual patch is to patch patch area. individual landing landing in in aa patch is proportional proportional to area. Interactions Interactions that that involve the parasitism parasitism term, do not not because because the the corcor involve two two individuals, individuals, such such as as the term, do responding expressions individuals rather rather than than areas. areas. When When responding expressions sum sum over over discrete discrete individuals -7 0), 0), itit will will be be important important to to have have we the continuous-space continuous-space limit limit later (w -+ w e take take the later (00 these terms correct. these terms correct. Alternatives Alternatives
This small, discrete in This derivation derivation is is heuristic, heuristic, starting starting from from small, discrete patches patches evolving evolving in discrete discrete time time and and then then taking taking straightforward straightforward limits limits to to reach reach the the continuous continuousspace, space, continuous-time continuous-time equations. equations. This This procedure procedure makes makes it it easier easier to to see see when when and (removable) singularities possible to and why why (removable) singularities occur occur in in the the equations. equations. It It is is also also possible to construct construct aa rigorous rigorous derivation derivation starting starting from from infinitesimal infinitesimal space space and and time time increments increments rather rather than than small small but but finite finite ones ones and and using using probability probability measures, measures, but but we we have have always always found found it it easier easier and and clearer clearer to to proceed proceed heuristically. heuristically. more rigor Mathematically can see see Barton Mathematically inclined inclined readers readers can Barton et et al. al. (2002) for for aa more rigorous continuous space ous approach approach to to stochastic stochastic population population dynamics dynamics in in continuous space in in aa pop population-genetic ulation-genetic context. context. The The predation predation process process just just described, described, where where predators predators turn turn their their prey prey into into aa single single newborn newborn predator, predator, models models aa parasitoid-host parasitoid-host interaction interaction where where only only one one egg egg can can emerge emerge from from aa host. host. This This kind kind of of parasitoid parasitoid model model is is the the simplest simplest to to analyze analyze because because only only two two spatial spatial locations locations are are involved involved in in the the interaction. interaction. If If we we wanted wanted to to allow allow more more than than one one newborn newborn predator predator per per host, host, we we could could define define aa natal natal dispersal dispersal kernel kernel according according to to which which the the newborn newborn predators predators jumped jumped away away from from the the host, host, but but we we would would then then have have to to keep keep track track of of three three spatial spatial locations locations (predator, (predator, prey, prey, and and newborn) newborn) at at aa time; time; to to calculate calculate the the expected expected change change in in predator-predator predator-predator covariances, covariances, we we would would need need to to estimate estimate the the probability probability of of aa four-point newborn, and four-point configuration configuration (predator, (predator, prey, prey, newborn, and neighbor neighbor predator). predator). We We could could mitigate mitigate this this problem problem by by turning turning the the prey prey into into an an incubating incubating prey, prey, which which would would be be represented represented as as aa different different class class of of individuals individuals in in the the equations equations and and which which could could then then be be immune immune from from superinfection. superinfection. At At aa constant constant rate, rate, the the
BENJAMIN M. BOlKER BENJAMIN M. BOLKER
54 54
incubating incubating prey prey transforms transforms into into one one or or more more newborn newborn predators predators ~ separating separating the time of birth from the time of predation eliminates the need to keep track of many locations locations (Fig. .1). of as as many (Fig. 33.1). Similar problems, problems, and similar solutions, arise iinn predator-prey predator-prey models. If newborn newborn predators predators disperse disperse away away either either from from the the location location of of the the prey prey or or from from the location of their parent, several spatial locations have to be tracked. Another analogous to Another possibility, possibility, analogous to incubating incubating prey, prey, is is to to say say that that predation predation turns turns aa hungry hungry predator predator into into aa sated sated predator. predator. The The sated sated predator predator would would "decay" "decay" at at aa constant constant rate rate to to aa pair pair of of predators, predators, aa parent parent and and aa newborn; newborn; if if sated cannot kill kill prey, model structure would also also give sated predators predators cannot prey, this this model structure would give rise rise to to aa handling time. Multiple stages of satiation could make the time lag between predation predation and predator reproduction reproduction gamma distributed distributed (Keeling and and Grenfell, 11998); 998); predator predator efficiency less than than one could could be handled handled by requir requiring experience multiple ing that that predators predators experience multiple prey prey events events or or by by assuming assuming there there is is aa probability less than one that satiated predator probability less than one that aa satiated predator reproduces reproduces when when it it becomes becomes hungry again. These problems problems of definition are implicit in all ecological models, from sim simnonspatial models to SEPMs. Lattice models often make slightly artificial ple nonspatial Durrett and rules to avoid having multiple individuals in the same lattice cell ((Durrett Levin, Levin, 2000); 2000); in in nonspatial nonspatial models, models, these these details details can can often often be be accounted accounted for for by by setting up proper proper functional and numerical responses (Keeling et a!., al., 2000; Cuddington and Yodzis, 2002). The parasitoids parasitoids in this model are stupid, stupid, neglecting any local cues that that could guide them to higher prey densities. We could introduce introduce adaptive adaptive for foraging rules by letting the probability of movement, or the length of moves, Upv
4,
Upv
(~
4,
G
(~
4, t o + At
o
to + At
to+At
II
4,
t o +t=
to + i
I
(a) Standard interaction: interaction: predator predator (parasitoid) immediately immediately converts converts prey into a single offspring. offspring.
o
+4,
(b) (b) Parasitism with with incubation: incubation: predator nto an predator converts converts prey prey iinto Incubating Incubating prey, which which later (after t,) tl) becomes predator predator offspring. offspring.
tG
i
(c) Predation: prey disappears, disappears, prey becomes Sated Sated (and pregnant), later (after gestation tG) tc) gives gives birth to a single single off offspring.
Fig. Predator-prey Fig. 3.1 3.1 Predator-prey transition transition rules. Predators are Po, P0, P1; P1; prey are V; V; incubating incubating prey (e.g., containing containing parasitoid eggs or larvae) are I; and sated, pregnant pregnant predators are S. S. At At denotes an instantaneous change, whereas whereas other time time lags t" tl, tG tG denote events that that occur after an exponentially distributed distributed lag time.
3. 3. CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
$$ 55
decrease decrease in in response response to to increasing increasing local local prey prey density. density. This This behavioral behavioral response response would result result in in an an aggregation aggregation of of parasitoids in areas areas of of high high prey prey density, density, would parasitoids in mitigating the prey in parasitoid mitigating the prey depletion depletion that that occurs occurs in parasitoid neighborhoods. neighborhoods. Another Another common common foraging foraging rule, rule, allowing allowing the the directions directions of of successive successive moves moves to 99 8 ) , to become become less less correlated correlated in in response response to to favorable favorable habitat habitat (Turchin, (Turchin, 11998), could could bbee incorporated incorporated iinn the the point point process process model model but but iiss harder harder ttoo preserve preserve in in moment moment equations. equations. We We could could also also add add effects effects of of environmental environmental heterogeneity heterogeneity simply simply by by making making some some of of the the parameters parameters be be functions functions of of location. location. Making Making death death or or movement movement rates although perhaps rates spatially spatially heterogeneous heterogeneous is is simpler simpler ((although perhaps less less interesting) interesting) than than making making predation predation or or birth birth rate rate spatially spatially heterogeneous heterogeneous because because making making the the predation predation rate rate heterogeneous heterogeneous induces induces aa triple triple spatial spatial interaction interaction in in the the process, which interaction when process, which will will turn turn into into aa four-way four-way interaction when the the covariance covariance is is calculated. calculated.
Compute Changes in and Pair Pair Densities Compute Expected Expected Changes in Singlet Singlet and Densities For For each each mean mean density, density, the the derivation derivation of of the the spatial spatial moment moment equations equations requires requires two two averages averages or or expectations. expectations. The The first first average average is is the the expected expected change change in in occupancy occupancy over over aa short short time time M At in in aa patch patch of of size size w ~o for for aa given given starting starting con configuration figuration of of neighbors; neighbors; this this is is the the standard standard expected expected change change calculated calculated in in deriving deriving most most ecological ecological models models and and is is denoted denoted by by an an overbar. overbar. In In general, general, this this expected expected rate rate of of change change is is �F(x) = i) [change in AF(x) = k(rate E(rate of of event event/)[change in F(x) F(x) if if event event ii occurs]�t occurs]At
(throughout (throughout this this derivation, derivation, E F, G, G, and and H H refer refer to to arbitrary arbitrary state state variables; variables; w, w, x, x, y, y, and and zz are are arbitrary arbitrary locations). locations). Once Once we we have have taken taken this this first first average average to to cal calculate culate the the expected expected amount amount of of change change that that will will occur occur at at aa particular particular location location starting from from aa particular particular configuration, configuration, we we need need to to average average all all possible possible con constarting figurations figurations across across the the ensemble ensemble to to find find the the expected expected amount amount of of change change that that will will occur occur at at the the location location starting starting from from all all configurations configurations with with particular particular mean mean densities densities and and correlations. correlations. This This second second expectation, expectation, denoted denoted by by angle angle brackets brackets 0, (.), reduces reduces the the information information on on the the right-hand right-hand side side of of the the equation equation from from all all of of the ust the the information information about about the the configuration configuration to to jjust the expected expected patch patch densities densities and and joint joint patch patch densities densities (doublets). (doublets). The The same same procedure procedure is is applied applied to to calculate calculate the the average average expected expected rate rate of of change change of of doublets, doublets, e.g., e.g., <�F(x)G(y) (AF(x)G(y)),, the the rate rate of of change change of of pairs pairs with with prey prey in in aa patch patch at at point point x x and and parasitoid parasitoid in in aa patch patch at at point point y; y; these these rates rates will will involve involve doublets doublets and and triplets. triplets. Alternatives
For point that can drop For simplicity, simplicity, we we assume assume at at this this point that �t At is is short short so so that that we we can drop terms happen with terms where where two two events events occur occur at at once once [which [which happen with probability probability pro pro2 ] . Retaining portional possible [leading portional to to (�t) (At)2]. Retaining finite finite time time steps steps is is also also possible [leading even eventually tually to to an an integrodifference integrodifference equation equation for for the the changes changes in in spatial spatial covariances covariances ((Lewis, Lewis, 2000)], models, it 2000)], although although as as usual usual with with discrete-time discrete-time models, it requires requires more more care care in in defining defining the the possible possible orders orders of of events. events. In In addition, addition, retaining retaining double double events events will will lead lead to to fourth-moment fourth-moment terms terms in in expressions expressions for for covariances, covariances,
BENJAMIN M. BOLKER BENJAMINM. BOLKER
56
which which will will have have to to be be dealt dealt with with by by some some extended extended form form of of moment moment closure closure (see (see below). below).
Assume Assume Spatial Spatial Homogeneity Homogeneity By isotropy or By assuming assuming spatial spatial homogeneity homogeneity (specifically (specifically homogeneity homogeneity and and isotropy or translational make all translational and and rotational rotational invariance), invariance), we we can can make all singlet singlet expectations expectations (F(x) = joint products F(x) G (y) ) depend into global averages, into global averages, (F(x)) - (F). (F). Similarly, Similarly, joint products (((F(x)G(y))) depend only points: (FG) (FG) ((Ix only on on the the distance distance between between two two points: - yl). Yl). Assuming Assuming that that space space Ix is algebra, allowing is homogeneous homogeneous considerably considerably simplifies simplifies the the algebra, allowing us us to to collapse collapse terms (e.g., «(AF(x)) ((AF(x)) F(y) F(x)(AF(y))) = 2 (AF(x)F(y) (AF(x)F(y))).). In addition, various terms F(y) + F(x) (AF(y))) = it expectations across it simplifies simplifies the the interpretation interpretation of of the the expectations across ensembles ensembles introduced introduced above, above, which which we we can can now now take take as as averages averages across across space space in in aa single single realization realization rather rather than than as as averages averages across across ensembles. ensembles. Alternatives Alternatives
Certain easier to Certain kinds kinds of of spatial spatial heterogeneity heterogeneity are are easier to incorporate incorporate than than others. others. As assume that point As long long as as we we assume that the the probability probability of of events events occurring occurring at at aa point depends only only on depends on the the occupancy occupancy of of that that point point and and of of its its neighborhood, neighborhood, rather rather than absolute position, position, we express the than on on its its absolute we can can still still express the dynamics dynamics in in terms terms of of moments - covariances between moments~covariances between population population densities densities and and environmental environmental parameter distributions predation rate. parameter distributions such such as as mortality mortality or or predation rate. At price of some algebraic relax the At the the price of some algebraic complexity, complexity, we we can can relax the assumption assumption of of spatial spatial homogeneity homogeneity still still further. further. For For example, example, Lewis Lewis (2000) (2000) and and Lewis Lewis and ala (2000) mean and invad and Pac Pacala (2000) analyzed analyzed the the mean and covariance covariance structure structure of of an an invading ing population population by by preserving preserving information information about about the the distance distance of of points points from isotropy (directionality) from the the wave wave front. front. As As another another example, example, non nonisotropy (directionality) in in some some processes processes could could be be studied studied by by allowing allowing joint joint product product expectations expectations to to depend G (y) = (FG) depend on on direction: direction: (F(x) (F(x)G(y))= (FG) ((IxI x -- YI, y[, 60).). Allowing Allowing for for some some degree rather than degree of of absolute absolute ((rather than relative) relative) spatial spatial heterogeneity heterogeneity also also opens opens the the possibility possibility of of modeling modeling covariance covariance dynamics dynamics as as aa function function of of position position along Edge effects along aa gradient gradient or or distance distance from from an an edge. edge. ((Edge effects are are not not easy easy to to incorporate dynamics in incorporate in in moment moment equations, equations, which which typically typically explore explore dynamics in aa hypothetical hypothetical infinite infinite landscape. landscape. Modeling Modeling dynamics dynamics as as aa function function of of dis distance one-dimensional habitat tance from from edge edge in in aa one-dimensional habitat or or from from an an infinitely infinitely long, long, straight straight edge edge in in aa two-dimensional two-dimensional habitat habitat might might be be feasible, feasible, but but edge edge effects landscape would would probably effects in in aa finite finite two-dimensional two-dimensional landscape probably only only be be cal calculable culable numerically. numerically.))
Product Product Expansion/Convert E x p a n s i o n / C o n v e r t Joint Joint Products Products to to Covarlances Covarlances This This step step first first expands expands joint joint and and triple triple products products in in terms terms of of means, means, covari covariances, and ances, and third third moments moments and and then then subtracts subtracts A(F)(G) A(F)(G) (the (the expected expected change change in in the mean densities) densities) from the product product of of the the mean from joint-product joint-product equations equations to to turn turn them them into covariance equations instead of into covariance equations instead of joint-product joint-product equations. equations. It It is is useful useful to to explicitly explicitly write write out out singular singular or or "delta "delta function" function" terms terms that that take take into into account account
3. CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS 3.
57 57
the the possibility possibility that that the the two two individuals individuals share share the the same same patch patch (when (when we we take take the the continuous-space continuous-space limit limit below, below, this this means means that that there there is is really really only only one one individ individual present). present). The The definition definition of of the the covariance covariance is is thus thus ual (FG)([x-
Yl)=
(F)(G)
CFG(IX-
+
Yl) + ~xyO'FG"
((3.1) 3.1 )
We We can can expand expand the the triple triple product product (FGH) (FGH) (x, (x, y, y, z) z) similarly, similarly, arriving arriving at at aa (fairly F), (G), (fairly ugly) ugly) formula formula containing containing products products of of the the mean mean densities densities (((F), (G), (H») (H)) and all all of of the the covariances, covariances, as as well well as as the the third third moment moment M3pCH(x, M3F~H(x, y, y, z). [We and z). [We specify third central moments M3 by all three locations involved in specify third central moments M3 by all three locations involved in the the triple: triple: with with spatial spatial homogeneity homogeneity and and isotropy isotropy they they are are fully fully determined determined once once we know know two two sides sides and and an an angle angle (so we could could quote quote them them as as M3 M3pcH(]x - Yy],I , we (so we pCH( l x Ix zj, 8x -xz)), but this notation is simpler.] In addition, the expansion Ix - z], 0xy-xz)), but this notation is simpler.] In addition, the expansion con cony tains aa series series of of singular singular third-moment third-moment terms terms that that result result when when two two or or more more tains points in in the the triangle triangle coincide. coincide. For For example, example, aa term term (H)apc (H)crFc + + CH,pd CH.Fc(]x -- zz]) lx points l) results when P and G are in same patch: results when F and G are in the the same patch: apc O'FG denotes denotes the the covariance covariance of of P F and G G within within the the same distance zero) zero),, and and CH,pc CH.FC denotes denotes the and same patch patch ((at at distance the covariance patch containing both P and G When we covariance of of H H with with aa patch containing both F and G.. When we scale scale to to continuous too small more than continuous space space below, below, and and patches patches become become too small to to contain contain more than one one individual, individual, the the singular singular terms terms will will express express the the case case where where one one individual individual plays plays two two roles roles at at once, once, e.g., e.g., as as both both neighbor neighbor and and competitor. competitor. The The within withinpatch covariance covariance apc crF~ will zero for for F F i:~ G G and and (F) (F) for for F F = = G, G, which which we we will equal equal zero patch can become can express express by by writing writing it it as as o8F~ (F) (or (or o8Fc (G)); CF,CH CF.cH will will similarly similarly become Pc (G»); Pc (F) 0CHC 8cHCFc. One can can sometimes sometimes justify justify dropping dropping the the third-moment third-moment singular singular PC. One terms for for analytical analytical simplicity simplicity in in the the limit limit of of long-range long-range interactions interactions ((Bolker terms Bolker and Pacala, 1999). and Pacala, 1 999). The that we to The only only delicate delicate part part of of these these expansion expansion is is that we have have to to remember remember to drop nothing really really happens: happens: for drop cases cases where where nothing for example, example, if if the the event event "parasitoid "parasitoid moves to y the same points xx == y, y, then then no no move move really really takes takes moves from from x x to y"" involves involves the same points place and we should term. place and we should omit omit the the singular singular term. Alternatives Alternatives
The covariance is is only one way way of of expressing dependence. We We The covariance only one expressing spatial spatial dependence. (FG) or could also leave all all expressions in terms terms of of joint joint densities could also leave expressions written written in densities (FG) or densities, scaled scaled we could could write write the the spatial spatial dynamics dynamics in in terms terms of of conditional we conditional densities, or muhiplicative multiplicative covariances covariances (neglecting ( neglecting singular singular covariances, correlations, or covariances, correlations, terms): terms) : Conditional Conditional density density Scaled Scaled covariance covariance Correlation Correlation Multiplicative Multiplicative covariance covariance
(F I G) G) = = ~Fc) � (FI (C) f.:EG - (FC) CoFG __ (FG) -- (F) (F) (C) (G) SCPC = (P I G) G) -- (F) (F) SC;G ~c) = - (c) - (FI (C) -(C) (FC) ...il'.QL (F) (C) _ CFG CFG = ( F G ) (F)(G) __ (FG) rC~A PC - (F) (C) 1 (C) - - 1 ~ F G - - (F) (G) (F) (F) (G) (F) (G) (C) - (F) _
(FG) ...il'.QL (F) (G) (C) (F)
_
-
'
= CCP + 11 F GC + --
These forms forms are are all all equivalent, equivalent, but but have have advantages advantages in in different different situations: situations: These for for example, example, using using conditional conditional densities densities makes makes itit easy easy to to take take the the limit limit where where one one species species is is invading invading at at low low densities. densities.
BENJAMIN BENJAMIN M. M. BOLKER BOlKER
58 58
Moment Closure M o m e n t Closure Moment closure closure is is the the key key step step in in the the derivation, derivation, but but also also (potentially) (potentially) Moment one of of the the simplest. simplest. One One way way or or another, another, we we have have to to deal deal with with the the presence presence of of one third-moment terms, terms, which which incorporate incorporate the the information information about about the the spatial spatial pattern pattern third-moment that cannot cannot be be fully fully captured captured by by pairwise pairwise approximations. approximations. The The simplest simplest solution solution that is power-1 power-l closure, closure, which which assumes assumes that that the the probability probability of of aa given given triangular triangular concon at location location x, x, type type G G at at location location y, y, type type H H at at location location z) z) figuration (e.g., (e.g., type type FF at figuration is described described adequately adequately by by taking taking the the probability probability of of pairs pairs and and assuming assuming that that the the is third point point in in the the triangle triangle is is independent. independent. This This corresponds corresponds to to the the relationship relationship third (FGI-I) (FGH) = = (FG) (FG) (t-t) (H) + + (GH) (GH) (F) (F) + + (FH) (FH) (G) (G) -- 22 (F) (F> (G) (G) (t-I). (H). If If we we expand expand this this relationship in in terms terms of of central central moments, moments, we we find find happily happily that that the the power-1 power-1 relationship assumption corresponds corresponds to to M3FcH(x, M3FCH(X, y, y, z) z) -= O, 0, so so we we simply simply drop drop third-moment third-moment assumption terms. terms. Alternatives Alternatives
Different moment moment closures closures are are possible possible (see (see Box Box 3.1). 3 . 1 ) . In In some some cases, cases, it it may may Different actually actually be be simpler simpler to to substitute substitute the the relationships relationships between between noncentral noncentral moments earlier earlier in in the the derivation: derivation: the the author's author's habit habit of of waiting waiting until until this this point point moments comes from from using using power-1 power-1 closures, closures, which which are are simplest to apply apply at at this this step step by by comes simplest to setting moments to setting central central moments to zero. zero. Criteria for choosing aa moment closure are are analytical Criteria for choosing moment closure analytical simplicity, simplicity, stability, stability, and and numerical accuracy. Different better for for different kinds of Different closures closures may may work work better of numerical accuracy. problems, for for example, in competitive competitive communities communities vs vs predator-prey predator-prey or epidemic problems, example, in or epidemic systems. There is even even aa difference difference between between different regions of of phase phase space space systems. There is different regions within within the the same same model. model. For For example, example, power-1 power-1 closure closure is is unstable unstable for for calculating calculating invasion rates in in aa simple simple epidemic model m in in the the limit limit of of small small densities densities of of invasion rates epidemic model infectives, infectives, the the equations equations simply simply blow blow up up m but but actually actually predicts predicts equilibrium equilibrium den densities sities more more accurately accurately than than the the power-2 power-2 closure closure (Brown, (Brown, 2001 2001).). Power-1 Power-1 closures closures are unstable, but 999); power-2 are unstable, but analytically analytically tractable tractable (Bolker (Bolker and and Pacala, Pacala, 11999); power-2 clo closures sures may may be be the the best best overall overall choice choice for for stability, stability, although although power-3 power-3 closures closures per perform form better better under under some some conditions conditions (Law (Law et et aI., al., 2003). 2003). Filipe Filipe has has discussed discussed aa variety of of more more complex, complex, more more accurate accurate closure closure schemes; schemes; while while these these are are dis disvariety cussed cussed in in aa lattice lattice context, context, they they could could easily easily be be adapted adapted to to continuous-space continuous-space mod models 999; Filipe 1 ; Filipe Maule, 2003). els (Filipe, (Filipe, 11999; Filipe and and Gibson, Gibson, 200 2001; Filipe and and Maule, 2003). (These (These closures closures are are intended, intended, and and are are probably probably more more useful, useful, for for cases cases where where accurate accurate numeric solutions solutions are are required required rather rather than than for for cases cases where where one one will will attempt attempt to to numeric analyze analyze the the moment moment equations.) equations.) Hybrid Hybrid closures, closures, which which scale scale between between two two dif different ferent closures closures in in different different regions regions of of phase phase space space (e.g., (e.g., when when infection infection is is rare rare or or common), 1999). The common), have have been been used used in in aa nonspatial nonspatial context context by by Dushoff Dushoff ((1999). The choice choice of of closures closures is is still still more more of of an an art art than than aa science. science. At At present, present, recommendations recommendations are are ((1) 1 ) use use power-1 power-1 closures closures for for analytically analytically tractable tractable results, results, except except possibly possibly in in inva invasion sion cases; cases; (2) (2) use use power-2 power-2 closures closures otherwise; otherwise; and and (3) (3) explore explore the the accuracy accuracy of of aa particular particular closure closure with with simulations simulations and and always always check check key key results. results.
Take Take Continuum C o n t i n u u m limits Limits Now Now we we let let the the patch patch size size w 03 and and the the time time step step !::. Att both both become become infinitesi infinitesimally small. small. Prior Prior to to taking taking the the limits limits we we rescale rescale the the mean mean densities densities (F) (F) to to den denmally sities clw2• sities ff = = (F)/w, (F)/03, and and the the covariances covariances CFC CFc to to covariance covariance densities densities CFC cFc == =- CF CFd03 2.
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
59 59
BOX 3.1 Verbal, Analytic, and Graphical Descriptions of Different Moment Closures
Each moment closure breaks the probability of a triangular configuration (FGI-I) down in a different way: while power-l assumes that each point is independent of the remain ing pairs, power-2 and power-3 deal with pairs only. Power-2 considers pairs (sides) two at a time, while power-3 considers all three sides simultaneously. For each geometric assumption, we can write out the relationship between the triple and the pairs and points (noncentral moments), expand it in terms of central moments, and solve it for M3. Dieckmann and Law (2000) give more details on the consistency conditions that must be met by any moment closure rule.
Closure Power-l Power-2 (asymmetric) Power-3
Noncentral moments
Central moments
(FG) (H) + (GH) (F) + (FH) (H) - 2 (F) (G) (H) (FG)(FH)
o
«FGH»
(M3)
(F)
(eFG (Iy - xl) eFH (Ix - zl) eGH (Iy - zl)
(FG)(FH)(GH) (F)(G)(H)
+
+
+
eFG (Ix - y l) eFH (Ix - Zl) (G)(H) eFc (lx - y l) eGH (ly - zl) (F)(H)
eFH (Ix - zl) eGH (Iy - zl) (F)(G»
I « F)(G)(H»
The power-2 asymmetric closure shown here multiplies only one pair of sides «FG) and (FI-I) . It can be made symmetric by including terms for the other two pairs of sides, but it is useful in the asymmetric form when a triangle naturally contains a focal indi vidual, which breaks the symmetry (Murrell and Law, 2000). Symmetric power-2 clos ures require a slightly counterintuitive correction term in order to make them consistent (Dieckmann and Law, 2000). Power-2 closure is analogous to the usual pair approxi mation used in lattice models; it represents an assumption of independence between neighbors of a focal individual. I n the graphical representation below, lines between circles indicate dependence; juxtaposition indicates multiplication (combining independent probabilities). • •
10 :.0 -0 -2 0 +
+.
•
.0
power-1 power-2
power-3
BENJAMIN M. BOlKER BENIAMIN M. BOLKER
60 60
In In this this limit, limit, summations summations become become integrals integrals and and delta delta functions functions become become Dirac Dirac delta delta functions, functions, but but all all Dirac Dirac delta delta functions functions are are found found inside inside integrals integrals and and can can be be extracted extracted by by the the rule rule that that Jf 3x ~x d dxx = - 11.. We We can can also also simplify simplify variance variance terms terms (0" there can only be be zero zero or patch, (FF) (or;2)) because because when when there can only or one one individual individual per per patch, (FF) = = (F) (this is (F) = = 00 or or 11 and and (0" (or;2)) = = (F) (F) (this is equivalent equivalent to to assuming assuming Bernoulli Bernoulli statistics statistics within within each each patch) patch).. Alternative Alternative
If preferred, we can retain sizes, although although in this case case we may If preferred, we can retain finite finite patch patch sizes, in this we may have time evolution within have to to write write out out separate separate equations equations for for the the time evolution of of the the withinpatch variances variances 0" cr;2 ((Bjornstad Bolker, 2000) 2000) and and the the within-patch within-patch Bjornstad and and Bolker, patch covanances covariances O"pc. O'FG.
Simplify Delta Functions and Convolutions Simplify Delta Functions and Convolutions The The last last step step is is just just algebraic algebraic tidying tidying and and involves involves no no new new assumptions assumptions or or conceptual leaps. leaps. We values out conceptual We take take delta delta functions functions and and average average values out of of integrals integrals (y) dy ); write ((all all kernels kernels K normalized so /C are are normalized so that that JK f/C(y) dy = 11); write [y ] y- x[ x I = r; r; and y[) dz and write write terms terms of of the the form form Jf K([z /C([z-- x[)cPc x])cFo ([z ( ] z- y]) dz as as convolutions, convolutions, (K· (r) (Bolker also rewrite (/C*cFc)(I x - Yy[) l ) == (K*cpc) (/C*cFG)(r) (Bolker et et ai., al., 2000). 2000). We We can can also rewrite the the · cPc)([x average ( [y - x[)cpc([y a v e r a g e covariances c o v a r i a n c e s that that appear appear in in the the mean mean equations, equations, JK f/C(]yx])cFo(]y(we can K omit in the subscript when it is clear from context x[)dy, x[)dy, as as CK,P e~,FG (we can /C omit in the subscript when it is clear from context C which kernel is weighting term). which kernel is being being used used as as aa weighting term). All All the the algebra algebra results results in in the the following following mean mean equations equations for for the the parasitoid parasitoid model: model:
��dv = fvv - f..LVV - a(v2 + dt
@ dp dt dt
- fw
-
~w
- ~ ( v 2 + cuvv' CU~v, vv) vv) - "'( ~ ((vp vp + + cUpv, -~u~, vp) ~p)
= (vp ++ Cc-upv, ..Lpp - "'( ~l(vp v p )- f ~PP Um vp)
(3.2) (3.2)
_
These These are are the the covariance covariance equations: equations:
ac a c vyy v ((r) r) Dvy( (r)v = 2[f 2 [ f vyD r)v + + f f vy((D D vy**ccvyy v ) ()(r) r)-'--'-- = at 8t
- f..Ll~vCvv(r) yc yy(r)
2Uyy(r) ~((g4vv -- a(( Uyy * cCvv)(r) yy)(r) ++ vv2Ltvv(r) --~l(v(blnv*Cnv)(r) Pcvv(r)) "'( (v(Upy *cpy)(r) ++ pc yy(r)) *
ac yp(r) Ocvn(r) 8t at
+ + Uyy b l v v ((r)c r ) c vyy v + + cCvv(r)) yy(r))
(3.3) (3.3)
(pcyp(r) ++ v( Cpp)(r) + + Up Unv(r)vp) yp)( r ) -- f..LI~vCvn(r) Upy ** cpp)(r) - f fv(Z)v v(Ltnv v(Dy ** cCvp)(r) ---'--'- = y(r)vp) vc yp(r) -- "'(~l(pcvn(r) pc yp(r) + (v( Up y ** cvp)(r) + "'( ~l(v(blnv Cvn)(r) + + pc Pcvv(r)) - f..L I~nCvp(r) yy(r)) -- mC m c vynp(r) (r) + + m(Dp m(g)n * cCvn)(r) yp)(r) -- a(vc a(VCvn(r) Uyy ** cCvn)(r)) yp)(r)) yp(r) ++ ((blvv
(3.4) (3.4)
acpp(r) Ocnn(r) (pc p(r) ++ v( = 2 2[~l(pcvp(r) v(blpv Cpp)(r) + + U b lpeyv(r)vp) ( r ) v p ) -- f p~pCpp(r) Upy "* cpp)(r) -..Lpcpp(r) aOtt--'- = ["'( y mcpe(r) + + m(Dp m(7)p '*:. cpp)(r)] Cpe)(r)] -- mcpp(r)
(3.5) (3.5)
•.
3. 3.
CONTINUOUS-SPACE MODELS MODELS CONTINUOUS-SPACE
3.4 3.4
61 61
NUMERICAL N U M E R I C A L RESULTS RESULTS We can evaluate the spatial dynamics of of the parasitoid-prey parasitoid-prey system stochas stochastically, by running an individual-based individual-based simulation, or deterministically (and approximately), approximately), by integrating the moment equations numerically. As suggested previously, it is important when beginning to work equations to when beginning work with moment moment equations check check their their accuracy accuracy (possibly (possibly for for aa variety variety of of different different closures) closures) over over aa wide wide range of chapter focuses range of conditions. conditions. Because Because this this chapter focuses on on the the derivation derivation and and meaning meaning of of moment moment equations, equations, the the following following section section shows shows only only aa brief brief overview overview of of the the numerical numerical results results from from the the parasitoid-prey parasitoid-prey system. system. Starting Starting from from the the monocul monoculture ture equilibrium equilibrium of of the the prey prey (running (running only only the the equations equations for for V V and and Cvv Cvv to to equi equilibrium) and then introducing librium) introducing a small density of of parasitoids, the system shows shows (In fact, damped damped oscillations oscillations to to an an equilibrium equilibrium (Fig. (Fig. 3.2). 3.2). (In fact, because because moment moment equa equations tions represent represent the the dynamics dynamics of of the the average average neighborhood neighborhood across across an an infinitely infinitely
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N umerical solutions Numerical solutions of of parasitoid-prey parasitoid-prey moment m o m e n t equations. equations 9 Parameters Parameters (nonspatial): (nonspatial): tv fv = = 2.0, 29
fv .0, a .0. All All kernels fv = = 11.0, c~ = = 0.1 0.1,, fLp l~p = = 0.2, 0.2, 'Y 7 = = 0.1 0.1.. Parasitoid Parasitoid movement movement rate rate m mpp = = 11.0. kernels exponential exponential (sym (symmetric Wm), with and competition = 0.2, metric exponential), exponential), K /C = = 221m / m exp(exp(-Irl/m), with prey prey dispersal dispersal and competition scale scale mDv mDv = = muvv muvv-0.2,
.0. Solutions parasitoid parasitoid dispersal dispersal and and parasitism parasitism scale scale mupv mupv = = mvp m~p = = 11.0. Solutions by by the the Euler Euler method method with with total total spatial spatial length 25, 25, Ax Ax = = 0.25, 0.25, total total time time = 80, 80, At At = = 0.025. 0.025. length =
62 62
BENJAMIN BENJAMIN M. M. BOLKER BOLKER
extended extended stochastic stochastic system, system, they they should should always always converge converge to to an an equilibrium: equilibrium: without without global global interactions, interactions, there there is is no no way way that that parasitoid-prey parasitoid-prey populations populations can can stay stay in in synchrony synchrony across across an an infinite infinite landscape.) landscape.) For For the the parameters parameters chosen chosen in Fig. 3.2, the dynamics (Fig. 3.2a) dynamics of the mean mean densities in the spatial model (Fig. are are nearly nearly identical identical to to the the dynamics dynamics obtained obtained from from the the analogous analogous nonspatial nonspatial model (Fig. .2c and 3.2d) show (Fig. 3.2b). However, the covariances (Fig. (Fig. 33.2c show interest interesting patterns: the parasitoid-prey parasitoid-prey covariance at short short spatial lags quickly becomes negative itoids deplete deplete the becomes negative as as paras parasitoids the prey prey in in their their immediate immediate neigh neighborhood, driving the average parasitoid prey covariance covariance negative (Fig. (Fig. 3.2d). This This zone zone of of depletion depletion grows grows rapidly rapidly up up to to about about tt = = 55 with with aa maximum maximum corre correlation 12 units scale of lation length length of of 12 units and and then then shrinks shrinks back back to to aa length length scale of approxi approximately 33 units. prey mately units. The The average average covariance covariance between between parasitoids parasitoids and and prey (cvp (Cvp = =f fUvp(r)cvp(r)dr), which describes describes the the net net effect effect of of spatial spatial segregation segregation on on Uvp(r)cvp(r)dr), which the (Fig. 3.2d): the parasitism parasitism rate, rate, is is always always negative negative (Fig. 3.2d): the the spatial spatial structure structure always always reduces rate. The reduces the the effective effective parasitism parasitism rate. The average average intraspecific intraspecific covariance covariance of cvv = fflgvv(r) cvv(r) dr) of prey( prey(evv dr) is is always always positive positive ~ prey prey experience experience intraspe intraspeUvv(r) Cvv(r) cific raises their - but cific crowding crowding that that raises their effective effective density density~ but drops drops as as parasitoids parasitoids reduce also experience experience intraspecific reduce the the prey prey density. density. Parasitoids Parasitoids also intraspecific aggregation aggregation (cpp (epp = ffblvp(r) Cpp(r) dr dr > > 0), 0), but but this this aggregation aggregation has has no no direct direct effect effect on on the the Uvp(r) cpp(r) dynamics mean densities. dynamics of of the the mean densities. What What about about the the effects effects of of (for (for example) example) changing changing parasitoid parasitoid scales? scales? Figure Figure 3.3 3.3 compares compares the the results results of of simulations simulations with with numerical numerical integration integration of of the parasitoid the moment moment equations equations for for aa broad broad range range of of parasitoid parasitoid scales scales ((parasitoid dispersal scale were equal in dispersal scale scale and and parasitism parasitism scale were set set equal in all all cases) cases).. For For large large parasitoid relatively accurate; parasitoid scales, the moment moment equations equations are relatively accurate; the prey den density spatial case; para sity converges converges on on the the expected expected density density in in the the non nonspatial case; and and the the parasitoid density is slightly below the nonspatial case, the only nonspatial density. In this case, effect enhance intraspecific competition among effect of space is to enhance intraspecific competition among the prey. As in many many simple simple parasitoid-prey parasitoid-prey models, models, the the lowered lowered productivity productivity of of the the prey prey base base is is reflected reflected in in aa lowered lowered density density of of the the parasitoids parasitoids rather rather than than in in the the prey prey density density itself. itself. As As the the parasitoid parasitoid scale scale decreases, decreases, the the intraspecific intraspecific aggregation aggregation of of the the prey prey increases ) . The increases (from (from Cvv evv ~ 0.2 0.2 to to = ~ 11). The much much larger larger effect, effect, however, however, is is the the large large increase increase in in magnitude magnitude of of the the spatial spatial segregation segregation between between parasitoids parasitoids and and prey prey (cv p). p). In (ev In effect, effect, this this spatial spatial segregation segregation lowers lowers the the efficiency efficiency of of the the parasitoids parasitoids by simultaneous increase by making making it it harder harder for for them them to to find find prey prey nearby. nearby. The The simultaneous increase in in both both parasitoid parasitoid and and prey prey densities densities with with increasing increasing spatial spatial segregation segregation may may seem seem surprising. surprising. However, However, we we can can understand understand it it by by seeing seeing that that in in the the nonspatial nonspatial ana analogue logue of of the the current current model, model, decreasing decreasing parasitoid parasitoid effectiveness effectiveness increases increases both both parasitoid and prey density; limiting parasitoid parasitoid effectiveness increases the pro productivity ductivity of of the the prey prey base, base, which which counterintuitively counterintuitively feeds feeds through through to to increase increase the the parasitoid parasitoid density density as as well. well. (If (If we we were were modeling modeling the the evolutionary evolutionary dynamics dynamics of probably see of this this system, system, we we would would probably see that that the the evolutionary evolutionary stable stable strategy strategy for for parasitoids and interaction parasitoids of of unlimited unlimited movement movement and interaction lowers lowers the the average average density density of looking at Eq. (3.2), of the the population.) population.) By By looking at Eq. (3.2), we we see see that that we we can can encapsulate encapsulate all all of of the the effects effects of of spatial spatial structure structure by by rescaling rescaling the the parasitoid parasitoid effectiveness effectiveness to to 2 in gen 'V [1 ~'' = = 'V ~ [[11 - cvp/(vp)] evp/(Vp)] and and the the prey prey competition competition to to a c~'' = = a ~ [1 - cvv/(v evv/(v2)]; )]; in general, eral, 'V y'' < < 'V y and and a c~'' > > a. c~. The The mean mean equations equations tell tell us us that that parasitoid parasitoid density density will will begin begin to to drop drop if if 'V ~/'' falls falls below below aa critical critical value value (fv ( f v -- fLv)/(2afLp), ~v)/(2oL~p), but but for for the the =
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Fig. 3.3 Comparisons Comparisons of moment equations. in Fig. 3.3 of simulations simulations with with numerical numerical integration integration of of moment equations. Parameters Parameters as as in Fig. 3.2, 3.2, but but with and parasitoid Fig. with varying varying (equal) (equal) scales scales of of parasitism parasitism (mupv) (mupv) and parasitoid dispersal dispersal (mDp)' (m~p).
parameter ",' < - the parameter values values given given earlier earlier that that would would require require ~/' < 0.4", 0.4~/m the effects effects of of spatial seen here here decrease spatial structure structure seen decrease ",' ~/' only only to to aa minimum minimum of of 0.84",. 0.84~/. How How accurate accurate are are the the moment moment equations equations at at capturing capturing the the observed observed changes changes in in the the covariances? covariances? In In this this case, case, the the simple simple power-1 power-1 closure closure that that we we have have captures captures the the qualitative qualitative changes changes in in the the means means and and covariances, covariances, although although it it underpredicts underpredicts all of spatial structure. all of the the effects effects of of spatial structure. It It does does the the worst worst job job predicting predicting parasitoid parasitoid clustering clustering (luckily, (luckily, parasitoid-parasitoid parasitoid-parasitoid covariance covariance is is the the one one covariance covariance term term that that has has no no direct direct effects effects on on mean mean density). density). Other Other symmetric symmetric closures closures (power-2 (power-2 and and power-3; power-3; not not shown) shown) give give very very similar similar results results to to the the power-1 power-1 closure, closure, but but an an asymmetric asymmetric power-2 power-2 closure closure fails fails badly, badly, predicting predicting zero zero parasitoid-prey parasitoid-prey covari covariance. (but spatial ance. In In general, general, we we expect expect the the moment moment equations equations to to work work well well (but spatial struc structure ture to to be be irrelevant) irrelevant) for for large large scales scales of of movement movement or or interaction interaction and and to to fail fail at at extremely scales. We based on kernel extremely short short scales. We can can define define aa neighborhood neighborhood size size based on the the kernel shape exponential kernel shape and and the the scale scale parameter parameter (Bolker, (Bolker, 1999). 1999). For For the the exponential kernel used used here, here, the the neighborhood neighborhood size size equals equals 4m. 4m. In In the the past, past, we we have have found found that that moment moment equations equations give give accurate accurate predictions predictions of of mean mean densities densities when when individuals individuals average average = 1 0-100 discrete 10-100 discrete neighbors neighbors within within their their interaction interaction neighborhood neighborhood (Bolker (Bolker and and Pacala, 997), although Pacala, 11997), although qualitative qualitative results results may may apply apply for for lower lower numbers. numbers.
BENJAMIN BENJAMIN M. BOlKER BOLKER
64
Predictions 3b) for Predictions of of prey prey density density are are reasonably reasonably accurate accurate (Fig. (Fig. 3b) for scales scales in in the the range 2, this range of scales corresponds corresponds to range 0.5-2. 0.5-2. Because Because prey prey density density is is = ~ 2, this range of scales to aa neighborhood population of 4 6, slightly below the expected range. Similarly, 4 to 116, for parasitoid density scales between 0.1 and 11.0 .0 work, which for a density of = ~88 gives gives aa range range of of 3.2 3.2 to to 32 32 neighbors. neighbors. A full analysis of this system would other changes in would explore the effects of other relative interaction interaction scales; for example, does changing parasitoid parasitoid movement and parasitism scales and parasitism scales independently independently have have any any interesting interesting effects? effects? [If [If one one sets sets all (prey dispersal, dispersal, parasitoid parasitoid movement, equal and all scales scales (prey movement, parasitism) parasitism) equal and varies varies them them simultaneously, simultaneously, the the effect effect of of prey prey aggregation aggregation dominates dominates by by lowering lowering prey parasitoid densities.] One prey productivity productivity and and hence hence lowering lowering equilibrium equilibrium parasitoid densities.] One could also look for simplified cases where some analysis is possible; as shown 1 999), one can find the equilibria of moment moment equations in Bolker and Pacala ((1999), analytically if the problem is restricted restricted to one or two two different scales or if some processes are given infinite (global) scales. In the parasitoid-prey parasitoid-prey case, one one might might get get quite quite simple simple results results if if prey prey competition competition were were global global and and para parasitoids did not not move, being able only to eat prey that that settled within within their their par parasitism neighborhood. neighborhood. Ultimately, strategies for simplifying simplifying and analyzing the equations, equations, and and connecting connecting the the results results to to the the more more general general results results from from numer numerical solutions and simulations, depend on the ecological questions of interest.
3.5 3.5
APPLICATIONS APPLICATIONS Spatial Spatial moment moment equations equations have have yet yet to to see see widespread widespread use use in in ecology, ecology, although a few groups of researchers have used them to answer a variety of questions section reviews questions in in spatial spatial ecology. ecology. This This section reviews briefly briefly the the (small) (small) literature literature of applications of spatial moment equations, making connections to nonspa nonspatial or noncontinuous noncontinuous models where appropriate. appropriate.
Single-Species Single-Species Dynamics Dynamics The The simplest simplest possible possible application application of of moment moment equations equations is is to to populations populations of of a single species; this system is called the spatial logistic model and is analogous nonspatial contact process on to both the classical non spatial logistic equation and the contact the lattice (Law et al., aI., 2003) 2003).. The model defines the fecundity and density densityindependent mortality rates of individuals, the strength of density dependence, and the kernels for dispersal and competition. competition. Competition can lower the fecundity of parents with increasing neighborhood neighborhood density; lower establishment probability of offspring with increasing density of the neighborhood where they land increase mortality land after after dispersing; dispersing; or or increase mortality probability probability of of adults. adults. The The results results are are simple: spatial structure structure can be either aggregated (positive average covariance) or even (negative average covariance). Aggregation occurs when individuals have short dispersal scales relative to competition scales and low intrinsic reproductive numbers (fecundity numbers (fecundity divided divided by by density-independent density-independent mortality mortality or or expected expected lifetime reproduction in the absence of competition). The overall strength of spatial effects is largest when neighborhood size, size, the number of neighbors with which an individual interacts, is small: the neighborhood size size is proportional to population spatial carrying population density density (determined (determined approximately approximately by by the the non nonspatial carrying
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
65 65
capacity). In the absence of interspecific competition or environmental hetero heterogeneity, the intrinsic rate of reproduction (fecundity minus density-independent mortality) and the spatial scale of dispersal merely set the temporal and spatial scales of the population without without having any qualitative effect on population dynamics ((Bolker Bolker and Pacala, 11997). 997). Heterogeneous environments widen the range of possibilities. Hetero Heterogeneity could affect fecundity, mortality, competition, or the probability of dispersal (or the scale of dispersal and competition, although this would lead to some thorny technical problems). Models Models with heterogeneous fecundity and mortality connect directly with applied questions of population viability in degraded and fragmented landscapes, as well as with basic questions about the evolution of dispersal (Rousset and Gandon, 2002). Models with variation about foraging in dispersal are connected more closely to classical questions about and aggregation in heterogeneous landscapes, but are also relevant to conser conservation. Either heterogeneous mortality or heterogeneous movement probabilities can lead to individuals aggregating in a good (low-mortality, low-movement) habitat (Murrell and Law, 2000; Bolker, Bolker, 2003). This habitat association can shield individuals from the worst effects effects of habitat degradation, but with sufficiently short dispersal scales the negative effects effects of intraspecific competition overwhelm the advantages of habitat association [and the moment equations break down; they do not capture capture the percolation phenomena that can lead to sudden sudden extinction in some continuous-space models (Bascompte and Sole, 997)]. Sol~, 11997)]. Finally, one can use single-species moment equations to predict invasions and to begin to understand the feedbacks among environmental heterogeneity, endogenous heterogeneity, and invasion speed at the edge of a spreading wave. Various pieces of the puzzle are present, but they have not been put together into a single picture. The only work with spatial moment equations analyzes the patchiness of invading populations and shows that patchiness and local crowd crowding slow invasion speeds below what would be expected from a continuous population (Lewis, 2000). Other work has been done with lattice models (Ellner et aI., 998) and diffusion equations (Shigesada and Kawasaki, 11997) 997) to explore al., 11998) the the effects effects of of endogenous endogenous and and exogenous exogenous variability, variability, respectively, respectively, but but these these phe phenomena have not been fully integrated into the elegant literature on dispersal and invasion speeds (Skellam, 11951; 95 1 ; Mollison, 1991; Clark et al., aI., 11998). 998).
Interspecific Interspecific Competition Spatial dynamics and spatial models have been explored extensively in research on competitive coexistence, particularly in plant communities (Tilman, 11994). 994) . The natural history of weedy and early successional species clearly suggests that that variations in spatial strategies could maintain diversity in environments where there is insufficient variation in resource availability for 993). Spatial niche separation to maintain diversity (Tilman and Pacala, 11993). coexistence can be interpreted as the exploitation of endogenous endogenous and exoge exogenous spatial covariance in resource availability. The basic mechanisms of spa spametapopulation tial coexistence have been explored in meta population ((Levins Levins and Culver, 1971; Tilman, 11994; 994; Pac ala and Rees, 11998; 998; Chesson, 2000a), lattice Pacala (Silvertown et aI., 992; Holmes and Wilson, 1998), and continuous-space al., 11992;
BENJAMIN BENJAMIN M. M. BOLKER BOLKER
66
models Gopasalmy, 11977a,b); 977a,b); more models ((Gopasalmy, more recent recent work work has has applied applied the the tools tools of of pair pair (Harada 994) and spatial moment (Harada and Iwasa, 11994) moment equations equations (Bolker and Pacala, 11999; 999; Bolker aI., 2003). Bolker et et aI., al., 2000; 2000; Dieckmann Dieckmann and and Law, Law, 2000; 2000; Bolker et et al., 2003). Chesson (2000a,b) (2000a,b) has developed an analytic framework framework to to quantify and test the processes by which coexisting species partition exogenous temporal and spatial variability. His framework framework largely neglects the dynamics of endogenous variation, although although it has recently begun to incorporate incorporate continuous space; the results on equilibrium equilibrium correlations correlations developed by Snyder and Chesson (2003) correspond correspond to the results one one would would get by deriving moment moment equations equations but neglecting the singular terms in the covariances. Finally, the evolutionary dynamics of dispersal and spatial coexistence remain largely unexplored unexplored (Hovestadt 1 ; Rousset and Gandon, 2002). A full accounting (Hovestadt et aI., al., 200 2001; accounting of spa spatial coexistence will have to incorporate incorporate both endogenous and exogenous vari variation and to recognize that that spatial traits such as dispersal represent just a subset of all plant plant competitive and life life history traits and that that the coevolution of these traits needs to be considered as a whole (Rees, 11996). 996).
Predator-Prey Predator-Prey The The spatial spatial dynamics dynamics of of predator-prey predator-prey interactions interactions are are currently currently an an area area of of active research, with many theoreticians theoreticians trying to understand understand whether differ differences between spatial and nonspatial models in stability and equilibrium densities are caused by diffusion limitation or other other spatial modifications modifications of the predation 993; Cuddington predation process (McCauley et aI., al., 11993; Cuddington and Yodzis, 2002; Hosseini, 2003). Keeling et al. (2002) used moment equations (for a metapopu metapopulation system) to understand understand the broad broad conditions under which spatial structure stabilizes or destabilizes a predator-prey predator-prey model. The The usual usual approaches approaches to to understanding understanding predator-prey predator-prey and and other other complex complex ecologies in continuous continuous space make use of simulation models and a variety of 995; Pascual and Levin, 11999) 999) to identify the scaling rules (Rand and Wilson, 11995; relevant scales at which these ecologies are best understood. understood. Looking at the problem from a very different perspective, spatial moment equations have a different set of advantages and disadvantages. Moment equations equations typically look at the average densities and covariance covariance densities of an entire system and, as a result, have some difficulty capturing local stability or instability. Capturing Capturing the effects of finite system size, which might be another another way to address the problem, problem, is similarly difficult because of the algebraic complexity of moment equations with spatial inhomogeneity. In addition, addition, some of the nonlinearities nonlinearities and resulting coherent coherent structures (e.g., spirals) that that arise in typical predator-prey predator-prey models may be poorly represented represented by second-order second-order moment 994). moment closures (Tainaka, 11994). However, spatial spatial moment moment equations equations also offer unique unique advantages. advantages. Other Other approaches approaches have explained the changes in predator-prey predator-prey dynamics from nonspatial nonspatial expectations expectations as a result of endogenous endogenous patchiness, patchiness, but have not not tried tried to predict predict the endogenous endogenous patchiness patchiness itself. By calculating calculating the shape and scale of equilibrium equilibrium covariances covariances within within and and between between species, we can understand understand how how different spatial and demographic demographic parameters parameters determine the the endogenous endogenous structure structure of of predator-prey predator-prey systems systems in in continuous continuous space space and, by extension, extension, the spatial dynamics of predator-prey predator-prey systems (M. Desai,
3. CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
67 67
unpublished moment equations can natnat unpublished manuscript). manuscript). Furthermore, Furthermore, spatial spatial moment equations can urally be include exogenous urally be extended extended to to include exogenous variation variation in in predation predation risk risk in in aa more general form than the patchy structure structure implied by metapopulation metapopulation models. models.
Epidemics Epidemics Epidemics are a subtype of a predator-prey system that that is of obvious prac practical and theoretical interest. Spatial models of epidemics go back to the panpan demic models of Kermack and McKendrick ( 1 927); since then, epidemics have demic models of Kermack and McKendrick (1927); since then, epidemics have been studied in metapopulation, lattice, and continuum models. Stochastic spatial models have always presented a challenge, however, and until recently, computational simulation models have been the main method of exploration (Duryea 999). Moment (Duryea et et al., al., 11999). Moment equations equations and and their their discrete discrete pair pair approxima approximation analogues have proved to be powerful tools for analyzing spatial epidemic models (SaW et 994; Keeling al., 11997; 997; Rand, 999; Keeling, 999a; models (Sato et al., al., 11994; Keeling et et al., Rand, 11999; Keeling, 11999a; Boots and Sasaki, 2000, 2002). Many of the analyses have used social contact Boots and Sasaki, 2000, 2002). Many of the analyses have used social contact networks spatial structure, networks as as the the underlying underlying spatial structure, which which is is probably probably more more appro appropriate for human diseases than a two-dimensional plane. For diseases of plants and wildlife, however, lattice or continuum models may be best. Some of the remaining challenges in using spatial moment equations for epidemiology are adapting some of the sophisticated lattice closures developed by Filipe and Gibson 1 ) to Gibson (200 (2001) to the the continuous-space continuous-space case; case; incorporating incorporating exogenous exogenous spatial spatial heterogeneity in suceptibility and infectivity; and combining disease transmis transmission with other ecological processes such as competition to explore the effects of disease in a broader ecological setting.
3.6 OUTLOOK 3.6 OUTLOOK Continuous-space Continuous-space models models represent represent aa different different approach approach to to spatial spatial ecology ecology than population models than the the meta metapopulation models discussed discussed throughout throughout most most of of this this book; book; they they are relevant for populations living in different habitats from the spatially dis disconnected, connected, temporally temporally persistent persistent patches patches that that are are the the main main concern concern of of meta population ecology. This chapter has reviewed continuous-space models metapopulation with with a strong focus on spatial moment equations, a relatively new analytical framework for reducing simple individual-based models to a set of integral equations for the densities and spatial covariances of interacting populations. Why Why would would one one choose choose continuous-space continuous-space models models (continuum, (continuum, lattice, lattice, or or spatial IBM) over patch occupancy or meta population models? There metapopulation There are two two basic reasons: first, one might be studying an organism or community that that inhabits a continuous habitat, where the patchy-habitat assumption is simply unrealistic. Second, one might be asking explicit questions about the scale and pattern of spatial structure structure that have no meaning in a discontinuous spatial model. Tactical questions and landscapes incorporating specific features such as barriers and complex boundaries between habitat types would favor spatial IBMs, IBMs, whereas whereas strategic strategic questions questions and and simple simple landscapes landscapes with with more more conti continuous nuous variation variation would would favor favor lattices lattices or or continuum continuum models. models.
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BENJAMIN M. BOLKER BENJAMIN M. BOLKER
Given Given aa continuous-space continuous-space model, model, how how does does one one choose choose among among the the different different options? options? As As described described earlier, earlier, spatial spatial IBMs IBMs are are flexible flexible and and realistic realistic but but hard hard to to generalize, generalize, whereas whereas lattice lattice and and continuum continuum models models are are more more general general and and tractable tractable but but make make different different kinds kinds of of unrealistic unrealistic assumptions. assumptions. Lattice Lattice models models assume assume aa particular, particular, unrealistic unrealistic spatial spatial structure structure that that complicates complicates the the connec connection tion with with measured, measured, individual-level individual-level data data [except [except in in cases cases where where data data are are gathered 1 994)]. Continuum gathered at at the the level level of of lattice lattice cells; cells; Silvertown Silvertown et et al. al. ((1994)]. Continuum mod models although it els eliminate eliminate demographic demographic stochasticity stochasticity ((although it can can be be reintroduced reintroduced by by adding adding noise noise with with the the appropriate appropriate scaling scaling properties) properties) and and tend tend to to reduce reduce the the complexity complexity of of individual individual movement movement rules rules to to diffusion diffusion and and advection advection terms. terms. Nevertheless, be Nevertheless, both both of of these these model model types types can can be be extremely extremely useful. useful. Both Both can can be solved numerically by simpler solved numerically by standard standard packages packages (lattice (lattice models models are are much much simpler computationally) computationally) and and both both are are some some what what tractable tractable analytically, analytically, although although most analyzed approximately approximately most nonlinear nonlinear spatial spatial models models of of any any type type can can only only be be analyzed or special cases invasion conditions. or for for special cases such such as as invasion conditions. The spatial moment equations emphasized emphasized in The spatial moment equations in this this chapter chapter represent represent aa dif different, relatively set of ferent, relatively new new set of compromises. compromises. Spatial Spatial moment moment equations equations have have two major disadvantages. two major disadvantages. First, First, because because they they depend depend on on aa heuristic heuristic approx approximation imation of of the the full full spatial spatial configuration configuration of of the the system, system, they they may may fail fail under under certain spatial certain conditions conditions ~ at at present, present, one one must must always always verify verify the the results results of of spatial moment equations with moment equations with simulations simulations for for at at least least aa few few sets sets of of parameters. parameters. Second, equations can be Second, the the detailed detailed derivation derivation and and analysis analysis of of moment moment equations can be daunting daunting for for the the average average ecologist ecologist or or evolutionary evolutionary biologist, biologist, although although some some automated automated tools tools have have been been developed developed to to help help the the process. process. Spatial Spatial moment moment equations equations do do have have strong strong counterbalancing counterbalancing advantages, advantages, however. however. Because Because their their parameters parameters are are expressed expressed directly directly in in terms terms of of individual individual behavior, behavior, moment moment equations equations are are useful useful for for studying studying the the connections connections between between individ individual Because their variables (spatial ual behavior behavior and and spatial spatial dynamics. dynamics. Because their state state variables (spatial cor correlations) preserve preserve information information about about the the shape shape and and scale scale of of spatial spatial pattern, pattern, relations) moment causes and spa moment equations equations are are useful useful for for understanding understanding the the causes and effects effects of of spatial pattern. Spatial moment equations equations can tial pattern. Spatial moment can preserve preserve some some aspects aspects of of demo demographic landscape heterogeneity, graphic stochasticity stochasticity and and of of landscape heterogeneity, although although they they cannot cannot typically typically handle handle movement movement orientation orientation by by individuals individuals or or landscape landscape struc structures, such flexibility than tures, such as as edges edges and and barriers, barriers, and and they they have have less less flexibility than spatial spatial IBMs IBMs for for incorporating incorporating details details of of individual individual state; state; at at present, present, these these state state dynamics (Fig. 33.1). .1 ). dynamics can can only only be be included included by by adding adding new new types types to to the the model model (Fig. Spatial Spatial moment moment equations equations can can bbee analyzed, analyzed, although although often often only only by by simplify simplifying models considerably. ing the the models considerably. Nevertheless, Nevertheless, spatial spatial moment moment equations equations can can form form aa bridge bridge between between complex complex spatial spatial IBMs IBMs and and simple, simple, analytically analytically tractable tractable models. models. Like Like metapopulation metapopulation models, models, moment moment equation equation approaches approaches can can be be used used to to explore population explore aa broad broad range range of of ecological ecological interactions. interactions. Also, Also, like like meta metapopulation models, models, moment moment equations equations work work best best in in communities communities with with aa particular particular kind kind of equations, these of underlying underlying spatial spatial geometry. geometry. In In the the case case of of moment moment equations, these are are communities communities where where both both population population densities densities and and environmental environmental characteristics characteristics change individual is predictable (at change relatively relatively smoothly; smoothly; where where the the fate fate of of an an individual is predictable (at least least on on average) average) from from the the characteristics characteristics of of its its neighborhood; neighborhood; and and where where that that neighborhood neighborhood is is large large enough enough to to encompass encompass at at least least aa dozen dozen or or so so interacting interacting organisms. Moment equations can shapes and scales of organisms. Moment equations can predict predict the the shapes and scales of population population
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
69 69
patterns patterns generated generated by by the the interaction interaction between between environmental environmental variability variability and and endogenous endogenous population population interactions, interactions, and these predictions predictions can sometimes sometimes be handled handled analytically. analytically. Equally Equally important, important, moment moment equations equations have have aa natural natural con connection nection with data data that that are collected at an individual individual scale, with individual-based individual-based computational computational models, models, and and ultimately ultimately with with the the life life and and death death of of individuals individuals in in ecological ecological communities. communities.
sdfsdf
Part II Metapopulation Ecology
sdfsdf
M FTAPO PU LATIO N PO PULATION 4 � META 4
DYN AMICS IN IG H LY DYNAMICS IN H HIGHLY FRAGMENTED FRAG M ENTED LANDSC A PES LANDSCAPES
Otso Ovaskainen and Ilkka Hanski
4.1 4.1
INTRODUCTION INTRODUCTION The ecological dynamics of meta populations have many facets, which can metapopulations hardly be all studied in any single investigation, nor analyzed with with a single metapopulation metapopulation model. Ecologists have constructed constructed particular models to 995; Smith et aI., investigate source-sink dynamics (Pulliam, 1988; 1988; Paradis, 11995; al., 11996; 996; Walters, 2001; Hels, 2002; Waiters, 2001; 2002; Chapter 16), 16), the the influence of immigration and emigration on the type of local population population dynamics (Gyllenberg et aI., al., 11993; 993; Rohani et aI., 11996; 996; Ruxton aI., 11997; 997; Saether et aI., 11999; 999; etal., Ruxton et etal., etal., Nachman, Nachman, 2000), 2000), models of habitat habitat selection influencing metapopulation metapopulation dynamics (Ray et al., aI., 1991; 1991; Doncaster, 2000; 2000; Etienne, 2000; 2000; Danchin et aI., al., 200 1 ), and 2001), and so forth. Models of individual movement behavior in heterogen heterogeneous landscapes (Hanski et aI., al., 2000; 2000; Ricketts, 2001; 2001; Gobeil and Villard, 2002; 2002; Ovaskainen and Cornell, 2003) 2003) provide building blocks for metapopu metapopulation models, and single-species meta population models in turn can be metapopulation extended to model the dynamics of metacommunities 992; metacommunities (Wilson, 11992; Holyoak, 2000; 2000; Klausmeier, Klausmeier, 2001; 2001; Holt, 2002). 2002). The purpose of the models ranges from attempts to clarify general principles, as described by Bolker in
Ecology, Ecology, Genetics, and Evolution of Metapopulations Metapopulations
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Copyright 2 004, Elsevier, 2004, Elsevier, Inc.
2-323448-4 0-12-323448-4 0-1
OTSO OVASKAIN EN AND ILKKA OVASKAINEN ILKKA HANSKI
74 14
Chapter Chapter 33 for spatial spatial continuum continuum models, to spatially extended population population viability analysis (Sjogren-Gulve (Sj6gren-Gulve and Hanski, Hanski, 2000; 2000; Akcakaya, 2000; 2000; Lindenmayer et aI., al., 2001; 2001; Ferreras et aI., al., 2001). 2001). This chapter chapter focuses on models that that are iinn the hard hard core ooff ecological metapopulation SPOM) and their metapopulation theory: stochastic patch patch occupancy occupancy models ((SPOM) approximations, including the mother mother of all ecological metapopu metapopudeterministic approximations, lation models, the Levins model. There are several reasons for devoting an entire chapter chapter to these models. The classic metapopulation metapopulation idea of a "popula "population" tion" of local populations populations is captured, captured, in its bare bare essence, by SPOMs. In a broader broader biological perspective, SPOMs belong to "island models," models," which have played an important important role not not only in population population and community ecology (MacArthur (MacArthur and Wilson, 1963, 1963, 1967), 1967), but also in population population genetics (Wright, 1931; 1931; Slatkin, 1977; 1977; Wade and McCauley, 1988; 1988; Whitlock and McCauley, 1990) and evolutionary studies (Wright, 1931, 1931, 1940). 1940). These models often 1990) allow a rigorous mathematical analysis, an obvious advantage for theory. However, there is yet another another reason for focusing on SPOMs in this chapter, a reason reason that that complements the advantages of the models for theoreticians: theoreticians: SPOMs are good good models for real metapopulations metapopulations living in highly fragmented landscapes, landscapes, to the extent extent that that they can be parameterized with empirical data (Chapter 5) 5) and turned turned into tools that that hold substantial promise for conserva conservation, landscape landscape planning, and management. Figure 4.1 4.1 gives an example of a highly fragmented landscape, landscape, in fact two two representations representations of it. The map on the left shows the 56 habitat habitat patches (dry meadows) suitable for reproduction reproduction by the Glanville fritillary butterfly (Melitaea land Islands in Southwest Finland (Melitaea cinxia) cinxia) in one part of the A .~land (Hanski, 11999b). 999b). The long-term project on the Glanville fritillary has played an
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A network of 56 an example throughout this chapter. This 56 habitat patches used used as an throughout this patch network is part of a larger network of habitat patches inhabited by the Glanville fritillary butterfly (Melitaea land Islands 999b; Nieminen (Melitaea cinxia) in the A ~,land Islands in Southwest Finland (Hanski, 11999b; et aI., al., 2004). 2004). Panel A shows a map of of the real real landscape, whereas panel BB shows the simplified view (network of circular habitat patches) assumed by the spatially realistic metapopulation theory. (network The areas areas of the the circles in BB are proportional to the the sizes sizes of the habitat patches, but but they have have circles in not been drawn to scale. The patch indicated by an arrow is analyzed in Fig. Fig. 4.7. 4.7.
Fig. 4.1
4.. 4
M ETAPOPULATION DYNAMICS METAPOPULATION DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
75 75
important important role in stimulating stimulating research on SPOMs, and and we will use the the results of empirical studies example in of our our empirical studies of of this this butterfly butterfly as as aa focal focal example in this this chapter. chapter. The The rest of the landscape landscape in Fig. 4.1 mostly consists of cultivated cultivated fields and and forests. The "highly fragmented The characteristic characteristic feature feature of of "highly fragmented landscapes" landscapes" is is that that the the pooled pooled area of suitable suitable habitat habitat for for the species of interest interest is limited limited to to say less than than 110% 0 % of the total total area of the landscape, landscape, and and the habitat habitat occurs in discrete 1 ) shows patches, none of which is very large. The map map on the right (Fig. 4. 4.1) shows the simplified view of the same patch patch network network in which which we only distinguish distinguish between between the the breeding breeding habitat habitat and and the the remaining remaining matrix matrix and and in in which which the the actual approximated by actual shapes shapes of of the the patches patches are are approximated by circles. circles. How How common common are are "highly "highly fragmented fragmented landscapes? landscapes?"" Nobody Nobody knows knows the the answer species, but answer for for aa comprehensive comprehensive sample sample of of species, but there there are are reasons reasons to to suspect suspect that that aa large large fraction fraction of of invertebrates invertebrates and and increasing increasing numbers numbers of of other other species species occur occur in highly fragmented fragmented landscapes. landscapes. Community Community ecologists ecologists have docu documented mented repeatedly repeatedly that that most most species species in in most most communities communities are are locally locally rare rare (Whittaker, 975; May, 975; Gaston, 994) . While While there there are reasons (Whittaker, 11975; May, 11975; Gaston, 11994). are many many reasons for of habitat. habitat. Furthermore, for local local rarity, rarity, one one likely likely reason reason is is aa limited limited amount amount of Furthermore, locally rare species tend tend to be rare also in the the surrounding surrounding landscape (Hanski, 11998b), 998b), suggesting frag suggesting that that for for such such species species the the landscape landscape is is often often highly highly fragmented. For the land Islands Islands it it has has been mented. For the Glanville Glanville fritillary fritillary in in the the A Aland been relatively relatively straightforward straightforward to document document the degree of fragmentation fragmentation empirically empirically ~ ca. 4000 0.6% of 4000 fragments fragments covering covering 0.6% of the the total total landscape landscape (Nieminen (Nieminen et et aI., al., 2004) 2004) - but 1 990) have argued but for most most other other species it is not. Murphy Murphy et aI. al. ((1990) argued that that species species with with small small body body sizes, sizes, high high rates rates of of population population increase, increase, short short genera generation tion times, times, and and specific specific resource resource requirements requirements are are predisposed predisposed to to have have aa meta population structure. structure. A fraction of insects possess metapopulation A large large fraction of insects possess these these attributes. attributes. There There are are also also very very large large numbers numbers of of species species that that live live in in discrete discrete "minor "minor habi habitats" 949), such but tats" (Elton, (Elton, 11949), such as as decaying decaying tree tree trunks, trunks, which which are are not not permanent permanent but change change in in time. time. Furthermore, Furthermore, large large numbers numbers of of parasites parasites inhabit inhabit aa highly highly frag fragmented once we that, from their perspective, host individu mented landscape landscape once we realize realize that, from their perspective, host individuals are Grenfell and 1 997). The metapopulation als are habitat habitat patches patches ((Grenfell and Harwood, Harwood, 1997). The metapopulation theory with due to the theory developed developed here here applies applies also also to to these these systems, systems, with due attention attention to the dynamics dynamics of of the the landscape. landscape. Finally, Finally, as as we we all all know, know, the the expanding expanding human human enterprise fragmentation of many habitat enterprise typically typically leads leads to to the the loss loss and and fragmentation of many habitat types types in of species highly in most most landscapes, landscapes, increasing increasing the the frequency frequency of species whose whose habitat habitat is is highly fragmented. fragmented.
Stochastic Stochastic Patch Patch Occupancy Occupancy Models Models Stochastic Stochastic patch patch occupancy occupancy models models are are based based on on two two major major simplifying simplifying assumptions. relates to assumptions. The The first first assumption assumption relates to the the structure structure of of the the landscape, landscape, which which is is assumed assumed to to consist consist of of discrete discrete patches patches of of breeding breeding habitat habitat surrounded surrounded by Fig. 4 .1 landscape is by the the matrix matrix as as shown shown in in Fig. 4.1 ~ the the landscape is highly highly fragmented. fragmented. The The second major assumption second major assumption concerns concerns the the description description of of population population dynamics; dynamics; SPOMs possible states SPOMs recognize recognize only only two two possible states for for each each habitat habitat patch, patch, which which can can be be either either occupied occupied by by the the focal focal species species or or unoccupied. unoccupied. Thus Thus the the sizes sizes and and struc structures explicitly accounted accounted for. tures of of local local populations populations are are not not explicitly for. While While these these simpli simplifications fications restrict restrict the the range range of of problems problems to to which which SPOMs SPOMs may may be be applied, applied, they they make make the the models models both both tractable tractable for for rigorous rigorous mathematical mathematical analysis analysis
76 16
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HANSKI
((Ovaskainen Ovaskainen and 2002, 2003a; and Hanski, Hanski, 2001 2001,, 2002, 2003a; Ovaskainen, Ovaskainen, 2003) 2003) and and turn turn them tools that used increasingly empirical studies them into into effective effective tools that are are used increasingly in in empirical studies (Moilanen, 999, 2000; 999b; Hanski (Moilanen, 11999, 2000; Hanski, Hanski, 11999b; Hanski and and Ovaskainen, Ovaskainen, 2000; 2000; Ter Ter Braak Braak and and Etienne, Etienne, 2003; 2003; Chapter Chapter 55).) . The The original original "island "island model" model" approach approach to to modeling modeling spatially spatially structured structured dynamics in population population biology assumes that that all the patches are identical. In the the case case of of SPOMs, SPOMs, this this assumption assumption also also means means that that all all the the patches patches and and all all the the local local populations populations occupying occupying these these patches patches are are equally equally connected connected to to each each other. We call models making this assumption assumption homogeneous SPOMs, which are patches obviously are described described in in Section Section 4.2. 4.2. Assuming Assuming identical identical patches obviously simplifies simplifies the models greatly, the models greatly, which which has has been been the the overriding overriding justification justification for for this this assumption. One might also ask what what could be gained by assuming spatial assumption. variation variation in in the the properties properties of of the the patches patches - - would would such such more more complex complex mod models provide any novel insight to the processes processes that SPOMs are used to study? that SPOMs The is. The The answer answer depends depends on on what what our our aim aim is. The greatest greatest advantage advantage of of employ employSPOMs with ing ing heterogeneous heterogeneous SPOMs with variation variation in in patch patch properties properties stems stems from from the the opportunity apply them opportunity to to meaningfully meaningfully apply them to to real real metapopulations metapopulations living living in in highly fragmented landscapes, either for the purposes purposes of management management and conservation or for the purpose of critically testing model predictions predictions for advancing understanding. A advancing scientific scientific understanding. A core core idea idea in in this this research research is is to to make make structural structural model assumptions assumptions about how the key landscape features, such as habitat habitat patch areas, qualities, and spatial locations, locations, will influence the two two key processes of classic classic metapopulation dynamics; local local extinction and recolon processes metapopulation dynamics; recolonization. We have called the combination combination of heterogeneous SPOMs (assuming dissimilar patches patches and hence patch-specific dissimilar and hence patch-specific transition transition probabilities) probabilities) with with these these structural structural model model assumptions assumptions as as the the spatially spatially realistic realistic metapopulation metapopulation theory ((SMT; SMT; Hanski, 1 b; Hanski Hanski and Hanski, 200 2001b; and Ovaskainen, Ovaskainen, 2003 2003).). The The assump assumppopulation processes to landscape structure are discussed in tions relating population Section models based Section 4.3, 4.3, where where we we also also describe describe the the two two most-studied most-studied models based on on this this theory: theory: the the spatially spatially realistic realistic Levins Levins model model and and the the incidence incidence function function model. model. The The stochastic stochastic theory theory of of heterogenous heterogenous SPOMs SPOMs poses poses mathematical mathematical prob probthat a meta metapopulation lems, which relate to the fact that population living in a large patch described by presence or network network with with n n patches patches and and described by the the presence or absence absence of of the the number of possible states, 2 nn.. We describe species in these patches has a huge number two two approaches to overcome these problems. First, for many purposes it is sufficient approximation of sufficient to to use use aa deterministic deterministic approximation of the the stochastic stochastic model; model; the the deterministic theory is described in Section 4.4. Second, Ovaskainen (2002a) deterministic theory has has described described an an effective effective approach approach for for analyzing analyzing the the stochastic stochastic model model for for aa heterogeneous patch network with the help help of an appropriate heterogeneous patch appropriate stochastic model model for for aa homogeneous homogeneous network, network, which which latter latter model model can can be be analyzed analyzed mathematically. mathematically. This This approach, approach, described described in in Section Section 4.5, 4.5, is is akin akin to to the the idea idea of of effective effective population population size size in population population genetics: construct an unstructured unstructured model that that behaves similarly to a structured model with respect to some model features of interest. Following the description description of deterministic deterministic and sto stochastic chastic theories theories of of SPOMs, SPOMs, Section Section 4.6. 4.6. presents presents some some comparisons comparisons with with other approaches other approaches to to modeling modeling metapopulation metapopulation dynamics, dynamics, particularly particularly focused focused on key issues issues in on one one of of the the key in ecological ecological metapopulation metapopulation dynamics, dynamics, the the extinc extinction Section 4.7 tion threshold threshold for for long-term long-term persistence. persistence. Section 4.7 discusses discusses the the current current
4.4.
METAPOPULATIONDYNAMICS DYNAMICS IIN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION N HIGHLY
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state of the theory and its applications, and Section 4.8 outlines the broader significance significance of of the the spatially spatially realistic realistic metapopulation metapopulation theory, theory, emphasizing emphasizing the the contributions contributions that that it it makes makes toward toward greater greater unification unification of of research research in in popula population biology. This This chapter chapter is is mostly mostly about about theory theory and and models, models, although although our our purpose purpose is is not not just to describe a mathematically rigorous body of theory but also to extract that stem extract biologically biologically relevant relevant messages messages that stem from from SMT. SMT. We We illustrate illustrate many many of of the the results results with with examples examples based based on on the the habitat habitat patch patch network network for for the the Glanville . 1 . We Glanville fritillary fritillary butterfly butterfly depicted depicted in in Fig. Fig. 44.1. We will will not not review review the the litera literature on on the the application application of of the the models models to to real real meta metapopulations, which is is cov covture populations, which 999b, in Chapter Chapter 55 (for (for more more ecologically ecologically oriented oriented reviews, reviews, see see Hanski Hanski 11999b, ered in 200 l b) . The 2001b). The kind kind of of models models examined examined here here are are employed employed in in some some of of the the analy analy20-22 in this volume. ses presented in Chapters 20-22
4.2 4.2
PATCH OCCUPANCY OCCUPANCY MODELS: MODELS: HOMOGENEOUS HOMOGENEOUS PATCH PATCH NETWORKS NETWORKS PATCH Mathematically, Mathematically, SPOMs SPOMs are are defined defined as as Markov Markov chains chains (discrete-time (discrete-time models) models) or Markov Markov processes processes (continuous-time (continuous-time models). models). However, However, most most of of the the existing existing or theory models, but theory does does not not relate relate to to the the full full stochastic stochastic models, but to to their their deterministic deterministic approximations, speaking, account approximations, which, which, mathematically mathematically speaking, account only only for for the the drift drift in in the dynamics dynamics while while ignoring ignoring the the variance variance due due to to stochastic stochastic fluctuations. fluctuations. This This the section considers considers models models that assume identical identical habitat habitat patches. section that assume patches. We We start start with with the the familiar model, which familiar Levins Levins model, which is is helpful helpful in in highlighting highlighting the the basic basic concepts concepts and and qualitative theory behind SPOMs. qualitative theory
Levins Metapopulation Metapopulation Model Model Levins Levins 1 969, 11970) 970) assumed habitat patches patches in patch Levins ((1969, assumed that that the the number number of of habitat in aa patch network is is infinite, network infinite, which which allowed allowed him him to to formulate formulate aa patch patch occupancy occupancy model model directly from the deterministic viewpoint. viewpoint. However, However, we we will will view directly from the deterministic view here here the the Levins the deterministic deterministic mean-field mean-field approximation of the the stochastic Levins model model as as the approximation of stochastic identical and and logistic (Box 4. 1 ) , which assumes aa finite network of of nn identical logistic model model (Box 4.1), which assumes finite network equally connected of which may be be either equally connected patches, patches, each each one one of which may either occupied occupied or or empty. If patch is is occupied, occupied, it it is is assumed assumed to to go go extinct extinct at at aa fixed fixed rate empty. If aa patch rate E E == e.e. The colonization colonization rate rate of of an an empty is assumed to depend on the the fraction fraction The empty patch patch is assumed to depend on of occupied occupied patches patches as as C C == ck/n, ck/n, where where cc is is aa colonization colonization rate rate parameter parameter and and of o ::; k-< k ::; nn isis the the number number of of occupied occupied patches. patches. The The reasoning reasoning behind behind this this 0-< expression for for the the colonization colonization rate rate is is that that kk occupied occupied patches patches are are assumed assumed to to expression produce emigrants emigrants that that disperse disperse randomly randomly to to any any of of the the habitat habitat patches patches (both ( both produce occupied and and empty). empty). As As described described in in Box Box 4.1, 4 . 1 , these these transition transition rates rates from from occupied empty to to occupied occupied patches patches and and vice vice versa versa define define the the stochastic stochastic logistic logistic model model empty as a Markov Markov process. process. If the the number number of of patches patches is is large, large, the the stochastic stochastic logistic logistic model model may may be be If approximated To do p(t) == k(t)/n k(t)/n approximated by by aa diffusion diffusion process. process. To do this, this, we we denote denote by by p(t) the fraction fraction of of occupied occupied patches patches at at time time t.t. The The diffusion diffusion process process is is determined determined the by E[dp 2] /dt of of by the the mean mean (or (or drift) drift) ~(p) j.L(p) == E[dp]/dt E[dp]/dt and and the the variance variance ~r2(p) (j2 (p) == E[dp2]/dt the the infinitesimal infinitesimal rate rate of of change change (Karlin (Karlin and and Taylor, Taylor, 1981), 1 9 8 1 ), which which are are given, given,
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI
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BOX 4.1
The Stochastic logistic Model
The stochastic logistic model has been studied in a variety of contexts, including pop ulation biology, chemistry, and sociology, and the dynamic behavior of the model is well understood (Kryscio and Lefevre, 1 989; Jacquez and Simon, 1 99 3; Ovaskainen, 2001 ). We will describe the model here as a SPOM describing the dynamics of a species inhab iting a network of n identical habitat patches. The stochastic logistic model is defined as a Markov p rocess, which assumes that an occupied patch turns empty at a fixed rate E e and that an empty patch turns occu pied at a rate C ck/n, where c is a colonization rate parameter and 0 s k s n is the number of occupied patches. To put the model into a mathematical setting, k(t) is the random variable giving the number of occupied patches at time t, and q(t) is the prob ability distribution describing the state of the metapopulation, with the component qlt) being the probability that the system is in state k(t) = i (i 0, ... ,n) at time t. By stand ard theory of Markov processes (e.g., Grimmet and Stirzaker, 2001 ), the probability dis tribution q evolves according to the forward equation dq(t)/dt q(t)P, where P is the (n + 1 )*(n + 1 ) matrix =
=
=
=
(1 )
In Eq. (1 ), the diagonal elements are defined as di ei + Ci, and Ci ci(n - i)/ n and ei give total (instead of patch-specific) colonization and extinction rates, assuming that presently i of the n patches are occupied. In the stochastic logistic model, eventual metapopulation extinction is certain, and the biologically most fundamental quantity that may be derived from the model is the time that the meta population is expected to persist. Assuming that the initial state of the metapopulation is drawn from the quasistationary distribution (Box 4.2), the mean time to extinction may be approximated by (Andersson and Djehiche, 1 998; Ovaskainen, 2001 ) ei
=
=
=
*
T
(b- exp( - np* ) -;; \j n ( 1 p* )n-l p*2' 1
=
(2)
_
where p is the deterministic equilibrium value for the fraction of occupied patches * given by p = 1 - e/c. Equation (2) shows that the time to extinction increases expo * nentially with the number of habitat patches (assuming that p is kept fixed). Before extinction, the process approaches a quasistationary distribution 'IT (Box 4.2), which * may be approximated by a normal distribution with mean p and variance (1 -p*)/n (Ovaskainen, 2001 ).
for the for the stochastic stochastic logistic logistic model, model, as as (Saether ( Saether et et al., aI., 1999; 1 999; Ovaskainen, Ovaskainen, 2002a) 2002a) ~(p) ep, p) -- ep, cp ( l -- p) � (p) == cp(1
cr2(p) a2 (p) ==
cp(1 + e p+ cp( l --p )p) nn
(4.1) (4. 1 )
ep .
.
(4.2) (4.2)
4 4..
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The Levins model model is The Levins is obtained obtained from from the the diffusion diffusion approximation approximation by by assum assuming ing that that the the number number of of patches patches is is so so large large that that stochastic stochastic fluctuations fluctuations may may be be ignored. ignored. In In this this case, case, the the model model reduces reduces to to the the familiar familiar deterministic deterministic differen differential equation equation
@ dp dt = = f.1(p) I~(p) dt
=
cp( 1 -- pp))
= cp(1
-
-
e ep..
p
(4.3) (4.3)
Figure 4.2 compares the behavior of the stochastic logistic model with the deterministic deterministic Levins model. Realizations of the stochastic model fluctuate fluctuate around around the deterministic prediction, and the deterministic equilibrium state P" p* corresponds corresponds to to the the peak peak in in the the quasistationary quasistationary distribution distribution of of the the stochastic stochastic model model (for (for the the definition definition of of quasistationarity quasistationarity see see Box Box 4.2). 4.2). A A more more throughout throughout analysis relationship between analysis of of the the relationship between the the deterministic deterministic Levins Levins model model and and the the stochastic logistic model stochastic logistic model has has been been presented presented by by Ovaskainen Ovaskainen (2001 (2001),) , Etienne Etienne (2002b), (2002b), and and Etienne Etienne and and Nagelkerke Nagelkerke (2002). (2002). Lande 1987, 1988b) Lande ((1987, 1988b) extended extended the the Levins Levins model model to to account account for for habitat habitat loss loss by by assuming assuming that that aa fraction fraction 11 - hh of of the the habitat habitat patches patches are are permanently permanently colonization. We destroyed, destroyed, and and thus thus only only the the fraction fraction h h remains remains suitable suitable for for colonization. We assume to discriminate assume that that the the migrating migrating individuals individuals are are unable unable to discriminate between between suit suitable able and and unsuitable unsuitable patches, patches, hence hence they they attempt attempt to to colonize colonize the the latter latter in in pro prois portion portion to to their their number. number. The The colonization colonization rate rate of of empty empty patches patches ((11 - p p) ) is thereby c p h , and and the the model model is is given given as as thereby reduced reduced to to cph, -
-
@ dp
--:it dt = - cph( cph(11 - p) p)
-
-
e e p ,,
p
(4.4) (4.4)
where where p p is is the the fraction fraction of of occupied occupied patches patches out out of of the the suitable suitable ones. ones. The The key key prediction population will prediction made made by by Lande's Lande's model model is is that that the the meta metapopulation will persist persist (there is a nontrivial nontrivial equilibrium state) if and only if the threshold threshold condition condition e e
hh > > - cc
(4.5)
B
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ij' c �
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0.05 0.04 0.03 0.02 0.01 0.2
0.4
0.6
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Occupancy state p
Fig. 4.2 model and and the Fig. 4 . 2 Comparison Comparison between between the the stochastic stochastic logistic logistic model the Levins Levins model. model. (A) (A) A A sin single gle simulation simulation realization realization of of the the stochastic stochastic logistic logistic model model compared compared to to the the deterministic deterministic predic prediction and its tion of of the the Levins Levins model. model. (B) (B) The The exact exact quasistationary quasistationary distribution distribution 'IT ~r and its normal normal distribution distribution [mean 00. [mean p*, p*, variance variance (1 (1 - p*)/n] approximation. approximation. Parameter Parameter values values ee = = 11,, c = = 2, 2, n n = 1100. =
OTSO OTSO OVASKAINEN OVASKAINENAND ILKKA ILKKAHANSKI HANSKI
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BOX 4.2 4.2
Quasistationary Quasistationary Distributions Distributions and and Extinction Extinction nmes Times
Mathematically, SPOMs SPOMs are defined as as Markov chains or Markov processes, processes, for which amount of general theory is available (e.g., Grimmet Grimmet and Stirzaker, Stirzaker, 2001 2001). ). We a large amount describe briefly here part of the theory that is relevant in the analysis analysis of SPOMS. SPOMS. As As described in Section 4.3, the state of of a general heterogeneous SPOM SPOM is specified =1, where the component by the random variable 0 O = {0/}7 {Oi}n=l, component 0/ Oi specifies specifies whether patch ii ). We assume is empty (0/ (Oi = 0) or occupied (O{ (Oi = 11). assume that these 22nn states states are ordered in some manner with with metapopulation extinction (0/ (Oi = = 0 0 for all i) being the first state. let (t) :s Let 11 :s _< Q Q(t) _< 2n 2 n denote the random variable describing the state of the metapopula metapopulaq� t)};;: , denote the probability distribution defined tion at time tt and let q(t) q ( t ) = {{q~[t)}/2=~l by q,(t) q,(t) = P[ P[Q(t) = i]. i]. By standard theory, the probability distribution q q evolves accord accordQ(t) = ing to the forward forward equation either as dq(t)ldt dq(t)/dt = q(t)P (for Markov processes) processes) or as q(t + 11)) = q(t = q(t)P q(t)P (for Markov chains). The 2n 2 n** 22nn matrix P P is called the generator matrix (Markov processes) processes) or the transition matrix (Markov chains) and is composed of transi transition rates rates (or probabilities) from any occupancy state to any other occupancy state (Grimmet 1 ). (Grimmet and Stirzaker, 200 2001). If a metapopulation happens to to go extinct, there are no occupied patches to to produce If migrants, and thus empty patches cannot be recolonized. Mathematically, metapopumetapopu lation extinction is an absorbing state, which the process process will eventually reach with with istribution of the Markov process probability 11,, and thus the stationary d distribution process (or chain) is concentrated on q, ql.. Although the stationary distribution is uninformative, it is still mean meaningful to SPOMs. This may be done in terms of to study the limiting behavior of SPOMs. of the quasistationary distribution 'IT, ~r, which is defined as the limiting distribution conditioned on nonextinction. In practical terms, a metapopulation that succeeds succeeds in persisting for a long time converges toward the quasistationary distribution. More precisely, we first condition the probability distribution q(t) q(t) on nonextinction as =
=
=
=
=
=
(t) m/(t) mj
= Pr( P r (Q Q (t) (t) = =
=
. t) '" ilQ(O ~ 1I )) IIQ(
= =
�(O
11 (t) - q, q1(t)
(1 (I))
after which the quasistationary distribution is defined as the limiting distribution, ~r = ilim ~ met). re(t). The quasistationary distribution 'IT ~r exists exists and is unique, provided that the dln pat etwork is irreducible, meaning that any patch is able to colonize (possibly patch network through through intermediate colonizations) any other patch. Technically, the quasistationary distribution 'IT ~r may be derived as the left leading eigenvector of matrix Po, P0, which is obtained by deleting the first row row and the first column from P. P. Furthermore, drawing the initial state from the quasistationary distribution, the expected time to metapopula metapopulation extinction is given by 'IT =
P
T
=
T=
{-liP 1 /(1 -
- 1/13 for Markov processes, processes, 1/(1 - p) p) for Markov Markovchains, chains,
(2) (2)
where p is the leading eigenvalue of Po 965, 11967). 967). P0 (Darroch and Seneta, Seneta, 11965,
is is satisfied. satisfied. In In other other words, words, the the species species is is expected expected to to persist persist only only if if the the amount amount (� = of of habitat habitat (h) (h) exceeds exceeds aa threshold threshold value value (8 = e/c), e/c), which which is is set set by by the the proper properties model, the ties of of the the species. species. In In the the original original Levins Levins model, the threshold threshold condition condition is is given given by by cc > > e, e, as as in in that that model model h h = = 11.. Based Based on on Eq. Eq. (4.5), (4.5), we we may may con conclude clude that that the the long-term long-term persistence persistence of of aa species species in in aa fragmented fragmented landscape landscape is is
4. 4.
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
81 81
facilitated by by increasing increasing amount amount of of suitable suitable habitat habitat (large (large h), h), aa small small risk risk of of local local facilitated e), and and aa good good colonization colonization ability ability (large (large c). c). extinction (small (small e), extinction The The most most fundamental fundamental message message from from these these simple simple models models for for ecology ecology and and conservation is is that that aa species species may may go go deterministically deterministically extinct extinct even even though though some some conservation suitable habitat habitat remains remains at at the the landscape landscape level, level, but but exactly exactly how how much much habitat habitat is is suitable needed for long-term persistence, persistence, and and how how could could we we estimate estimate this this value? value? The The needed for long-term p * the the fracfrac answer given given by by Lande's Lande's model model is is as as follows. follows. First, First, let let us us denote denote by by p* answer tion of of occupied in an an initial initial situation situation in in which which aa fraction fraction h0 ho of of the the landland tion occupied patches patches in scape consists consists of of suitable suitable habitat. habitat. Assuming Assuming that that the the species species is is at at its its scape population-dynamic equilibrium, equilibrium, we we may may solve solve the the value value of of the the species species paramparam population-dynamic eter 80 = eter e/c from from Eq. = e/c Eq. (4.4) (4.4) as as 80 == h0(1 ho( 1 -- p*), p". ), after after which which the the threshold threshold condition condition [Eq. (4.5)] This observation observation leads leads to to the the (4.5)] may may be be rewritten rewritten as as hh >> h0(1 ho( l -- p*). p * ). This [Eq. seemingly very very useful useful result, result, dubbed dubbed the the Levins Levins rule rule (Hanski (Hanski et et al., aI., 1996), 1 996), that that the the seemingly minimum minimum amount amount of of habitat habitat necessary necessary for for long-term long-term persistence persistence can can be be estimated estimated by just just recording recording the the amount amount of of empty empty habitat habitat while while the the species species is is still still common. common. by Carlson the Levins rule to to an an endangered endangered bird bird species in Carlson (2000) (2000) applied applied the Levins rule species in Sweden and Finland, leucotos). Sweden and Finland, the the white-backed white-backed woodpecker woodpecker (Dendrocopos (Dendrocopos leucotos). Using from the National Park Park in Poland, where where the the species species is Using data data from the Bialowieza Bialowieza National in Poland, is still still = 0.66 p* == 0.81, 0.81, from which the the extinction common, he estimated common, he estimated that that ho = 0.66 and and p* from which extinction This result result is is consistent consistent with with the of suitsuit threshold is estimated threshold is estimated as as 0.13. 0.13. This the amount amount of able habitat 0.12) and and in in Finland Finland (h (h = = 0.08), remaining in Sweden (h (h < < 0.12) able habitat remaining in Sweden 0.08), where where the populations have is encouraging, but we we consider the populations have declined declined severely. severely. The The result result is encouraging, but consider that and simplified the land that it it is is based based on on such such aa simplified simplified theory theory and simplified description description of of the landscape that the the Levins has primarily value. There are several factors scape that Levins rule rule has primarily pedagogic pedagogic value. There are several factors that are are not not included included in the model model but but which which are are likely likely to to influence influence metapopulametapopula that in the tion tion dynamics dynamics and and hence hence the the threshold threshold condition condition in in practice, practice, including including the the rescue rescue effect 996; Gyllenberg Hanski, 11997). 997). Furthermore, effect (Hanski (Hanski et et aI., al., 11996; Gyllenberg and and Hanski, Furthermore, as as the the models do take into account variation models do not not take into account variation in in the the properties properties of of the the patches, patches, any any best. The estimates estimates of of h h and and P p':" are are rough rough approximations approximations at at best. The same same comments comments and Lande models in genapply to the stochastic logistic model and the Levins and in gen eral: population dynamics, eral: they they provide provide qualitative qualitative insight insight to to classic classic meta metapopulation dynamics, but but they they have have limited limited value value for for aa quantitative quantitative metapopulation metapopulation analysis. analysis. To To be be fair, fair, they they were were not not meant meant to to do do that. that. To To follow follow our our interest interest in in developing developing predictive predictive metapopulation models landscapes, we metapopulation models for for highly highly fragmented fragmented landscapes, we next next turn turn to to heterogeneous SPOMs, SPOMs, which which may may be be used used more more readily readily in in the the study study of of real real heterogeneous metapopulations in in real real landscapes. landscapes. It It will will turn turn out, out, however, however, that that the the behavior behavior metapopulations of of the the spatially spatially heterogeneous heterogeneous models models may may often often be be best best understood understood by by studying studying their their homogeneous homogeneous counterparts, counterparts, which which leads leads to to resurrection resurrection of of the the Levins Levins model model via via aa new new interpretation interpretation of of the the model model parameters. parameters.
4.3 4.3
PATCH PATCH OCCUPANCY OCCUPANCY MODELS: MODELS: HETEROGENEOUS HETEROGENEOUS PATCH PATCH NETWORKS NETWORKS fundamental difference difference between homogeneous and heterogeneous heterogeneous The fundamental homogeneous and SPOMs SPOMs is is that that in in the the latter latter the the colonization colonization and and extinction extinction probabilities probabilities ((or or rates) rates) are are different different for for different different habitat habitat patches. patches. The The state state of of aa meta metapopulation living living in a heterogeneous heterogeneous network network of nn patches, patches, such as shown shown in population Fig. 1 , is Fig. 4. 4.1, is described described by by the the vector vector 0 O = {OJ}i=l' {Oi}in=l, where where the the component component OJ Oi
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HAN HANSKI
82 82
specifies OJ = specifies whether whether patch patch ii is is empty empty ((Oi = 0) 0) or or occupied occupied ((OJ 0 i= - - 11) ).. The The metapopulation metapopulation model model may may be be specified specified either either in in continuous continuous time, time, in in which which case case the the model model is is mathematically mathematically aa Markov Markov process, process, or or in in discrete discrete time, time, in in which which case case the the model model becomes becomes aa Markov Markov chain. chain. We We denote denote the the colonization colonization rate (for (for Markov Markov processes) processes) or or the the colonization colonization probability probability (for (for Markov Markov rate chains) of an empty patch i by Cj = Cj(O) and the extinction rate (or chains) of an empty patch i by Ci = Ci(O) and the extinction rate (or prob probpatch by Ej Ei = = Ej(O). Ei(O). ability) of an occupied patch Before Before turning turning to to specific specific SPOMs SPOMs for for heterogeneous heterogeneous networks, networks, it it is is worth worth noting that the qualitative theory presented in Box 4.1 for the stochastic logisnoting that the qualitative theory presented in Box 4.1 for the stochastic logis tic tic model model applies applies almost almost independently independently of of the the functional functional forms forms for for the the coloniza colonization tion and and extinction extinction processes processes Cj Ci and and Ej• Ei. Metapopulation Metapopulation extinction extinction is is an an absorbing absorbing state, state, which which the the process process will will eventually eventually reach reach with with probability probability one. one. Before Before the the inevitable inevitable extinction, extinction, the the process process converges converges toward toward aa quasistationary quasistationary distribution 'IT, which is given as the subdominant eigenvector distribution ~r, which is given as the subdominant eigenvector of of aa stochastic stochastic transition matrix transition matrix P (see (see Box Box 4.2 4.2 for for details). details). This This quasistationary quasistationary distribution, distribution, referred to as ""(stochastic) often referred (stochastic) metapopulation equilibrium" (Hanski, 11999b) 999b) or population ((Frank Frank and or as as the the "established "established phase" phase" of of the the meta metapopulation and Wissell, Wissell, 2002; 2002; Grimm 2004), is Grimm and and Wissel, Wissel, 2004), is of of great great importance importance for for ecological ecological applications applications of of the population and the theory. theory. It It relates relates directly directly to to the the average average size size of of the the meta metapopulation and is is needed population that needed for for determining determining the the extinction extinction risk risk of of aa meta metapopulation that has has already already persisted persisted for for some some time. time. As As the the quasistationary quasistationary distribution distribution does does not not account account for for transient not sufficient dynam transient dynamics, dynamics, however, however, it it is is not sufficient for for the the description description of of the the dynamics of of aa newly newly established established metapopulation metapopulation nor nor aa metapopulation metapopulation that that has has been been ics perturbed perturbed recently. recently. In In such such cases, cases, one one needs needs to to find find out out also also the the probability probability of of reaching reaching the the quasistationary quasistationary distribution, distribution, which which influences influences the the extinction extinction risk risk of the metapopulation al., 1991a; of the metapopulation within within a given given time time horizon horizon (Verboom et et aI., 1 99 1a; Stephan, 1 993; Ovaskainen Hanski, 2002; 2002; Grimm Grimm and and Wissel, Wissel, 2004). 2004). Stephan, 1993; Ovaskainen and and Hanski, We will will next formulate deterministic approximations of the full full stochastic stochastic We next formulate deterministic approximations of the = {pi}n_ {Pi}7= 11 aa vector vector with with the the component component ii givgiv SPOM. To this, we we denote by pp = SPOM. To do do this, denote by is occupied. occupied. A A deterministic deterministic version version of ing the probability that that patch ing the probability patch ii is of the the by the the SPOM may may be be obtained by replacing replacing the the vector vector of patch occupancies occupancies 0 SPOM obtained by of patch O by vector of of occupancy probabilities probabilities p ((Ovaskainen Ovaskainen and and Hanski, Hanski, 2001), 200 1 ),
dpi(t) dt pi(t + 1) - pi(t)
= Ci(p(t))(1 - p i ( t ) ) -
Ei(p(t))pi(t).
(4.6)
In this this equation, equation, the the upper upper formula formula relates relates to to continuous-time continuous-time models models and and the the In lower one one to to discrete-time discrete-time models. models. If If the the colonization colonization and and extinction extinction rates rates lower (probabilities) depend depend on on the the occupancy occupancy state state in in aa linear linear fashion, fashion, Eq. Eq. (4.6) (4.6) may may (probabilities) be derived derived from from the the stochastic stochastic model model in in the the same same way way as as the the Levins Levins model model was was be derived from from the the stochastic stochastic logistic logistic model. model. Note Note that that for for nonlinear nonlinear models, models, derived Eq. (4.6) (4.6) does does not not necessarily necessarily correspond to the the drift drift term, term, although although in in Eq. correspond exactly exactly to practice practice it often often gives a good good approximation. approximation. We will will put put flesh flesh to to the the skeleton skeleton of of SPOMs SPOMs by by introducing introducing two two examples examples of of We biologically reasonable reasonable models, models, which which have have been been dubbed dubbed the the spatially spatially realistic realistic biologically Levins model model (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2000) 2000) and and the the incidence incidence function function Levins
4. 4.
83 83
M ETAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
model (Hanski, (Hanski, 1994a). 1 994a). Before Before doing we have have to to add add aa critical critical compocompo model doing that, that, we nent to to the the models, models, the the set set of of assumptions assumptions that that relate relate the the colonization colonization and and nent extinction rates rates to to the the structure structure of of the the fragmented fragmented landscape. landscape. extinction
Metapopulation Theory Spatially Realistic Metapopulation A particular particular SPOM SPOM is is defined defined by by describing describing how colonization and and extinction extinction A how colonization rates (or (or probabilities) probabilities) depend depend on on the the structure structure of of the the landscape landscape and and on on the the prespres rates ent occupancy occupancy pattern. pattern. The The dependence dependence of of population population turnover turnover on on habitat habitat fragfrag ent ment areas areas and and spatial spatial locations, locations, and and possibly possibly on on other other quantities, quantities, such such as as habitat habitat ment quality, places the theory explicitly explicitly in a spatial context. context. As one the metapopulation metapopulation theory one may use use here here landscape landscape measures measures that that describe describe the the structure structure of of real real fragmented fragmented may landscapes, we we have have called called the the combination combination of of heterogeneous heterogeneous SPOMs SPOMs and and the the landscapes, assumptions mapping mapping population population turnover turnover to to landscape landscape structure structure the the spatially spatially assumptions realistic metapopulation metapopulation theory theory (Hanski, (Hanski, 2001b; 2001 b; Hanski Hanski and and Ovaskainen, Ovaskainen, 2003). 2003). realistic The starting point description of the landscape landscape structure structure as depicted in The starting point is is aa description of the as depicted in Fig. 4.1. 4.1. Thus Thus the the network network is is assumed assumed to Fig. to consist consist of of n circular circular habitat habitat patches. patches. We area of of patch patch i, by dij di; the the distance cen We denote denote by by AAii the the area i, and and by distance between between the the centroids of extinction and and colonization rates are are now now defined troids of patches patches ii and and j. j. The The extinction colonization rates defined as functions of di;, using using some specific arguments, arguments, of of which which the the followfollow as functions of AAii and and dij, some specific ing sections give two examples. examples. In most cases cases it it is is sensible to assume assume that that aa ing sections give two In most sensible to decreasing patch area increases the extinction the contribu decreasing patch area increases the extinction rate rate and and decreases decreases the contribution of the the respective population to to connectivity populations because because tion of respective population connectivity of of other other populations population sizes tend decreasing patch patch area. Likewise it tend to decrease with with decreasing local population sensible to to assume assume that that the the probability probability of of an an empty empty patch patch becoming colonis sensible becoming colon ized ized increases increases with with increasing increasing connectivity connectivity to to existing existing populations. populations. Figure Figure 4.3 4.3 gives gives an an example example for for the the well-studied well-studied Glanville Glanville fritillary fritillary butterfly. butterfly. We unites the metapopulation theory We will will note note in in passing passing that that SMT SMT unites the classic classic metapopulation theory ((CMT) CMT) based 1 969, 11970) 970) and based on on the the pioneering pioneering models models by by Levins Levins ((1969, and the the MacArthur and Wilson dynamic theory of island biogeography (DTIB) of MacArthur ((1963, 1 963, 11967) 967) (Hanski, because (Hanski, 2001b). 2001b). CMT CMT and and DTIB DTIB are are obviously obviously related related because the the expected expected number number of of species species on on an an island island or or in in aa habitat habitat fragment fragment (the (the
A 1 .?;:3 .Q 0.8 0.8
.m .0
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<1l .c . 0
a.
§ U O
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o. 0.6 ._o 0.4 .N c 0.2 t--
0.4 0.4
to
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Connectivity Iogl0S
4.3 The dependence dependence of extinction extinction probability on patch size (A) and the dependence dependence of Fig. 4.3 (B). Dots represent data collected for nine successive colonization probability on connectivity (S). network of ca. 4000 4000 habitat patches in the A Aland generations of the Glanville fritillary from a network land al., 2004). 2004). Lines Lines depict depict maximum maximum likelihood likelihood estimates based on the entire Islands (Nieminen et aI., 7, �em data data set. set. Parameter Parameter values values e = 0.38, 0.38, c = 0.083, Ci ~ = = 0.84, 0.84, �ex ~ex 0.1 0.17, ~em = 0.07 0.07 and and �im ~im = 0.30. 0.30. = --
--
--
OT50 OTSO OVA5KAINEN OVASKAINEN AND AND ILKKA ILKKA HAN5KI HANSKI
84 84
dynamic dynamic variable variable in in DTIB) DTIB) is is given given by by the the sum sum of of probabilities probabilities of of different different species species occurring occurring in in that that fragment fragment (the (the dynamic dynamic variable variable in in CMT). CMT). Assuming Assuming aa mainland identical and species, the basic DTIB mainland pool pool of of R R identical and independent independent species, the basic DTIB is is obtained by obtained by multiplying multiplying both both sides sides of of Eq. Eq. (4.6) (4.6) by by R R and and assuming assuming the the same same colonization species. However, colonization and extinction extinction rate parameters parameters for for all the species. and equally connected connected habitat habitat patches as in CMT, assuming a set of identical and Eq. the rate of change in the the probability of occupancy for each Eq. (4.6) (4.6) gives gives the rate of change in probability of occupancy for each frag fragment, which also the change in ment, which is is also the rate rate of of change in the the fraction fraction of of occupied occupied fragments. fragments.
Spatially Levins Model Spatially Realistic Realistic Levins Model (SRLM) (SRLM) Our Our first example example is the the spatially spatially realistic Levins model, model, which which is defined by the assumptions about the following following general general assumptions about the the colonization colonization and and extinction extinction rates: rates: Ci = Z cijOj , j4:i Ei = 8i~
(4.7) (4.7)
where where Cciiij denotes denotes the the contribution contribution that that patch patch jj (when (when occupied) occupied) makes makes to to the the words, the colonization colonization colonization rate of patch patch i.i. In other words, colonization rate of patch patch ii is aa linear occupancies of linear function function of of the the occupancies of potential potential source source patches, patches, whereas whereas the the extinction extinction rate rate is is independent independent of of the the occupancies occupancies of of the the other other patches. patches. Equation Equation (4.7) (4.7) is is structurally structurally similar similar to to the the stochastic stochastic logistic logistic model. model. If If the the patches are identical SRLM reduces patches are identical and and equally equally connected, connected, SRLM reduces to to the the stochastic stochastic logistic model. Our next task is to make specific assumptions assumptions as to how how colon colonization ization and and extinction extinction rates rates depend depend on on the the structure structure of of the the patch patch network. network. Local Local Extinction
Assuming Assuming that that all all the the patches patches are are of of equal equal quality, quality, it it is is reasonable reasonable to to assume assume that that the the carrying carrying capacity capacity of of aa patch patch is is proportional proportional to to its its area. area. Thus Thus number of individuals in local population Ki K i =k =k A A ii gives gives the the number of individuals in patch patch ii when when the the local population is is at at its its carrying carrying capacity, capacity, and and kk denotes denotes the the density density of of individuals. individuals. If If there there is is variation variation in in density density from from patch patch to to patch patch due due to to differences differences in in patch patch quality, quality, this in the as one this can can be be straightforwardly straightforwardly taken taken into into account account in the model, model, as as long long as one has has estimates estimates of of patch-specific patch-specific density. density. In In order order to to estimate estimate the the extinction extinction rate rate of assumptions on of aa local local population, population, we we need need to to make make assumptions on the the type type of of local local dynamics. dynamics. One possibility would would be to assume that that the local population population behaves according to the stochastic stochastic logistic model, model, now now viewed as a model for the number in aa local this case, the rate of local the number of of individuals individuals in local population. population. In In this case, the rate of local extinction extinction would would essentially essentially decrease decrease exponentially exponentially with with local local carrying carrying cap capK; (Box (Box 4. 1 ). The acity, acity, Exti E x t i ex: ~ r e -Ki 4.1). The exponential exponential scaling scaling arises arises from from demographic demographic stochasticity, stochasticity, which which is is of of major major concern concern in in very very small small populations populations but but not not in in larger ones. all but is larger ones. Typically, Typically, the the extinction extinction risk risk of of all but the the smallest smallest populations populations is dominated Chapter 14). Adding Adding environmen dominated by environmental environmental stochasticity stochasticity ((Chapter environmental stochasticity population dynamics stochasticity to models for for local population dynamics leads to the much much milder scaling (Lande, 993; Foley, 994) milder power-law power-law scaling (Lande, 11993; Foley, 11994)
Extj Exti = =
e e e = A!ex, K ~{e ex x - A {ex ' K 1 1
(4.8) (4.8)
4.
METAPOPULATION IN HIGHLY METAPOPULATION DYNAMICS DYNAMICS IN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
85 85
where where ee is is an an extinction extinction rate rate parameter. parameter. The The scaling scaling factor factor �~ex may be be inter interex may preted preted as as aa measure measure of of the the strength strength of of environmental environmental stochasticity, stochasticity, with with aa low low value x indicating that environmental stochasticity is of major importance value of of �Gx indicating that environmental stochasticity is of major importance e determining the extinction extinction risk (Hanski, 11998a). in determining 998a). Colonization Colonization
To To estimate estimate the the colonization colonization rate rate of of an an empty empty patch, patch, we we make make the the simplify simplifying assumption assumption that that all the occupied occupied patches patches are at their local carrying carrying capac capacity (or that that the sizes of the local populations populations are proportional proportional to the respective individuals produces carrying carrying capacities). capacities). We We assume assume that that each each of of the the Ki K i individuals produces propagules propagules in in aa continuous continuous fashion fashion with with aa rate rate ss and and that that the the propagules propagules are are randomly (and independently randomly (and independently of of each each other) other) redistributed redistributed according according to to aa radi radially symmetric Box 4.3) ally symmetric dispersal dispersal kernel kernel f(r) fir) ((Box 4.3).. Assuming Assuming that that each each propagule propagule has probability qq of colonization, it col has aa constant constant probability of successful successful colonization, it follows follows that that the the colonization onization rate of an empty empty patch patch ii is given by Ci(O) = ~a KjsqgqOj = c a i ~ -- Ajfldq)Oj j:/:i j~i
= ~ cqOj, j~i
(4.9)
where = f(dij)Ai where 'Vij ~///= f(dij)Ai is is the the fraction fraction of of propagules propagules produced produced by by an an individual individual in in patch patch jj that that are are expected expected to to migrate migrate to to patch patch i,i, and and the the colonization colonization rate rate parameter collate the parameter cc is is defined defined as as cc = = ks ksq. What this this formula formula does does is is to to collate the con conq . What tributions all source tributions of of all source populations populations (currently (currently occupied occupied patches) patches) to to the the colo colonization nization rate rate of of the the focal focal patch; patch; the the contribution contribution of of source source patch patch jj depends depends on on the the size size of of patch patch jj and and its its distance distance from from the the focal focal patch patch ii as as specified specified by by the the dis diskernel. persal kernei. Generally, Generally, both both the the source source and and the the target target patch patch areas areas may may influence influence indi individual vidual movements movements and and hence hence the the contribution contribution of of patch patch jj to to the the colonization colonization of of patch patch ii in in aa nonlinear nonlinear manner manner (Hanski (Hanski et et ai., al., 2000; 2000; Moilanen Moilanen and and Nieminen, Nieminen, 2002; Ovaskainen 2002; Ovaskainen and and Cornell, Cornell, 2003). 2003). Assuming Assuming aa power-law power-law relationship relationship for for both and immigration, may be be generalized to the both emigration emigration and immigration, Eq. Eq. (4.9) (4.9) may generalized to the form form Cij and Cij = - cA c A fi ~m iAm~emAf q'dt 1emf(dij) I~ ij!~ (Ovaskainen, (Ovaskainen, 2002b). 2002b). Values of the scaling factors factors �~im im and �em depend on ~em depend on the the biological biological processes processes determining determining immigration immigration and and emigra emigration empirical data tion rates and and may may either be estimated estimated from empirical data or be derived from from submodels for for immigration immigration and emigration. emigration. submodels The model is The deterministic deterministic mean-field mean-field model is obtained obtained by by replacing replacing the the vector vector of of patch occupancy probabilities Gyllenberg and patch occupancies occupancies by by the the vector vector of of occupancy probabilities ((Gyllenberg and Hanski, 997). For example, Hanski, 11997). example, Hanski Hanski and and Ovaskainen Ovaskainen (2000) (2000) assumed assumed a sim simple continuous-time e/Aii ple continuous-time deterministic deterministic SRLM SRLM given given by by Eq. Eq. (4.6) (4.6) with with Ei Ei = ---e/A where we have assumed that �ex and C IAjexp( - adij)pj, where and Cii = = ccEAjexp(-oLdij)pj, we have assumed that ~ex = = �em ~em = = 1 1,, an an exponential exponential dispersal dispersal kernel kernel (the (the normalization normalization constant constant is is included included in in the the � im = parameter parameter c), c), and and no no effect effect of of target target patch patch size size on on colonization colonization ((i~im = 00).) . The The exponential exponential dispersal dispersal kernel kernel used used here here iiss phenomenological phenomenological (does (does not not have have aa mechanistic mechanistic explanation) explanation) but but it it typically typically fits fits well well with with observa observations 999; Sutherland tions (e.g., (e.g., Conrad Conrad et et ai., al., 11999; Sutherland et et ai., al., 2000; 2000; Byrom, Byrom, 2002; 2002; Griffith Griffith and and Forseth, Forseth, 2002) 2002).. Note Note that that it differs differs from from the the dispersal dispersal kernel kernel introduced 1 ) in 4.3 by introduced in in Eq. Eq. ((1) in Box Box 4.3 by the the factor factor r. r. As As seen seen in in Section Section 4.4, 4.4, the the deterministic deterministic model model yields yields many many useful useful results results about about metapopulation metapopulation dynamics. dynamics.
86 8 6
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HAN SKI
BOX
4.3
Dispersal Kernels
In order to construct a SPOM, one needs to know the fraction 9ij of propagules pro duced by an individual in patch j that are expected to migrate to patch i. The fraction 9ij is often described with the help of a dispersal kernel 0, which specifies how propag ules are redistributed in a landscape. More precisely, assuming that the propagules are produced in the origin, the fraction of propagules that disperse to a region X k R2 is assumed to be given by fxO(x,y)dA, where dA denotes integration with respect to area. As the total number of propagules should be conserved, we always assume that the dis persal kernel is normalized as fR2 O(x,y)dA = 1 . In many cases, it is reasonable to assume that the dispersal kernel is radially symmetric, in which case it may be denoted by f(r) O(r,O), and the fraction of propagules that disperse to distance r with r, < r < r2 is given by 21rf��f(r)rdr. For example, if one assumes that propagules move toward a fixed (but random) direction u ntil they settle with a fixed rate, one has =
f(r)
=
271'r ex
(1 )
e-C1r,
where the parameter ex depends on the ratio of the speed of the individuals and the rate at which they settle. For another example, if one assumes that propagules move accord ing to random walk u ntil they settle with a fixed rate 8, one has f(r)
=
2!/O(lfr),
(2)
where Ko is a modified Bessel function of the second kind and a is the diffusion coeffi cient that may be calculated from the parameters of the random walk (Turchin, 1 998). Assuming that the landscape is highly fragmented, i.e., that the patches are small with respect to the dispersal distances, it follows that 9ij may by approximated by
(3) Deriving 9ij from a dispersal kernel 0 is best justified for passive propagule dispersal, whereas dispersal strategies with active search behavior may not correspond exactly to any dispersal kernel. For example, if the propagules move according to random walk but bias their movement toward the habitat patches, the patches are, in a sense, competing for the migrants. In such a case, the dispersal rate between any two patches does not depend just on the areas of and the distance between the two patches, but also on the spatial config uration of the entire patch network (Ovaskainen and Cornell, 2003). Further, strong het erogeneity of the dispersal habitat (e.g., dispersal corridors or barriers) may prevent the use of dispersal kernels, although in general they provide a reasonable first approximation.
The SRLM may also be formulated as a discrete-time model by assuming that that the the time time step step of of the the model model is is sufficiently sufficiently small. small. In In this this case, case, general general expres expresextinction probabilities are given by sions for the colonization and extinction
{
-- exp ( - � CiPj) , exp(-.j~/cijOj), Ei -e ) E i = 11 - exp( exp(-ei). Ci Ci = = 1l
i
.
(4.10) (4.10)
ETAPOPULATION DYNAMICS N HIGHLY 4. 4. M METAPOPULATION DYNAMICS IIN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
87 87
As As continuous-time continuous-time SRLM SRLM is is structurally structurally simple simple and and thus thus analytically analytically tractable, tractable, we we use use it it later later for for developing developing much much of of the the mathematical mathematical theory. theory. However, connect the using the as However, we we will will connect the theory theory to to data data using the discrete-time discrete-time SRLM, SRLM, as the rest the latter latter is is more more appropriate appropriate for for the the Glanville Glanville fritillary fritillary butterfly. butterfly. For For the the rest of model 4 . 1 0 with the of this this chapter, chapter, we we will will call call the the model 4.10 with parameters parameters estimated estimated for for the Glanville fritillary butterfly model, whereas Glanville fritillary butterfly (Fig. (Fig. 4.3) 4.3) as as the the Glanville Glanville fritillary fritillary model, whereas SRLM realistic SRLM will will always always mean mean the the continuous-time continuous-time version version of of the the spatially spatially realistic Levins model. model. Levins
Incidence Incidence Function Function Model Model (IFM) (IFM) The The incidence incidence function function model model is is aa discrete-time discrete-time SPOM SPOM that that was was introduced introduced by Hanski ((1994a) 1994a) and been applied empirical studies by Hanski and has has since since been applied widely widely in in empirical studies Ovaskainen, 2002a) (Hanski, 999b, 2001b; (Hanski, 11999b, 2001b; Chapter Chapter 5). 5). An An extended extended version version ((Ovaskainen, 2002a) of of the the incidence incidence function function model model is is defined defined as as Ci =
z Sz S i + l/c'
Ei -- m l "n ( ~ e x ] ) ( 1
(4. 11) (4.11) -- Ci' r
i
where where
m ·�(' d··II ) OI· s SiI = A ~�im i m xX � E A �e gem = A ..:::.. A Ii 1/if(dij)Oj I ij:/=i *i
(4.1 ( 4 . 1 22) )
iiss the the metapopulation metapopulation dynamic dynamic connectivity connectivity ooff patch patch ii to to extant extant local local popula populations. tions. In In comparison comparison with with the the linear linear SRLM, SRLM, the the IFM IFM is is more more complex, complex, as as it it has has two two structural structural parameters. parameters. First, First, parameter parameter zz relates relates to to the the assumptions assumptions behind behind the the colonization colonization process, process, values values greater greater than than zz = = 11 reflecting reflecting the the presence presence of of an 1994a) assumed an Allee Allee effect. effect. In In the the original original version version of of the the IFM, IFM, Hanski Hanski ((1994a) assumed aa could follow relatively strong Allee effect relatively strong Allee effect (z (z = = 2), 2), which which could follow from from the the interaction interaction of immigrants at colonization in a sexually reproducing species. of immigrants at colonization in a sexually reproducing species. Second, Second, parameter parameter rr measures measures the the strength strength of of aa rescue rescue effect, effect, describing describing aa reduced reduced extinction size. Hanski extinction risk risk due due to to immigrants immigrants enhancing enhancing the the local local population population size. Hanski ((1994a) 1994a) assumed that r = 1 . As with the SRLM, the deterministic assumed that r = 1. As with the SRLM, the deterministic approxima approximation tion is is obtained obtained by by replacing replacing the the vector vector of of patch patch occupancies occupancies with with aa vector vector of of occupancy probabilities in Eq. (4. 12). occupancy probabilities in Eq. (4.12).
4.4 4.4
DETERMINISTIC DETERMINISTIC THEORY THEORY This be This section section focuses focuses on on three three ecologically ecologically interesting interesting issues issues that that may may be addressed First, we kind of addressed by by the the deterministic deterministic theory. theory. First, we examine examine what what kind of thresh threshold SPOMs predict old conditions conditions SPOMs predict for for metapopulation metapopulation persistence persistence and and use use these these conditions SPOMs into classes. Second, conditions to to classify classify SPOMs into three three qualitatively qualitatively distinct distinct classes. Second, we we ask ask about about the the "values" "values" of of individual individual habitat habitat patches patches in in the the sense sense of of the the con contributions population dynamics tributions that that the the patches patches make make to to meta metapopulation dynamics and and persistence. persistence.
88 88
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HANSKI
Third, Third, we we explore explore transient transient dynamics, dynamics, the the specific specific aim aim being being to to quantify quantify the the length length of of time time it it takes takes for for aa metapopulation metapopulation to to return return to to equilibrium equilibrium follow following a perturbation.
Metapopulation Capacity Capacity Metapopulation The question that The most most fundamental fundamental question that may may be be addressed addressed by by aa patch patch occu occupancy model model is is whether whether aa given given species species is is expected expected to to persist persist in in the the long long term term pancy in aa given given fragmented fragmented landscape. landscape. In In stochastic stochastic models, models, where where eventual eventual extinc extincin tion tion is is always always certain, certain, it it is is natural natural to to assess assess the the risk risk of of extinction extinction by by analyzing analyzing the the time time it it takes takes for for the the metapopulation metapopulation to to go go extinct. extinct. In In the the deterministic deterministic framework, framework, aa metapopulation metapopulation may may be be classified classified as as viable viable if if the the model model pos possesses �. , toward sesses aa stable stable nontrivial nontrivial equilibrium equilibrium state state pp*, toward which which the the deterministic deterministic model model converges converges asymptotically. asymptotically. The The deterministic deterministic perspective perspective is is useful useful espe espemetapopulations for for which which the role of stochastic fluctuations is cially for large metapopulations not as pronounced pronounced as for for small metapopulations. metapopulations. not In In the the Levins-Lande Levins-Lande model, model, the the deterministic deterministic threshold threshold condition condition for for persist persistis given given by by hh > > 8, 8, where where h h is is the the amount amount of of suitable suitable habitat habitat and and 88 = -- e/c e/c is is ence is parameter defined defined by the properties properties of the species (Section 4.2). 4.2). Hanski and a parameter Ovaskainen Ovaskainen (2000) (2000) extended extended this this result result to to the the spatially spatially realistic realistic Levins Levins model, model, in in threshold condition for for deterministic persistence is given by which the threshold
(4.13) (4.13)
~.M > g
Here Here A.M ~M is is called called the the metapopulation metapopulation capacity capacity ooff the the fragmented fragmented landscape, landscape, and and 88 == e/c is aa species species parameter in the the original Levins model. model. Comparing Comparing e/c is parameter as as in original Levins we note that in in the spatially realistic realistic model, model, the the metapopumetapopu Eqs. Eqs. (4.5) (4.5) and and (4.13), (4.13), we note that the spatially role of of the the amount amount of of habitat habitat (h) in the nonspatial lation capacity lation capacity (A.M) (~M) plays plays the the role (h) in the nonspatial model. A.M also takes into into account how well well the the patches patches are are connected model. However, However, ~.M also takes account how connected A.M is given as the to each the meta population capacity to each other. other. Mathematically, Mathematically, the metapopulation capacity X M is given as the leading eigenvalue eigenvalue of matrix M M with elements mii and leading of the the n*n n*n matrix with elements mii = 00 and =
+ �i~imA~emf(d#) mAi�em/t:(' d1/ ) .· mijII-- = 8cij/ei - - il�ex + A~ex = A m = 8c II-le-I =
(4.14) (4.14)
The gives the that patch patch jj makes makes to the colonization colonization The element element mij mij gives the contribution contribution that to the rate multiplied by the expected lifetime mij thus thus measthe expected lifetime of of patch patch i.i. mij meas rate of of patch patch i,i, multiplied ures fraction of were the would be be occupied occupied if if patch patch jj were the only only ures the the fraction of time time that that patch patch ii would source of of immigrants immigrants (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2000; 2000; Ovaskainen Ovaskainen and and source Hanski, Hanski, 2001). 2001). The analogy analogy between between ~M A.M and and hh extends extends beyond beyond the the threshold threshold condition. condition. In In The the the Levins-Lande Levins-Lande model, model, the the equilibrium equilibrium fraction fraction of of occupied occupied patches patches is is given given by p* p * == 11 - ~/h, where pp t 8/h, whereas whereas in in the the SRLM SRLM itit is is given given by by ppt~ == 11 - ~/XM, 8/A.M' where by the weights being i, = ki iiss aa weighted weighted fraction fraction ooff occupied occupied patches, patches, Px ~iWiPi, the weights Wi Wi being Wi Pi.. P defined by patch for details). defined patch values (see later later for details). Figure 4.4 gives an empirical example example of the extinction extinction threshold threshold using using the the Figure 4.4 gives an empirical of the Glanville structure of Glanville fritillary fritillary model. model. Due Due to to the the nonlinear nonlinear structure of the the discrete-time discrete-time from the the other other model model model, itit is is not not possible possible to to extract extract aa species species parameter parameter ~8 from model, parameters, and and consequently consequently the the threshold threshold condition condition for for persistence persistence is is given given parameters, A.M >> 11 (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2001). 2001). Although Although there there is is aa in this this model model as as kM in -
-
=
4. 4.
889 9
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
B
A
or{'
'* (D iii
0.8 0.8
>- 0.6 0 .6 >,
� 0.4
r<.l t-
o13
cc. ~
0o o<.l
9
0.2
9 .
..
_O _ . . . L
~o .....
0C;:
i
, . o. "..o: .. . 9: o ~ I 9 9� : . .
. ' . "" , .. : . . ooO . .. ~ , oe 9 .el 9 , .
.
.
.
.
0.8 0.8
•• go • • O0 O0 " 9 '
oo
>, >- 0.6
� 0.4 <.l
e-
o8
c. �
0o o
.
-0.25 0 0.25 0.5 -1 .25 -1 -1 -0.75-0.5 -0. 75 -0.5 -0.25 -1.25
Metapopulation capacity capacity Ioglo 109,0 ;~M Metapopulation AM
.
9
• 9• ,.. • •• • • 9 O'Oo0 • •• 9
o•
• 9 9•
0.2
•
-1 -1
.O
~.,_
.L
.
~,
-0.5 -0.5
9
-~
-!
0
0.5
~-
9 . ..... . 9.
g~O0
.•o•• . •
•
9
•
- - =_ . . . . .
1
1 .5 1.5
Total area area IogloA 1091QA Total
Fig. 4.4 4.4 An empirical empirical example example of of the the treshold condition condition for for persistence persistence using the the Glanville Glanville Fig. fritillary model parameters from Fig. 4.3. Each dot dot depicts depicts 11 of of 59 59 semi-independent semi-independent patch patch fritillary model with with parameters from Fig. networks like the the one one illustrated illustrated in in Fig. Fig. 4.1. 4.1 . The The horizontal horizontal axis axis shows shows the the metapopulation metapopulation capacapa networks like city (A) or the the total area (B) (B) of of the the network, network, and and the the vertical vertical axis axis shows shows the the weighted weighted average of city (A) or total area average of the observed observed probability probability of of patch patch occupancy occupancy during during the the 9-year 9-year study study (weights (weights patch patch values values in in A A the and patch areas areas in in B). B). The The dashed dashed line line in in A A depicts depicts the the threshold threshold for for persistence persistence (kM (AM = 1). 1 ). and patch =
variation in in patch patch quality quality and to regional sto lot of lot of scatter, scatter, much much of of it it due due to to aa variation and to regional stochasticity, it is is evident metapopulation capacity is able rank the the netnet chasticity, it evident that that the the metapopulation capacity is able to to rank works in in terms terms of presence or the species more accurately works of the the presence or absence absence of of the species more accurately than than the total area area of of habitat habitat in the the patch network ((Fig. the total patch network Fig. 4.4B). 4.4B). Note Note that that many many of of AM == 11 have, have, howhow the the threshold the patch patch networks networks that that fall fall below below the threshold condition condition aM ever, had substantial substantial metapopulations metapopulations during the study study period. period. The ever, had during the The explan explanation is twofold. twofold. First, First, the the different different networks networks are are not not completely completely ation for for this this result result is independent of thus aa large large metapopulation meta population may may keep keep aa nearby nearby independent of each each other, other, and and thus small small patch patch network network occupied, occupied, although although the the small small network network would would alone alone be be below below the the threshold threshold condition condition for for long-term long-term persistence. persistence. Second, Second, the the parameter parameter estimates estimates (Fig. (Fig. 4.3) 4.3) are are based based solely solely on on the the observed observed annual annual transi transitions, tions, and and they they thus thus ignore ignore the the occupancy occupancy state state observed observed in in the the first first year year ((see see Chapter Chapter 5). 5). As As there there has has been been aa general general decline decline in in the the occurrence occurrence of of the the butterfly I. butterfly during during the the study study period period (which (which may, may, however, however, be be only only temporary; temporary; I. Hanski, Hanski, unpublished unpublished results), results), the the parameter parameter estimates estimates are are likely likely to to be be biased biased in in the the direction direction of of overestimating overestimating the the critical critical amount amount of of habitat habitat needed needed for for meta population persistence metapopulation persistence (Moilanen, (Moilanen, 2000). 2000). Correcting Correcting for for the the bias bias would would move move the the vertical vertical line line in in Fig. Fig. 4.4A 4.4A somewhat somewhat toward toward the the left. left.
Classification of Classification o f SPOMs SPOMs The The most most fundamental fundamental classification classification of of metapopulations metapopulations in in the the determinis deterministic tic context context is is into into those those that that are are beyond beyond the the extinction extinction threshold threshold and and persist persist (have (have aa stable stable nontrivial nontrivial equilibrium equilibrium state state p p*" > > 0) 0) and and those those that that are are below below the the extinction extinction threshold threshold and and go go extinct. extinct. A A more more refined refined classification classification consid considers ers in in addition addition whether whether the the trivial trivial equilibrium equilibrium state state p p*':' = 00 corresponding corresponding to to meta population extinction unstable, aa small metapopulation extinction is is stable stable or or unstable. unstable. If If it it is is unstable, small meta population has metapopulation has aa deterministic deterministic tendency tendency to to grow, grow, and and thus thus the the metapopu metapopulation lation may may be be expected expected to to be be able able to to invade invade an an empty empty patch patch network network success successfully. fully. The The threshold threshold condition condition for for successful successful invasion invasion may may be be written written as as AI k1 > > 8, 8, where where AI kI is is called called the the invasion invasion capacity capacity of of the the network network (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2001). 2001). In In the the SRLM, SRLM, the the metapopulation metapopulation capacity capacity AM kM coincides coincides with with
90 90
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKAHAN HANSKI
the the invasion invasion capacity capacity }.J, ~.I, but but this this is is not not the the case case in in general. general. Three Three types types of of sit situations uations may may be be distinguished distinguished based based on on the the relationship relationship between between metapopula metapopulation 1 ). Models tion and and invasion invasion capacities capacities (Fig. (Fig. 4.5; 4.5; Ovaskainen Ovaskainen and and Hanski, Hanski, 200 2001). Models for for which which }.[ X1= = }.M XMare are called called Levins-type Levins-type models. models. Models Models for for which which }.[ ]kI < < }.M )kM possess population level possess aa meta metapopulation level Allee Allee effect, effect, meaning meaning that that although although aa metapopu metapopulation lation could could persist persist in in aa network, network, aa single single small small local local population population cannot cannot invade invade an population level an empty empty network. network. A A meta metapopulation level Allee Allee effect effect leads leads to to multiple multiple equi equilibria population dynamics. libria in in meta metapopulation dynamics. In In mechanistic mechanistic terms, terms, this this may may be be caused caused by by an an Allee Allee effect effect in in the the colonization colonization process, process, by by aa rescue rescue effect effect in in the the extinc extinction process, process, or or by by aa combination combination of of the the two two ((Ovaskainen and Hanski, Hanski, 2001 2001).). tion Ovaskainen and While While the the SRLM SRLM belongs belongs to to Levins-type Levins-type models, models, the the IFM IFM is is an an example example of of aa model with a strong metapopulation level Allee effect ((Fig. Fig. 4.5). Multiple equilibria equilibria in in meta metapopulation dynamics are are difficult difficult to to test test because because Multiple population dynamics this this requires requires data data for for several several independent independent patch patch networks. networks. The The long-term long-term study land Islands study of of the the Glanville Glanville fritillary fritillary in in the the A Aland Islands has has produced produced the the most most convincing 995a), including convincing example example so so far far (Hanski (Hanski et et aI., al., 11995a), including aa demonstration demonstration of Hanski, 11999b). 999b). A of the the rescue rescue effect effect on on local local extinction extinction ((Hanski, A signature signature of of multi multiple ple equilibria equilibria is is aa bimodal bimodal distribution distribution of of patch patch occupancy occupancy frequencies frequencies (or, (or, more more properly, properly, of of p}... px in in the the case case of of heterogeneous heterogeneous networks) networks).. Putative Putative exam examples Hanski, 11982) 982) have ples of of bimodal bimodal "core-satellite" "core-satellite" distributions distributions ((Hanski, have been been described 999b). The described for for aa wide wide range range of of taxa taxa (Hanski, (Hanski, 11999b). The fundamental fundamental message message
:is
0.8 1l 0.8 .13
25 0.8
K~. 0.6 0.6 O
._o 0.4 0.4 ,g e.c:
0.2 .�E 0.2 O o O -0 o()
B B
A
�
"no Allee effect /
~
weak Allee effect
s 0.6 o. 0.4
strong Allee effect
._~ 0.2
N
. . . . . . . . . .
"~
0.2 0.4 0.6 0'.8' ' :1 Incidences the other Incidences inin the other patches patches 88
AI = AM
a. 6 o_ 6
t~ a: "Oi~ 4 a:
FI
�
22
0.2 0.4 0.6 0.8 0.4 0.6 0.2 0.8 1 the other other patches patches Incidences inin the Incidences
Levins-type model Levi ns-typem odel
~ "~~~---;q /
rescue rescue effect effect
c C
~M
4
uJ
no no rescue rescue effect effect
weak Allee effect effect weak Allee
~ ~ . ~ A l l e e estron f f e cgt Allee effect
0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 0.2 Incidences inin the the other other patches patches Incidences Fig. 44.5 One-dimensional illustration illustration of of three three qualitatively qualitatively different different metapopulation metapopulation models. models. Fig. . 5 One-dimensional (A) The The colonization colonization probability probability of of patch patch ii with with no no Allee Allee effect, effect, with with a weak weak Allee Allee effect, effect, and and with with (A) with and and without without aa rescue effect. effect. strong Allee Allee effect. effect. (B) (6) Extinction Extinction probability probability of of patch patch ii with a strong p)g(p)/p for for Levins-type Levins-type models models and and for for models models possessing possessing aa weak weak (C) The The principal principal map map (1 (1 - p)g(p)/p (C) (2001) for for discussion discussion and and for for the the definition definition strong Allee Allee effect. effect. See See Ovaskainen Ovaskainen and and Hanski Hanski (2001) or or aa strong of of the the principal principal map map g9 (modified (modified from from Ovaskainen Ovaskainen and and Hanski, Hanski, 2001). 2001). -
4. 4.
91 91
METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
for conservation conservation is is that that aa metapopulation metapopulation may may crash crash unexpectedly unexpectedly to to extincextinc for tion from from aa state state of of commonness. commonness. This This happens happens when when the the metapopulation metapopulation tion crosses the the unstable unstable internal internal equilibrium equilibrium due due to to perturbation perturbation or or when when the the pospos crosses itive internal internal equilibrium equilibrium is is lost lost due due to to an an environmental environmental change. change. itive
Patch Values Values Patch
While AM characterizes characterizes the the capacity capacity of of an an entire network to to support support While ~.M entire patch patch network viable metapopulation, metapopulation, one one might might often wish to to assess assess the the contributions contributions that that aa viable often wish particular patches patches make make to to the the persistence persistence of of aa metapopulation. metapopulation. For For example, example, particular in metapopulation metapopulation management, management, limited may force one to to decide decide in limited resources resources may force one which of of two two patches patches is is more more valuable valuable and and should should be be conserved conserved (Lindenmayer (Lindenmayer which and Possingham, Possingham, 1996a,b; 1 996a,b; Moilanen Moilanen and and Cabeza, Cabeza, 2002; 2002; Chapter Chapter 22). 22). In In the the and context of of metapopulation meta population dynamics, dynamics, the the value value of of aa habitat habitat patch patch depends depends not not context only on on the the size size and and the the quality quality of of the the patch, patch, but but also also on on its its connectivity connectivity to to the the only remaining network. Hanski (2003a) examined the remaining network. Ovaskainen Ovaskainen and and Hanski (2003a) examined the concept concept of of "patch value" by particular patches patches make make "patch value" by quantifying quantifying the the contributions contributions that that particular to population dynamics. the value to meta metapopulation dynamics. They They concluded concluded that that the value of of aa particular particular patch assessed properly without specifying what exactly is meant meant by by patch cannot cannot be be assessed properly without specifying what exactly is aa "contribution Table 4.1 four biologically biologically "contribution to to metapopulation metapopulation dynamics. dynamics."" Table 4.1 lists lists four meaningful alternatives. meaningful alternatives. the dynamic value highlight one Wi, termed We will We will here here highlight one particular particular measure, measure, Wi, termed the dynamic value of More precisely, precisely, Wi Wi is is defined the long-term that of the the patch. patch. More defined as as the long-term contribution contribution that patch makes to to the colonization events events in in the the network. network. To define W W= = {Wi}n= patch ii makes the colonization To define l, { Wi}i= 1, we into account that patch patch jj makes makes to to aa colcol we first first take take into account b bii, the direct direct contribution contribution that ij, the onization population is onization event event in in patch patch i.i. Assuming Assuming that that the the meta metapopulation is at at equilibrium, equilibrium, it it is is natural natural to to define define bb~iji as as (Ovaskainen, (Ovaskainen, 2002a) 2002a)
bij = kiPj* TABLE TABLE 4. 4.11
dCol(p * + 8ej) de.
(4. 15) (4.15)
Four Used to Patch Val ues Four Measures Measures Used to Characterize Characterize Patch Values i
Target Target quantity quantity
Perturbation Perturbation measures
Metapopulation capacity Metapopulation capacity �M kM
dAM vSi- dAi
Colonization Colonization events events Metapopulation Metapopulation size size SS
dS US = � U~ = dA; dAi u~== Ss -- Ssii UL d d logT logT t1 = ~- � dAi t7 = log(TIT) t~= log(T/T)
Dynamic Dynamic measures
Appropriate for for Appropriate Rare species
Wi Wi (see (see text) text)
Common species Common species
1
Time Time to to extinction extinction T
1
Rare species
aa Superscripts Superscripts SS and and L L refer refer to to small small and and large large perturbations, perturbations, respectively. respectively. Metapopulation Metapopulation size size SS is is defined defined as as population SS = = 2,iS • i s i piP * ,j, where where Si si isis the the weight weight given given to to patch patch i.i. The The quantities quantities �k, k~, Si, Si, and and T T denote denote the the meta metapopulation i
i
capacity, population size, capacity, the the meta metapopulation size, and and the the time time to to extinction extinction in in aa network network from from which which patch patch i has has been been removed removed (modified (modified from from Ovaskainen Ovaskainen and and Hanski, Hanski, 2003a). 2003a).
92 92
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HAN HANSKI
at at e~ = = 0, 0, where where eeji is is the the jth jth unit unit vector vector and and the the scaling scaling factor factor kkii is is chosen chosen so so 5), the 'j measures that that '!. Eij bbiiij = = 1. 1. In In Eq. Eq. (4.1 (4.15), the term term pp')" measures the the fraction fraction of of time time that that patch jj may may possibly possibly contribute contribute to to the the colonization colonization rate rate of of patch patch i,i, whereas whereas patch the remaining remaining term term measures measures how how sensitive sensitive the the colonization colonization rate rate of of patch patch ii is is the to the the contribution contribution that that patch patch jj makes. makes. For For example, example, in in the the SRLM, SRLM, bb ij0 is is to given as b//=
Picij
(4.16) (4.16)
.
E kP ~gik While While bbiiij measures measures the the direct direct contribution contribution that that patch patch jj makes makes to to aa coloniza colonizapatch i,i, we would would ultimately like to measure measure the long-term con contion event in patch tribution tribution by by including including the the full full chain chain of of colonization colonization events events through through the the network. network. This This may may be be done done by by raising raising the the matrix matrix B to to an an infinite infinite power, power, which is which is equivalent equivalent to to defining defining W W as as the the left left leading leading eigenvector eigenvector of of matrix matrix B = 7, = ( Ovaskainen and Hanski, 2003a). Doing so, W measures = {b {bii}in, i=l (Ovaskainen and Hanski, 2003a). Doing so, W i measures j ij} j l the colonization events the long-term long-term contribution contribution of of patch patch jj to to colonization events in in the the entire entire network network and and is is thus thus independent independent of of the the target target patch patch i.i. Furthermore, Furthermore, as as '!. = 1 , W represents the relative value of patch j. Figure 4.6 illustrates E/W/= 1, W/represents the relative value of patch j. Figure 4.6 illustrates j jWj the behavior of Glanville fritillary the behavior of the the measure measure W W using using the the Glanville ffitillary model model in in the the net network shown shown in in Fig. Fig. 4.1. 4.1. Note Note that, that, in in this this example, the patch patch values values are are dis disexample, the work tributed Fig. 4. 1B). This tributed more more evenly evenly than than patch patch areas areas ((Fig. 4.1B). This is is not not the the case case in in general, general, but but it it happens happens in in the the Glanville Glanville fritillary fritillary model model as as the the patch patch area area scal scal2002b; Ovaskainen ing ing factor factor �~ = �ex ~ex + + �e ~em "at-�~im is less less than than 11 (Ovaskainen, (Ovaskainen, 2002b; Ovaskainen m+ im is and Hanski, and Hanski, 2003a). 2003a). =
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Fig. 4 4.6 An illustration illustration of of the the patch patch value value measure measure W W in in the the patch patch network network shown shown in in Fig. Fig. 4.1. 4. 1 . Fig. . 6 An (A) (A) Sizes Sizes of of the the dots dots are are proportional proportional to to the the values values of of the the habitat habitat patches, patches, and and contour contour lines lines indiindi cate the the relative relative value hypothetical patch would attain attain ifif added added to to aa particular particular location location cate value that that aa hypothetical patch would within within the the network. network. (B) (6) Values Values of of the the habitat habitat patches patches with with respect respect to to patch patch areas. areas. The The slope slope of of the the fitted fitted regression regression line line is is 0.31. 0.31 . The The figure figure is is based based on on the the Glanville Glanville fritillary fritillary model model with with paramparam eters e --= 0.30 0.30 and and cc == 0.13 0.1 3 estimated estimated from from data data restricted restricted to to the the network network shown shown in in Fig. Fig. 4.1 4.1 and and eters the the remaining remaining structural structural parameters parameters estimated estimated from from the the entire entire metapopulation metapopulation (Fig. (Fig. 4.3). 4 .3).
4. 4.
93 93
METAPOPULATION IN HIGHLY FRAGMENTED LANDSCAPES METAPOPULATION DYNAMICS IN LANDSCAPES
Transient Dynamics
Metapopulation Metapopulation capacity capacity A.M ~M and and patch patch values values refer refer to to the the equilibrium equilibrium state state
p" relate to p*,, and and they they thus thus relate to the the long-term long-term behavior behavior of of the the metapopulation metapopulation with with-
out population happens out any any reference reference to to its its present present state. state. If If the the meta metapopulation happens to to be be far far away from away from its its population population dynamic dynamic equilibrium, equilibrium, it it may may be be of of great great importance importance to about the to be be able able to to say say something something about the transient transient dynamics. dynamics. For example, consider For example, consider aa species species that that persists persists initially initially very very well well so so that that most most of of the the habitat habitat patches patches would would be be occupied occupied most most of of the the time. time. Assume Assume then then that that due habitat loss, loss, the due to to habitat the situation situation changes changes rapidly rapidly so so that that many many of of the the patches patches are metapopulation capacity are lost lost from from the the network. network. As As the the metapopulation capacity of of the the network network declines, the population is declines, the meta metapopulation is expected expected to to move move to to aa lower lower occupancy occupancy state, state, or or it it may may even even go go extinct. extinct. However, However, this this does does not not happen happen instantaneously, instantaneously, and and the the length length of of the the transient transient period period is is often often of of great great interest. interest. A A transient transient may may also also occur occur in in the the opposite opposite direction, direction, as as is is the the case case if if aa species species invades invades an an initially initially empty empty network. network. Figure Figure 4.7 4.7 illustrates illustrates that that the the length length of of such such transient transient periods periods may may well well be be up up to to 5-1 5-100 generations generations in in the the Glanville Glanville fritillary fritillary model. model. Ovaskainen Ovaskainen and and Hanski Hanski (2002) (2002) investigated investigated the the transient transient time time in in the the SRLM SRLM by model is able to approximate by first first demonstrating demonstrating that that the the original original Levins Levins model is able to approximate the the transient transient behavior behavior of of the the SRLM SRLM and and then then calculating calculating the the transient transient time time for for the task, we the former. former. To To accomplish accomplish the the first first task, we denote denote the the parameters parameters of of the the Levins Levins model model by by c T and and Ii ~" so so that that the the model model is is defined defined as as
ap dp c p ( 1 -- pp)) -- � ep. dt = =� 2"p(1 ~p. dt
(4.1 7) (4.17)
This model approximates following trans This model approximates the the behavior behavior of of the the SRLM SRLM given given the the following transformations formations (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2002). 2002). First, First, one one has has to to interpret interpret the the Pi, the Eq. (4.1 7) as variable variable p p in in Eq. (4.17) as p" px = = kjWj ]~iWiPi, the weighted weighted fraction fraction of of occupied occupied patches. patches. Second, Second, the the parameter parameter Ii ~ is is interpreted interpreted as as the the effective effective extinction extinction rate, rate,
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in the model in in the Fig. 4.7 4.7 Transient Transient behavior behavior in the Glanville Glanville fritillary fritillary model the network network shown shown in in Fig. Fig. 4.1 4.1 with with parameter values as in Fig. Fig. 4.6. In panel A all the patches are initially assumed to to be occupied, whereas in panel B, only the patch indicated by an arrow 4.1 is initially occupied. The thick arrow in Fig. Fig. 4.1 line shows Thin lines mean of 000 simu line shows the the prediction prediction of of the the deterministic deterministic model. model. Thin lines show show the the mean of 11000 simulations, lations, the the lower lower line line derived derived from from all all replicates replicates and and the the upper upper one one ignoring ignoring such such replicates, replicates, which extinct by that time. which have have gone gone extinct by that time. Dashed Dashed lines lines show show 95% 95% confidence confidence intervals intervals derived derived from all simulation from all simulation replicates.
94 94
OTSO LKKA HAN SKI OTSO OVASKAINEN OVASKAINEN AND AND IILKKA HANSKI
defined defined by by e ~ = = 1/(ki 1/(EiWi/Exti). Third, the the parameter parameter c T is is interpreted interpreted as as the the W/Ext;). Third, "AM/a. effective colonization effective colonization rate, rate, defined defined by by c 7" = =e ~" kM/8. Using the SRLM to Using the aforementioned aforementioned transformation transformation of of SRLM to the the Levins Levins model, model, the the length period, defined defined as time it length of of the the transient transient period, as the the time it takes takes for for the the metapopu metapopulation close the lation to to return return from from its its initial initial state state to to aa state state close the new new equilibrium, equilibrium, may may be Ovaskainen and be written written as as the the product product of of four four factors factors ((Ovaskainen and Hanski, Hanski, 2002). 2002). First, First, the the length length of of the the transient transient period period increases increases with with the the distance distance between between the the present state. Second, transient is present state state and and the the equilibrium equilibrium state. Second, the the length length of of the the transient is longer span) than longer for for species species that that have have slow slow dynamics dynamics (e.g., (e.g., due due to to aa long long life life span) than for species with dynamics. Third, transient period for species with fast fast dynamics. Third, the the length length of of the the transient period is is longer longer in in aa patch patch network network that that has has few few large large patches patches than than in in aa network network with with many many small small patches, patches, as as the the turnover turnover rate rate is is expected expected to to be be slower slower for for larger larger patches. transient period patches. Fourth Fourth and and most most important, important, the the length length of of the the transient period is is expected to species that close to their extinction expected to be be especially especially long long for for species that are are close to their extinction threshold threshold following following perturbation. perturbation. The The fourth fourth conclusion conclusion has has the the important important implication implication for for conservation conservation that that many many rare rare species species living living in in recently recently deteriorated deteriorated landscapes landscapes may may be be "doomed" "doomed" to to extinction. extinction. They They still still exist exist because because they they have have not not had had time time to to go go extinct extinct yet, yet, and and the the time time it it takes takes to to go go extinct extinct is is especially especially long long in in the the case case of of species species whose whose long-term number of long-term persistence persistence is is most most precarious. precarious. The The number of species species that that are are pre predicted habitat loss loss and dicted to to ultimately ultimately go go extinct extinct due due to to past past habitat and fragmentation fragmentation rep represents 994; Hanski resents the the extinction extinction debt debt in in the the community community (Tilman (Tilman et et a!., al., 11994; Hanski and and Ovaskainen, Ovaskainen, 2002). 2002). The The extinction extinction debt debt is is paid paid either either by by letting letting the the species species go go extinct extinct or or by by improving improving the the quality quality of of the the landscape landscape sufficiently sufficiently for for the the species species that debt. Hanski Hanski and that constitute constitute the the extinction extinction debt. and Ovaskainen Ovaskainen (2002) (2002) discuss discuss an an example example of of extinction extinction debt debt in in beetle beetle species species living living in in boreal boreal forests forests in in Finland. Finland.
4.5 4.5
STOCHASTIC STOCHASTIC THEORY THEORY Recall analysis of occupancy models models is Recall that that the the analysis of stochastic stochastic patch patch occupancy is difficult difficult because space is heterogeneous network because the the size size of of the the state state space is 2n 2 n for for aa heterogeneous network of of n n patches. This develops approximation approximation methods patches. This section section develops methods that that take take advantage advantage of of possible possible links links between between deterministic deterministic and and stochastic stochastic frameworks. frameworks. In In particular, particular, this section shows values may heterogeneous this section shows that that patch patch values may be be used used to to transform transform aa heterogeneous meta population to metapopulation to its its homogeneous homogeneous ("ideal" ("ideal")) counterpart, counterpart, which which behaves, behaves, in in some some relevant relevant respects, respects, similarly similarly as as the the original original heterogeneous heterogeneous metapopulation. metapopulation.
Effective Effective Metapopulation Metapopulation Size Size Let Let us us start start with with the the SRLM. SRLM. As As stated stated in in the the previous previous section, section, both both the the equi equilibrium librium state state and and the the transient transient behavior behavior of of the the deterministic deterministic SRLM SRLM can can be be approximated approximated by by the the one-dimensional one-dimensional Levins Levins model model by by replacing replacing the the original original model parameters parameters with with the the effective effective colonization rate c F~ and and the the effective effective model colonization rate extinction extinction rate rate e. T. These These factors factors account account for for the the deterministic deterministic drift drift (growth) (growth) in in the model. The the model. The main main difference difference between between deterministic deterministic and and stochastic stochastic models models is is that that the the latter latter account account for for stochastic stochastic fluctuations fluctuations around around the the mean mean dynamics, dynamics, which which arise arise due due to to the the finite finite size size of of the the network. network. We We may may ask ask whether whether one one
4. 4.
995 5
METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
could also also transform transform the the size size of of aa heterogeneous heterogeneous metapopulation meta population into into an an effeceffec could tive metapopulation metapopulation size size ~', ii, which which would would control control for for the the amount amount of of stochastic stochastic tive fluctuations. To To incorporate incorporate the the latter latter into into the the model, model, we we require require that that the the fluctuations. (T22 in in Eq. Eq. (4.2) (4.2) be be the the same same for for the the weighted weighted fraction fraction of of infinitesimal variance variance (r infinitesimal occupied patches patches Px p" in in the the SRLM SRLM and and for for the the simple simple fraction fraction of of occupied occupied occupied patches pp in in the the one-dimensional one-dimensional Levins Levins model. model. Equating Equating the the two two variances variances at at patches the equilibrium leads to to the equilibrium state state p': p ':- leads n' n~ = =
cp '� ( l - p t )
ep '� Exti( P':- )P/' ] ZL W2[C~ W;[ Co[i(p" )(l -- Pi':) Pi', ) + Exti(p*)pi*] i +
Tp'~(1 - p'~) + ~p'~ ----- .
(4.18) (4. 1 8 )
t
As expected, expected, the the effective effective number number of of habitat habitat patches patches increases increases with with the the real real As number of of patches patches and and with with decreasing decreasing variance in the values of of these patches. number variance in the values these patches. If all all patches are identical, the effective effective number number coincides coincides with the actual actual numnum If patches are identical, the with the ber. the SRLM SRLM is in Fig. 4.8. The line corcor ber. Transformation Transformation of of the is illustrated illustrated in Fig. 4.8. The dashed dashed line responds to to the transformed model, model, whereas whereas the responds the analytically analytically transformed the continuous continuous line line corresponds fitted model. corresponds to to aa numerically numerically fitted model. In nonlinear models, such such as fritillary model, model, or IFM, it In nonlinear models, as the the Glanville Glanville fritillary or the the IFM, it may not be be possible possible to to derive derive analytical expressions for the effective coloniza may not analytical expressions for the effective colonization and extinction rates rates or for for the effective metapopulation tion and extinction metapopulation size. size. These quanquan tities may still still be be determined fitting aa structurally tities may determined numerically numerically by by fitting stcucturally similar similar homogeneous model to of the the heterogeneous model homogeneous model to the the drift drift and and the the variance variance of heterogeneous model (Ovaskainen, 2002a). Figure example for the Glanville Glanville fritillary fritillary (Ovaskainen, 2002a). Figure 4.9 4.9 gives gives an an example for the model. The homogeneous model model consists 50 patches, which is is somewhat model. The homogeneous consists of of 50 patches, which somewhat less 56 patches. less than than the the actual actual number number of of 56 patches. The The difference difference between between the the effective effective number number of of habitat habitat patches patches and and the the actual actual number number of of habitat habitat patches patches is is explained (Fig. 4.6). example, the explained by by aa variation variation in in patch patch values values (Fig. 4.6). In In this this example, the distri distribution bution of of patch patch values values is is relatively relatively even even and and hence hence the the difference difference is is not not very very great. great.
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Fig. model in Fig. 4.8 4 . 8 Transformation Transformation of of the the SRLM SRLM to to the the stochastic stochastic logistic logistic model in the the network network shown shown in sample of 00 randomized in Fig. Fig. 4.1 4.1.. Dots Dots represent represent aa sample of 1100 randomized occupancy occupancy states states from from the the quasista quasistationary tionary distribution distribution obtained obtained by by simulation. simulation. Lines Lines represent represent fitted fitted models; models; the the continuous continuous line line isis based based on on numerical numerical fitting, fitting, and and the the dashed dashed line line is is based based on on the the analytical analytical transformation transformation given 1 8). The given by by Eq. Eq. (4. (4.18). The two two panels panels show show (A) (A) infinitesimal infinitesimal mean mean f.L I~ and and (B) (B) infinitesimal infinitesimal variance variance a2. ~2. Parameter Parameter values values as as in in Fig. Fig. 4.6 4.6 except except ee = = 11,, Cc = = 0.5. 0.5.
96 96
OTSO OVASKAINEN AND ILKKA ILKKA HANSKI B B
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Fig. 4.9 in Fig. Fig. 4.8 but for model with values as 4.9 As As in 4.8 but for the the Glanville Glanville fritillary fritillary model with parameter parameter values as in in Fig. Fig. 4.6. 4.6. Single realizations of of the the full heterogeneous model model (C) and of of the transformed transformed homogeneous homogeneous model (D).
Spatially Spatially Correlated Correlated and and Temporally Temporally Varying Varying Environmental Environmental Conditions Conditions Metapopulation Metapopulation models models typically assume assume that the dynamics of local popula populations are independent independent of each other other and that that the environmental environmental conditions remain constant. In reality, both assumptions are violated to a smaller or greater extent ((Baars Baars and Van Dijk, 11984; 984; Ims and Steen, 11990; 990; Hanski, 1999b; Lundberg Lundberg et et aI., al., 2000; 2000; Nieminen Nieminen et et aI., al., 2004). 2004). Spatially Spatially correlated correlated and and tempo tempoenvironmental conditions present a challenge for for meta metapopulation rally varying environmental population theory from the viewpoint of both model analysis (Heino et al., aI., 11997; 997; Engen et aI., al., 2002; Ovaskainen, 2002a) and parameter parameter estimation (Thomas, 1991; Hanski and Woiwod, 11993; 993; Bjornstad et aI., al., 1999, Williams and Liebhold, al., 2002). Viewing the metapopulation metapopulation as a population population of 2000; Peltonen et aI., populations, populations, stochasticity stochasticity in in patch patch occupancy occupancy dynamics dynamics in in aa constant constant environ environment ment is is analogous analogous to to demographic demographic stochasticity, stochasticity, whereas whereas temporal temporal variability variability in dynamics leads to variability that that is analogous to environmental stochastic stochasticity. Hanski ((1991) 1991 ) has termed these two population-level stosto two forms of meta metapopulation-level chasticities regional stochasticity, chasticities as as extinction-colonization extinction-colonization stochasticity stochasticity and and regional stochasticity, respectively. In a finite network network of habitat habitat patches, regional stochasticity leads to both spatially correlated correlated and temporally temporally varying parameter parameter values; these two two phenomena phenomena actually actually represent represent the the two two sides sides of of the the same same coin. coin. The The effective effective metapopulation metapopulation size size approach approach described described earlier earlier can can be be extended extended to situations in allowing one include to situations in which which the the parameters parameters vary vary temporally, temporally, allowing one to to include regional model. This regional stochasticity stochasticity into into the the model. This is is illustrated illustrated in in Fig. Fig. 4.10, 4.10, which which is is otherwise identical to Fig. 4.9 but now now with the parameter parameter values for for the Glanville fritillary butterfly estimated separately for the eight annual transitions temporal variation to the model present in our dataset. As expected, adding temporal increases the variance Var[Llp;.J (Figs. 4.9B and 4.10B) and thus the amplitude Var[APx] (Figs.
4.
97 91
METAPOPULATION METAPOPULATION DYNAMICS IN HIGHLY HIGHLY FRAGMENTED LANDSCAPES LANDSCAPES E[Ap~] E[l1p J
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f-----------""'-<3 00 O 3000 4 400 00 30 00 5500 260 300 360 400 460 500 $60 00 2200 1lo0 00 200 1100 Time Time tt Time t Fig. 4.9 but with Fig. 4.1 4 . 1 00 As in Fig. 4.9 with temporally temporally varying varying parameters. parameters. The parameter parameter distributions distributions e = (0.24, 0048, 34, 0.1 7, 0.30, 0.53, 0.04) and c = 1 4, 0.1 0, 0. 1 0, 0.33, 0.08, 0.08, 0.48, 0.33, 0. 0.34, 0.17, 0.04)and = (0. (0.14, 0.10, 0.10, 0.06, 0.22) have been estimated separately from 3), from the eight eight annual transitions transitions (see (see Fig. Fig. 4. 4.3), whereas the remaining remaining parameters have been kept fixed. o o
of stochastic fluctuations (Figs. 0C). Figure 4. 1 1 compares (Figs. 4.9C and 4.1 4.10C). 4.11 compares the statistical properties of the two two models and their homogeneous homogeneous counterparts. counterparts. As variation flattens As expected, expected, temporal temporal variation flattens the the quasistationary quasistationary distribution distribution (Fig. 4.1 1A), but note that (Fig. 4.11A), that it also adds especially low-frequency fluctuations fluctuations to the behavior of the model (Fig. (Fig. 4.1 1B). As illustrated by Figs. 4.9, 4.10, and 4.11B). 4.1 1, the effective metapopulation 4.11, metapopulation size approach approach gives a very accurate description of the Glanville fritillary model in all the senses investigated here. The example in Fig. 44.10 . 1 0 demonstrates cor demonstrates that that temporal temporal variation (spatial correlation) in parameter values increases the extinction risk of a meta population metapopulation by amplifying stochastic fluctuations, fluctuations, which which may be restated restated by observing that that temporal variation decreases the effective number of habitat patches. B
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0.1
98 98
OTsO LKKA HANSKI HAN SKI OTSO OVAsKAINEN OVASKAINEN AND AND IILKKA
Ovaskainen Ovaskainen and and Hanski Hanski (2003b) (2003b) made made this this statement statement explicit explicit in in the the context context of of the the stochastic stochastic logistic logistic model. model. They They showed showed that that if if extinctions extinctions and and coloniza colonizations tions are are correlated, correlated, with with aa correlation correlation coefficient coefficient p, p, the the effective effective number number of of patches patches is is reduced reduced to to
nF/ nnee = = ----- 9 ((nn -- 11 )p ) p ++ 11
(4. 19) (4.19)
Such changes the model in Such aa correlation correlation changes the qualitative qualitative behavior behavior of of the the model in several several ways. Most metapopulation extinc ways. Most importantly, importantly, for for pp > > 00 the the mean mean time time to to metapopulation extinction increase exponentially habitat patches, patches, as tion does does not not increase exponentially with with the the number number of of habitat as predicted . 1 , but predicted by by Eq. Eq. (2) (2) in in Box Box 4 4.1, but it it now now grows grows according according to to the the power-law power-law T Ovaskainen and T IX oc nlfp nl/~ ((Ovaskainen and Hanski, Hanski, 2003b). 2003b).
4.6 4.6
COMPARISON COMPARISON WITH W I T H OTHER OTHER MODELING MODELING APPROACHES APPROACHES This This section section compares compares stochastic stochastic patch patch occupancy occupancy models models with with other other mod modeling eling approaches approaches that that have have been been used used in in metapopulation metapopulation studies. studies. The The other other approaches include individual-based simulation models, models structured approaches include individual-based simulation models, models structured by by population size, lattice models, and spatial moment moment equations, population size, lattice models, and spatial equations, which which were were described Chapters 11 and 3. This described briefly briefly in in Chapters and 3. This section section compares compares the the different different modeling modeling approaches approaches in in the the context context of of extinction extinction thresholds thresholds and and their their relation relation to landscape structure, issues in population to landscape structure, which which is is one one of of the the key key issues in meta metapopulation dynamics dynamics for for both both research research and and management. management. To To facilitate facilitate this this comparison, comparison, we we start start by by summarizing summarizing three three ecologically ecologically significant significant messages messages that that have have been been discussed discussed in in this this chapter chapter in in the the context context of of SPOMs. SPOMs. 9 First First and and most most important, important, there there is is aa critical critical amount amount of of habitat habitat below below which which aa species species is is expected expected to to go go deterministically deterministically extinct. extinct. In In highly highly frag fragmented landscapes, extinction threshold depends not mented landscapes, the the extinction threshold depends not only only on on the the total also on configuration of total amount amount of of habitat, habitat, but but also on the the spatial spatial configuration of the the habi habitat population capacity, tat patch patch network. network. As As characterized characterized by by meta metapopulation capacity, metapopulation by increasing metapopulation persistence persistence is is facilitated facilitated by increasing connectivity connectivity among the habitat patches. among the habitat patches. • population 9 Second, Second, the the contribution contribution that that individual individual patches patches make make to to meta metapopulation persistence persistence depends depends not not only only on on their their area area and and quality, quality, but but also also on on their their position network. Well-connected patches make make generally position within within the the network. Well-connected patches generally aa greater greater contribution contribution than than isolated isolated patches. patches. • population extinction 9 Third, Third, meta metapopulation extinction is is aa stochastic stochastic event, event, which which depends depends not population dynamics, not just just on on the the deterministic deterministic drift drift in in meta metapopulation dynamics, but but also also on on fluctuations fluctuations around around the the drift. drift. Stochastic Stochastic fluctuations fluctuations increase increase with with aa decreasing decreasing number number of of habitat habitat patches, patches, and and thus thus extinction-colonization extinction-colonization stochasticity stochasticity increases increases the the extinction extinction risk, risk, especially especially in in small small metapopu metapopuFurthermore, spatially correlated local dynamics or or temporally lations. Furthermore, varying varying environmental environmental conditions conditions amplify amplify stochastic stochastic fluctuations fluctuations and and thus thus increase increase the the risk risk of of metapopulation metapopulation extinction. extinction. •
The The first first message message has has been been studied studied extensively extensively in in the the literature. literature. Fahrig Fahrig (2002) (2002) reviewed reviewed aa number number of of lattice-based lattice-based modeling modeling studies studies attempting attempting to to disentangle disentangle
LANDSCAPES D SCAPES 4. METAPOPULATION DYNAMICS IN HIGHLY FRAGMENTED LAN
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fragmentathe effects of habitat loss (fraction of destroyed habitat) and habitat fragmenta tion (spatial configuration of the remaining habitat). She categorized the studies modinto two classes based on the type of models used, either patch occupancy mod (colonization-extinction models; Hill and Caswell, 11999; With and King, els (colonization-extinction 999; With 11999a) 999a) or individual-based simulation models ((birth-immigration-death-birth-immigration-death- emigration Bevers, 2002; emigration models; models; Flather Flather and and Bevers, 2002; Fahrig, Fahrig, 2001 2001).). Fahrig Fahrig (2002) (2002) drew drew three major conclusions, which are all well in line with the results derived from SPOMs. SPOMs. First, First, all all of of the the modeling modeling studies studies she she considered considered predicted predicted aa threshold threshold amount metapopulation persistence. Second, a smaller amount of available habitat for metapopulation total population persistence total amount amount of of habitat habitat was was required required for for meta metapopulation persistence if if the the habi habiThird, the spatial configuration configuration of the remaining habi habitat occurred in clusters. Third, remained tat had the strongest effect if only a small fraction of the landscape remained suitable for the species. Similar conclusions were reached by Hiebeler (2000) (2000) and and Ovaskainen et al. (2002) (2002) using a lattice-based patch occupancy model. distinction between patch patch occupancy occupancy models and individual-based individual-based A major distinction simulation simulation models models reviewed reviewed by by Fahrig Fahrig (2002) (2002) was was that that the the latter latter predicted predicted only only habitat fragmentation fragmentation on extinction extinction threshold, threshold, which prediction prediction a mild effect of habitat with much much of the empirical literature literature (Harrison and Bruna, 11999; is in line with 999; Fahrig, 2002). 2002). Fahrig (2002) (2002) argued that that the difference between the two two types metapopulation of models occurs because the underlying reasons decreasing meta population persistence persistence with with increasing increasing amount amount of of habitat habitat loss loss are are different. different. In In patch patch occu occupancy pancy models, models, the the colonization colonization rate rate of of empty empty patches patches decreases decreases with with increasing increasing habitat loss, whereas in individual-based individual-based simulation simulation models, an increasing habitat number number of individuals individuals spend their time in the landscape landscape matrix, matrix, where repro reproduction duction is is not not possible possible and and mortality mortality is is assumed assumed to to be be high. high. However, However, as as migra migration mortality mortality is one of the major reasons why why a reduced reduced colonization colonization rate with tion major reasons rate with distance is assumed assumed in extinction-colonization extinction-colonization dynamics dynamics (Hanski, 11999b), distance 999b), the difference between the the two causes for extinction threshold threshold is is somewhat somewhat difference between two causes for the the extinction superficial. parameterize a patch superficial. Indeed, one could parameterize patch occupancy model using data from simulation simulation of an individual-based which case the the two two moddata individual-based model, in which mod should give more more or less identical identical predictions. eling approaches approaches should predictions. We suggest that that the difference difference between the two approaches is not two modeling approaches not fundamental fundamental and and on quantitative quantitative that the differences differences observed by Fahrig (2002) that (2002) were were based on model assumptions. Most Most importantly, to model assumptions. importantly, the lattice-based lattice-based studies are sensitive to spatial (instead of "patches" "patches" consisting consisting of of spatial scale, especially if single lattice cells (instead a cluster cluster of assumed to support independent independent local populations. The of cells) are assumed to support populations. The (2002) did did not not include include environmental environmental simulation reviewed by Fahrig simulation studies studies reviewed Fahrig (2002) stochasticity at at the the level of of habitat habitat patches, patches, which, which, in the the case of of lattice lattice models, models, stochasticity would amount to to spatially correlated stochasticity stochasticity in in clusters clusters of of lattice lattice cells cells would amount spatially correlated comprising a single habitat patch (Gu ( Gu et al., aI., 2002). 2002). In SPOMs SPOMs and and in reality, comprising habitat patch the patch patch area-dependent area-dependent extinction extinction rate rate leading leading to to aa high high rate rate of of extinction extinction of of the small populations populations in in small small patches patches comprises comprises an an important important reason reason why why the the spaspa small configuration of of the the habitat habitat may may greatly influence the the extinction threshold tial configuration greatly influence extinction threshold in highly fragmented fragmented landscapes. landscapes. Concerning our our second second message, which which relates relates to to the the value value of of individual individual Concerning habitat patches, patches, most most of of the the previous previous literature literature has has been been based based on on simulation simulation habitat studies. For For example, example, Lindenmayer Lindenmayer and and Possingham Possingham (1996a,b) ( 1 996a,b) developed developed aa studies. simulation simulation model model to to assess assess the the persistence persistence of of Leadbeater's Leadbeater's possum possum in in southsouth eastern Australia. Australia. In In line line with with our our results, results, they they concluded concluded that that extinction extinction eastern
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probability probability was was influenced influenced both both by by the the size size and and by by the the spatial spatial arrangement arrangement of of habitat patches. They found that in some circumstances circumstances the probability probability that that metapopulation is extant extant may be a good meas measa patch is occupied while the metapopulation for metapopulation metapopulation viability. The habitat habitat patch networks used ure of its value for from a single 300-ha reserve reserve to twelve twelve 25-ha reserves. in their studies varied from As As expected, expected, metapopulations metapopulations in in networks networks of of many many small small reserves reserves were were vul vulnerable to extinction extinction due to demographic demographic and environmental environmental stochasticities. stochasticities. Conversely, a metapopulation metapopulation inhabiting inhabiting the the single large reserve was suscep susceptible to extinction extinction in a catastrophic catastrophic wildfire, highlighting highlighting the need for for several scattered reserves. more scattered Other 1 997), Other studies studies that that have have addressed addressed patch patch values values include include Keitt Keitt et et al. al. ((1997), who used percolation percolation theory to quantify quantify connectivity at multiple scales and and assigned conservation conservation priorities to habitat habitat patches based on their their contribu contributions to connectivity. In addition, 1 997) addition, using sensitivity analysis, Keitt et al. ((1997) identified identified critical critical "stepping "stepping stone" stone" patches patches that, that, when when removed removed from from the the land landscape, would 19 9 8 ) would cause large changes changes in connectivity. connectivity. Moilanen Moilanen et al. ((1998) constructed constructed aa somewhat somewhat aadd hoc hoc measure measure ooff patch patch values values while while analyzing analyzing dynamics dynamics in a classic metapopulation metapopulation of the American American pika using the incidence function function model. Their measure measure was an attempt attempt to define the patch patch value in the the sense Moilanen et 1998 ) considered sense of of our our measure measure W, W, although although Moilanen et al. al. ((1998) considered only only aa two-generation two-generation impact of habitat habitat patches patches on colonizations. colonizations. Verboom et al. (200 1 ) emphasized (2001) emphasized the the importance importance of of large large "key "key patches" patches" with with aa stabilizing stabilizing role role in in habitat habitat patch patch networks. networks. Patch Patch values values have have also also been been addressed addressed analyt analytpopulation models ically in the context context of meta metapopulation models structured structured by the sizes of local populations. 1 994) and Gyllenberg and 1 997) noted populations. Lawton Lawton et al. ((1994) and Hanski Hanski ((1997) noted that down due due to that the the Levins rule (Section ( Section 4.2) may may break break down to a variation variation in habihabi tat destruction of high-quality patch has a greater impact tat patch patch quality, as destruction of a high-quality patch has impact destruction of low-quality patch. on metapopulation metapopulation size than than destruction of a low-quality patch. Our which is concerned interplay between Our third third message, which concerned with with the the interplay between stochas stochastic and dynamics, has been studied studied by Casagrandi and Gatto Gatto and deterministic deterministic dynamics, Casagrandi and ((1999, 1999, 2002a,b) 2002a,b) using using metapopulation metapopulation models models structured structured by by local local population population size. They They assumed of local populations negative-binomially assumed that that the sizes of populations are negative-binomially distributed to with our results, Casagrandi Casagrandi distributed to facilitate facilitate the the model model analysis. analysis. In line with our results, and parameter space into and Gatto Gatto classified the the parameter into four four regions, of which which one extreme the other to extreme corresponded corresponded to deterministic deterministic extinction extinction and and the other one to metapopulation persistence. persistence. Stochasticity Stochasticity was was shown shown to to create create intermediate intermediate metapopulation which either either demographic demographic stochasticity stochasticity or or environmental environmental catastrocatastro zones, in which phes imposed imposed a substantial substantial risk risk of of extinction, extinction, even though though the the metapopulation metapopulation phes was was predicted predicted to to persist persist in a deterministic deterministic model. model. Many Many studies studies have have concon cluded that that the the risk risk of of metapopulation metapopulation extinction increases with with increasing increasing cluded extinction increases regional stochasticity, stochasticity, that that is, increasing increasing the the scale of of environmental environmental variability. variability. regional For For example, example, Palmqvist Palmqvist and and Lundberg Lundberg (1998) ( 1 998) used used a coupled coupled map map lattice lattice model density-dependent dynamics dynamics and model consisting consisting of of local local populations populations with with density-dependent and density-independent migration. migration. They They found found that that the the major major determinant determinant of of the the density-independent risk of between local of metapopulation metapopulation extinction extinction is the the balance balance between local population population variability synchrony in local extent variability and and synchrony local population population fluctuations. fluctuations. To what what extent regional stochasticity stochasticity increases increases the the extinction extinction risk risk of of real real metapopulations metapopulations is a regional quantitative question, question, which which cannot cannot be be answered answered by by just just demonstrating demonstrating that that quantitative there there is some some degree degree of of spatial spatial correlation correlation in local dynamics. dynamics.
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES 4. METAPOPULATION 4.
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CURRENT STATE STATE OF OF THE THE THEORY THEORY A AND APPLICATIONS 4.7 CURRENT 4.7 N D APPLICATIONS Metapopulation theory theory based based on on stochastic stochastic patch patch occupancy occupancy models models has has its its Metapopulation roots in in the the Levins Levins model model (Levins, (Levins, 1969), 1 969), which which was was originally originally developed developed to to roots study alternative alternative pest pest eradication eradication strategies. strategies. In In the the 1970s, 1 970s, models models stemming stemming study from the the Levins Levins model model were were constructed constructed to to study study group group selection selection (Levins, (Levins, from 1 970; Levitt, Levitt, 1978) 1 978) and and competitive competitive (Levins (Levins and and Culver, Culver, 1971; 1971; Horn Horn and and 1970; MacArthur, 1972; 1 972; Slatkin, Slatkin, 1974; 1 974; Hanski, Hanski, 1983) 1 98 3 ) and and predator-prey predator-prey interinter MacArthur, actions (Maynard (Maynard Smith, Smith, 1974). 1 974) . Patch Patch occupancy occupancy models models for for two two or or more more actions species were were investigated investigated further further in in the the 1990s 1 990s (Nee (Nee and and May, May, 1992; 1 992; Tilman Tilman et et species al., 1994; 1 994; Lei Lei and and Hanski, Hanski, 1997; 1 997; Nee Nee et et al., al., 1997; 1 997; Holt, Holt, 1997). 1 997). These These studies studies al., have largely remained remained at at the the theory theory level, level, although although Hanski Hanski and and Ranta Ranta (1983) ( 1983) have applied the the Levins model model extended extended to to three three competitors competitors to to study study the the dynamics dynamics applied water fleas fleas in in rock rock pools. A few few other other studies studies have have employed employed of Daphnia of Daphnia water homogeneous SPOMs SPOMs in the the study study of of actual actual metapopulations metapopulations (Lande, 1988b; 1 9 88b; homogeneous As et al., 1997; Kindvall, 1996; 1 996; As 1 997; Doncaster Doncaster and Gustafsson, 1999; 1 999; Carlson, Carlson, Kindvall, and Gustafsson, 2000). However, given the the typically marked marked heterogeneities heterogeneities in real habitat habitat 2000). patch networks, these applications applications are best best interpreted interpreted as attempts attempts to to increase increase patch networks, these qualitative about metapopulation metapopulation dynamics. Quantitative predictions predictions qualitative insight insight about dynamics. Quantitative are best best based based on on heterogeneous heterogeneous SPOMs SPOMs and and the the spatially spatially realistic realistic metapopumetapopu are lation lation theory theory that that has been developed developed since the first first incidence function function models (Hanski, 1 992, 1994a). 1 994a). (Hanski, 1992, advantage of of SMT in comparison of the other The great advantage comparison with with much much of other theory in spatial ecology is that applied to real metapopulations metapopulations for that the models can be applied purposes research, management, management, and conservation. purposes of research, conservation. The key here is the set of assumptions assumptions that that these models make on the influence of landscape structure on population population processes. SMT prescribes research tasks for ecologists engaged in empirical work work in terms of testing model assumptions assumptions and predictions. The incidence 994a) has been employed widely in the incidence function model (Hanski, 11994a) study of classic metapopulations metapopulations of insects, frogs, small mammals, and birds (reviews in Hanski, 11999b, 999b, 2001 b ) . Statistically rigorous methods have been 2001b). developed to estimate model parameters of both SPOMs and their mean-field approximations, including maximum likelihood (Moilanen, 11999, 999, 2000) and Bayesian estimation estimation (O'Hara (O'Hara et al., 2002; Ter Braak and Etienne, 2003; Chapter Chapter 5). 5). Although metapopulation theory based oonn SPOMs has advanced greatly in recent years, years, the theory is still still in a state of active development and major steps forward forward can be expected to be taken in the near future. We list here some of the challenges for for further research. First, it would be interesting to integrate SPOMs SPOMs with with metapopulation metapopulation models models structured structured by by the the sizes sizes of of local local popula populations Gyllenberg and 992; Gyllenberg 997; Casagrandi tions ((Gyllenberg and Hanski, Hanski, 11992; Gyllenberg et et al., al., 11997; Casagrandi and and Gatto, 11999). 999). Among other other things, things, this would allow allow one one to extend extend mechanis mechanistic Etienne, 2000, 2002b) tic models models for the rescue effect ((Etienne, 2002b) to heterogeneous heterogeneous patch patch networks. networks. Second, and related related to the former, not accounting accounting for for individuals largely prevents prevents the extension extension of of SPOMs to address population genetic and evolutionary questions. questions. However, independently of of the ecological ecological research on SPOMs, 1 997) have SPOMs, Whitlock Whitlock and and Barton Barton ((1997) have started started to to develop develop population population genetic genetic theory theory that that takes takes into into account account the the influence influence of of spatial spatial heterogeneity heterogeneity at at the the landscape landscape level, level, and and Heino Heino and and Hanski Hanski (2001 (2001)) have have modeled modeled the the influence influence
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of of landscape landscape structure structure on on the the evolution evolution of of migration migration rate rate (see (see also also Parvinen, Parvinen, 2002; 2002; Parvinen et aI., al., 2003 2003).). There There is much potential in these directions. Third, Third, SPOMs apply best to highly fragmented fragmented landscape, landscape, although although the theory theory has also been applied to has also been applied to landscapes landscapes described described as as lattices lattices by by interpreting interpreting the the regularly spaced lattice cells as habitat patches ((Ovaskainen Ovaskainen et aI., al., 2002; 2002; Pakkala Pakkala et aI., al., 2002). 2002). In this context context in particular, the the standard standard assumption assumption of independent independent local dynamics dynamics becomes becomes very questionable, questionable, and ways have to be devised of incorporating incorporating spatially correlated correlated dynamics dynamics into the model (for one approach aI., 2002). approach with with potential potential for for some some practical practical applications, applications, see see Gu Gu et et al., 2002). Finally, it may be possible to to combine combine SPOMs with with other other techniques techniques used in spatial Chapter 3). spatial ecology, such such as spatial spatial moment moment equations equations ((Chapter
4.8 4.8
LINKS WITH LINKS W I T H LANDSCAPE LANDSCAPE ECOLOGY, ECOLOGY, EPIDEMIOLOGY, EPIDEMIOLOGY, AND AND MATRIX MATRIX POPULATION POPULATION MODELS MODELS Incorporating population Incorporating the the influence influence of of landscape landscape structure structure on on meta metapopulation processes in the manner manner that that this is done in SMT has led to the realization realization that that more substantial substantial links can be forged forged between between meta metapopulation and more population ecology and than has been perceived previously (Hanski, 2001 2001b; related disciplines than b; Hanski Ovaskainen, 2003) 2003).. In the first place, SMT contributes contributes to the conceptual conceptual and Ovaskainen, unification metapopulation ecology landscape ecology. unification of of metapopulation ecology and and landscape ecology. Merging Merging of of these been anticipated these two two fields fields has has been anticipated for for aa long long time time (Hanski (Hanski and and Gilpin, Gilpin, but not not much much has has happened happened until until now, now, as as can can be be seen seen from from aa lack lack of of 11991), 99 1 ) , but Chapter 11,, Fig. 11.2). .2). Metapopulation papers that that refer to both disciplines ((Chapter Metapopulation ecology ecology and and landscape landscape ecology ecology have have largely largely adhered adhered to to their their own own research research tra traditions, to the extent extent that that shared key concepts concepts such as connectivity connectivity are used in aa different different manner manner (Tischendorf (Tischendorf and and Fahrig, Fahrig, 2000a; 2000a; Moilanen Moilanen and and Hanski, Hanski, 200 1 ) . Nonetheless, 2001). Nonetheless, many landscape landscape ecologists ecologists have the same goal in their research spatial structure research as as we we do, do, to to take take account account of of the the influence influence of of the the spatial structure of of the other processes processes (Turner 1 ) . So the landscape landscape on on population population and and other (Turner et et aI., al., 200 2001). So far, far, most of these on simulations Chapter 2). SMT rep most of these studies studies have have been been based based on simulations ((Chapter 2). SMT represents aa coherent coherent body resents body of of theory theory that that should should go go aa long long way way in in answering answering the the needs of those landscape needs of those landscape ecologists ecologists who who are are interested interested in in population population and and metapopulation processes. metapopulation Second, spatially spatially realistic meta population models metapopulation models are closely related to matrix matrix population population models models for for age-structured age-structured and and size-structured size-structured populations populations ((Caswell, Caswell, 200 1 ) . As 2001). As aa matter matter of of fact, fact, much much of of the the mathematical mathematical theory theory is is the the same Ovaskainen and 1 ) . Just same ((Ovaskainen and Hanski, Hanski, 200 2001). Just as as traditional traditional matrix matrix models models divide divide populations populations into age classes, SMT SMT divides metapopulations metapopulations into individual individual habitat habitat patches. patches. Important Important mathematical mathematical similarities similarities include include the the role role of of the the dominant dominant eigenvalue eigenvalue and and eigenvector eigenvector of of the the respective respective population population matrices matrices in in determining population growth persistence. There determining population growth rate rate and and persistence. There are are also also significant significant differences. differences. For For example, example, in in SMT, SMT, transitions transitions are are possible possible between between any any pairs pairs of of "classes", "classes", whereas whereas in in age-structured age-structured models, models, individuals individuals move move from from one one class class to to another another in in aa predictable predictable manner. manner. Another Another difference difference is is in in the the manner manner in in which which density density dependence dependence typically typically enters enters into into the the models. models. Age-structured Age-structured models drives intraspecific models assume assume that that the the effective effective population population density density that that drives intraspecific competition competition (e.g., by increasing mortality) is the overall density, calculated calculated as
4. 4. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
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the weighted sum of individuals in the different age classes. This simplification does models, where does not not apply apply to to spatially spatially structured structured models, where one one needs needs to to know know exactly which patches are occupied (local as opposed to global density) to cal calculate culate the colonization colonization rate rate of an empty patch. Third, SPOMs are closely related to epidemiological models (see (see Chapter 20 20 and references therein). The archetypal metapopulation metapopulation model, the Levins model, is identical to a basic epidemiological model, the susceptible-infected-susceptible (SIS) (SIS) model. model. Ovaskainen Ovaskainen and and Grenfell Grenfell (2003) (2003) utilized utilized the the correspondence correspondence between metapopulation and between metapopulation and epidemiological epidemiological models models in in using using the the patch patch 4.4 in the study of the effectiveness of various value measures discussed in Section 4.4 intervention intervention scenarios for sexually transmitted transmitted diseases. Further cross crossfertilization of these two populations two fields has much much potential, potential, including meta metapopulations inhabiting dynamic landscapes versus disease transmissions in dynamically changing changing contact contact networks, networks, the the evolution evolution of of migration migration rate rate and and other other life life history history traits versus the evolution of virulence, and predator-prey population mod predator-prey meta metapopulation models versus SIR models. With such links as described earlier developing among meta population metapopulation ecology, landscape ecology, epidemiology, and general population population ecology, a new era of exciting research is in the horizon.
sdfsdf
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A PPLICATION OF APPLICATION OF STOCH ASTIC PATCH STOCHASTI C PATC H OCCUPANCY OCCUPANCY MODELS TO RE AL MODELS TO REAL META PO PULATIONS M ETAPO PU LATIO N S Rampal S. Etienne, Cajo J.F. J.F. ter Braak, and Claire C. Vos
5 .1 5.1
INTRODUCTION INTRODUCTION Consensus has emerged about about the generally detrimental effects of habitat loss populations (Hanski, loss and and habitat habitat fragmentation fragmentation on on the the viability viability of of meta metapopulations (Hanski, 11999). 999). Habitat Habitat loss and fragmentation both affect the balance between extinc extinction populations and recolonization of tion of of local local populations and recolonization of empty empty patches: patches: patch patch occupancy occupancy decreases and the meta population is more prone to extinction. Simple metapopulation Simple statistical models models for for patch patch occupancy occupancy have have been been used used to to ascertain ascertain the the existence existence of of these these detrimental detrimental effects effects in in real real metapopulations metapopulations (Van (Van Dorp Dorp and and Opdam, Opdam, 1987; 1987; Merriam, 11988), 988), but their predictive power to show the magnitude of these effects limited, as incorporate the effects is is very very limited, as they they do do not not incorporate the mechanisms mechanisms underlying underlying metapopulation dynamics ((local local extinction and colonization) colonization).. The Levins ((1969, 1969, 11970) 970) model, the prototype metapopulation model that is based on these these mechanisms, mechanisms, is is biologically biologically too too unrealistic, unrealistic, whereas whereas size-structured size-structured and and individual-based metapopulation models are mostly too complex to be para parameterized with with available data. These data almost never consist of accurate
Ecology, Ecology, Genetics, Genetics, and and Evolution Evolution of of Metapopulations Metapopulations
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estimates of sizes. At best, they comprise observations estimates of population population sizes. At best, they comprise observations of of patches patches in in aa patch being occupied occupied or single year patch network network being or empty empty for for aa single year (snapshot (snapshot data; data; consecutive, years (which give Hanski, Hanski, 1994) 1994) or or for for several, several, not not always always consecutive, years (which give infor information mation of of the the population population turnover). turnover). There There is, is, however, however, one one class class of of metapopulation metapopulation models models that that can can both both capture biological detail easily parameterized capture sufficient sufficient biological detail and and be be relatively relatively easily parameterized with with these data: data: the patch occupancy see Chapter these the stochastic stochastic patch occupancy models models (SPOMs, (SPOMs, see Chapter 4). 4). Given Given the the current current occupancy occupancy states states of of all all patches patches in in aa patch patch network, network, SPOMs SPOMs predict predict for for each each patch patch the the probability probability that that it it will will be be occupied occupied at at any any time time in in the local extinction the future. future. These These probabilities probabilities depend depend on on the the probabilities probabilities of of local extinction colonization, which, in turn, turn, may be constant constant (resulting in a stochastic and colonization, Levins depend in variety of Levins model) model) or or depend in aa variety of ways ways on on the the quantities quantities that that are are con considered relevant, relevant, such quality and (or spatial sidered such as as patch patch quality and connectivity connectivity (or spatial cohesion; cohesion; Opdam Opdam et et aI., al., 2003 2003).). Perhaps Perhaps the the simplest simplest such such quantities quantities are are the the area area of of the the patch patch and and the the distance distance between between patches, patches, but but other other quantities quantities are are sometimes sometimes preferable because because area preferable area and and interpatch interpatch distance distance are are not not always always the the best best pre predictor 1b). This dictor variables variables (Thomas (Thomas et et aI., al., 200 2001b). This chapter chapter uses uses as as an an example example aa model model for for aa tree tree frog frog (Hyla (Hyla arborea arborea)l metapopulation metapopulation with with area area and and interpatch interpatch distance as well two predictor variables variables (Vos distance as well two other other predictor (Vos et et aI., al., 2000) 2000).. For For another another 1996). example, ai. ((1996). example, see see Hanski Hanski et et al. Application Application of of aa SPOM SPOM to to aa real real metapopulation metapopulation consists consists of of four four parts. parts. First, First, patch SPOM must patch occupancy occupancy data data must must be be collected. collected. Second, Second, the the SPOM must be be formulated. formulated. Here Here questions questions such such as as "what "what processes processes do do II want want to to describe describe and and what what vari variables do ables do II need need and and what what processes processes and and variables variables can can II dispense dispense with?" with?" must must be be answered. mathematical representation answered. Third, Third, given given the the mathematical representation of of the the SPOM SPOM that that results model must parameterized using results from from the the second second step, step, the the model must be be parameterized using the the dataset dataset at parameters. Fourth, at hand hand and/or and/or independent independent information information about about these these parameters. Fourth, pre predictions model by scenarios. The reli dictions can can be be made made with with the the model by considering considering different different scenarios. The reliability of ability of these these predictions predictions must must be be assessed assessed through through uncertainty uncertainty analysis. analysis. This assumes the SPOM to This chapter chapter assumes the dataset dataset and and the the SPOM to be be given given and and concen concentrates trates on on the the third third part part by by reviewing reviewing the the various various methods methods that that have have been been developed SPOMs using developed to to parameterize parameterize SPOMs using snapshot snapshot and/or and/or turnover turnover data. data. It It then focuses on some of still remaining, their consequences then focuses on some of the the problems problems still remaining, and and their consequences for populations; these for the the fourth fourth part, part, making making model model predictions predictions for for real real meta metapopulations; these problems are frog example. example. We We problems are illustrated illustrated with with the the afore-mentioned afore-mentioned tree tree frog show to determine show that that the the model model predictions predictions are are useful useful to determine the the best best conservation conservation strategies metapopulation. For strategies for for the the real real metapopulation. For this this we we compare compare the the distribution distribution pattern 2002 with pattern in in 2002 with the the distribution distribution pattern pattern predicted predicted from from the the model model cali cali1-1983 and brated brated with with data data from from 198 1981-1983 and 1986. 1986. The The chapter chapter ends ends with with aa dis discussion application of cussion of of the the four four parts parts in in the the application of SPOMs SPOMs and and an an overview overview of of the the insights application of insights gained gained by by the the application of SPOMs SPOMs as as reported reported in in the the literature. literature.
5.2 5.2
STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS Stochastic Stochastic patch patch occupancy occupancy models models have have been been formulated formulated in in discrete discrete time time 994) and ((Day Day and Possingham, 1995; and Possingham, 1995; Hanski, Hanski, 11994) and continuous continuous time time (Verboom (Verboom 998; Etienne, et 9 9 1 ; Frank Frank and et aI., al., 11991; and Wissel, Wissel, 11998; Etienne, 2002; 2002; Etienne Etienne and and
5. 5.
11 007 1
STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS STOCHASTIC
Nagelkerke, 2002; 2002; Chapter Chapter 4). 4). Because Because census census data data are are usually usually separated separated by by Nagelkerke, or several several years, years, itit seems seems natural natural to to use use aa discrete-time discrete-time model model with with aa time time 11 or step of of i1 year; year; in in this this way way difficulties difficulties with with extinction extinction and and colonization colonization probprob step abilities not not being being constant constant during during the the year year can can be be avoided. avoided. Consider Consider now now aa abilities patch network network of of N N patches. patches. If If aa patch patch ii contains contains aa population, population, this this popupopu patch lation can go go extinct extinct in in one one time time step probability Ei, Ei, and and if if the the patch patch lation can step with with probability is empty, empty, it it can can be be colonized colonized with with probability probability Ci Ci (Ei (Ei and and Ci Ci are are denoted denoted by by is Exti and and COli in Chapter Chapter 4, 4, but but we we adhere adhere to to the the symbols symbols introduced introduced here here Exti Coli in for better comparison comparison with with the the literature literature cited). cited) . Let Let us us denote denote the the state state of of for better at time time tt by by X/(t), Xi(t), which is aa binary binary variable: variable: we we have have for for an an occuoccu patch patch ii at which is = 1 1 and and for an empty empty patch patch Xi(t) Xi(t) == 00 (Xi (Xi is is denoted denoted by by Oi 0i pied patch patch Xi(t) Xi(t) = pied for an in total state state of in Chapter Chapter 4). 4 ) . The The total of the the metapopulation metapopulation at at time time t,t, X(t), X(t), can can then then N ones ones and and zeros. zeros. The The colonization colonization be described described by by aa vector vector containing containing N be at time time tt + + 11 usually usually depends depends on on the the state state X(t) X(t) probability of of patch patch ii at probability because this this state state determines determines the the number number of of dispersers dispersers being being sent sent out that because out that may such as patch i,i, but may end end up up in in patch but not not on on earlier earlier states states such as X(t X(t - 11). ) . The The extincextinc tion probability probability is considered to be independent independent of X(t), but but when when tion is usually usually considered to be of X(t), there is Brown and and Kodric-Brown, Kodric-Brown, 1977) 1 977) populations on on there is aa rescue rescue effect effect ((Brown - - populations the brink brink of are rescued extinction by by immigrants immigrants i- the the the of extinction extinction are rescued from from extinction extinction on the probability and thereby extinction probability probability depends depends on the colonization colonization probability and thereby on X(t). on X(t). The the SPOM by aa single single formula, formula, valid valid The dynamics dynamics of of the SPOM can can thus thus be be described described by for each each patch patch i, i, that that specifies specifies the the probability the state of patch patch ii at time for probability of of the state of at time tt + the meta population at at time time t: + 11 conditional conditional on on the the state state of of the metapopulation t: - Ei
P[Xi(t + 1)IX(t)]
=
li i
i,ci
Ci
if if Xi(t) Xi(t)== if if Xi(t) Xi(t) = = = if if Xi(t) Xi(t) = if if Xi!t) Xi( t) = =
11 and Xi(t + and Xi(t + 11 and and Xi(t Xi(t + + 00 and Xi(t + and Xi(t + 00 and and Xi(t Xi( t +
11 )) == 11)) = = 11)) = = 11)) = =
11 00 00 11
((s.1) 5. 1 )
in in which which Ei E i and and Ci C i are a r e shorthand shorthand for for EM) Ei(t ) and and Ci(t), Ci(t), the the extinction extinction and and colonization colonization probabilities probabilities for for the the transition transition from from time time tt to to time time tt + + 11.. These These may may depend depend on on species species and and landscape landscape characteristics characteristics that that can can be be treated treated either either as as observed observed variables variables or or known known or or unknown unknown parameters parameters in in the the model. model. Because Because SPOMs SPOMs possess possess the the Markov Markov property property that that the the probability probability distri distribution bution for for the the state state at at time time tt + + 11 is is completely completely determined determined by by the the state state at at time time t, t, Markovian Markovian theory theory can can be be invoked invoked to to state state various various characteristics characteristics of of the the metapopulation. metapopulation. One One such such characteristic characteristic is is that that the the metapopulation, metapopulation, left left undisturbed population not undisturbed and and conditional conditional on on the the meta metapopulation not having having gone gone extinct extinct untimely, untimely, will will settle settle in in aa pseudoequilibrium pseudoequilibrium in in which which the the probability probability that that it it is in in aa certain certain state state (e.g., (e.g., all all patches patches occupied) occupied) no no longer longer changes changes in in time. time. This This is pseudoequilibrium pseudoequilibrium is is also also called called the the quasistationary quasistationary state state (see (see further further Darroch Darroch and 965; Gyllenberg 994; Gosselin, 998; and Seneta, Seneta, 11965; Gyllenberg and and Silvestrov, Silvestrov, 11994; Gosselin, 11998; Ovaskainen, Ovaskainen, 2001; 2001; Chapter Chapter 4). 4). This This is is aa property property used used frequently frequently in in param parameterization eterization methods, methods, as as discussed discussed later. later. We We will will first first give give aa few few examples examples of of SPOMs. SPOMs.
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
08 1 08
Hanskl's Hanski's Incidence Incidence Function Function Model Model In 1 994) and In probably probably the the best-known best-known SPOM, SPOM, described described by by Hanski Hanski ((1994) and better better known incidence function known as as the the incidence function model model (IFM), (IFM), the the extinction extinction probability probability is is set set proportional to proportional to the the inverse inverse of of patch patch area: area:
)
(5.2) (5.2)
ex Ei = min 1, A-~.
where where ee and and x x are are parameters parameters (e is is sometimes sometimes written written as as A5 A~ where where Ao A0 is is the the minimum minimum possible possible patch patch area area in in which which aa viable viable population population can can exist, exist, x x is is 4) and denoted denoted by by �~ex in Chapter Chapter 4) and the the colonization colonization probability probability is is aa saturating saturating ex in function measure S: function of of connectivity connectivity measure S: (5.3) (5.3)
Ci-- $2i + y2
where where y y is is aa parameter parameter and and Si Si is is given given by by aa sum sum of of contributions contributions from from all all occupied occupied patches patches weighted weighted by by their their distance distance to to patch patch i:i: Si(t) = ~.~ j ~ i X j ( t ) A i e x p ( - ~ d i
((5.4) 5 .4)
i)
where where ex e~ is is aa parameter parameter that that can can be be interpreted interpreted as as the the inverse inverse of of the the mean mean dispersal 19 94) dispersal distance distance of of the the species species under under consideration. consideration. This This is is Hanski's Hanski's ((1994) model model without without the the rescue rescue effect. effect. Incorporating Incorporating the the rescue rescue effect effect means means multiplying multiplying the the extinction extinction probability probability by by 11 - Ci• Ci. In In later later work, work, Hanski Hanski and and co-workers co-workers added added the the area area of of the the destination destination patch patch Ai A i to to the the connectivity connectivity and and parameters parameters that that determine determine the the strength strength of of the the contribution contribution of of the the patches patches of of origin 996; Hanski, 998a,b; Moilanen origin and and destination destination (e.g., (e.g., Wahlberg Wahlberg et et aI., al., 11996; Hanski, 11998a,b; Moilanen and 998; Chapter Moilanen and and Hanski, Hanski, 11998; Chapter 4). 4). See See Moilanen and Nieminen Nieminen (2002) (2002) for for aa discussion Formulas for extinction discussion of of different different measures measures of of connectivity. connectivity. Formulas for the the extinction and some mechanistic basis as and colonization colonization probabilities probabilities have have some mechanistic basis as pointed pointed out out by by Hanski 1 998a,b). Hanski ((1998a,b). The 1 994) derived The model model is is called called IFM IFM because because Hanski Hanski ((1994) derived an an incidence incidence function colonization probabilities. function from from the the extinction extinction and and colonization probabilities. For For this, this, he he first first defined defined the the incidence incidence ]; Ji of of patch patch ii as as the the probability probability that that it it is is occupied. occupied. He He then then assumed population is quasistationary state, assumed that that the the meta metapopulation is in in the the quasistationary state, which which he he interpreted time. This interpreted as as ]; Ji being being independent independent of of time. This leads leads to to -
J.1 == J·(l Ji Ji(1 - E Ei)1 ) + + ((11 - J J·)G i )1 C i1 ==> ~ JJ.i1 == 1 -
-
c Ci 1
Ci Ci + + E Eii
(5.5) (,.5.:5)
For For example, example, for for the the IFM IFM with with rescue rescue effect effect [but [but without without the the cutoff cutoff at at 11 of of Ei Ei in Eq. Eq. (5.2)], (5.2)], the the incidence incidence function function becomes becomes in 1
Ji =
ey2 1+
Axs 2
(5.6) (5.6)
5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS 5.
1109 09
The The Extended Extended IFM IFM for f o r Tree Tree Frog Frog Data Data Vos et al. ai. (2000) extended the IFM to model a tree frog metapopulation metapopulation in aa pond southwest The pond network network in in southwest The Netherlands, Netherlands, for for which which occupancy occupancy data data have 98 1-1983 and 986 (see (see Box . 1 ) . In model, the have been been collected collected in in 11981-1983 and 11986 Box 55.1). In their their model, the form replacing the form of of the the function function for for colonization colonization probability probability is is relaxed relaxed by by replacing the exponent (as already 994) and exponent of of 2 2 by by parameter parameter Zz (as already suggested suggested by by Hanski, Hanski, 11994) and by by allowing allowing the the contribution contribution of of patch patch area area in in the the connectivity connectivity measure measure to to be 996; bb is be determined determined by by parameter parameter bb (Wahlberg (Wahlberg et et aI., al., 11996; is denoted denoted by by �~em in em in
BOX 5.1 The Tree Frog Metapopulation in Zealand Flanders, The Netherlands The tree frog metapopulation exists in an agricultural landscape in southwest The Netherlands (Fig. B5. 1 ). The suitable habitat forms approximately 1 .5% of the total landscape cover and is separated by intensively used agricultural fields that are unsuit able for the species. (Semi)natural vegetation can be found on dikes, in coastal sand dunes, and in meadows with cattle-drinking ponds, the aquatic habitat used by the tree frog. The terrestrial habitat of the tree frog consists of shrubs, bushes, and vegetation of high herbs. The tree frog distribution has declined sharply in The Netherlands since the 1 950s as a result of intensification of agriculture. Most (semi)natural elements such as cattle-drinking ponds and hedgerows have been cleared. Zealand Flanders
ftAst
•
o
4
IO km
L' _ --'_ ---' ,
Fig. 85.1
(2000).
The position of the tree frog study area. Reprinted with permission from Vos et al.
11 11 00
RAM PAL S. RAMPAL S. ETIENNE ETIENNE ET ET Al. AL.
From 1 98 1 until 1 986 the distribution pattern of the tree frog in Zealand Flanders was monitored. Statistical analysis showed that the occupation probability of a pond increased with the number of other ponds and. the a rea of high herbs and bushes in the surrounding (Vos and Stumpel, 1 996). Analysis of turnover data from 1 98 1 to 1 983 showed that local extinctions were related to spatial features of the landscape: extinction probability decreases with pond size (Vos et a I ., 2000). Pond size encom passes both pond area and suitable terrestrial habitat within a radius of 250 m around the pond . Also, colonization events were correlated to connectivity as well as habitat quality factors. Comparing dispersal distances with distances between ponds shows that the habitat network is still connected by dispersing individuals (Fig. B5 .2). Analysis of dispersal events showed that occupied ponds were preferred over empty
.-
, '.
"
t
, , . .. "
"
� " '.
": : ":-
, ,
,
"
,
,
"
"
" ,
, .'
, ..
,
-
:
"
.
• •
#
.
. . i::,:
# .' -.
, .. '
.
.
.
.. ... "
roT; , -;,!..!; ,:;-"� "': 4...A .•
-:' , --,.
pond
dispersal between ponds
land habitat
•
4 km
»
Fig, 85,2 Overview of dispersal events registered by capture-recapture techniques in Zealand Flanders during the period 1 98 1 -1 989. Reprinted with permission from Vos et al. (2000).
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1111 11
ponds, although many empty ponds were present at close ranges. This implies the importance of conspecific attraction and a rescue effect in this metapopulation pond system. In the next step toward spatially explicit connectivity measures the permeability of the landscape matrix is incorporated. Results of a radio telemetry study 0/os, 1 999), show that moving tree frogs have a high preference for hedgerows, that they avoid arable land, and that pasture takes an intermediate position. Incorporating these char acteristics in the connectivity measure of the extended IFM model will improve the model further. The results seem to justify the application of the metapopulation concept for guide lines for optimal spatial habitat configuration for this species. The protection policy plan for the tree frog in The Netherlands is actually based on meta population concepts. Apart from the necessary improvements of habitat quality, the size and connectivity of the habitat networks of the tree frog are being improved (Crombaghs and Lenders, 2001 ). Results for a tree frog meta population in Twente, a region in the east of The Netherlands, already show that the species reacts positively to these restoration meas ures (Braad, 2000).
Chapter Chapter 4). 4). Furthermore, Furthermore, three three variables variables were were introduced: introduced: B;j Bii,' H Hi,i, and H H2, 2,ii.' I ,;, and describes the the quality of the the matrix matrix habitat habitat between and j, j, and describes quality of between patches patches ii and and its value was all combinations = 00 for for aa total total barrier, barrier, its value was set set for for all combinations of of ii and and jj (e.g., (e.g., Bij = water Bij Bii == 11 for for no no barrier, barrier, and and B;j Bii == 0.5 0.5 for for aa semibarrier). semibarrier). HI,i Hl,i isis the the water conductivity, percentage cover cover of both conductivity, and and H H2,i the percentage of the the water water vegetation; vegetation; both 2,; isis the were each pond. deter were measured measured for for each pond. Two Two additional additional parameters, parameters, q qll and and q q2, 2 , deterextinction and mine Hl,i and H2,i respectively. respectively. The The extinction and colonization colonization mine the the effect effect of of H I ,i and probabilities probabilities are are now now given given by by
Bij
(
)
eH 1,i ql) eHt 1
E 1 ,, ax , Ei1· = = ((11 - C)min Ci)min/1 1 AX 1
__
(5.7) (5.7)
and and
sf CCii == -----'-- y Szi ++ HYq2 sT q2,i H22,i
(5.8) (5.8)
with with
SSi(i (t)t) = L ~ Xj(t)A Xj( t)A~Bq e x p (-ad - (xd/j) fB ij exp( ij) jjri=- i
(5.9) (5.9)
(Note (Note that that y y in in this this model model is is not not the the same same as as y y in in Hanski's Hanski's IFM.) IFM.) We We mention mention this this model, model, not not only only as as an an example example of of aa SPOM SPOM with with other other variables variables than than patch patch area area and and interpatch interpatch distance, distance, but but also also because because it it is is used used as as an an illustration illustration of of some some of of the the problems problems in in parameter parameter estimation. estimation.
11 11 22
5.3 5.3
RAMPAL S. ETIENNE RAMPAL S. ETIENNE ET ET AL. AL.
METHODS METHODS TO TO ESTIMATE ESTIMATE PARAMETERS PARAMETERS OF OF STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS Because Because SPOMs SPOMs contain contain few few parameters, parameters, it it is is attractive attractive to to attempt attempt to to esti estimetapopulation data, that that is, data data on the occupancy occupancy of some mate them using meta population data, or years. Several or all all of of the the patches patches in in aa network network for for 11 or or several several years. Several methods methods have have for this purpose. They use only data data on transitions been developed for transitions between occupancies colonizations, noncolonizations), occupancies (extinctions, (extinctions, nonextinctions, nonextinctions, colonizations, noncolonizations), data data from a snapshot year (which typically shows spatial clusters of snapshot of a single year shows spatial occupied occupied patches patches as as the the result result of of the the history history of of extinctions extinctions and and colonizations colonizations patch network), network), or both. This section section reviews these methods. methods. Table 55.1 in the patch .1 summarizes summarizes their their main main features and and shortcomings. shortcomings.
Hanski 1 994): Regression Hanski ((1994): Regression of of Snapshot Snapshot Data Data In his classic paper, Hanski ((1994) 1 994) not not only introduced introduced the IFM, but but also showed showed how how it it can can be be parameterized parameterized in in aa simple simple way way using using snapshot snapshot data data of of patch presented by Hanski ((1991) 1991 ) patch occupancies, occupancies, elaborating elaborating ideas ideas presented by Peltonen Peltonen and and Hanski and Hanski Hanski ((1992). 1 992). He proposed proposed nonlinear regression to obtain obtain point point estimates nonlinear regression parameters, with with standard errors, by maximizing maximizing the logarithmically logarithmically of the parameters, transformed likelihood, i.e., the loglikelihood, loglikelihood, transformed
E[Xi lnJi +
(1 -
Xi)ln(1 - Ji)]
(5.10) (5.10)
l
Note Note from Eq. (5.6) that that in the IFM without without rescue effect, e and and y form a single composite parameter so that they cannot be estimated separately composite parameter that cannot separately from snapshot data. Independent Independent information information on the minimum minimum sustainable patch snapshot size separate ee and size Ao A0 can can help help to to estimate estimate ee and and thus thus separate and y. y. out that that Eq. (5.6) ( 5 . 6 )- the IFM with with rescue Ter Braak et al. ((1998) 1 998) pointed out effect linearized by logit effect and and with with all all patches patches larger larger than than Ao A 0- can can be be linearized by the the logitIi giving transformation of transformation of Ji giving 10g log
0g( Si ) Ji)) == �o [30 + �l ~311og(Ai)+ 132 log( Ai ) + � (� 2 1log(S/) li~ji -j;
(5.11) (5.1 1)
with - log(ey2 ), �131 1= with �o 130 = =-log(ey2), = x, x, and and � ~ 22 = z 2 2.. Therefore, Therefore, available available data data being being binary values, point point estimates parameters (with approximate standard binary values, estimates for for the the parameters (with approximate standard errors) can be obtained obtained easily by using logistic regression (McCullagh (McCullagh and NeIder, - at Nelder, 1989) 1989) of of the the occupancies occupancies Xi on on 10g(Ai) log(Ai) with with offset offset 210g(Si) 21og(Si)at least least when when the the values values of of a cx and and bb in in the the definition definition of of Si Si are are given. given. (An (An offset offset is is aa . ) Although predictor with a regression coefficient coefficient of 11.) Although this is mathematically mathematically equivalent equivalent to Hanski's nonlinear nonlinear regression, it is computationally computationally much much more more efficient. It is historically 1 994) already historically interesting interesting that that Eber and Brandl ((1994) performed occupancies without performed such such logistic logistic regression regression of of patch patch occupancies without knowing knowing the link to the incidence function function model (which had had not not been published yet); they they just just used used aa statistical statistical model. model. The The extended extended IFM IFM can can be be fitted fitted to to snapshot snapshot data data using using aa logistic logistic regression regression of of occupancies occupancies Xi on on 10g(Ai), log(A/), 10g(Si), log(S/), log( log(Hi,i) H1,i) a and b by logistic regression by calculating and ,i)' We can estimate and log(H log(H2,i). We can estimate oL and b by logistic regression by calculating 2
>,
e-
0
0 E
2~ 0
0
0
Turnover only
No (ignored)
Yes
Logistic regression
Likelihood optimization (non-linear regression)
Likelihood optimization (MC simulation)
<
Approximations of standard errors
Approximations of standard errors
Approximations of Full uncertainty confidence intervals distribution
Full uncertainty distribution
Z
No
No
No
Z
Yes
Yes
Fast / very fast
Fast
Fast
Slow
Very slow
Slow
gO
Ter Braak and Etienne (2003) i_
Not at all (turnover only) Yes
o~
Spatial structure and turnover Correctly Yes
~.j r
. ~0
0..~
0
Bayesian MCMC (MetropolisHastings) 9 ,.~
.
~
r
r
c,.) .~,
0 .,,~
9- ~ 9- ~
0 .,.~
.~
9
9-0
Bayesian MCMC (Gibbs sampling)
9
0 0
O'Hara et al. (2002) s_
O~
0
- - ~ o~
Z
z-'~
~
Not at all (turnover only) No
0
r
Turnover only
m
O~
o l
Z
~
r ~.~
r
o
.,..~
0""
~ ~
0
r
~
0
No (ignored/ considered empty) Likelihood optimization (nonlinear regression)/ with rescue effect: logistic regression Approximations of standard errors No
oO O~ O~
Spatial structure and turnover Correctly
Spatial structure only Approximation
r
o
Is uncertainty analysis of model predictions possible? Computational efficiency?
E
Spatial structure and turnover Approximation
What type of information is used? How is quasistationarity incorporated? Are missing data taken into account?
How is uncertainty in estimates specified?
E
Moilanen (1 999)
Hanskl (1994)
Method (reference)
What method are parameter estimates based on?
,1,,,o
Ter Braak et al. (1 998)/ Vos et al. (2000)
~
o9~ =
Verboom et al. (l99l)/SjtigrenGulve and Ray (1996)
. m
0
,i,,,o
0 c-
4-o
. m
x
4-o
0
i1
0
0
o~
Overview o f Features o f Existing Methods t o Estimate Parameters o f SPOMs from Occupancy Data
u~
...1
TABLE 5.1
9
o0
0
0
113
. . IN
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
4 1 114
the regression over values of locating where the regression over aa grid grid of of values of ex 0~ and and bb and and by by locating where the the log log likelihood [Eg. (5.1 0)] is likelihood [Eq. (5.10)] is maximum. maximum. There 998; Ter There are are five five caveats caveats in in the the method method (Ter (Ter Braak Braak et et aI., al., 11998; Ter Braak Braak and and Etienne, Etienne, 2003 2003).). First, First, the the condition condition of of nonextinction nonextinction of of the the metapopulation metapopulation (required (required for for quasistationarity) quasistationarity) is is omitted omitted in in Eq. Eq. (5.5 (5.5).) . This This is is perhaps perhaps aa valid valid approximation approximation for for large large networks networks that that are are unlikely unlikely to to go go extinct extinct quickly, quickly, but but because population models because meta metapopulation models are are often often used used to to explore explore scenarios scenarios intended intended to threatened meta populations, this need not to conserve conserve threatened metapopulations, this need not always always be be the the case. case. Second, Second, it it is is assumed assumed in in Eq. Eq. (5.5) that that the the extinction extinction and and colonization colonization prob probabilities abilities of of patch patch ii are are constant constant in in time. time. This This is is not not true, true, for for they they both both depend depend on on the the evolving evolving states states of of the the other other patches. patches. Third, Third, the the connectivity connectivity S S is is calculated calculated using using the the patch patch occupancies occupancies of of the the same same year year and and not not using using the the patch patch occupancies occupancies of of the the previous previous year year as as it it should should according according to to the the model model [Eq. reason is data. [Eq. (5.4)]; (5.4)]; the the reason is of of course course that that these these are are unknown unknown for for snapshot snapshot data. Fourth, likelihood [Eq. [Eq. ((5.10)], 5 . 10)], patch Fourth, in in the the log log likelihood patch occupancies occupancies are are assumed assumed to to be be statistically statistically independent. independent. However, However, this this assumption assumption is is not not met met because because the the patch patch occupancies occupancies are are regressed regressed on on the the connectivity connectivity SS that that is is calculated calculated from from the same data. 5 . 1 0 ) is best aa pseudo-log the same data. Therefore, Therefore, Eq. Eq. ((5.10) is at at best pseudo-log likelihood. likelihood. Fifth, Fifth, in in the the derivation derivation of of the the incidence incidence function function Eq. Eq. (5.6), the the cutoff cutoff at at 1I of of Ei Ei in Ao. in Eq. Eq. (5.2 (5.2)) is is neglected. neglected. This This is is warranted warranted only only if if all all areas areas are are larger larger than than A0. Missing Missing data data (occupancies (occupancies are are unknown unknown for for some some patches) patches) cannot cannot be be dealt dealt with with properly properly in in this this method. method. They They are are basically basically ignored. ignored. It It is is apparent apparent from from Eq. Eq. (5.4) (5.4) that that this this may may affect affect the the connectivity. connectivity. By By ignoring ignoring patches patches of of which which no no data data are are available, available, one one effectively effectively assumes assumes them them to to be be empty empty because because they they do contribute to do not not contribute to connectivity. connectivity. If If the the patches patches were were actually actually occupied, occupied, the the colonization colonization probability probability will will be be underestimated. underestimated.
Verboom al. (1 991 ), Sjogren-Gulve and Ray 996), Verboom et et al. (1991), Sj6gren-Gulve and Ray (1 (1996), and Eber Eber and and Brandl 996): Regression and Brandl (1 (1996): Regression of of Turnover Turnover Data Data While While Hanski Hanski ((11994) 9 9 4 ) oonly n l y considered considered snapshot snapshot data, data, Verboom Verboom et et ai. al. ((1991), 1 991 ), Sjogren-Gulve 1 996), and, Sj6gren-Gulve and and Ray Ray ((1996), and, less less commonly commonly cited, cited, Eber Eber and and Brandl 1 996) looked 1 99 1 ) Brandl ((1996) looked only only at at turnover turnover data data [actually, [actually, Verboom Verboom et et ai. al. ((1991) also also looked looked at at the the frequency frequency that that aa patch patch is is found found occupied, occupied, see see later]. later]. They They realized that 5 . 1 ) are realized that Ei and and Ci in in Eq. Eq. ((5.1) are conditional conditional probabilities probabilities that that can can be be fit fitted logistic regression regression (McCullagh ted to to data data by by logistic (McCullagh and and Nelder, Nelder, 1989). 1989). To To parame parame)] with terize possible pairs Xj(t + 11)] terize Ei Ei,, they they created created aa dataset dataset with with all all possible pairs [X;(t), [Xi(t), Xi(t with and then then applied applied logistic logistic regression regression with with Xi(t Xi(t + 11)) as as the the response response Xi(t) = 11 and variable. Similarly, variable. Similarly, Cj Ci was was parameterized parameterized by by applying applying another another logistic logistic regres regression all possible )] with Xj(t + 11)) as sion to to all possible pairs pairs [Xj(t), [Xi(t), Xi(t + 11)] with X;(t) Xi(t) = = 00 and and with with Xi(t as the the response response variable. variable. They They thus thus assumed assumed that that the the probabilities probabilities of of extinction extinction and and colonization colonization behave behave logistically: logistically: Ei Ei = --
1
----1 + exp( - Ue,i )
1 + exp(-ue,i) 11 Cj Ci = -- ---11 + + eexp( xp(-u cUc,i , i))
(5.12) (5.12)
5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS 5.
11 11 55
where where UUee and and UUcc are are linear linear functions functions of of the the variables variables of of importance, importance, such such as as patch area area and and connectivity, connectivity, the the logarithm logarithm of of which which are are treated treated as as explanatory explanatory patch variables variables in in logistic logistic regressions. regressions. The The parameters parameters in in Ue Ue and and UUcc are are fitted fitted from from the the ""extinction extinction dataset" dataset" and and the the "colonization "colonization dataset," dataset," respectively, respectively, giving giving point point estimates estimates and and standard standard errors. errors. They They found found that that extinction extinction is is significantly significantly related related to to patch patch area area (and (and not not connectivity) connectivity) and and colonization colonization to to connectivity connectivity area).. ((and and not to area) Sjogren-Gulve 1 996) used Sj6gren-Gulve and and Ray Ray ((1996) used point point estimates estimates in in subsequent subsequent computer computer simulations simulations of of the the discrete-time discrete-time SPOM SPOM to to predict predict the the long-term long-term trend trend in in 1991 ) had occupancy, occupancy, whereas whereas 55 years years earlier earlier Verboom Verboom et et al. al. ((1991) had missed missed that that opportunity, opportunity, partly partly because because they they adhered adhered to to their their continuous-time continuous-time SPOM. SPOM. Note Note that that extinction extinction probabilities probabilities do do not not have have the the logistic logistic form form in in the the IFM IFM [Eq. [Eq. (5.2)] (5.2)] and and they they are are even even linked linked to to the the colonization colonization probabilities probabilities in in the the ((extended) extended) IFM IFM with with aa rescue rescue effect effect [see [see Eq. Eq. (5.7)]. (5.7)]. We We must must therefore therefore resort resort to nonlinear nonlinear regression regression (Vos (Vos et et aI., al., 2000) 2000) where where the the log log likelihood likelihood to to be be to maximized is
2: t+l 1 )))) InE ~ [X [ X ii((t)(1 t ) ( 1 -X - X i (it( + lnEii + + X X ii(( tt)X ) X ii(( tt+ + l )1l)nIn( ( 11 - E Ei)] i)]
i, (5.13) (5.~31 ++ � ~ [ ([(1 1 - X X ii(( tt))X ) ) X ii((tt + + l1)lnCi Xi(t))(1 X ii(( tt+ + l 1) )))l nln(1 (1 - Ci)] Ci)] ) lnCi ++ ((11 -- X i( t))(l -- X tI
Ter Ter Braak Braak et et al. al. (1998) (1998) showed showed that that this this is is equivalent equivalent to to maximizing maximizing the the extinction extinction and and colonization colonization parts parts of of the the log log likelihood likelihood separately, separately, provided provided Ei Ei and independent parameters, parameters, so and Ci Ci have have different, different, independent so giving giving aa formal formal justification justification approach by Verboom et al. Sj6gren-Gulve and Ray (1996). (1996). 1991 ) and Sjogren-Gulve of the approach al. ((1991) Although this method is theoretically sound are no technical difficulties Although this method is theoretically sound (there (there are no technical difficulties as using only turnover data has some some shortcomings. shortcomings. First, First, as with with snapshot snapshot data), data), using only turnover data has it requires data collection in at at least least 2, 2, but but preferably years. it requires data collection in preferably several several consecutive consecutive years. Second, while snapshot data many extinctions extinctions and coloniza Second, while snapshot data are are the the result result of of many and colonizations in tions in the the history history of of the the metapopulation metapopulation and and are are therefore therefore considered considered to to con contain information, turnover turnover data data provide provide little tain a lot of of information, little information information if if turnover turnover that data data show only extinctions and and colonizations colonizations [most inforis slow so ::hat only a few extinctions [most infor mation would be provided provided if the number number of mation would be if the of extinctions extinctions (cq. (cq. colonizations) colonizations) and and the were equal]. the number number of of nonextinctions nonextinctions (cq. (cq. noncolonizations) noncolonizations) were equal]. Third, Third, miss missing resulting in the same bias as ing data data are are again again ignored, ignored, resulting in the same bias as for for snapshot snapshot data. data.
Ter Ter Braak Braak et et al. al. (1998) (1998) and and Vos Vos et et al. al. (2000): (2000): Combining Combining the the Previous Previous Approaches Approaches To make make full full use use of of data, data, both both the the historical information contained To historical turnover turnover information contained in dataset and the turnover in the the first first year year of of aa dataset and the turnover information information in in the the following following (1991) not not only only applied applied years should should be be extracted. extracted. Interestingly, Interestingly, Verboom Verboom et et al. al. (1991) years logistic regression regression to to turnover turnover events events as as discussed discussed earlier, earlier, but but also also applied applied logistic logistic regression to to the the frequency frequency that that aa patch is found found occupied the logistic regression patch is occupied during during the years of of survey. survey. For For aa dataset dataset of of only only 11 year, year, the the latter latter is is formally formally equivalent equivalent years (1994). For For data data of of several several years, years, the the to the the snapshot snapshot data data analysis analysis of of Hanski Hanski (1994). to method also also takes takes turnover turnover events events into into account, account, but but extinction extinction and and colonizacoloniza method tion considered equal 1 1 1 000 sequence sequence is is considered equal to to 101010). 101010). tion are are not not separated separated (a (a 111000
11 11 66
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
Verboom 1991 ) must Verboom et et al. al. ((1991) must have have realized realized that that there there is is more more information information in in 1 994, 11996) 996) certainly aa dataset dataset than than only only turnover. turnover. Eber Eber and and Brandl Brandl ((1994, certainly realized realized this, approaches. this, but but they they did did not not combine combine the the two two approaches. It 1998) who It was was Ter Ter Braak Braak et et al. al. ((1998) who mentioned mentioned aa pragmatic pragmatic method method to to combine likelihood of combine the the two two previous previous approaches. approaches. For For this, this, the the likelihood of the the dataset dataset is incidence dataset" is partitioned partitioned in in three three parts: parts: the the ""incidence dataset" from from the the first first year year of of aa dataset, dataset, the the "extinction "extinction dataset," dataset," and and the the "colonization "colonization dataset" dataset" from from the the following (see also following years. years. The The likelihood likelihood to to be be maximized maximized (see also later) later) is is the the sum sum of 5 . 1 3 ), so of Eq. Eq. (5.10) and and Eq. Eq. ((5.13), so Hanski's Hanski's incidence incidence function function model model [Eq. [Eq. (5.6)] is used used for for the the first first year. year. This This combined combined approach, approach, first first applied applied in in Vos Vos et et al. al. is (2000), suffers suffers of of course course from from the the same same problems problems as as its its constituents: constituents: it it uses uses aa pseudo-likelihood pseudo-likelihood for for the the first first year year data, data, it it yields yields only only point point estimates estimates of of the the mean mean and and standard standard error error of of the the parameters, parameters, and and cannot cannot handle handle missing missing data data properly. properly.
Moilanen 999) : Monte Moilanen (1 (1999): Monte Carlo Carlo Simulation Simulation Moilanen 1 999) provided Moilanen ((1999) provided the the first first solution solution to to problems problems involving involving the the pseudo-likelihood based on pseudo-likelihood and and missing missing data data in in aa new new approach approach based on maximum maximum likelihood estimation using Carlo simulations. Because of likelihood estimation using Monte Monte Carlo simulations. Because of the the Markov Markov property SPOM, the probability of values property of of the the SPOM, the probability of aa dataset dataset X of of T years, years, given given values for for the the parameters parameters 0 O,, can can be be written written as as
P[X P[XO]=P[X(1)]P[X(2)IX(1)]...P[X(t+I)X(t)]...P[X(T)X(T-1) (5.14) . 14) [ 0 ] = P[X( 1 )]P[X( 2 ) [X( 1 )] . . . P[X( t + 1 ) [X( t ) ] . . . P[X( T) [ X( T- l ) ]] (5 with, with, for for each each year year t of of this this sequence, sequence,
PP[X(t+l)X(t)]=i-iP[Xi(t+l)X(t) [X( t + 1 ) [X( t ) ] = IT P [Xi( t + 1 ) [ X( t)]]
. 15) (5 (5.15)
I
because because the the states states are are independent independent conditional conditional on on the the state state of of the the system system in in the the previous between spatial in previous year. year. Note Note the the separation separation between spatial information, information, contained contained in P[X(1)], P[X(l )], and the turnover turnover information, contained in the remaining conditional probabilities, al. ((1998). 1 998). Equation probabilities, as as noted noted by by Ter Ter Braak Braak et et al. Equation (5.14) (5.14) is is the the true true likelihood that likelihood that needs needs to to be be maximized. maximized. Instead 1 999) approximated 1 )] by Instead of of using using Eq. Eq. (5.6), (5.6), Moilanen Moilanen ((1999) approximated P[X( P[X(1)] by Monte Monte Carlo Carlo simulation. simulation. From From an an arbitrary arbitrary state, state, the the IFM IFM is is simulated simulated until until the the quasi quasistationary stationary equilibrium equilibrium is is considered considered to to be be reached reached and and then then for for another another L L time time steps steps to to obtain obtain simulated simulated states states X Xuu ((uu =- L.L). 1...L). The The approximation approximation is is then then
1 u~lP[X ( 1 )]Xu]
(5.1 6) (5.16)
_
for some large large number for some number L. L. This This Monte Monte Carlo Carlo approximation approximation derives derives from from the the equation (Ter equation (Ter Braak Braak and and Etienne, Etienne, 2003 2003)) K K
= k2~,P[Yk]P[X(1)IYk]= 2: P[Yk ]P[X(l ) [ Yk ] Pe[x(1)] [X( l )] = k=l
(5.17)
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1 11 77
where the summation summation is over all possible states Y Yk. probabilities P[Y P[Yk] drop k] drop k • The probabilities out simulation series out of of Eq. Eq. (5.17) because because the the simulation series X Xuu is is self-weighing, self-weighing, that that is, is, each each Xuu is generated by the simulation simulation with probability probability proportional proportional to P[X P[Xu]. X u]' The usually huge reason for Carlo simulation simulation [Eq. [Eq. (5.16)] is reason for the the Monte Monte Carlo is that that K K is is usually huge so so that summation in cannot be that the the summation in Eq. Eq. (5.17) cannot be carried carried out out in in practice. practice. Missing years in data data can be handled handled by simulation simulation as well. If, for for example, example, data data for for year year 2 are are missing, missing, we we require require in in the the likelihood likelihood [Eq. (5.14)] the the term term
(1 )]P[X(3)IXk(2)] X(1)])] = = k~ X(1)]P[X(3)Xk(2)] PP[X(3) [X(3)IX(1 P[Xk(2) IX ±]=~P[xk(2) K
1i = 1
(5. 18) (5.18)
where where the the summation summation iiss over over all all possible possible states states X Xk(2). To approximate approximate k (2). To Eq. (5.18), the (M being large) from the state state in in year year 2 is is simulated simulated M M times times (M being large) from the the state state in Carlo approximation in year year 1. The The Monte Monte Carlo approximation is is then then [analogously [analogously to to Eq. (5.16)] e[x(3)lx(1)] ~
1
M
-c;~,I'[X(3)Xu(2)] IVl u=~-I
(5.19)
Although 1 999) does Although Moilanen Moilanen ((1999) does not not mention mention it it explicitly, explicitly, the the same same procedure procedure can can be be applied applied if, if, instead instead of of aa complete complete year, year, only only aa few few occupancies occupancies are are miss missing in a single year. Evidently, only the missing data data are simulated. simulated. Being 1 999) approach Being aa maximum maximum likelihood likelihood method, method, Moilanen's Moilanen's ((1999) approach only only produces produces point point estimates, estimates, although although estimates estimates of of confidence confidence limits limits can can be be com computed but require puted ((but require aa lot lot of of computing computing time). time).
O'Hara O'Hara et et al. al. (2002): ( 2 0 0 2 ) : Bayesian Bayesian MCMC M C M C on on Turnover Turnover Data Data To To obtain obtain aa full full joint joint probability probability distribution distribution of of the the model model parameters parameters instead instead of of mere mere point point estimates, estimates, the the maximum maximum likelihood likelihood method method must must be be abandoned. abandoned. O'Hara et al. (2002) were O'Hara et al. were the the first first to to adopt adopt aa Bayesian Bayesian approach approach to to parameter parameter estimation. estimation. The The central central idea idea in in Bayesian Bayesian theory theory (e.g., (e.g., Gelman Gelman et et aI., al., 1995) is is that that our value of our knowledge knowledge of of the the value of aa parameter parameter can can be be represented represented by by aa probability probability distribution containing new information about this parameter distribution and and that that data data containing information about parameter can be be used used to to adjust adjust this this probability probability distribution. distribution. The The probability probability distributions distributions before been used called prior before and and after after data data have have been used to to update update our our knowledge knowledge are are called prior and posterior probability distributions. Bayes' formula posterior probability distributions. Bayes' formula describes how how the prob probability distribution distribution of model model parameter parameter E> | is adjusted adjusted using data data X:
P[E>IX] /'[olx] = =
P[XIE>]P[E>] e[xlo]e[o] e[x] P[X]
(5.20) (5.20)
The can The posterior posterior probability probability distribution distribution P[E>IX] P[| can often often be be approximated approximated through Markov Markov chain Monte Carlo through chain Monte Carlo (MCMC) (MCMC) simulation simulation with with the the Metropolis-Hastings Metropolis-Hastings algorithm. algorithm. The The Metropolis-Hastings Metropolis-Hastings algorithm algorithm consists consists of of the following steps. First, arbitrary values values of the model parameters parameters are chosen. New New values values of of the the model model parameters parameters are are then then drawn drawn from from aa probability probability distri distribution called The form of this jumping distribution " bution called the the jumping jumping distribution distribution ]Ju. The form of this jumping distribution U is is arbitrary, arbitrary, but but aa smart smart choice choice will will facilitate facilitate calculations calculations and and convergence convergence of of the the
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
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simulation simulation to to the the posterior posterior distribution. distribution. These These new new values, values, denoted denoted collectively collectively by by e | , are are now now accepted accepted with with probability probability "
, P[|
X]Ju[| u-110.]
r = min(1 p - [ ~ - i ~ 7 ~ ) u - - - 1 ] )
(5.2 1) (5.21)
where where e | u -1 represent represent the the previous previous values values of of the the model model parameters. parameters. This This pro procedure cedure of of drawing drawing new new values values and and accepting accepting or or rejecting rejecting them them is is iterated iterated many many (u denotes times times (u denotes the the iteration iteration number) number) and and creates creates aa Markov Markov chain; chain; acceptance acceptance parameter values u only depends depends on the values accepted in of new parameter values in iteration iteration u iteration (a sam iteration u u - 11.. The The set set of of values values e | u generated generated in in this this way way constitutes constitutes (a samThe ple from) from) the the posterior posterior distribution distribution p[eIX]. P[| The first first half half of of the the iterations iterations is is ple often needs some some time often discarded discarded because because the the simulation simulation needs time (called (called the the burn-in burn-in period) to period) to converge converge to to the the stationary stationary distribution distribution of of the the Markov Markov chain chain (which (which has has nothing nothing to to do do with with the the Markov Markov property property of of the the SPOM). SPOM). 1 ), the When When Eq. Eq. (5.20) is is inserted inserted into into Eq. Eq. (5.2 (5.21), the probabilities probabilities P[X] P[X] cancel cancel and and only only the the prior prior probability probability distribution distribution pre] P[| and and the the probability probability of of data data remain. Note conditional on conditional on the the model model parameters parameters p[Xle] P[XI| remain. Note that that p[Xle] P[XI| is is the the likelihood. prior can based on prior knowledge knowledge of model likelihood. The The prior can be be chosen chosen based on prior of the the model parameters, given by SPOM itself, is, parameters, and and the the likelihood likelihood p[Xle] P[XI| is is given by the the SPOM itself, that that is, Eq. and (5.1). O'Hara O'Hara et al. (2002) used only turnover turnover Eq. (5.14) with Eqs. Eqs. (5.15) and data case the data in in which which case the state state in in the the first first year year is is considered considered to to be be given given so so that that P[X(1)] P[X( l)] drops out of Eq. (5.14). For the jumping distribution they used a nornor mal mean e matrix k (to be chosen arbi mal distribution distribution with with mean | u and and covariance covariance matrix E (to be chosen arbitrarily, trarily, but but aa smart smart choice choice speeds speeds up up convergence). convergence). As 1 999) approach, As in in Moilanen's Moilanen's ((1999) approach, the the problem problem of of missing missing data data can can be be tack tackled led by by simulating simulating them, them, but but in in aa different different way. way. In In the the Bayesian Bayesian context, context, missing missing data are are in in fact fact treated treated as as parameters; parameters; the the MCMC thus also also yields yields posterior posterior data MCMC thus probability probability distributions distributions for for these these missing missing data. data. The The Metropolis-Hastings Metropolis-Hastings algor algorithm alternate sampling ithm allows allows alternate sampling of of (sets (sets of) of) parameters, parameters, that that is, is, they they do do not not need need to joint jumping to be be drawn drawn from from aa single single joint jumping distribution distribution simultaneously. simultaneously. It It is is most most convenient to sample the parameters and convenient to sample the set set of of model model parameters and the the set set of of missing missing data data chose to sample missing missing data in al. (2002) chose in turn. turn. O'Hara O'Hara et et al. to sample data for for each each patch patch in in 5.2 on each each year year separately separately (see (see Box Box 5.2 on Gibbs Gibbs sampling). sampling). As As O'Hara O'Hara et et al. al. (2002) only only considered considered turnover turnover data, data, they they could could not not use use all all information information in in aa dataset. dataset. This This problem problem was was solved solved by by Ter Ter Braak Braak and and Etienne Etienne (2003). -
Ter Ter Braak Braak and and Etienne Etienne (2003): (2003): Bayesian Bayesian MCMC MCMC o onn the the Full Full Dataset Dataset While al. (2002) were analysis of While O'Hara O'Hara et et al. were working working on on their their Bayesian Bayesian analysis of also developing turnover data, Ter Braak and Etienne (2003) were turnover data, Ter Braak and Etienne were also developing aa Bayesian Bayesian method. method. This This method method turned turned out out to to generalize generalize the the approach approach by by O'Hara O'Hara et et al. al. (2002) on on two two main main points. points. First, First, Ter Ter Braak Braak and and Etienne Etienne (2003 (2003)) were were able able to to also also exploit exploit the the informa informamissing preyears, tion tion in in the the first first year. year. Their Their idea idea was was to to extend extend data data with with L L missing preyears, with with L L aa large large number, number, and and to to choose choose arbitrary arbitrary fixed fixed states states for for the the year y e a r --L. L. The data, given chosen states The likelihood likelihood of of extended extended data, given the the chosen states in in year y e a r --L, L , is is simply simply aa product with the product of of L L + T T - 1 transition transition probabilities probabilities [compare [compare Eq. Eq. (5.14) (5.14)with the -
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5. 5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
BOX 5.2 Simulating Missing Patch Data One at a Time by Gibbs Sampling When some patch states are unknown (Le., with missing data), the expression for the likelihood [Eq. (5 . 1 4)] cannot be calculated, even if P[X(l )] were known, because some transition probabilities of Eq. (5. 1 ) are then unknown. The trick is to fill in missing data from the correct conditional distribution. O'Hara et al. (2002) achieved this by simulat ing each missing patch state in turn, starting from initial guessed states and initial model parameters. After each missing state is simulated once (or more than once), new model parameters are proposed [and accepted with the acceptance probability of Eq. (5.21 )] . With the then current model parameters, each missing state is simulated again. Next, new model parameters are proposed and so on until convergence (i.e., when the dis tribution of the model parameters and simulated states does not change any more). To simulate a single missing patch, we need the probability Pi that patch i at time t is occupied, given all other patch states, denoted by X_i' This probability can be calcu lated by Pi = P[X,(t) = l 1X-;] =
1
".
� f;'
(1 )
with f; the odds ratio (McCullagh and Neider, 1 989) fi =
P[Xi(t) = l IX-i] P[X;(t) = 1 , X_a = , P[Xi(t) = 0ILi] P[Xi(t) = 0, X-I]
(2)
The second equality in Eq. (2) follows from the rule for conditional probability, P[AIB] = P[A,B]/P[B], and by noting that P[X-;] drops out. By applying Eqs. (5.' 4) and (5 .1 5) to the numerator and denominator of Eq. (2) and observing that all terms cancel except those involving years t - 1 , t and t + 1 we obtain f r
P [Xr· (t) -= X(_ =_'_ t)_ -_ , )] P[_ I,X( t-_ i ( t)] ' X_ + ' )-,--, I Xi(_ t --_ --:l _ _ = -:-:�-:-: - : --: P[Xi(t) = 0 IX(t - 1 :-::)] -::)r P[X(t + 1 ) IXi(t) = 0, X- i(t
(3)
In this method, a 1 is filled in for the unknown state with probability Pi and a 0 with probability 1 - Pi' Simulating the missing values in this way is known as Gibbs sampling because we sample from the exact conditional distribution. The acceptance probability then equals 1 . Equation (3) is not cheap to calculate, as the second term in both the numerator and the denominator of fi involves the multiplication of N + , transition probabilities, as is evident from Eq. (5.1 5). If many patch states are missing for a partic ular year, it is computationally more efficient to simulate them jointly using the Metropolis-Hastings algorithm of Box 5.3. If a state is missing in' the last year, the corresponding term can simply be removed from the likelihood because the other states do not depend on it. Equivalently, the miss ing state is simulated as the other missing states but with Pi as defined by Eq. (1 ). O'Hara et al. (2002, personal communication) used Metropolis-Hastings to each missing patch in turn with proposals derived from Eq. (5.1 ), i.e., without conditioning on X(t + 1 ). This is less efficient than Gibbs sampling because Eq. (3) needs to be calculated for the acceptance probability.
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
0 1 220
first many missing missing values first term term dropped]. dropped]. There There are are very very many values in in the the extended extended data, data, but but apart apart from from that, that, there there is is nothing nothing to to prevent prevent aa standard standard Bayesian Bayesian analysis analysis with simulation. The with MCMC MCMC simulation. The validity validity of of choosing choosing arbitrary arbitrary fixed fixed states states for for year the Markov of aa SPOM the probabil year -L -L is is guaranteed guaranteed by by the Markov property property of SPOM that that the probability ity of of the the system system being being in in aa certain certain state state does does not not depend depend on on the the state state of of the the system system in in the the infinite infinite past. past. In In formula, formula, [X( - L)] P[X(1)] = lim l i m PP[X( [ X ( 1l) )] ] iP P[X(-L)] P[X( l )] = <x L---~oc L�
(5.22) (5.22)
X(-L). Thus, it is wise to take L large. As Eq. (5.1 (5.17), for any state X( -L). Thus, 7), Eq. (5.22) requires requires the the assumption assumption of of quasistationarity. quasistationarity. Second, each patch patch in each year Second, instead instead of of sampling sampling missing missing data data for for each in each year separ separately, ately, Ter Ter Braak Braak and and Etienne Etienne (2003) (2003) chose chose to to sample sample missing missing data data for for all all patches patches in (see Box in aa single single year year simultaneously, simultaneously, thus thus only only alternating alternating between between years years (see Box 5.3). 5.3). This This turned turned out out to to be be much much faster faster than than the the method method of of O'Hara O'Hara et et al. al. (2002). (2002).
BOX 5.3 Simulating Missing Patch Data by the Metropolis-Hastings Algorithm In the Metropolis-Hasting algorithm of Ter Braak and Etienne (2003), missing patch data for a particular year t are simulated jointly by proposing values for the missing states (proposals) and accepting the proposals with an acceptance probability r [Eq. (5.2 1 ) with e and X interchanged]. Years are updated in turn. If the metapopulation is believed to have low turnover probabilities (low E; and Cj), it is easy to generate sensible proposals for all missings in year t, given the current states in the year before and after. If the states of a patch i n the years before and after the miss ing value are the same, propose this state with high probability, say 0.99; if the states differ, choose "1 " with probability 1f2• This rule is applied to all missing patches in year t, yielding an N vector of proposed states denoted by X*(t) to distinguish it from the cur rent state. Of course, nonmissing data are not simulated, so that for these patches the proposed state and the current state are identical. The so-generated proposal X*(t) is accepted with acceptance probability r and X(t) is retained if the proposal is not accepted. To calculate the acceptance probability r, we need the ratio of the proposal distributions and the ratio of the likelihoods of x*(t) and X(t) [compare Eq. (5.21 )]. Because proposed patch states are generated independently, the proposal distribu tion Ju is the binomial probability .
Ju[X* (t) IX(t)]
=
N
rrp�i (t)(l ;= 1
•
Pi)(1 -Xi {t)) •
-
(1 )
J
with Pi the probability in the proposal scheme that a 1 is filled in, i.e., Pi is 0.99, 0.01 , or 0.5. The proposal distribution Ju[X(t)IX*(t)] is obtained as in Eq. (1 ) with X(t) and X*(t) interchanged. The ratio of the probability (likelihood) of X*(t) over that of X(t), given the states of all patch states in other years, is
P[X* (t)IX- tl
P[ X *(t),X-d
f = P[X(t)IX-t] = P[X(t),X- d =
P[X* (t)IX(t - 1 )] P[X(t + 1 JIX* (t)] P[X(t)lxCt
-
1 )] P[X(t + 1 ) IX(t)]
(2)
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1121 21
and depends only on the states in the neighboring years t - 1 and t + 1 (see Box 5.2). In Eq. (2) X-t denotes the states of all patches in all years except year t. The acceptance ratio r is r =
f
lu_ < t)_ ] )IX [X_ ( t....;. _ _ . *_ 1u[X * (t)I X( t)]
(3)
In the proposal scheme of Ter Braak and Etienne (2003), the {Pi} are not as simple as below Eq. (1 ). Instead, Pi is calculated as in Eq. (1 ) of Box 5.2 but with the odds-ratio f; simplified to f..
tX....,. P[_ (_ )] (t_ _ _ _ '_ ) ] P[ t)_ = _'-,_X i (_ IX -,(,_ t_ +_ )I X i_ = ,_ , X_ __ '· ( t_ )_ '-,....,. ,= P[Xi ( t) = 0IX( t - ' )] P[X,(t + ' )IXi ( t) = O,X-i ( t)]'
(4)
which involves four instead of 2(N + 1 ) transition probabilities and is thus inexpensive to calculate. The proposal X*(t) is not generated with the correct conditional distribu tion, but that does not matter because the theoretical validity of the method hinges on the correct acceptance probability r, which is calculated via Eqs. (1 )-(3). Note that Eq. (1 ) depends on X(t) because Pi depends on X(t) via fi in Eq. (4). It is instructive to see which proposals are generated when Ei and Ci would be con stant over time (Le., when the connectivity of a patch remains the same over time). If X(t - ' ) 0 and X(t + ' ) 0, then fi C;f;/(l - Q2, which will be a small value if Ei and/or Ci is small so that Pi is also close to 0 so that with only a small probability a 1 is proposed in year t. Similar considerations show that if X(t - ' ) 1 and X(t + 1 ) 1 , Pi is close to 1 so that with a large probability a 1 is proposed in year t. If X(t - 1 ) *' X(t + 1 ) 0, then f; (1 - Ei)/(1 - Q so that if Ci and E; are equal or both small, Pi is approx imately one-half. The proposal mechanism is thus very similar to the one begun in this Box. It has the advantage of yielding good proposals for more variable metapopulations and does not require a tuning parameter such as the value 0.99 in our initial scheme. Software to implement the full Bayesian analysis is available in the archives of Ecology: http://www.esapubs.org/archive/ecol/E084/005. =
=
=
=
=
5.4 5.4
=
=
REMAINING REMAINING PROBLEMS PROBLEMS FOR FOR PREDICTIONS: PREDICTIONS: A A CASE CASE STUDY STUDY (2003)) uses the spatial information information The method of Ter Braak and Etienne (2003 of of the the first first year year of of aa dataset, dataset, reflecting reflecting the the history history of of the the metapopulation, metapopulation, as as well well as as the the turnover turnover information information in in the the following following years, years, it it can can handle handle missing missing data, data, and and it it provides provides aa joint joint probability probability distribution distribution of of the the model model parameters. parameters. Because Because missing missing data data are are treated treated as as parameters parameters that that need need to to be be estimated, estimated, aa probability points is probability distribution distribution of of the the occupancy occupancy at at the the missing missing data data points is also also provided. provided. This This is is very very convenient convenient if if one one wants wants to to make make predictions, predictions, as as shown shown later. Bayesian approach later. Furthermore, Furthermore, the the Bayesian approach allows allows for for many many extensions extensions of of the the dataset ((Moilanen, method, such as misclassifications of occupancies in the dataset Moilanen, 2002) and and temporal temporal stochasticity (regional stochasticity in the terminology of of 2002) 991). We Hanski, Hanski, 11991). We refer refer to to the the discussion discussion for for their their possible possible consequences consequences (see (see also also Ter Ter Braak Braak and and Etienne, Etienne, 2003 2003).). However, However, in in addition addition to to technical technical difficul difficulties ties of of the the MCMC MCMC approach approach [When [When has has the the MCMC MCMC converged? converged? How How many many
1122 22
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
preyears added? Is perhaps an preyears (L) (L) should should be be added? Is there there perhaps an even even more more efficient efficient algor algorithm ithm than than the the one one presented presented by by Ter Ter Braak Braak and and Etienne Etienne (2003)?], (2003)?], there there are are (at (at least) least) three three fundamental fundamental difficulties difficulties that that need need to to be be considered considered before before cali calibrated SPOMs can brated SPOMs can be be used used for for prediction. prediction. First, First, Ter Ter Braak Braak et et al. al. (2003) (2003) invoked invoked the the assumption assumption of of quasistationarity quasistationarity to be able use occupancy The question to be able to to use occupancy datasets datasets to to their their fullest fullest extent. extent. The question is is whether this this assumption is warranted warranted and and what what impact impact it it has. even whether assumption is has. Second, Second, even if model parameters are precisely precisely known, known, predictions predictions of if the the model parameters are of occupancy, occupancy, turnover, will be turnover, or or the the time time to to extinction extinction will be uncertain uncertain due due to to the the inherent inherent sto stochasticity of of the model (termed (termed extinction-colonization chasticity the model extinction-colonization stochasticity stochasticity in in Hanski, 99 1 ) . If contribution of inherent stochasticity stochasticity to Hanski, 11991). If the the contribution of the the inherent to the the total total uncertainty predictions is more than uncertainty in in model model predictions is large, large, data data collection collection for for more than aa few few years be rather is possible years may may be rather fruitless. fruitless. Third, Third, in in making making predictions predictions it it is possible to to take into account take into account any any planned planned changes changes in in network network structure structure (habitat (habitat creation, creation, barrier barrier removal). removal). Indeed, Indeed, comparing comparing the the outcome outcome of of several several scenarios scenarios is is often often the main purpose al. 2002a). the main purpose (e.g., (e.g., Wahlberg Wahlberg et et al. 2002a). However, However, unpredictable unpredictable changes plans) may changes in in the the landscape landscape (habitat (habitat turnover, turnover, unknown unknown management management plans) may make reliable predictions make reliable predictions rather rather difficult. difficult. These These problems problems are are discussed discussed in in order order using using the the tree tree frog frog SPOM SPOM mentioned mentioned earlier earlier on on the the tree tree frog frog dataset dataset and and on on aa simulated simulated dataset dataset for for the the same same network network with with the the same same pattern pattern of of miss missing ing values values as as the the real real dataset. dataset.
The Quasistationarity Quasistationarity Assumption quasistationarity allowed allowed Hanski Hanski ((1994) to estimate estimate model model 1 994) to Assuming quasistationarity parameters parameters from from aa single single year year of of occupancy occupancy data data only, only, and and it it allowed allowed Ter Ter Braak Braak et 1998) and et al. al. ((1998) and Ter Ter Braak Braak and and Etienne Etienne (2003 (2003)) to to extract extract information information in in aa dataset about the dataset about the history history of of the the metapopulation, metapopulation, as as reflected reflected by by the the first first year year of of aa dataset. dataset. However, However, how how do do we we know know that that the the metapopulation metapopulation is is in in the the quasi quasistationary not? To stationary state state or or not? To answer answer this this question, question, we we need need to to know know more more about about the metapopulation (Moilanen, the history history of of the the metapopulation (Moilanen, 2000), 2000), or or we we can can attempt attempt to to find find the answer using dataset itself. itself. Assuming Assuming that information about past the answer using the the dataset that information about the the past history is history is usually usually unavailable, unavailable, it it seems seems worthwhile worthwhile to to explore explore the the dataset dataset for for information about the information about the presence presence or or absence absence of of quasistationarity. quasistationarity. If If we we can can find find such also avoid avoid the such information, information, we we also the objection objection that that assuming assuming quasistationarity quasistationarity makes predictions worthless because it makes predictions worthless because it is is aa self-fulfilling self-fulfilling prophecy. prophecy. We start by well as the We start by comparing comparing parameter parameter estimates estimates as as well as predictions predictions for for the complete dataset dataset assuming assuming quasistationarity quasistationarity (which (which is is denoted denoted by by QS6), QS6), the complete the first year 1 ) , and and the first year only only dataset dataset assuming assuming quasistationarity quasistationarity (QS (QS1), the turnover turnover dataset Figures 5.1 cumulative posterior posterior distributions distributions for dataset (TO). (TO). Figures 5.1 and and 5.2 5.2 show show cumulative for the tree frog Figure 5.1 shows that the tree frog SPOM SPOM and and dataset. dataset. Figure 5.1 shows that posterior posterior distributions distributions of the the model model parameters for QS1 QS1 are are wider wider than than those those for for QS6 and TO, TO, which which of parameters for QS6 and are but note )' This does not necessar are quite quite alike alike ((but note the the differences differences for for x x and and Q q2). This does not necessar2 ily ily mean mean that that the the predictions predictions must must also also be be different different because because the the model model param parameters eters may may be be highly highly correlated. correlated. Indeed, Indeed, Fig. Fig. 5.2A 5.2A shows shows that that the the variance variance in in distributions QS11 and distributions of of the the occupancy occupancy after after 100 100 years years is is similar similar for for QS and TO, TO, indi indicating about the cating that that there there is is nearly nearly as as much much information information about the occupancy occupancy in in QS1 QS1 as as in in TO. TO. Combining Combining them them in in QS6 QS6 gives gives aa slightly slightly narrower narrower distribution distribution (i.e., (i.e., steeper .2A. steeper cumulative cumulative distribution) distribution),, as as shown shown in in Fig. Fig. 55.2A.
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The The variances variances may may be be similar, similar, the the medians medians are are different, different, with with QS6 QS6 (0.129) (0.129) being 1 ) and (0.158). The being intermediate intermediate between between QS1 QS1 (0.094 (0.0941) and TO TO (0.158). The occupancies occupancies in in the 1 9 8 1-1983, 1986) 1 1, 0.1 1 1, 0.141, the 4 4 data data years years ((1981-1983, 1986) are are 0.1 0.111, 0.111, 0.141, and and 0.147, 0.147, so so TO TO seems seems to to reflect reflect this this upward upward trend. trend. This This is is not not accidental. accidental. Moilanen Moilanen (2000) (2000) ascertained ascertained that that parameter parameter estimation estimation based based on on aa few few years years of of simulated simulated A A
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turnover data data often often leads to the the prediction of aa substantially substantially higher higher or or lower lower turnover leads to prediction of occupancy, SPOM was occupancy, even even if if the the simulated simulated SPOM was in in the the quasistationary quasistationary state. state. It It remains to whether this change in remains to be be seen seen whether this change in occupancy occupancy is is still still noteworthy noteworthy when when the the full full posterior posterior distribution distribution is is known. known. To To illustrate illustrate this this change change in in occupancy, occupancy, we we simulated simulated the the tree tree frog frog system system using median values QS6 posteriors . 1 for 000 time using the the median values of of the the QS6 posteriors in in Fig. Fig. 55.1 for 11000 time steps steps to quasistationary state additional time to reach reach the the quasistationary state and and for for 66 additional time steps steps to to generate generate aa dataset similar to real dataset dataset similar to the the real dataset (see (see Fig. Fig. 5.3). 5.3). We We then then estimated estimated the the model model parameters TO. Figures parameters for for QS6, QS6, QS1, QS1, and and TO. Figures 5.4 5.4 and and 5.5 5.5 show show posterior posterior distri distributions corresponding predictions butions for for the the model model parameters parameters and and the the corresponding predictions of of patch patch occupancy similar to . 1 and occupancy and and turnover. turnover. These These figures figures are are similar to Figs. Figs. 55.1 and 5.2, 5.2, respec respectively, places in tively, except except that that TO TO and and QSl QS1 have have traded traded places in Fig. Fig. 5.5; 5.5; medians medians are are 0.158 1 ) , 0.139 QS6), and 0.158 (QS (QS1), 0.139 ((QS6), and 0.124 0.124 (TO). (TO). Again, Again, TO TO reflects reflects the the trend trend in in data, . 1 53, 0.1 33, 0.129). data, which which are are downward downward in in this this case case (0.158, (0.158, 00.153, 0.133, 0.129). Nevertheless, Nevertheless, because because the the posteriors posteriors of of the the occupancies occupancies for for QSl QS1 and and TO TO largely largely overlap, overlap, the the assumption assumption of of quasistationarity quasistationarity is is not not refuted refuted by by the the dataset. dataset. Therefore, TO, appears Therefore, the the use use of of QS6, QS6, which which contains contains more more data data than than TO, appears to to be be warranted. warranted. This This warrant warrant becomes becomes even even stronger stronger when when we we calculate calculate the the median median occupancy 00 iterations (Fig. 5.3), occupancy in in the the simulations simulations after after 1100 iterations (Fig. 5.3), which which is is 0.139, 0.139, precisely the the median median value value for for QS6. QS6. Obviously, Obviously, for for the the real real dataset dataset we we cannot cannot precisely perform perform this this check, check, but but there there too too we we find find aa large large overlap overlap of of the the posteriors posteriors for for QSl the assumption of quasi stationarity cannot QS1 and and TO, TO, so so the assumption of quasistationarity cannot be be refuted, refuted, and and using using QS6 QS6 seems seems the the best best choice. choice. If If the the posteriors posteriors have have little little overlap, overlap, we we can can interpret population may interpret this this as as aa sign sign that that the the meta metapopulation may not not be be in in the the quasistation quasistationary ary state, state, and and we we should should perhaps perhaps refrain refrain from from using using QS6 QS6 and and use use TO TO instead. instead. Until Until now, now, we we have have only only looked looked at at the the predictions predictions of of the the occupancy occupancy after after 100 100 years. years. Predictions Predictions of of the the turnover turnover in in 100 100 years, years, pictured pictured in in Figs. Figs. 5.2B 5.2B and and
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S.SB, QSl contains 5.5B, show show that that QS1 contains little little information information about about turnover. turnover. Common Common sense sense tells us that there there is is no turnover in snapshot data, tells us that no real real information information about about turnover in snapshot data, i.e., i.e., in QS1, QS1, and and that that any any found an artifact of the the model. In in found information information must must be be an artifact of model. In preliminary posteriors for for QSl preliminary MCMC M C M C simulations, simulations, we we sometimes sometimes found found posteriors QS1 con coninformation about turnover. It It turned out that that this this disappeared disappeared taining some some information about turnover. turned out taining when increased the the value value of when we we carried carried out out more more iterations iterations or or increased of L. L. Hence, Hence, if if contain information information about turnover, this this should be the posteriors posteriors for for QSl QS1 contain about turnover, should be the B B
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RAMPAL S. ETIENNE RAMPAL S. ETIENNE ET ET AL.
interpreted interpreted as as aa sign sign that that the the MCMC MCMC simulations simulations have have not not yet yet converged converged or or that that L L is is not not taken taken large large enough. enough. As As already already mentioned mentioned earlier, earlier, differences differences in in the the posteriors posteriors of of the the model model parameters parameters do do not not necessarily necessarily entail entail differences differences in in model model predictions predictions because because parameters parameters may may be be highly highly correlated. correlated. We We remarked remarked that that in in the the IFM IFM with with res rescue cue effect, effect, parameters parameters ee and and y y cannot cannot be be distinguished distinguished by by snapshot snapshot data; data; they they appear appear as as aa product product in in Eq. Eq. (5.6). (5.6). Although Although this this model model is is mathematically mathematically not not completely completely exact, exact, this this high high correlation correlation between between ee and and y y is is still still to to be be expected expected for Because the al. (2000) contains contains additional for QS1 QS1.. Because the extended extended IFM IFM of of Vos Vos et et al. additional , we need to correct for these parameters to observe this parameters parameters q q ll and and qq2, we need to correct for these parameters to observe this 2 correlation. correlation. The The appropriate appropriate transformation, transformation, relating relating Eq. Eq. (5.7) (5.7) to to Eq. Eq. (5.2) (5.2) and and Eq. Eq. (5.8) (5.8) to to Eq. Eq. (5.3), (5.3), is is ' (ql )log H log log ee' = = log log ee + + (ql - Ql ~l)log H1l ' q2 log log yy' = = log log yy + + ((q2 - Q2)log/4 )log H2
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Inherent Inherent Stochasticity Stochasticity Evidently, more data reflected in in Evidently, the the more data available, available, the the better better the the predictions, predictions, as as reflected steeper and 5.5). These cumulative steeper cumulative cumulative posteriors posteriors (Figs. (Figs. 5.2 5.2 and 5.5). These cumulative posteriors posteriors will, however, however, never never be be vertical vertical regardless regardless of of the the amount amount of of data data available available will, because because the the model model is is stochastic. stochastic. This This inherent inherent stochasticity stochasticity may may contribute contribute much much more more to to the the total total uncertainty uncertainty than than the the uncertainty uncertainty in in the the model model param parameters, useless. Hence, eters, which which would would make make further further data data collection collection rather rather useless. Hence, the the question arises how needed. Figure question arises how much much data data are are needed. Figure 5.7 5.7 shows shows predictions predictions for for occupancy occupancy and and turnover turnover for for the the simulated simulated tree tree frog frog system system when when the the model model
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parameters parameters are are known known exactly exactly compared compared to to the the predictions predictions based based on on QS6, QS6, shown shown earlier in Fig. 5.2 (real dataset) dataset) and and Fig. 5.5 (simulated (simulated dataset). As far as gained if more data as occupancy occupancy is is concerned, concerned, not not much much would would be be gained if more data became became available. However, available. However, more more data data would would convey convey more more information information about about turnover. population is turnover. This This is is important important if if the the meta metapopulation is at at risk risk of of extinction. extinction. A A metapopulation turnover will will go than aa metapopulation with with high high turnover go extinct extinct much much faster faster than meta population with low turnover and the same occupancy metapopulation occupancy (i.e., the same basic population capacity; basic reproduction reproduction number number Ro R0 or or meta metapopulation capacity; Ovaskainen Ovaskainen and and Hanski, 1 ) . Hence, 5.7B suggests that Hanski, 200 2001). Hence, Fig. 5.7B that data data overestimated overestimated the turnover turnover and population extinction and thus thus underestimated underestimated the the meta metapopulation extinction time, time, that that is, is, the the true true time parameterized with time to to metapopulation metapopulation extinction extinction is is larger larger than than the the model model parameterized with available available data data predicts. predicts.
Changes in Environment C h a n g e s in Environment The aforementioned aforementioned model model predictions predictions all assumed assumed that that the system e.g., using deci remained unaltered. To weigh several management management options options ((e.g., decision theory, Possingham, 11996, 996, 11997; 997; Possingham Possingham et aI., al., 2001 2001),), one needs to be be able able to to change change system system properties properties by, by, for for example, example, introducing introducing patches, patches, increasing increasing patch patch quality, or reducing reducing the effective interpatch interpatch distance distance by build building corridors. corridors. Because these changes do not not affect the model model parameters, parameters, their their effect effect can can be be studied studied easily easily with with model model simulations. simulations. Problems Problems enter, enter, however, however, when when unknown unknown changes changes need need to to be be dealt dealt with. with. In In our our tree frog frog example, example, a large proportion proportion of the ponds ponds have disappeared disappeared and and been been created created since since 1986 1986 because because of of changes changes in in land land use use and and succession. succession. Precise Precise data data are are unavailable. unavailable. We We only only know know that that in in the the year year 2002, 2002, 23 23 patches patches are are occupied 9 8 1 , 11982, 982, occupied (in (in comparison, comparison, the the number number of of occupied occupied patches patches in in 11981, 1983, 986 is 10 patches 1983, and and 11986 is 22, 22, 22, 22, 28, 28, and and 29, 29, respectively) respectively) of of which which just just 10 patches existed (and 88 occupied) 9 8 1-1986. So 13 new existed (and occupied) in in 11981-1986. So there there are are at at least least 13 new patches, patches, but but perhaps perhaps many many more more that that are are empty. empty. This This makes makes it it impossible impossible to to compare compare
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the occupancy found the predictions predictions of of the the model model and and the the occupancy found in in 2002 2002 quantitatively, quantitatively, let alone 98 1-1986 data to cali alone that data for 2002 2002 can be combined combined with 11981-1986 calibrate brate the the model model even even better. better. Qualitative Qualitative comparison comparison is, is, however, however, still still possible. possible. 5.8A shows shows the spatial occupancy occupancy pattern pattern in 11986 location of Figure 5.8A 986 and the location the the 23 23 occupied occupied patches patches in in 2002. 2002. In 11986, 986, eight pond population pond clusters were occupied. occupied. In 2002 2002 the meta metapopulation retreated to only three of these clusters (clusters I, II, and III). III). It is apparent that the weakest weakest clusters have become become extinct extinct in 2002. 2002. These are the relatively small and isolated VIII) isolated clusters (clusters V, VI VI,, VII, and the one-pond one-pond cluster VIII) and a cluster with occupied pond in 1986 with only one occupied 1986 (cluster IV) IV).. Let us exam examine ine whether whether the the model model is is able able to to predict predict this this spatial spatial pattern. pattern. Figure Figure 5.8B 5.8B shows shows the patches being the probability probability of of patches being occupied occupied in in 2002 2002 based based on on simulations simulations start starting with the pattern in 1986 1986 and using posteriors posteriors obtained obtained from the real dataset assuming QS6) . The eight clusters assuming quasistationarity quasistationarity ((QS6). clusters (except (except for clus cluster ter VIII) VIII) are are the the only only parts parts of of the the metapopulation metapopulation where where ponds ponds with with aa prob probability of being occupied occupied of more than 0.25 0.25 occur, which which is consistent consistent with with the 2002 data. 2002 data. Clusters Clusters II and and III, III, which which remained remained occupied occupied in in 2002, 2002, are are the the only only clusters clusters that that contain contain ponds ponds with with aa probability probability of of more more than than 0.50 0.50 or or even even 0.75 0.75 of of being occupied. occupied. Thus the model is able to locate the parts of of the metapopu metapopulation with the highest survival potential. The fact that clusters V, VI, VII, and
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986 and 2002: Fig. 5.8 5.8 Occupancy patterns for the tree frog system. (A) Comparison of 11986 2002: patches empty in 11986 986 (gray dots), patches occupied 986 (black dots), patches occupied occupied in 11986 occupied 2002 (black asterisks), asterisks), patches empty in 11986 occupied in 2002 2002 (gray aster asterin 11986 986 and 2002 986 but occupied isks), and patches occupied in 2002 (B) Probability PP isks), 2002 that did not exist in 11986 986 (open circles). (8) of occupancy in 2002 2002 as predicted 986 and using predicted from simulation simulation starting starting in 11986 using the model parameters of QS6: 0.5 (black dots), 0.5 Q S 6 : 0O :s -< P P< < 0.25 0.25 (gray dots), 0.25 0.25 :s --- P P< < 0.5 0.5 :s - P P< < 0.75 0.75 (open squares), 75 :s squares), and 0. 0.75 -< P P :s -< 11 (open circle).
5. 5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
11 29 29
VIII model predictions, VIII have have become become extinct extinct is is in in accordance accordance with with the the model predictions, as as in in these clusters clusters only ponds with than 0.25 these only one one or or two two ponds with aa probability probability of of more more than 0.25 were were present present (the (the probability probability of of occupancy occupancy of of the the pond pond in in cluster cluster VIII VIII is is even even smaller smaller than than 0.25) 0.25).. However, However, in in contrast contrast to to data, data, the the model model does does not not neces necessarily sarily predict predict the the retreat retreat of of the the tree tree frog frog from from cluster cluster IV, IV, as as four four ponds ponds in in this this cluster probabilities larger cluster have have occupancy occupancy probabilities larger than than 0.25. 0.25. At At least least one one occupied occupied pond pond would would have have been been expected. expected. It It is is apparent, apparent, however, however, that that cluster cluster IV IV shows shows the recolonization, as the best best potential potential for for recolonization, as the the cluster cluster is is relatively relatively large large and and has has many ponds with many ponds with aa high high probability probability of of occupancy. occupancy. To To increase increase colonization colonization probability, probability, the the spatial spatial cohesion cohesion with with cluster cluster II should should be be improved. improved. This This is is in in accordance Crombaghs and accordance with with the the actual actual protection protection policy policy in in the the region region ((Crombaghs and Lenders, 1 ). Thus, Lenders, 200 2001). Thus, the the model model is is aa useful useful tool tool to to determine determine the the optimal optimal spa spatial tial strategy strategy to to conserve conserve the the tree tree frog frog metapopulation. metapopulation.
5.5 5.5
DISCUSSION DISCUSSION SPOMs SPOMs were were introduced introduced as as metapopulation metapopulation models models that that combine combine simplicity simplicity with with sufficient sufficient realism realism (particularly (particularly pertaining pertaining to to spatial spatial structure), structure), such such that that they populations and they are are ideal ideal for for assessing assessing the the current current condition condition of of real real meta metapopulations and for for predicting predicting the the effect effect that that human human intervention intervention has has on on metapopulation metapopulation per persistence. sistence. Hence, Hence, it it is is interesting interesting to to see see what what the the literature literature teaches teaches us us about about the the four four parts, parts, mentioned mentioned earlier, earlier, of of applying applying SPOMs SPOMs to to real real metapopulations, metapopulations, that population data that is, is, to to see see what what meta metapopulation data have have been been used, used, what what SPOMs SPOMs have have been been developed, developed, what what parameterization parameterization method method has has been been used, used, and and whether whether the SPOMs were the SPOMs were useful useful and and successful successful in in their their predictions. predictions. Perhaps metapopulation is Perhaps the the best-documented best-documented metapopulation is the the Glanville Glanville fritillary fritillary (Melitaea land Islands Islands in (Melitaea cinxia) cinxia) metapopulation metapopulation on on the the A ~lland in southwest southwest Finland Finland (Hanski, 994; Hanski aI., 11996; 996; Moilanen 998). There (Hanski, 11994; Hanski et et al., Moilanen and and Hanski, Hanski, 11998). There are are some 4000 habitat 0some 4000 habitat patches, patches, forming forming several several subnetworks subnetworks of of which which one one 550patch network has been been studied has been been modeled the patch network has studied in in more more detail. detail. It It has modeled with with the IFM method of 1 994) . Because IFM and and parameterized parameterized with with the the incidence incidence method of Hanski Hanski ((1994). Because the the model model describes describes data data very very well, well, the the two two variables variables of of the the IFM, IFM, area area and and interpatch interpatch distance, distance, have have been been declared declared as as the the main main explanatory explanatory factors. factors. Other Other factors (Moilanen and 99 8 ) . factors do do not not appear appear to to play play aa significant significant role role (Moilanen and Hanski Hanski 11998). Eber 1996), studying Eber and and Brandl Brandl ((1996), studying the the tephritid tephritid fly fly Urophora Urophora cardui cardui iinn north northeastern eastern Bavaria, Bavaria, came came to to the the same same conclusion conclusion based based on on their their logistic logistic regression regression on turnover turnover data, data, although although distance distance did did not not matter matter much much because because distances distances on between between patches patches were were smaller smaller than than the the average average dispersal dispersal distance. distance. However, However, several authors authors have true. For several have suggested suggested that that it it is is not not generally generally true. For example, example, Sjogren-Gulve 1 996), applying Sj6gren-Gulve and and Ray Ray ((1996), applying their their logistic logistic regression regression method method to to aa dataset of pool frog dataset of aa pool frog (Rana (Rana lessonae) lessonae) metapopulation metapopulation on on the the Baltic Baltic coast coast of of east central central Sweden, Sweden, considered east considered additional additional variables variables (mean (mean water water temperature, temperature, presence presence of of forestry) forestry) that that were were necessary necessary to to obtain obtain aa good good fit. fit. Fleishman Fleishman et et al. al. (2002), (2002), applying applying logistic logistic regression regression to to an an Appache Appache silverspot silverspot butterfly butterfly (Speyeria nokomis nokomis apacheana) apacheana) metapopulation metapopulation in in the the Toiyable Toiyable Range, Range, (Speyeria Nevada, Nevada, even even found found that that patch patch area area and and isolation isolation are are not not important important at at all. all. Other Other factors factors that that are are better better indicators indicators of of habitat habitat quality quality than than area area had had much much more 1 999) used more explanatory explanatory power. power. Lindenmayer Lindenmayer et et al. al. ((1999) used the the IFM IFM to to model model
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meta populations of South metapopulations of four four marsupials marsupials in in Buccleuch Buccleuch State State Forest, Forest, New New South Wales, common brushtail moun Wales, Australia: Australia: common brushtail possum possum (Trichosurus (Trichosurus vulpecula), vulpecula), mountain Trichosurus caninus), tain brushtail brushtail possum possum ((Trichosurus caninus), common common ringtail ringtail possum possum (Pseudocheirus (Pseudocheirus peregrinus), peregrinus), and and greater greater glider glider (Petauroides (Petauroides volans). volans). They They 1 994) incidence method. For parameterized the parameterized the model model with with Hanski's Hanski's ((1994) incidence method. For two two of of these these four four species, species, the the parameterization parameterization produced produced unreliable unreliable results, results, which which 1 999) blamed blamed on absence of Lindenmayer et al. al. ((1999) on the the absence of patch patch quality quality in in the the Lindenmayer et model. These These case case studies demonstrate that model. studies demonstrate that the the important important variables variables to to be be put put in the the model model should should be be determined on aa case by case basis, using expert eco ecoin determined on case by case basis, using expert logical knowledge logical knowledge and and statistical statistical techniques. techniques. While While in in these these examples examples model model structure structure is is altered altered by by additional additional variables, variables, processes. Several structure can can also also be be affected affected by by different different processes. Several authors authors model structure report interesting structure in report interesting consequences of their data analysis for model structure this al. (200 1 ) applied maximization on this respect. respect. Crone Crone et et al. (2001) applied likelihood likelihood maximization on snapshot snapshot data metapopulation on 71 islands data of of aa field field vole vole (Microtus (Microtus agrestis) agrestis) metapopulation on 71 islands in in the the Tvarminne archipelago concluded that Tvfirminne archipelago in in the the Gulf Gulf of of Finland Finland and and concluded that colonization colonization and and extinction extinction are are highly highly correlated: correlated: when when aa population population is is on on the the brink brink of of extinction, extinction, the the voles voles disperse disperse en en masse masse to to other other islands. islands. Although Although the the source source population then population then goes goes extinct, extinct, other other populations populations can can benefit benefit from from immigration, immigration, the 977). Peltonen Hanski the rescue rescue effect effect (Brown (Brown and and Kodric-Brown, Kodric-Brown, 11977). Peltonen and and Hanski ((1991) 1991 ) compared species of compared three three species of shrews shrews (Sorex (Sorex araneus, araneus, Sorex Sorex caecutiens, caecutiens, and and Sorex Sorex minutus) minutus) on on islands islands in in three three Finish Finish lakes lakes and and on on the the mainland. mainland. They They found, 1 994) approach, found, using using aa predecessor predecessor of of the the Hanski Hanski ((1994) approach, that that the the coloniza colonization tion rates rates of of these these species species were were identical, identical, but but with with different different contributions contributions from from the of settlement. the probability probability of of arrival arrival and and the the probability probability of settlement. Hence, Hence, the the simi similarity larity of of the the colonization colonization rates rates is is perhaps perhaps coincidental, coincidental, in in which which case case these these dif different should really really be ferent contributions contributions should be modeled modeled explicitly. explicitly. Wahlberg Wahlberg et et al. al. (2002), who 1 999) Monte who employed employed Moilanen's Moilanen's ((1999) Monte Carlo Carlo simulation simulation method method on on snapshot 14 snapshot data data of of the the marsh marsh fritillary fritillary butterfly butterfly (Euphydryas (Euphydryas aurinia) aurinia) in in 1114 patches in Joutseno region region of southern Finland, made yet another import patches in the the Joutseno of southern Finland, made yet another important discovery: if landscape is ant discovery: if the the landscape is very very dynamic dynamic itself itself (patches (patches appear appear and and dis disappear) represent this. this. If dynamics are appear),, the the model model must must represent If patch patch dynamics are due due to to succession succession or or due due to to metapopulation metapopulation behavior behavior of of the the patches patches themselves themselves [e.g., [e.g., if if the the patches patches are are host host plants plants to to aa moth, moth, (Nieminen, (Nieminen, 1996) or or host host butterflies butterflies modeled in to (Lei and to aa parasitoid parasitoid (Lei and Hanski, Hanski, 1997)], this this can can be be modeled in aa fairly fairly straightforward case study, straightforward way, way, but but if, if, as as in in our our case study, patches patches disappear disappear because because of of human activities, any human activities, any model model may may be be as as good good as as any any other. other. 1 999) raised about model Moilanen Moilanen ((1999) raised perhaps perhaps the the most most important important point point about model structure. (regional stochasticity structure. Temporal Temporal stochasticity stochasticity (regional stochasticity in in Hanski, Hanski, 1991) in in model i.e., for model parameters, parameters, i.e., for each each transition transition between between states states there there are are different different values for is values for the the model model parameters, parameters, may may determine determine whether whether aa metapopulation metapopulation is expected to extinct or expected to go go extinct or not, not, so so it it is is crucial crucial whether whether it it should should be be incorporated incorporated in model. It 1 999) did in the the model. It is is relatively relatively straightforward straightforward to to do do so: so: Moilanen Moilanen ((1999) did this this by by multiplying multiplying patch patch area area by by aa normally normally distributed distributed factor factor with with variance variance (J cr22,, but putting aa certain variance (J� but it it may may also also be be done done by by putting certain white white noise noise e~ with with variance (r2 on on the the model model parameters. parameters. The The latter latter is is advocated advocated by by Carlsson Carlsson and and Kindvall Kindvall (2001 who used used Moilanen's 1 999) approach the grasshopper (2001),), who Moilanen's ((1999) approach on on the grasshopper Stauroderus Stauroderus scalaris scalaris in in 158 patches patches on on Oland C)land Island, Island, southeast southeast Sweden, Sweden, and and concluded concluded that that it it is is not not mechanistic mechanistic enough enough to to generate generate the the patterns patterns they they
5. 5.
STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1131 31
found. It also straightforward straightforward to found. It is is also to estimate estimate the the corresponding corresponding parameters parameters (0" approaches. However, (or or or O"E (r~)) in in the the Monte Monte Carlo Carlo simulation simulation and and Bayesian Bayesian approaches. However, the given the Many years the estimates estimates may may not not be be very very accurate, accurate, given the available available data. data. Many years of of data data are are needed needed to to provide provide information information on on the the variability variability of of the the model model parameters. parameters. Because Because the the transitions transitions between between years years are are then then treated treated relatively relatively independently, independently, each each transition transition is is likely likely to to yield yield aa different different trend trend (Moilanen, (Moilanen, 2000), 2000), potentially potentially leading leading to to an an overestimate overestimate of of temporal temporal stochasticity. stochasticity. This This bias bias may may again again be be reduced reduced by by the the assumption assumption of of quasistationarity. quasistationarity. Now Now we we are are back apply? back at at the the problem problem mentioned mentioned in in the the case case study: study: does does this this assumption assumption apply? Because Because of of temporal temporal stochasticity, stochasticity, it it may may no no longer longer be be able able to to tell tell from from data data whether whether quasistationarity quasistationarity applies applies or or not, not, and and perhaps perhaps we we have have no no other other choice choice than than to to resort resort to to independent independent information information to to decide decide about about quasistationarity quasistationarity (Moilanen, (Moilanen, 2000) 2000).. Further Further study study is is evidently evidently needed. needed. Things Things may may even even get get worse worse when when data data themselves themselves contain contain errors. errors. Moilanen Moilanen (2002) (2002) mentioned mentioned three three errors errors that that can can be be made made in in data data collection. collection. First, First, patch patch areas areas can can be be estimated estimated wrongly. wrongly. Second, Second, unknown unknown habitat habitat patches patches may may be be located located within within or or around around the the study study area. area. Third, Third, occupancy occupancy observations observations may may be be false: false: truly truly occupied occupied patches patches may may have have been been observed observed as as being being empty empty (false (false zeros). zeros). The The other other possibility possibility is is that that truly truly empty empty patches patches are are observed observed as as occu occupied pied (false (false ones). ones). This This may may happen happen if if observation observation of of aa few few individuals individuals is is inter interpreted preted as as an an entire entire population. population. Moilanen Moilanen (2002) (2002) argued argued that that wrongly wrongly estimated seriously affect estimated patch patch areas areas can can seriously affect extinction extinction risk, risk, whereas whereas missing missing patches patches cause cause an an overestimation overestimation of of migration migration distances distances and and colonization colonization abil ability ity of of the the species. species. False False zeros zeros can can have have aa severe severe effect effect on on both both extinction extinction and and colonization colonization components, components, and and hence hence on on metapopulation metapopulation persistence. persistence. Like Like tem temporal poral stochasticity, stochasticity, such such data data errors errors can can be be dealt dealt with with in in the the Monte Monte Carlo Carlo simu simulation lation and and Bayesian Bayesian parameterization parameterization methods. methods. Moilanen Moilanen (2002) (2002) gives gives explicit explicit formulae formulae for for the the case case of of false false zeros, zeros, but but again, again, the the parameter parameter estimates estimates will will lose lose some some reliability. reliability. The The great great advantage advantage of of the the Bayesian Bayesian approach approach is is that that this this loss loss of of reliability reliability will will show show up up in in posterior posterior distributions distributions of of the the parameters parameters and and it it can can be be translated translated immediately immediately into into uncertainties uncertainties about about the the predictions. predictions. However, However, no no ready-made ready-made computer computer programs programs exist exist as as yet yet for for Bayesian Bayesian analy analysis sis of of these these extended extended models. models. Considering Considering all all these these allegations allegations at at the the information information richness richness of of data, data, we we have have to to face face the the important important question question of of how how much much data data (how (how many many patches patches and and how 1 99 8 ) how many many years) years) are are needed needed for for reliable reliable predictions. predictions. Ter Ter Braak Braak et et al. al. ((1998) showed showed bbyy simulation simulation that that the the curves curves for for extinction extinction and and colonization colonization [Eqs. [Eqs. (5.2) (5.2) and and (5.3)] (5.3)] are are discomfortingly discomfortingly variable variable when when estimated estimated from from 2 2 years years of of data 1 994) . data with with 50 50 patches, patches, thus thus mimicking mimicking data data and and analysis analysis of of Hanski Hanski ((1994). Tyre 0 1 ) conducted Tyre eett al. al. (20 (2001) conducted maximum maximum likelihood likelihood regression regression oonn snapshot snapshot data data of of the the hydrobiid hydrobiid snail snail Fonscochlea Fonscochlea zeidleri zeidleri in in only only 99 patches patches in in Bopeechee Bopeechee Springs, Springs, Australia, Australia, with with only only 33 patches patches occupied. occupied. Their Their objective objective was was to to com compare pare parameter parameter estimates estimates for for two two sets sets of of snapshot snapshot data: data: before before and and after after (trampling by impact (trampling by stock stock and and water water extraction) extraction).. They They con conhuman-driven impact cluded cluded that that the the method method has has too too little little power power to to detect detect this this impact. impact. The The reason reason they they mentioned mentioned is is that that the the number number of of patches patches is is too too small. small. However, However, the the rea reason son may may also also be be that that the the number number of of years years is is too too small. small. Thomas Thomas et et al. al. (2002a) (2002a) applied applied maximum maximum likelihood likelihood regression regression on on snapshot snapshot data data of of the the silver-studded silver-studded 33 patches in blue butterfly blue butterfly (Plebejus (Plebejus argus) argus) in in 33 patches on on the the Creuddyn Creuddyn Peninsula Peninsula in
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RAMPAL RAMPAL S. S. ETIENNE ETIENNE ET ET AL. AL.
north Wales. Wales. They They considered considered four four snapshots snapshots spread spread fairly fairly evenly evenly over over aa north period of 27 years and compared results obtained from one, two, three, or studfour snapshots. Their message was that predictions based on short-term stud ies may may considerably considerably underestimate underestimate turnover turnover and and caution caution must must therefore therefore be be ies taken when when extrapolating extrapolating to to long-term long-term dynamics. dynamics. They They found found themselves themselves sup suptaken ported 1 998), ported by by the the only only other other long-term long-term study study known, known, by by Moilanen Moilanen et et al. al. ((1998), population of the 66 patch patch meta metapopulation of the the American American on four snapshots in 20 years of the pika (Ochotona (Ochotona princeps) princeps) in in Bodie, Bodie, California. California. Although Although both both studies studies used used pika Hanski's 1 994) incidence Hanski's ((1994) incidence approach approach on on snapshot snapshot data data and and therefore therefore ignored ignored temporal temporal correlation correlation among among these these snapshots, snapshots, results results of of the the American American pika pika study were were recovered recovered by by Moilanen with his his Monte Monte Carlo Carlo simulation simulation study Moilanen ((1999) 1 999) with method method that that does does appropriately appropriately deal deal with with temporal temporal correlation. correlation. Hence, Hence, the the warning warning must must be be taken taken seriously. seriously. It It is is interesting, interesting, however, however, to to note note that that it it is is opposite to to Moilanen's Moilanen's (2000) (2000) warning warning that that short-term short-term data data may may indicate indicate aa opposite false trend. Still, both advocate the use of many years of data. These These results results justify justify the the question question of of whether whether there there is is some some rule rule of of thumb thumb about the the minimum minimum number number of of patches patches and and the the number number of of years years needed. needed. This This about question question is is not not easy easy to to answer answer for for (at (at least) least) five five reasons. reasons. First, First, there there is is obvi obviously aa trade-off trade-off between between the the number number of of patches patches and and the the number number of of years years ously needed. needed. Second, Second, consecutive consecutive years years may may contain contain aa different different amount amount of of infor information than than an equal number years with missing years years in between. Third, mation an equal number of of years with missing in between. Third, it matters matters whether whether the the assumption assumption of of quasistationarity quasistationarity is is made. Fourth, the the it made. Fourth, information information content content of of datasets datasets with with the the same same number number of of patches patches and and years years can can vary vary widely. widely. Fifth, Fifth, the the answer answer depends depends heavily heavily on on what what prediction prediction is is required to to be be reliable. reliable. For For example, example, our our results results in in the the case case study study (202 (202 patches, patches, required years, 11 additional year after missing years) years) show show that 3 consecutive consecutive years, additional year after 2 2 missing that both both patches, 11 year, and TO (202 QS1 (202 (202 patches, year, assumption assumption of of quasistationarity) quasistationarity) and TO (202 patches, 4 fairly reliable patches, 4 transitions) transitions) give give fairly reliable predictions predictions of of the the patch patch occupancy, occupancy, but turnover turnover is badly predicted predicted by by QSl only capable of statstat but is badly QS1.. Perhaps Perhaps we we are are only capable of ing a rule of thumb about what what datasets will not not give reliable predictions: thumb about predictions: netnet works with less than 25 patches patches and less than years of will most most works with less than 25 and less than 33 years of data data will probably be hopeless to parameterize. same time, we note note that the only probably be hopeless to parameterize. At At the the same time, we that the only way data you and to carry out way to be sure is to to simulate data you expect expect to to collect collect and to carry out the the parameter uncertainty analysis num parameter estimation estimation with with uncertainty analysis of of the the predictions predictions for for aa number of of simulated The Bayesian Bayesian approach is most most suited suited for for this ber simulated datasets. datasets. The approach is this purpur pose, demonstrated in O'Hara et pose, as as demonstrated in O'Hara et al. al. (2002). (2002). Once sufficiently sufficiently good good data data have have been been found found to parameterize the the model, model, we we Once to parameterize are ready for for one of the main purposes purposes of of developing developing the model: prediction, prediction, are ready one of the main the model: not not only only of of what what would would happen happen when when the the system system is is left left to to itself, itself, but but particuparticu larly larly to to assess assess the the impact impact of of conservation conservation strategies. strategies. For For example, example, O'Hara O'Hara et patches by et al. al. (2002) (2002) explored explored the the effects effects of of reducing reducing the the area area of of all all patches by some some factor. Cabeza Cabeza and and colleagues colleagues (Chapter (Chapter 22) 22) used used aa SPOM SPOM for selection factor. for optimal optimal selection of of sites sites to to conserve. conserve. In In our our case case study, study, we we studied studied the the consequences consequences of of insertinsert patches near near the the viable viable subnetworks. subnetworks. Whatever Whatever the the scenarios scenarios to to be be studied, studied, ing ing patches we which difwe urge urge the the use use of of uncertainty uncertainty analysis analysis because because it it is is the the only only way way in in which dif ferent ferent scenarios scenarios can can be be compared compared properly. properly.
FROM 6~ FROM META PO M ETAP O PULATIONS PU LATI O N S TO META COMMUNITIES TO M ETACO M M U N ITI ES Mathew and T homas E. Miller Mathew A. Leibold and Thomas
6. 6.11
INTRODUCTION INTRODUCTION Metapopulation thinking (Hanski and Gilpin, 11997, 997, Chapter 1) 1) has led to aa remarkable remarkable change change in in the the way way that that population population ecologists ecologists view view population population dynamics but is only now beginning to have similar effects on how ecologists view view community community dynamics. dynamics. In In population population biology, biology, the the shift shift in in scale scale from from con considering local population dynamics to population persistence in a network network of connected connected habitat habitat patches patches has has enabled enabled ecologists ecologists to to understand understand the the relative relative importance of different factors, such as habitat quality, species interactions, and migration in a scale-dependent context. One of the earliest and perhaps most obvious extensions of meta population theory was to pairs of interacting metapopulation 997; Hanski, 11999), 999), such as competitors (Levins and species (Nee et aI., al., 11997; Culver, 1971; Horn 972; Hastings, 1980) or predator Horn and MacArthur, 11972; predator and prey ((Caswell, Caswell, 11978). 978). The development from one- to two-species metapopula metapopulation tion models models is is important important in in that that it it adds adds aa significant significant level level of of complexity complexity and and realism: realism: the the colonization colonization and and extinction extinction rates rates of of the the focal focal species species become become functions of both habitat characteristics and the presence of the other species (Holt, 11993, 993, 11997). 997). Although (Holt, Although continued continued work work on on pairs pairs of of species species is is important, important, aa bigger bigger challenge challenge is addressing addressing patterns patterns in in community community structure structure involving involving more complex systems. To date, work work on metacommunities with this higher level of complexity is sparse and somewhat simplistic, but important important progress
Ecology, Ecology, Genetics, Genetics, and and Evolution Evolution Metapopulations of Metapopulations
33 1 33
Copyright Copyright 2004, Elsevier, Elsevier, Inc. Inc. 0-12-323448-4
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
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has has been been made made toward toward understanding understanding patterns patterns of of biodiversity biodiversity at at different different scales associated phenomena population, community, scales and and associated phenomena at at the the population, community, and and ecosys ecosystem tem levels. levels. It It is is at at this this level level of of complexity complexity that that the the perspective perspective of of metacommunities metacommunities becomes just aa simple simple extension metapopula becomes substantially substantially different different from from just extension of of metapopulation 992; Holt, Holt, 11993). 993). We tion ecology ecology (Wilson, (Wilson, 11992; We use use the the term term metacommunity metacommunity to to refer communities (sets species that refer to to sets sets of of communities (sets of of species that interact interact with with each each other) other) occurring Gilpin and occurring at at discrete discrete sites, sites, linked linked by by migration migration ((Gilpin and Hanski, Hanski, 1991). 1991). The The crucial sites at crucial elements elements are are multiple multiple potentially potentially interacting interacting species, species, multiple multiple sites at which which such such interactions interactions might might occur, occur, and and migration migration by by individuals individuals of of at at least least some some of of the the species species to to link link species species interactions interactions among among sites. sites. In In its its simplest simplest form, form, aa metacommunity metacommunity may may consist consist of of networks networks of of discrete discrete habitats habitats such such as as islands that linked by all species species among islands that are are linked by migration migration of of some some or or all among the the habitats. habitats. This community will This kind kind of of meta metacommunity will have have aa structure structure that that emerges emerges from from the the dynamics metapopulations, dynamics of of the the interconnected interconnected habitats. habitats. However, However, as as with with metapopulations, our situations, including our definition definition can can fit fit aa broader broader array array of of situations, including mainland-island mainland-island metacommunities, metacommunities, where where migration migration from from the the mainland mainland may may largely largely determine determine community patterns on community patterns on islands. islands. Nevertheless, Nevertheless, this this chapter chapter focuses focuses on on meta metacommunities independent entities entities with communities as as independent with their their own own emergent emergent structure. structure. A between A community community perspective perspective creates creates an an important important distinction distinction between meta population and metapopulation and metacommunity metacommunity views. views. The The most most basic basic issue issue in in metapopu metapopulation theory lation theory is is to to address address what what determines determines the the persistence persistence of of aa metapopula metapopulation habitat patches Chapter 4), tion in in aa system system of of connected connected habitat patches ((Chapter 4), whereas whereas the the most most basic issue in basic issue in metacommunity metacommunity studies studies is is to to address address what what regulates regulates coexistence coexistence of of multiple multiple species species in in such such aa system. system. Excluding Excluding the the case case of of exact exact ecological ecological equivalence equivalence among among species species (Hubbell, (Hubbell, 2001 2001),), long-term long-term coexistence coexistence of of species species requires requires that that they they show show some some trade-off trade-off among among important important aspects aspects of of their their biology scale (see Chesson, 2000). biology at at some some scale (see Chesson, 2000). Theoretical Theoretical work work to to date date in in meta metacommunities shows communities shows that that dynamics dynamics involving involving trade-offs trade-offs are are surprisingly surprisingly constrained reasons. We constrained for for complex complex reasons. We also also examine examine how how our our current current empirical empirical knowledge knowledge is is related related to to the the theory, theory, particularly particularly in in relation relation to to the the role role of of different regulate metacommunities. different possible possible trade-offs trade-offs that that might might regulate metacommunities.
6.2 6.2
FOUR FOUR BASIC BASIC PERSPECTIVES PERSPECTIVES ON ON METACOMMUNITIES METACOMMUNITIES We We can can identify identify four four basic basic perspectives perspectives on on metacommunities metacommunities that that have have gen generally set community thinking thinking to erally set the the context context for for meta metacommunity to date. date. These These simplistic simplistic frameworks communities differ frameworks for for meta metacommunities differ from from one one another another in in the the degree degree of of spa spatial heterogeneity tial heterogeneity among among patches patches and and in in the the degree degree of of connectedness connectedness among among the relative to species (Table .1). the patches patches relative to the the migration migration rate rate of of the the component component species (Table 66.1). This This section section also also relates relates these these perspectives perspectives ttoo the the traits traits ooff the the component component species species and and how how these these traits traits covary covary to to produce produce trade-offs trade-offs at at different different scales. scales.
The The Patch-Dynamic Patch-Dynamic Perspective Perspective This species version This view view is is the the one one that that corresponds corresponds most most closely closely to to aa multi multispecies version of generally seen seen as homogeneous, with of "classic" "classic" metapopulations. metapopulations. Patches Patches are are generally as homogeneous, with local populations populations subject subject to local extinctions. local to local extinctions. Colonization Colonization of of patches patches by by new new
FROM METAPOPULATIONS METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES 6. FROM
1 35 135
TABLE 6.1 6.1 TABLE
Four Simplified Simplified Perspectives Perspectives on on Metacommunities M etacommunities Arranged Arranged Four According to to whether whether Migration Migration Is Sufficient Sufficient to to Alter Alter Local Local Population Population Abundances Abundances According and whether whether Local or Patches Patches Are Are Heterogeneous Heterogeneous and Local Sites or
Migration Migration sufficient sufficient to to alter alter local abundances
Homogeneous patches Homogeneous
No Yes Yes
No
Yes
Species Species sorting Patch dynamics dynamics
Mass effect Neutral Neutral models models
species is is thought thought to to occur occur on on aa long long time time scale scale (through (through low low migration migration rates), rates), species so local local population population sizes sizes are are not not regulated regulated by by immigration immigration or emigration. so or emigration. Persistence the metacommunity thus depends depends on on how how colonization colonization Persistence of of species species in in the metacommunity thus of the the species species can can be be balanced against extinctions either to to interactions with of balanced against extinctions due due either interactions with other species or to to disturbance disturbance events events that that occur occur at at the the scale scale of of entire entire patches patches other species or ((Hanski, Hanski, 1999). 1 999). Local Local communities communities are are frequently "undersaturated" in in terms frequently "undersaturated" terms of number and/or to invasions invasions by by other other members members of the metameta of species species number and/or open open to of the community. If disturbance are sufficiently high, some species may may persist persist community. If disturbance rates rates are sufficiently high, some species in "fugitive species" being poor poor competitors. competitors. in such such metacommunities metacommunities as as "fugitive species" despite despite being To also have have sufficiently high migration migration rates rates to to allow allow them them To do do so, so, they they must must also sufficiently high to in the the metacommunity metacommunity with with superior superior competitors 1 980). to coexist coexist in competitors (Hastings, (Hastings, 1980). The perspective is is perhaps perhaps the the earliest view of com The patch-dynamic patch-dynamic perspective earliest view of aa meta metacommunity closely connected pioneer species species and munity and and is is closely connected to to ideas ideas about about fugitive fugitive or or pioneer and trade-offs abilities. Early trade-offs between between colonization colonization and and competitive competitive abilities. Early theoretical theoretical species coexistence work demonstrated demonstrated that that patch patch dynamics dynamics can can contribute contribute to to species coexistence work ((Caswell, Caswell, 11978; 978; Yodzis, 978). Recent Yodzis, 11978). Recent theoretical theoretical work work related related to to this this per perspective 1994), Kinzig al. ((1999), 1 999), Yu spective includes includes studies studies by by Tilman Tilman ((1994), Kinzig et etal. Yu and and Wilson Wilson (2001 (2001),) , Levine Levine and and Rees Rees (2002), (2002), and and Holt Holt (2002). (2002). The The primary primary result result of colon of these these studies studies is is that that although although aa trade-off trade-off between between competitive competitive and and colonabilities can (and potentially can allow allow for for coexistence coexistence of of species species (and potentially many many of of ization abilities them), them), the the range range of of parameters parameters that that allow allow coexistence coexistence by by this this mechanism mechanism is is surprisingly surprisingly limited. limited. The The patch-dynamic patch-dynamic perspective perspective is is also also associated associated with with certain certain types types of of ecosystems: ecosystems: the the concept concept of of habitat habitat patches patches connected connected by by migration migration underlies underlies our our understanding understanding of of hard-substrate hard-substrate intertidal intertidal systems systems (Paine 9 8 1 ) , the Connell, 11978; 978; (Paine and and Levin, Levin, 11981), the role role of of gaps gaps in in tropical tropical forests forests ((Connell, Denslow, 980), and Denslow, 11980), and effects effects of of small small mammals mammals on on certain certain plant plant communities communities (Platt 988; Huntley, 99 1 ) . (Platt and and Weis, Weis, 1977; 1977; Goldberg Goldberg and and Gross, Gross, 11988; Huntley, 11991).
The T h e Species-Sorting S p e c i e s - S o r t i n g Perspective Perspective If If connected connected habitat habitat patches patches are are heterogeneous heterogeneous in in some some important important charac character, ter, such such that that different different species species are are favored favored in in different different patches, patches, then then increased increased migration sorting" or migration can can lead lead to to aa ""sorting" or matching matching of of species species with with their their favored favored habitats. habitats. This This species-sorting species-sorting view view focuses focuses on on the the role role of of patch-type patch-type hetero heterogeneity geneity and and examines examines how how species' species' distributions distributions among among patches patches are are related related to to their their relative relative abilities abilities to to exist exist and and interact interact successfully successfully interact interact with with other other
11 336 6
MATHEW THOMAS E. MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS E. MILLER MILLER
species in the larger metacommunity Clements, 11916; 9 1 6; Gleason, 9 1 7; Tilman, metacommunity ((Clements, Gleason, 11917; Tilman, 1980). number of issues in 1980). A A number of issues in ecology ecology take take on on different different interpretations interpretations when when viewed viewed in in aa metacommunity metacommunity framework. framework. Some Some of of these these are are highlighted. highlighted. For For species species sorting sorting to to occur, occur, the the migration migration of of species species must must be be sufficient sufficient to to distrib distribute ute species species among among patches patches but but also also be be insufficient insufficient to to affect affect local local population population abundances. abundances. In In this this view, view, metacommunity metacommunity structure structure emerges emerges as as aa simple simple cumu cumulative outcome outcome of all local interactions, interactions, and and colonization-extinction colonization-extinction dynamics are only important important for determining determining the transient transient dynamics dynamics of the system until steady-state behavior metacommunity, it reaches a steady state. The steady-state behavior of the metacommunity, including including the distribution distribution of species among among patches, patches, is thus thus unrelated unrelated to rates of of colonization colonization and and extinction. extinction. Under Under species species sorting, sorting, immigration immigration may may have have within-patch diversity but but will be important important for for deter deterlittle effect on local, within-patch mining species identity identity within within patches. Work on this view includes that that by Tilman ala ((1993), 1 993), Holt 1 993, 11997), 997), and Leibold ((1996, 1 996, 1998). Tilman and and Pac Pacala Holt ((1993, and Leibold 1998).
The Mass Effects" The ""Mass Effects" Perspective Perspective In In this this view, view, patch-type patch-type heterogeneity heterogeneity is is also also important, important, but but migration migration is is suf sufficiently are ficiently high high that that local local population population abundances abundances of of component component species species are affected net emigration affected by by net emigration (producing (producing "source" "source" populations) populations) and and net net immigra immigrapopulations").) . Under Under these conditions, conditions, metacommunity metacommunity tion (yielding "sink populations" structure structure is is affected affected in in two two ways. ways. First, First, species species that that might might be be expected expected to to do do well closed community emigration or well in in aa closed community (with (with no no emigration or immigration) immigration) can can be be driven driven to extinction from superior by to extinction from local local patches patches where where they they are are competitively competitively superior by strongly strongly subsidized subsidized "sink "sink populations" populations" supported supported by by immigration immigration from from other other patches. Second, Second, such patches. such competitively competitively superior superior species species can can have have reduced reduced compet competitive itive abilities abilities in in metacommunities metacommunities because because of of the the emigration emigration of of individuals individuals (equivalent (equivalent to reduced local birth birth rates) as well. The overall result is that that mass effects can increase local diversity diversity (Mouquet and Loreau, 2002) 2002) by such "sub "subsidized" sidized" population population dynamics. dynamics. The The issue issue is is aa bit bit more more complicated, complicated, however, however, because require very emigration that itself because such such strong strong subsidy subsidy can can require very strong strong emigration that can can itself make local populations populations vulnerable to extinction. Thus Thus these mass effects effects can reduce reduce the the diversity diversity over over broader broader spatial spatial scales scales (the (the metacommunity metacommunity as as aa whole) whole) and reduce local local diversity mass effects and therefore therefore reduce diversity despite despite strong strong mass effects (Mouquet (Mouquet and and Loreau, 1 993), Loreau, 2002). 2002). Research Research work work on on this this perspective perspective includes includes that that by by Holt Holt ((1993), Amarasekare Amarasekare and Nisbet Nisbet (2002), and Mouquet Mouquet and Loreau (2002).
The The Neutral Neutral Perspective Perspective Of course, all presumes that Of course, all of of this this theory theory presumes that species species differ differ from from one one another another important ways, including their responses to local conditions conditions (patch-type in important specialization specialization and/or and/or competitive competitive superiority) superiority) and to the landscape landscape structure structure (migration (migration and background background extinction rates). One possible view is that that species not differ differ in any important important respect in any of these ways (Hubbell, 11997, do not 997, 200 1 ) . Although 2001). Although unlikely unlikely to to be be universally universally true true (see (see Zhang Zhang and and Lin, Lin, 1997; 1997; Yu Yu et aI., 998), this et al., 11998), this view view does does highlight highlight how how important important stochastic stochastic processes processes can can be be in in structuring structuring metacommunities metacommunities and and may may be be important important in in explaining explaining some some important important aspects aspects of of metacommunities, metacommunities, including including abundance abundance distributions distributions and and incidence aI., 2002). incidence functions functions (Hubbell, (Hubbell, 2001 2001;; see see Chave Chave et et al., 2002).
6. 6.
FROM M ETAPOPULATIONS TO TO METACOMMUNITIES METACOMMUNITIES FROM METAPOPULATIONS
11 337 7
A Missing Missing Synthesis Synthesis A Obviously, all all of of these these perspectives perspectives are are simplifications, simplifications, and and real real metacommetacom Obviously, munities are are likely likely to to be be structured structured by by some some mixture mixture of of the the processes processes that that are are munities by each each of of them. them. The The issue issue is is made made more more complex complex because because species species highlighted highlighted by in real real communities may interact interact with each other fundamen in communities may with each other despite despite having having fundamentally different migration rates. rates. For For example, example, aa predator predator may may have have aa migration migration tally different migration rate sufficiently high that that itit has has aa metapopulation metapopulation structure and significant significant rate sufficiently high structure and local population population regulation regulation due due to to migration, migration, while while itit interacts interacts with with prey prey that that local have much much lower lower migration migration rates rates and and in in which which local local population population regulation regulation is is have completely determined determined by by local local processes processes [see [see Holt Holt (1997) ( 1 997) and and van van Nouhuys Nouhuys completely and Hanski Hanski (2002) (2002) for for an an empirical empirical example]. example] . To To date, date, there there is is little little underunder and standing of of how how to to synthesize synthesize these these various various perspectives perspectives into into aa general general frameframe standing work, even even though though there there is is some some recognition recognition that that such such aa synthesis synthesis is is important important work, (Holt, 1997). 1 997). Surprisingly Surprisingly little little work work is is done done at at the the interface interface of of the the speciesspecies (Holt, sorting and patch-dynamics perspectives (but ( but see Holt, 1997; 1 997; Levine Levine and and Rees, Rees, sorting and patch-dynamics perspectives see Holt, 2002; Shurin Shurin et 2003 ). Work Work by by Amarasekare Amarasekare and and Nisbet Nisbet (200 1 ) and by 2002; et aI., al., 2003). (2001) and by Mouquet and and Loreau examines the interface between between species species sorting sorting Mouquet Loreau (2002) (2002) examines the interface and mass effects. It is is as yet unclear unclear how how to to integrate integrate these these various various perspecperspec and mass effects. It as yet tives with the stochastic processes described by Hubbell Hubbell (200 1 ), although although tives with the stochastic processes described by (2001), work by by Chave Chave et ai. (2002) (2002) indicates indicates that that the the effects can be be subtle. work et al. effects can subtle.
6.3 6.3
THEORETICAL METACOMMUNITIES THEORETICAL ISSUES ISSUES ABOUT ABOUT METACOMMUNITIES
The The Importance Importance of of Trade-Offs Trade-Offs about trade-offs trade-offs from a more conventional conventional It is helpful to start thinking about basis that basis that ignores ignores metacommunities. metacommunities. Traditional Traditional ecological ecological theory theory about about closed closed communities communities has has drawn drawn attention attention to to trade-offs trade-offs involving involving relative relative biotic biotic abilities. Within abilities. Within aa local local site, site, this this view view involves involves conventional conventional mechanisms mechanisms of of resource resource partitioning partitioning and and associated associated ecological ecological traits traits such such as as susceptibility susceptibility to to predators and and so forth. forth. At the local scale, the nature of these trade-offs trade-offs will predators determine species combinations stability of determine which which species combinations are are possible, possible, what what the the stability of the the system be, and responds to environmental change. system might might be, and how how the the community community responds to environmental change. Such Such differences differences in in responses responses and and impacts impacts on on resources resources and and other other fitness fitnessrelated related factors factors (this (this suite suite of of traits traits is is subsumed subsumed into into aa general general term term we we call call ""biotic biotic ability " ) can ability") can also also lead lead to to habitat habitat (patch-type) (patch-type) specialization specialization (involving (involving coexistence coexistence at at the the larger larger regional regional scale scale even even if if there there is is no no local local coexistence). coexistence). We species sorting" We can can think think of of this this process process as as one one of of ""species sorting" that that regulates regulates how how well well matched matched species species are are to to their their local local environments. environments. Although Although ecologists ecologists are are familiar familiar with with this this conventional conventional approach approach to to niche niche dynamics, dynamics, they they do do not not always always recognize recognize that that trade-offs trade-offs can can also also be be important important in in regulating regulating meta metacommunity community dynamics. dynamics. This This chapter chapter reviews reviews theory theory and and empirical empirical work work that that show show how how important important such such relative relative habitat habitat specialization specialization is is in in regulating regulating meta metacommunity community dynamics dynamics even even when when other other important important processes processes are are also also operating. operating. We We can can also also approach approach questions questions about about trade-offs trade-offs from from the the perspective perspective of of meta population theory metapopulation theory based based on on single single species. species. This This theory theory draws draws attention attention to to the the tension tension between between colonization colonization ability ability and and susceptibility susceptibility to to stochastic stochastic extinction extinction in in regulating regulating the the persistence persistence of of populations populations in in patchy patchy landscapes landscapes where where aa variety variety
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
11 38 38
allow species to persist persist in a given given landscape ((Chapter of strategies can allow Chapter 4). Similarly, population theory Similarly, meta metapopulation theory oriented oriented toward toward small small numbers numbers of of interacting interacting species species has has shown shown that that trade-offs trade-offs involving involving colonization colonization ability ability and and biotic biotic ability ability within 980; Tilman, 994; Taneyhill, within patches patches can can also also be be important important (Hastings, (Hastings, 11980; Tilman, 11994; Taneyhill, 2000; 2000; but but see see Klausmeier, Klausmeier, 2001; 2001; Yu Yu and and Wilson, Wilson, 2001; 2001; Levine Levine and and Rees, Rees, 2002). 2002). These These dynamics dynamics affect affect species species sorting sorting in in two two opposite opposite ways: ways: increased increased competi competitive tive ability ability can can be be related related to to increased increased specialization, specialization, but but high high rates rates of of migration migration may decrease the degree to which local sites are occupied by species that are best matched matched to to local local conditions conditions (see (see Mouquet Mouquet et et aI., al., 2002). 2002). Thus Thus connectivity connectivity in in aa metacommunity metacommunity can can change change conditions conditions from from those those that that favor favor habitat habitat specializa specialization and coexistence to those that favor species with traits related to overall biotic ability ability in in the the metacommunity metacommunity as as aa whole whole (including (including dispersal dispersal as as well well as as average average extinction rates and biotic ability over all patches). The The aforementioned aforementioned considerations considerations suggest suggest that that three three broad broad categories categories of of species species differences differences are are all all potentially potentially important important in in regulating regulating the the distribution distribution of species in of species in metacommunities: metacommunities: patch patch specialization specialization by by means means of of biotic biotic ability, ability, migration migration among among patches, patches, and and likelihood likelihood of of local local extinctions. extinctions. These These admit admittedly broad categories and lead lead to tedly broad categories could could probably probably be be subdivided subdivided further further and to even even more process of could be more possible possible traits; traits; for for example, example, the the process of migration migration could be dissected dissected component (Chapter (Chapter 113). 3). into an emigration and an immigration component now consider consider multiple patch patch types where these species differences can If we now effects, then we can consider traits and and trade-offs trade-offs that that may be have various effects, related to these species related to these species differences. differences. For For example, example, expression expression of of each each of of the the two different types of patches produces three categories in two produces six traits or ways that species can differ that influence their persistence and coexistence: that 1. 1. 2. 2. 3. 4. 5. 5. 6.
patch-type 11 Biotic ability in patch-type Biotic ability in patch-type patch-type 2 Migration ability into into patch-type patch-type 1 Migration ability into patch-type 2 Susceptibility to stochastic extinction in patch-type patch-type 1 to stochastic stochastic extinction in patch-type Susceptibility to patch-type 2
It is is important important to to understand understand that biotic ability that can can lead lead to to extincextinc It that any any biotic ability that tions (traits (traits 11 and 2) in in many many metacommunity metacommunity models models is is substantially distinct tions and 2) substantially distinct from from susceptibility to to stochastic stochastic extinctions extinctions (traits (traits 55 and and 6). 6). The The latter latter is is viewed as as being being independent independent of of biotic biotic interactions, interactions, due due either either to to stochastic stochastic viewed effects effects acting acting on on the the demography demography of of small small populations populations or or (perhaps (perhaps just just as as commonly) commonly) to to environmental environmental change change within within local local sites, sites, including including disturbances disturbances that affect large that can can affect large populations populations as well. Taken in in pairwise pairwise combinations, combinations, this this list list of of six six traits traits can can lead lead to to 15 15 possible possible Taken trade-offs between between species, species, and and more more of of them them involve involve niche niche axes axes that that operate at trade-offs operate at the the larger larger regional scale. This This perspective perspective results in many many more more mechanisms mechanisms that that can can regulate regulate coexistence coexistence and and metacommunity dynamics than than are considconsid ered ered in in local-scale local-scale models. models. To To date, date, metapopulation metapopulation theory theory has has addressed addressed only only some of with any degree of of these possibilities possibilities with any degree of effort. effort. Current work work on on metacommunities metacommunities reveals reveals important important interdependent interdependent concon Current straints on on each each of of these these possibilities. possibilities. Two Two features features are are important. important. First, First, the the straints behavior of of the the system system in in relation relation to to these these traits traits can can be be complex complex because because the the behavior behavior behavior of of metacommunities metacommunities may may vary vary in in response response to to these these trade-offs trade-offs in in ways ways
6. 6.
FROM METAPOPULATIONS M ETAPOPULATIONS TO METACOMMUNITIES FROM TO METACOMMUNITIES
11 339 9
that are are not not always always consistent. consistent. For For example, example, the the colonization-rate-competitivecolonization-rate--competitive that trade-off can can be important important in allowing allowing species to to coexist in a metacommetacom ability trade-off munity when when local populations populations are are subject to to stochastic stochastic extinctions, extinctions, thus thus munity allowing competitively subordinate subordinate species to to coexist with with competitive competitive dominant dominant allowing species. Under Under this this scenario, scenario, the the constraint constraint for for coexistence coexistence in in the the metacommunity metacommunity species. is that that the the colonization colonization rate rate of of the the subordinate subordinate exceed exceed some some critical critical value value that that is allows it it to to exist exist as as aa fugitive fugitive species species in in the the metacommunity metacommunity (i.e., (i.e., that that the the migramigra allows tion rate rate time time scale scale be be much much shorter shorter than than that that at at which which species species interactions interactions lead lead tion to extinction). extinction). The The colonization-rate-competitive-ability colonization-rate--competitive-ability trade-off trade-off can can also also allow allow to species to to coexist coexist when when colonization rates are are high high enough enough to to alter local abunabun species colonization rates alter local dances (through mass mass effects). effects). However, However, under this scenario, colonization by by the the dances (through under this scenario, colonization subordinate species species must must be be low low enough enough to to prevent prevent it it from from subsidizing subsidizing large sink subordinate large sink populations in in patches patches occupied by competitive competitive dominants dominants (Amarasekare (Amarasekare and and populations occupied by Nisbet, 2001). 200 1 ) . Nisbet, Second, models models based based o one mechanism mechanism can can also bbee strongly strongly altered altered bbyy the the Second, onn one presence of mechanisms involving other trade-offs. trade-offs. For example, involving other example, Law and Leibold (in press) have found found that that a metacommunity metacommunity of species with with nontrannontran sitive sitive assembly rules rules (species (species A invades and and excludes excludes species species B, B, species B invades and and excludes species C, and species C invades and and excludes species A) shows cycles in in the the frequency frequency of of patch patch types types (a (a metacommunity metacommunity becomes becomes domdom shows cycles turn, by sites containing and then then inated, in turn, containing species A, species B, species C, and background stochastic extinctions, species A again). In the absence of background extinctions, the amplitude of these cycles in patch patch occupancy varies in a neutral neutral manner manner (such amplitude as neutral limit of the the Lotka-Volterra Lotka-Volterra predator-prey models) . as the the neutral limit cycles cycles of predator-prey models). However, in in the such extinctions, the cycles to aa stable stable However, the presence presence ooff such extinctions, the cycles converge converge to point point at at which which the the frequency frequency of of occupancy occupancy states states is is determined determined by by the the relative relative colonization and colonization and stochastic stochastic extinction extinction probabilities probabilities of of the the three three species. species. Thus Thus the (under nontransitive the outcome outcome of of aa model model of of species species interactions interactions (under nontransitive assembly) assembly) is background extinc is strongly strongly modified modified by by the the presence presence of of patch patch dynamics dynamics ((background extinctions tions unrelated unrelated to to species species interactions). interactions).
The The Role Role of of Migration: Migration: From From Patch Patch Dynamics Dynamics to to Species Species Sorting Sorting to to Source-Sink Source-Sink Relations Relations A broader broader review of the theory on metacommunities metacommunities is beyond the scope of of this this chapter chapter (such (such aa review review is is under under preparation preparation by by Holyoak, Holyoak, Leibold, Leibold, and and Holt), but two two important important conclusions can be drawn about the dynamics of metacommunities. Loreau and metacommunities. First, First, the the role role of of migration migration is is not not simple simple ((Loreau and Mouquet, 999; Mouquet Mouquet, 11999; Mouquet and Loreau, 2002; Leibold and Norberg, Norberg, 2003) 2003):: Fig. . 1 presents Fig. 66.1 presents aa hypothetical hypothetical relationship relationship between between migration migration rate rate and and diversity. diversity. Clearly, Clearly, when when there there is is no no migration, migration, then then the the metacommunity metacommunity per perspective spective is is not not needed, needed, the the dynamics dynamics of of local local communities communities are are entirely entirely inde independent pendent of of one one another, another, and and diversity diversity in in local local communities communities is is expected expected to to be be relatively relatively low low due due to to migration migration limitation. limitation. At At minimal minimal levels levels of of migration, migration, every every species species can can potentially potentially get get to to every every patch, patch, but but the the order order of of colonization colonization and and relative relative competitive competitive abilities abilities become become important important in in determining determining local local community community structure. structure. The The result result is is some some degree degree of of species species sorting sorting in in which which migration may not enhance diversity but can allow significant species replace replacement ment and and some some specialization. specialization. In In particular, particular, "fugitive "fugitive species" species" are are possible possible
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
140 140 i
9
i en en w Z I o 0: en w (3 W 0... en
migration migration limitation limitation
f.
species species sorting sorting
i :
mass mass effects effects
i i
metacommunity metacommunity homegenization homegenization
--�--
i
9
9
i
MIGRATION MIGRATION RATE RATE 6.1 A hypothetical between migration Fig. hypothetical relationship relationship between migration rate and species richness illustrating illustrating Fig. 6.1 the major major mechanisms mechanisms by which migration affects community community structure. structure9 the which migration
under these conditions, conditions, species that that exist in the metacommunity metacommunity by virtue of under temporary temporary existence existence in patches that that have not yet been colonized colonized by patch-type specialists (Hastings, 11980). 980). In contrast, as migration increases until all species can disperse relatively quickly to all patches, the local assembly of communities colonization ((because because the communities is much much less affected by the the order order of of colonization the assembly process equilibrium behavior behavior relatively quickly, process can reach reach its final final equilibrium typically uninvasible community community but but also also possibly possibly aa pattern pattern of of repeated repeated typically an an uninvasible heteroclinic cycles), and heteroclinic and classic fugitive species are excluded by patch-type specialists. At yet higher rates, migration migration plays a different different role, being high enough to contribute growth rates. This allows species contribute significantly to local growth that inferior competitors that are locally inferior competitors to to persist through through mass effects (i.e., Mouquet and Loreau, 2002), 2002), increasing local diversity (Fig. (Fig. 6.1). 6 . 1 ) . At Mouquet and Loreau, At this migration may act to decrease diversity of the entire meta point, increased migration entire metacommunity, with consequential effects on local diversity as well. In the community, with consequential when migration migration is so high that that local populations populations are are completely completely extreme, when mixed, metacommunity metacommunity diversity (and thus thus local diversity) collapses into into that that mixed, expected expected in a single closed community. community.
The Importance Importance of of Habitat Habitat Heterogeneity Heterogeneity and Patch-Type Specialization The second second important conclusion is that habitat heterogeneity, heterogeneity, when The important conclusion that habitat when it is present, almost almost always always plays an an important important role role in maintaining maintaining diversity in present, metacommunities. Heterogeneity Heterogeneity allows allows habitat habitat specialization, specialization, which which can can metacommunities. provide a local local refuge refuge for for some some specialist specialist species and and increase increase metacommunity metacommunity provide (although not not local) local) diversity. At At higher higher rates rates of of migration, migration, such such refuges refuges can can (although act act as sources sources that that help help maintain maintain populations populations in less favorable favorable patches patches and and
ETAPOPULATIONS TO 6. 6. FROM FROM M METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES
141
increase increase both both local local and and metacommunity metacommunity richness richness (Mouquet (Mouquet and and Loreau, Loreau, 2002). In addition, addition, metacommunity metacommunity dynamics dynamics involving stochastic stochastic extinctions extinctions at local scale at the the local scale prevent prevent the the occurrence occurrence of of alternative alternative stable stable states states in in local local communities Shurin et communities in in the the absence absence of of patch-type patch-type heterogeneity heterogeneity ((Shurin et aI., al., 2003 2003),), eliminating eliminating another another potentially potentially important important mechanism mechanism that that might might maintain maintain biodiversity in homogeneous homogeneous patch-type patch-type metacommunities. metacommunities. However, in high biodiversity the the presence presence of of patch-type patch-type heterogeneity, heterogeneity, which which can can prevent prevent exclusion exclusion at at the the metacommunity metacommunity scale, scale, such such alternative alternative stable stable states states can can act act to to increase increase compositional compositional diversity even among among patches patches of the same type. In both both cases, diversity diversity is increased only if patch-type patch-type heterogeneity heterogeneity is sufficient to allow the "source" populations sink" populations. "source" populations that that support support ""sink" populations. Further, even under under these these conditions, conditions, migration migration is is constrained constrained in in direct direct relation relation to to the the degree degree of of habitat habitat heterogeneity heterogeneity (Amaresekare (Amaresekare and and Nisbet, Nisbet, 2001 2001;; Mouquet Mouquet and and Loreau, Loreau, 2002; 2002; Levine Levine and and Rees, Rees, 2002 2002).) . Therefore, Therefore, even iiff we limit ourselves ttoo considering considering only the six types of metacommunity-related metacommunity-related traits listed earlier, we obtain obtain a complex complex set of results about about metacommunity metacommunity dynamics dynamics and and the the ways ways they they alter alter patterns patterns of of coexist coexistence ence and and biodiversity. biodiversity. This This theoretically theoretically rich rich area area of of understanding understanding the the effects effects of trade-offs trade-offs at different spatial scales is only now now being addressed, addressed, and a number number of of important important results results may may soon soon be be forthcoming. forthcoming.
AND EXPERIMENTAL 6.4 OBSERVATIONAL OBSERVATIONAL AND EXPERIMENTAL EVIDENCE EVIDENCE 6.4 ABOUT ETACOMMUNITIES ABOUT M METACOMMUNITIES Conventional Conventional Analyses Analyses of of Species Species Distributions Distributions in in Relation Relation to to Patch-Type Patch-Type Heterogeneity Heterogeneity Empirical work on metacommunities metacommunities predates predates the metacommunity metacommunity concept concept most basic types of data data that that describe metacommunities metacommunities can itself. One of the most be summarized incidence matrix," matrix," a matrix summarized as an ""incidence matrix describing which which species are Bray and 957; Leibold are found found at at each each of of aa variety variety of of sites sites ((Bray and Curtis, Curtis, 11957; Leibold and and Mikkelson, Mikkelson, 2002). 2002). There There are are sophisticated sophisticated ways ways of of analyzing analyzing such such data, data, often often in null models models ((Gotelli Gotelli and 996) or in relation relation to to null and Graves, Graves, 11996) or environmental environmental gradi gradients 998). Although ents (see, (see, e.g., e.g., Legendre Legendre and and Legendre, Legendre, 11998). Although these these data data have have strong strong empirical roots roots and have long been implicitly related related to mechanisms mechanisms important important in in metacommunity metacommunity dynamics, dynamics, the the link link between between incidence incidence matrices matrices and and metacommunities metacommunities has has generally generally not not been been made made explicit. explicit. A good summary summary of this large body of work work is beyond beyond the scope of this chapter, but but a few points points relevant relevant to the metacommunity metacommunity concept concept are highlighted. First, this body body of work work shows shows that that species associations associations in local highlighted. habitats aI., 2002 2002 and habitats are are typically typically not not random random (for (for summaries, summaries, see see Gotelli Gotelli et et al., and Leibold and Mikkelson Mikkelson 2002), 2002), suggesting that that species interactions interactions or similar similarity ity of of biotic biotic requirements requirements affects affects species species distributions distributions and and could could affect affect larger larger scale scale (metacommunity) (metacommunity) dynamics. dynamics. Second, Second, environmental environmental variability variability in in abiotic abiotic factors factors (heterogeneity (heterogeneity of of patch patch types) types) is is often often important important in in explaining explaining the the dis distribution 957; Whitaker, tribution of of species species among among sites sites (see, (see, e.g., e.g., Bray Bray and and Curtis, Curtis, 11957; Whitaker, abundance of individual individual species or suites of species can be shown shown 11975). 975) . The abundance to to covary covary significantly significantly with with factors factors such such as as nitrogen nitrogen availability availability or or soil soil
MATHEW MATHEW A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER
14Z 142
moisture. moisture. Nevertheless, Nevertheless, some some spatial spatial effects effects may may be due purely purely to effects of of migration aI., 11992; 992; Pinel aI., 11995). 995) . migration (Borcard (Borcard et et al., Pinel Alloul Alloul et et al., Important Important descriptive descriptive studies studies ooff biodiversity biodiversity aatt different different scales indicate indicate a between local and regional complex relationship relationship between regional patterns of of diversity. For example, Shurin (Shurin, 2000; 1) example, 2000; Shurin et aI., al., 2000; 2000; Shurin and Allen, 200 2001) has shown shown that commonly commonly observed observed linear relationships relationships between between local and regional uninvasible local com regional diversity diversity were not necessarily associated with with uninvasible communities, but rather could explained as resulting from could be explained from a complex complex assembly process process in in metacommunities metacommunities involving involving multitrophic-Ievel multitrophic-level food food webs webs (see (see also also Srivastava, 999; Mouquet aI., 2003 Srivastava, 11999; Mouquet et et al., 2003).). In In another another example, example, Chase Chase and and Leibold (2002) (2002) showed showed that local and regional regional diversity diversity had qualitatively qualitatively different different relationships relationships with with productivity; productivity; the local relationship relationship was was unimodal unimodal and regional relationship relationship increased increased monotonically (Fig. 6.2 and the the regional monotonically (Fig. 6.2).) . Steiner Steiner and and 3 ) showed Leibold (200 (2003) showed that this situation situation can result from from metacommunity metacommunity dynamics nontransitive invasion invasion cycles cycles (see 997) dynamics in in which which nontransitive (see Morton Morton and and Law, Law, 11997) are 1 997, 200 1 ) has are more more common common at at high high productivity. productivity. Hubbell Hubbell ((1997, 2001) has also also a a
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Fig. 66.2 Fig. . 2 Results Resultsfrom a survey of pond species diversity relative to in in situ situ primary productiv productivity at local local and regional scales. scales. (Top) Producers (vascular plants and macroalgae). (Bottom) (Bottom) Benthic animals (insects, crustaceans, amphibians, and so on). on). (a) Local, Local, within-pond, within-pond, species 30). Both relationships are significantly unimodal diversity (N (N = 30). unimodal (P (P < < 0.05). 0.05). The line represents the estimated quadratic function. (b) (b) Regional, within-watershed, within-watershed, species species diversity. For both producers (regression: N 0, RR22 = 74, PP < 0, N = = 110, = 0. 0.74, < 0.001 0.001)) and benthic animals (regression: N N = = 110, 2 RR2 = 0.75, ), regional species species diversity was related linearly to primary productivity. 0.75, P P< < 0.001 0.001), From Chase and Leibold (2002). (2002).
FROM M ETAPOPULATIONS TO ETACOMMUNITIES 6. FROM METAPOPULATIONS TO M METACOMMUNITIES 6.
143 143
reinterpreted reinterpreted patterns of diversity using a novel neutral metacommunity per perspective in which which species have equal effects on one another another and the habitat habitat is homogeneous 1 ). The ability of Hubbell's models homogeneous (see (see also Bell Bell 2000, 2000, 200 2001). to predict predict patterns of relative abundances abundances suggests that that many community patterns patterns are created by random random "sampling" of larger species pool and cannot cannot be used to imply mechanisms related to species interactions. interactions.
Observational Explicit Metacommunity Observational Studies Studies with with Explicit Metacommunity Approaches Approaches A few studies of unmanipulated unmanipulated systems have specifically used a metacom metacommunity framework for the interpretation interpretation of regional patterns. One of the more important aI. (2001 important of these is work by Cottenie et al. (2001)) on zooplankton zooplankton metacom metacommunity structure structure in a set of interconnected interconnected ponds. They found found that that individual ponds showed strong habitat habitat heterogeneity related to predator predator composition; some ponds ponds had dense planktivorous planktivorous fish populations populations that may have led to high through trophic cascades and other ponds had lower planktivo planktivoalgal densities through rous fish densities leading to higher densities of large grazers and consequent consequent lower algal densities. More importantly, they quantified quantified the amount amount of migra migraheterogeneous ponds to determine whether whether migration increases tion among heterogeneous diversity through through source-sink dynamics. Despite measuring high rates of migra migrapond to the other), they only rarely found found evi evition (via (via stream flow from one pond dence for such source-sink dynamics. This work work indicates that the strength of species sorting between these different patch types (pond types) is remarkably strong and cannot cannot be affected easily by the homogenizing effects of migration. Other Other patterns patterns in natural natural systems may take on novel significance when interpreted interpreted in the context of metacommunity metacommunity theory, including those related to abundance patterns patterns (Hubbell, 2001 2001;; Chave and Leigh, 2002), 2002), to similarity and coexistence (see 998), and to ecosystem attributes (see Leibold, 11998), attributes (see (see Mouquet Mouquet et al., aI., 2002; Leibold et al., aI., 11997). 997). Nevertheless, Nevertheless, using metacommunity metacommunity theory theory interpret natural natural patterns patterns of variation variation is speculative speculative until this body of to interpret theory and its principal mechanisms have been evaluated evaluated by means of experi experimental manipulations. A tremendously tremendously rich array of work work remains to be done, work in which which experiments experiments are specifically designed to test mecha mechaespecially work nisms of metacommunity dynamics.
Experimental Experimental Work Work with with Metacommunities Metacommunities A variety of experimental studies studies have have been been conducted on metacommuni metacommuni1 95 8 ) work ties, going back to Huffaker's ((1958) work with predator-prey predator-prey systems of relamites maintained on increasingly complex arrays of oranges. However, rela tively few such manipulative manipulative studies have been conducted conducted with with more than than two species in true metacommunities, metacommunities, where a closed system of local communities is linked by migration. Yet such studies are necessary for appropriate appropriate tests of factors identified identified by models as being important important for predicting species persist persistence and diversity in metacommunities. metacommunities. The theoretical theoretical work work described earlier indicates indicates that that metacommunity metacommunity dynamics dynamics are especially likely to result result from complex interactions among migration (or isolation), habitat habitat disturbance disturbance (or other mechanisms for local stochastic extinctions), patch-type heterogeneity, and the interrelation among species traits that might result in trade-offs.
MATHEW A A.. LEIBOLD LEIBOLD AND THOMAS EE.. MILLER MILLER
114141 44
Migration Migration among local communities communities is perhaps the hallmark of metacom metacommunities. Several theoretical studies explicitly predict that that increasing rates of migration will lead to increasing local diversity until the migration migration rate is so high that local variation among sites is swamped ((Caswell, Caswell, 11978; 978; Caswell and 991; Loreau and Mouquet, 1999; Mouquet Cohen, 11991; Mouquet and Loreau, 2002). Increased migration can increase diversity by at at least two mechanisms. If species are migration migration limited, then increased migration migration may allow species to gain access to communities where they were previously absent (Tilman, 11994; 994; Shurin, 2001; Miller et aI., al., 2002). Alternatively, migration may act to to augment or or replace local reproduction reproduction for some species, allowing them to persist in communities from which they would otherwise be excluded. At extremely high rates of migration, however, both local and regional (metacommunity) diversity will decline, decline, as regionally dominant dominant species exclude others. Several laboratory studies have varied migration rates among local communities, communities, with with mixed effects on local diversity. Warren ((1996a,b) 1996a,b) investigated the role of migration in two two laboratory studies of protist metacommunities, enforcing migration at various rates by moving fluid between separate local habitats. In both studies, diversity did increase with with increasing migration rate, but the effects were relatively small and depended on the magnitude magnitude of other factors, such as disturbance disturbance and habitat habitat size (Fig. (Fig. 6.3). Similarly, Forbes and Chase (2002) varied the degree of connectivity among zooplankton zooplankton communities in experimental microcosms and found that increased migration had no effect on local diversity but decreased regional diver diversity through a homogenization of local communities. Gonzalez and co-workers ((1998; 1 998; Gonzalez and Chaneton, 2002) experi experimentally communities of mentally varied varied migration migration rates rates in in fragmented fragmented communities of microarthro microarthropods that that occur in dense moss growth. They manipulated migration by arranging moss in isolated fragments fragments (very low migration), in fragments connected to larger areas by moss "corridors, "corridors,"" and in continuous continuous areas of moss (Fig. (Fig. 6.4). Lower migration resulted in the loss of rare species, and corridors corridors alleviated the effects of fragmentation greatly. This result indicates that that metacommunity metacommunity dynamics involving colonization-competition colonization-competition trade-offs
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6.3 Effects Fig. 6.3 Effects of disturbance and migration rate on local protist species species richness richness in meta metacommunities consisting of eight experimental 1100-ml 00-ml microcosms. Dispersal Dispersal took place in 2-ml aliquots of of fluid moved at 3-day intervals between either 6 (low dispersal, open bars) bars) or 24 (high dispersal, shaded bars) randomly chosen pairs of microcosms. Disturbance consisted of placing either 1I (low disturbance) or 2 (high disturbance) randomly chosen microcosms in hot water baths every 3 days to kill the protozoa (data adapted from 996a). from Warren, 11996a).
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Effect Effect of of fragmentation fragmentation on on species species richness richness in in aa moss moss invertebrate invertebrate community. community. Local Local communities continuous with communities were were either either continuous with surrounding surrounding moss, moss, isolated isolated but but connected connected with with moss moss corridors, corridors, or or isolated isolated with with corridor. corridor. Figure Figure adapted adapted from from Gonzalez Gonzalez and and Chaneton Chaneton (2002). (2002).
were probably not small were probably not important. important. Kneitel Kneitel and and Miller Miller (2003) (2003) studied studied the the small communities communities of of protists protists and and rotifers rotifers found found in in the the water-filled water-filled leaves leaves of of pitcher pitcher plants. plants. They They moved moved small small volumes volumes of of water water among among suites suites of of leaves leaves at at three three different different rates, rates, while while also also manipulating manipulating the the presence presence of of predators predators and and the the input input of resources. Local of resources. Local diversity diversity was was highest highest at at intermediate intermediate levels levels of of migration, migration, although predators reduced reduced the although the the presence presence of of predators the effects effects of of migration. migration. Metacommunity Metacommunity dynamics dynamics were were therefore therefore important important in in regulating regulating local local diver diversity, habitats must understood in sity, but but the the effects effects of of migration migration among among habitats must be be understood in light light of local biotic biotic factors. of local factors. Disturbance open" Disturbance can can immediately immediately reduce reduce densities, densities, but but it it can can also also create create ""open" habitats habitats that that are are available available to to all all species species for for colonization colonization and and may may allow allow fugitive Connell, 11978). 978). Disturbance fugitive species species to to persist persist in in metacommunities metacommunities ((Connell, Disturbance is is often often considered considered in in continuous continuous communities communities in in which which smaller smaller scale scale disturb disturbances ances create create open open patches patches in in aa larger larger community community matrix. matrix. Although Although not not usually usually described described as as doing doing so, so, such such disturbances disturbances create create aa metacommunity metacommunity scenario scenario for for the the community community of of species species that that depend depend on on disturbances disturbances for for persistence persistence in in the the community. propagules to community. Such Such species species must must persist persist by by moving moving or or dispersing dispersing propagules to new new locally locally disturbed disturbed sites sites before before they they are are excluded excluded from from their their current current sites. sites. disturbance in this type of metacommunity should should result in a The effects of disturbance maximum diversity of maximum diversity of species species at at some some intermediate intermediate rate rate of of local local habitat habitat disturbance 984). For 1 996a) labora disturbance (Sousa, (Sousa, 11984). For true true metacommunities, metacommunities, Warren's Warren's ((1996a) laboratory tory protist protist studies studies varied varied the the rate rate of of destruction destruction and and reestablishment reestablishment of of local local communities communities and and showed showed that that disturbance disturbance (local (local patch patch destruction) destruction) resulted resulted only species loss only in in species loss from from the the metacommunity, metacommunity, but but disturbance disturbance was was important important protist richness (Fig. 6.3). in determining the effects of migration on protist Heterogeneity Heterogeneity among among local local habitat habitat patches patches could could be be caused caused by by aa variation variation in in biotic biotic or or abiotic abiotic conditions conditions and and is is predicted predicted to to increase increase metacommunity metacommunity diversity local communities diversity by by increasing increasing the the variety variety of of local communities and and forming forming refuges refuges for habitat specialists Loreau, 2002). for habitat specialists (Mouquet (Mouquet and and Loreau, 2002). Only Only two two studies studies have have directly Forbes and and directly manipulated manipulated heterogeneity heterogeneity within within aa metacommunity. metacommunity. Forbes Chase Chase (2002) (2002) varied varied the the arrangement arrangement of of experimental experimental zooplankton zooplankton communi communities ties to to influence influence the the probability probability of of moving moving from from similar similar to to dissimilar dissimilar habitats: habitats: arrangement arrangement had had no no effect effect on on local local or or regional regional diversity. diversity. Miller Miller et et al. al. (2003 (2003))
46 11 46
MATHEW N D THOMAS MATHEW A A.. LEIBOLD LEIBOLD A AND THOMAS EE.. MILLER MILLER
used used suites suites of of small small laboratory laboratory microcosms microcosms with with protist protist communities communities linked linked by by weekly weekly migration. migration. Two Two types types of of heterogeneity heterogeneity were were maintained: maintained: either either all all local local communities community were communities composing composing aa given given meta metacommunity were maintained maintained at at the the same same medium medium resource resource level level or or equal equal numbers numbers of of local local communities communities provided provided low, low, medium, and high high resource and Chase medium, and resource levels. levels. Like Like Forbes Forbes and Chase (2002), (2002), they they found found that variation had local diversity: diversity: species that such such variation had no no effect effect on on local species were were found found to to spe specialize cialize on on different different resource resource levels levels and and migration migration among among different different resource resource lev levels els had had little little effect effect on on establishment. establishment. Heterogeneity Heterogeneity did did lead lead to to higher higher regional regional diversity, because aa greater diversity, however, however, because greater regional regional diversity diversity of of resources resources allowed allowed lowpersist in low- and and high-resource high-resource specialists specialists to to persist in appropriate appropriate communities. communities. If If we we consider consider all all these these experimental experimental studies studies to to date, date, two two somewhat somewhat sur surprising prising findings findings emerge. emerge. First, First, migration migration may may play play less less of of aa role role in in structuring structuring communities thought. Indeed, some experiments communities than than previously previously thought. Indeed, some experiments found found no no effects effects of of migration migration on on local local diversity diversity (e.g., (e.g., Forbes Forbes and and Chase, Chase, 2002) 2002) and and other other studies results do studies found found only only minimal minimal effects. effects. Such Such results do not not preclude preclude the the possibil possibility ity of of important important effects effects of of migration migration on on community community composition composition through through species Tilman and 993, Leibold, 9 9 8 ) or that species sorting sorting (e.g., (e.g., Tilman and Pacala, Pacala, 11993, Leibold, 11998) or it it may may be be that that migration rates been sufficiently that experimental experimental migration rates have have not not been sufficiently high high enough enough to to affect mass effects (Fig. 66.1). . 1 ) . Second, number of affect diversity diversity through through mass effects (Fig. Second, aa number of other other fac factors, such regional heterogeneity, tors, such as as trophic trophic interactions, interactions, disturbance, disturbance, and and regional heterogeneity, must be incorporated of metacommunities; must be incorporated into into our our understanding understanding of metacommunities; in in particu particular, interactions with lar, these these factors factors may may have have important important interactions with migration. migration.
Evidence of Evidence of the the Role Role of of Trade-Offs Trade-Offs In in Metacommunltles Metacommunities Of Of greatest greatest interest interest to to us us in in this this chapter chapter are are studies studies that that looked looked explicitly explicitly at at trade-offs trade-offs in in species species traits traits and and how how these these trade-offs trade-offs may may affect affect species species persist persistence ence and and coexistence coexistence at at local local and and metacommunity metacommunity scales. scales. To To date, date, remarkably remarkably few few such such studies studies have have been been conducted conducted and and most most deal deal with with aa very very small small number number of species. Yu 1 ) investigated of coexisting coexisting species. Yu et et al. al. (200 (2001) investigated factors factors contributing contributing to to the the coexistence coexistence of of two two ant ant genera genera that that specialize specialize on on the the same same plant plant host. host. Census Census and and experimental experimental data data demonstrated demonstrated that that Azteca Azteca sp. sp. queens queens were were better better dis dispersers, persers, whereas whereas Allomerus Allomerus cf. cf. demerarae demerarae colonies colonies were were more more fecund, fecund, presum presumably ably making making Allomerus Allomerus aa better better competitor competitor on on any any given given host host plant. plant. As As aa result, result, species species success success was was aa function function of of interplant interplant distance: distance: at at low low plant plant densities, densities, the the Azteca Azteca sp. sp. was was dominant, dominant, whereas whereas at at high high densities, densities, the the better better competitor, competitor, Allomerus, prevailed. Spatial heterogeneity host-plant densities densities allowed allowed Allomerus, prevailed. Spatial heterogeneity in in host-plant regional coexistence regional coexistence due due to to this this trade-off trade-off in in migration migration and and competitive competitive abilities. abilities. This possible "niche This result result is is intriguing intriguing because because it it highlights highlights aa novel novel possible "niche axis" axis" that that might communities, heterogeneity might be be important important in in meta metacommunities, heterogeneity in in the the local local density density of of patches community (Yu 1 ) . Lei patches within within aa meta metacommunity (Yu and and Wilson, Wilson, 200 2001). Lei and and Hanski Hanski ((1998) 1998) documented documented aa similar similar trade-off trade-off between between competition competition and and colonization colonization in parasitoids attacking in aa system system of of parasitoids attacking the the Glanville Glanville fritillary fritillary butterfly butterfly (Melitaea (Melitaea cinxia). local density large role community, as cinxia). Again, Again, local density may may play play aa large role in in this this meta metacommunity, as changes changes in in host host density density result result in in aa decline decline in in the the proportion proportion of of populations populations attacked attacked by by the the better better parasitoid parasitoid competitor competitor but but have have little little effect effect on on the the pro proportion (Fig. 6.5). portion of of populations populations attacked attacked by by the the better better colonizer colonizer (Fig. 6.5). Working Working on on the the community community of of herbivores herbivores on on ragwort ragwort (Senecio (Senecio jacobea), jacobea), Harrison Harrison et et al. al. ((1995) 1 995) found found weak weak evidence evidence for for aa colonization-competition colonization-competition trade-off trade-off and and
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Changes parasitoids, Cotesia Cotesia melitaearum melitaearum and Hyposoter Hyposoter Changes in the abundances abundances of two parasitoids, As the number of host host populations hortico/a, horticola, and their host, host, the butterfly butterfly Melitaea Mefitaea cinxia. cinxia. As declines, populations attacked attacked by the better competitor, competitor, C. declines, the proportion proportion of populations C. melitaearum, melitaearum, declines, proportion attacked attacked by the better colonizer, H. hortico/a, relatively declines, whereas the proportion colonizer, H. horticola, is relatively Fig. 6.5 Fig. 6.5
unaffected. From (1 998). From Lei Lei and Hanski Hanski (1998).
argued trade-off was was present, it was was not not important important in in regulating regulating argued that, that, even even if if aa trade-off present, it metacommunity They hypothesized hypothesized that that other mechanisms (possibly (possibly metacommunity dynamics. dynamics. They other mechanisms related related to to patch patch type type specialization specialization and and species species sorting) sorting) were were most most likely likely allow allowing ing for for coexistence coexistence in in this this system. system. Two Two studies studies have have more more directly directly tested tested for for the the importance importance of of trade-offs trade-offs that that operate spatial scales. scales. Amarasekare Amarasekare (2000) para operate at at different different spatial (2000) studied studied two two egg egg parasitoids insect. She sitoids that that both both use use the the same same host host insect. She hypothesized hypothesized that that aa trade-off trade-off at at either either of of two two different different scales scales might might explain explain the the coexistence coexistence of of the the two. two. At At the the inferior competitor competitor might might be better better able to find unparasitized unparasitized local scale, the inferior hosts predation. At scale, the hosts and and thus thus avoid avoid intraguild intraguild predation. At the the regional regional scale, the inferior inferior competitor allowing it persist in competitor may may have have higher higher migration migration abilities, abilities, allowing it to to persist in host host patches patches isolated isolated by by distance. distance. Manipulative Manipulative studies studies and and field field surveys surveys were were used used to to demonstrate demonstrate that that the the loss loss of of the the dominant dominant competitor competitor was was related related not not to to host host isolation, isolation, but but to to host host productivity. productivity. Coexistence Coexistence was was therefore therefore best best explained explained by by aa trade-off trade-off between between competitive competitive ability ability and and the the ability ability to to find find unparasitized unparasitized hosts hosts at at the the local local scale scale and and not not by by larger larger scale scale variation variation in in abundance. Again, Again, species species sorting sorting by by local local patch-type patch-type attributes attributes was was host abundance. surprisingly surprisingly important important in in this this system, system, which which might might at at first first have have appeared appeared to community structured to correspond correspond more more closely closely to to aa meta metacommunity structured by by aa coloniza colonization-biotic-ability tion-biotic-ability trade-off. trade-off. The The communities communities of of bacteria bacteria and and invertebrates invertebrates found found in in the the water-filled water-filled leaves of leaves of pitcher pitcher plants plants have have the the advantage advantage of of occurring occurring at at discrete discrete scales: scales: within individual local) and among individual leaves ((local) among leaves within within a population population (regional (regional or or metacommunity). metacommunity). Kneitel Kneitel (2003) (2003) proposed proposed to to quantify quantify the the strengths strengths of of trade-offs trade-offs that that may may operate operate at at these these two two different different scales. scales. He He quan quantified tified the the growth growth and and colonization colonization rate rate of of protists protists and and rotifers rotifers found found in in the the
1148 48
MATHEW MATHEW A A.. LEIBOLD LEIBOLD AND AND THOMAS THOMAS EE.. MILLER MILLER
midtrophic-Ievel midtrophic-level assemblage of these inquiline communities. To To quantify trade-offs, he also determined the competitive ability and tolerance of preda predation for each of five species using a combination combination of laboratory and field experi experiments. ments. Identical Identical experiments experiments were were conducted conducted at at each each of of two two resource resource levels, levels, representing different types of local patches. There was little evidence of a trade-off species interactions trade-off between colonization colonization ability and traits related to species (competitive (competitive ability ability or or predator predator tolerance). tolerance). Predator Predator tolerance tolerance and and competitive competitive ability generally showed showed a strong negative relationship relationship (Fig. 6.6). Further, this negative relationship strongest between relationship was strongest between competitive competitive ability at one resource level predator tolerance indicating that resource level and and predator tolerance at at the the other, other, indicating that this this trade-off trade-off specialization. According to Kneitel, these data sug sugcould lead to patch-type specialization. gest that species are specialized to do well under different biotic conditions: these species coexist coexist because of of a trade-off trade-off between competitive competitive ability and among-leaf scale. predator tolerance that operates at the among-leaf Clearly, Clearly, any any strong strong conclusions conclusions about about how how metacommunity metacommunity dynamics dynamics work work in natural systems are premature. Few studies have really tackled the difficult issues involved involved in evaluating evaluating complex complex interactions interactions among among migration, migration, patch heterogeneity, heterogeneity, and disturbance in naturally existing regional species assem assemblages. been done relate any blages. Further, Further, even even less less has has been done to to relate any of of these these results results to to the the pat pattern of of covariation covariation among among species traits (or to determine how how this covariation covariation possible trade-offs) trade-offs).. Nevertheless, Nevertheless, a few few generalizations warresults in possible generalizations may be war ranted. First, migration can have large effects effects on local and regional diversity, evibut its role may depend strongly on other factors. Second, some of the evi indicates that patch-type heterogeneity heterogeneity and its effect effect on species sorting dence indicates not always be the case (Hubbell, (Hubbell, 2001; 2001; Bell, are often very strong. This may not 2000), depending from some systems is interpreted 2000), depending on how how evidence evidence from some of of these systems (see Chave et aI., long history al., 2002; 2002; Chave Chave and Leigh, 2002) 2002).. Finally, despite a long
44 3.5 3.5
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c.. 13_
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Relationship between between the the average competitive competitive effect of inquiline Fig. 66.6 Fig. . 6 Relationship of pitcher pitcher plant inquiline species and predation. The The competitive and (a) (a) colonization rate and (b) (b) tolerance to to predation. competitive effect effect was the the species in suppressing suppressing the the growth growth of of competitors competitors in pairwise pairwise competicompeti average effect of each species tion. Predator Predator tolerance rate of each species in the tolerance is the the standardized standardized growth growth rate the presence of of larlar of the the mosquito mosquito Wyeomyia Wyeomyia smithii. smithii. The The colonization colonization rate rate was determined determined by by the the rate rate of of vae of migration migration of of each species species into into unoccupied unoccupied pitchers in the the field. field. Four Four of of the the species species are protoproto zoans (BO, Bodo; CO, CO, Colpoda; Co/poda; CI, Cyclidium; Cyclidium; and and CH, chrysomonad chrysomonad sp.); sp.); the the fifth fifth species is a (BO, Bodo; rotifer (Habrotochus rosea) rotifer (Habrotochus rosea) from from Kneitel (2003). (2003).
FROM METAPOPULATIONS METAPOPULATIONS TO TO METACOMMUNITIES METACOMMUNITIES 6. FROM
49 1149
coloin ecological theory, surprisingly little evidence supports the view that a colo nization-competition trade-off is important important in allowing species species coexistence or nization-competition otherwise influencing diversity. A lack of evidence should not necessarily be interpreted interpreted as as aa general general pattern; pattern; instead, instead, it it should should be be aa call call for for further further study study on the role of trade-offs operating at different spatial and temporal scales.
6.5 6.S
DISCUSSION DISCUSSION In an intriguing theoretical 1 992) theoretical paper paper on metacommumtles, metacommunities, Wilson ((1992) predicted predicted that that aa surprisingly surprisingly rich rich diversity diversity of of behaviors, behaviors, including including reciprocal reciprocal relations between local population population dynamics and metacommunity metacommunity dynamics, communities in metacommunities, metacommunities, and the persistence of alternative stable communities that might might affect the nature nature of ecosystems. Wilson ((1992) higher level selection that 1 992) showed that that these phenomena phenomena were possible under at least some metacommun metacommunframeworks, but but he did not not identify the conditions conditions that that would would influence ity frameworks, occurrence. Work done since then has explored only some of the possi possitheir occurrence. much remains remains to be done. bilities, and much Nonetheless, Nonetheless, research research in in this this area area is is of of great great potential potential importance importance because because it it that many aspects aspects of population population biology, community community is increasingly obvious that structure, structure, and and ecosystem ecosystem dynamics dynamics are are influenced influenced by by metacommunity metacommunity processes. processes. Furthermore, Furthermore, numerous numerous environmental environmental issues issues can can be be related related directly directly to these processes. Habitat Habitat fragmentation, fragmentation, for example, example, creates patches and isolation ((and therefore interpatch interpatch migration). Conversely, humans humans affects their isolation and therefore facilitate facilitate the the movements movements of of other other organisms, organisms, leading leading to to novel novel associations associations of of metacommunities. Furthermore, Furthermore, humans humans alter alter environmental environmental condi condispecies in metacommunities. tions in local patches patches in a number number of ways that that alter alter the local fitness fitness of organ organisms. local scales scales and isms. Finally, Finally, humans humans also also influence influence extinctions, extinctions, both both on on local and over over and global scales. These effects are just as likely to alter meta metalarger regional and community dynamics dynamics as they are to to affect local ecological processes. community This has tried metacommunity concepts This chapter chapter has tried to to emphasize emphasize the the growth growth of of metacommunity concepts from their their origins in meta metapopulation theory to current relevance in reevalupopulation theory to their current reevalu community theory. In particular, particular, we argue that that although although ating many many aspects of community metapopulation 4), metapopulation theory theory emphasizes emphasizes understanding understanding species species persistence persistence (Chapter (Chapter 4), metacommunity species coexist metacommunity approaches approaches are are directed directed toward toward explaining explaining species coexistence, diversity, and their their consequences. consequences. Understanding coexistence entails entails ence, diversity, and Understanding coexistence understanding trade-offs in in species' species' responses responses to to different different environmental environmental facfac understanding trade-offs tors, both both biotic and and abiotic. abiotic. Although Although trade-offs trade-offs have long served as the the corcor tors, community theory, theory, metacommunity metacommunity dynamics draw draw attention attention to to nerstone of community traits that that operate operate at at larger scales and and especially to to migration migration and and the the heterohetero traits geneity of of local local habitats. habitats. Among-habitat Among-habitat migration migration is is one one of of the the defining traits geneity defining traits of metacommunities and and may frequently to diversity (if of metacommunities may act act frequently to increase increase not not only only diversity migration rates rates are greater than than extinction extinction rates), rates) , but community composition composition migration are greater but community through species sorting. sorting. Habitat heterogeneity influences through Habitat heterogeneity influences the the scale at at which which species trade-offs trade-offs are important: important: homogeneity homogeneity of of habitats habitats may may require that that trade-offs allow coexistence within within local habitats, habitats, whereas whereas heterogeneity of of trade-offs habitats expands expands the the scale scale at at which which coexistence coexistence may may occur. habitats At this this stage, stage, we we are are still still learning learning much much about about the the roles roles of of altered altered migramigra At tion, tion, local local adaptation, adaptation, and and extinctions extinctions in metacommunities. metacommunities. Our Our knowledge knowledge to to
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MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
date of of these these effects effects shows shows that that they they can can be be complex complex and and multivaried, multivaried, but but studstud date ies have have also also produced produced aa surprising surprising richness richness of of results results that that we we are are only only beginbegin ies ning to to perceive. perceive. Nonetheless, Nonetheless, we we also also think think that that aa couple couple of of generalities generalities are are ning beginning to to emerge emerge from from both both aa theoretical theoretical and and an an empirical empirical perspective. perspective. beginning Experimental work work on on migration migration has has suggested suggested its its importance, importance, but but itit may may play play Experimental larger role role in in community community composition composition than than in in community community diversity. diversity. In In addiaddi aa larger tion, habitat habitat heterogeneity heterogeneity appears appears to to be be very very important important for for metacommunity metacommunity tion, dynamics, but but very very little little experimental experimental work work has has been been done done in in this this area. area. Finally, Finally, dynamics, very few few studies studies of of metacommunities metacommunities have have discussed discussed or or quantified quantified species species very trade-offs, aa clear clear call call for for further further experimental experimental work work in in this this area. area. trade-offs, A metacommunity metacommunity perspective, perspective, in in which which scale scale is is incorporated incorporated directly directly into into A our thinking thinking in in the the form form of of local local habitats habitats linked linked by by migration, migration, is is not not aa new new our view for for ecologists, ecologists, but but recent recent theoretical theoretical and and experimental experimental work work in in this this area area view demonstrates significant significant promise. promise. A A metacommunity metacommunity view view is is clearly important, demonstrates clearly important, not just just in in explaining explaining diversity diversity at at the the landscape landscape level level (something (something that that has has been been not obvious for for a long time), but but also also in constraining constraining the role that that disturbance, disturbance, obvious long time), the role migration, and and heterogeneity heterogeneity can can play play in regulating regulating biodiversity biodiversity at at local levels. migration, local levels. We that this may also also make make strong strong contributions to underunder We feel feel that this approach approach may contributions to standing community community composition composition through through understanding understanding traits traits that allow standing that allow coexistence at at different different spatial spatial scales. scales. A A rich rich array array of of issues issues awaits awaits both both theortheor coexistence etical and and empirical empirical investigations. etical investigations.
Part III Metapopulation Genetics
sdfsdf
O
SELECTION AND SELECTION AND DRIFT IN IN META PO PULATIONS M ETAPO PU LATIO N S Hanski and G aggiotti, by by Michael Michael C. Whitlock Whitlock Hanski and Gaggiotti,
7.1 7.1
INTRODUCTION INTRODUCTION The distribution distribution of of aa species species over over space space has has many many interesting interesting and and important important The evolutionary population genetic evolutionary consequences. consequences. All All of of the the basic basic population genetic forces forces m drift, drift, selection, recombination spatially selection, migration, migration, mutation, mutation, and and recombination m act act differently differently in in aa spatially structured enhanced or structured population. population. Genetic Genetic drift drift can can be be enhanced or diminished diminished relative relative to to aa panmictic population of panmictic population of the the same same total total size. size. Selection Selection can can be be more more or or less less effec effective. tive. Migration Migration is is impossible impossible without without aa spatial spatial context; context; the the consequences consequences of of muta mutations tions tend tend to to be be lowered, lowered, and and the the effective effective recombination recombination rate rate is is reduced. reduced. This This chapter chapter reviews reviews some some of of the the effects effects of of population population structure, structure, in in particular particular focus focusing ing on on how how selection selection and and drift drift are are changed changed by by the the fact fact that that species species exist exist in in space. space. This This chapter chapter takes takes aa heuristic heuristic and and largely largely nonmathematical nonmathematical look look at at these these issues, issues, trying trying to to express express intuitively intuitively some some recent recent results results in in spatial spatial population population genetics. genetics. This chapter chapter focuses focuses on on the the dynamics dynamics of of aa single single locus, locus, whereas whereas the the topics topics of of multi multiThis 1, and locus locus selection selection and and quantitative quantitative genetics genetics are are discussed discussed in in Chapters Chapters 9, 9, 111, and 12. One One very very important important summary summary statistic statistic about about the the effects effects of of population population structure structure turns turns out out to to be be one one of of the the oldest: oldest: Wright's Wright's FST• FST. There There are are several several ways ways to to define define FST, Fsv, but but they they all all are are standardized standardized measures measures of of the the genetic genetic differentiation differentiation among among populations. populations. Here Here let let us us define define FST Fsw as as the the variance variance in in allele Vamong ), standardized allele frequencies frequencies across across populations populations ((Vamong), standardized by by the the mean mean allele allele frequency frequency (p) (p):: FST Fsy == V Vamong/p(1 - pl. p). FST FST has has several several key key features features that that make make amongfp( 1 -
enetics, and Ecology, Ecology,G Genetics, and Evolution Evolution of of Metapopulations Metapopulations
53 1 53
evier, Inc. Copyright Copyright 2004, Els Elsevier, Inc. 0-12-323448-4 0-12-323448-4
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 54 54
it it useful useful and and interesting interesting for for the the study study of of evolution evolution in in structured structured populations. populations. FsT has has the the same same expectation expectation for for all all neutral neutral autosomal autosomal loci, loci, although although First, FST even neutral neutral loci loci can can vary vary substantially substantially around around this this expectation. expectation. Moreover, Moreover, even this this expectation expectation is is determined determined by by the the demographic demographic properties properties of of the the species, species, such such as as migration migration rates, rates, local local population population sizes, sizes, and and geography geography FST FsTcan can there therefore encapsulate encapsulate a lot of useful information information about about the demographic demographic history history of fore species. FST FsTtends tends to to be be larger larger if if local local populations populations are are not not connected connected by by high high aa species. rates rates of of migration migration and/or and/or if if local local population population sizes sizes are are small. small. Finally, Finally, FST FsT is is readily readily measurable measurable from from easily easily obtained obtained data data on on real real populations, populations, and and there there is is already already aa lot lot of of information information about about FST FsT in in nature. nature. There There are are other other useful useful ways ways to to view view the the information information conveyed conveyed by by FST FsT beyond beyond its its use use as as aa measure measure of of genetic genetic variance variance among among populations. populations. FST FsT is is indication of the amount amount of relatedness among individuals in the same also an indication demes. demes. If If FST FsT is is high, high, then then individuals individuals in in the the same same demes demes are are highly highly related related to to one one another; another; in in other other words, words, they they share share many many alleles. alleles. All All else else being being equal, equal, this this also pertains to alleles within aa diploid diploid individual: individual: if if FST FsT is is high, high, individuals individuals also pertains to alleles within to be homozygous homozygous than are more more likely to than would would be predicted predicted by by Hardy-Weinberg various interpretations Hardy-Weinberg frequencies. frequencies. These These various interpretations and and implications implications FSTare are useful useful in in interpreting interpreting the the results results that that follow. follow. It It turns turns out out that that because because of FST FsTrepresents represents both both the the relatedness relatedness of of individuals individuals within within aa de deme and the the excess excess FST me and homozygosity, homozygosity, it it is is often often the the only only extra extra parameter parameter needed needed to to describe describe how how population structure structure changes the pace of evolution. evolution. population This This chapter chapter reviews reviews the the effects effects of of spatial spatial population population structure structure on on the the amount amount response to selection. The greater greater part part of the chapter chapter then then of genetic drift and the response uses these these results results to to discuss discuss basic basic evolutionary evolutionary genetic genetic quantities quantities in in structured structured uses populations, such balance point point between between mutation mutation and and selection, selection, mutamuta populations, such as as the the balance inbreeding depression, depression, the probability of fixation tion load, load, inbreeding the probability fixation of of new new alleles, alleles, and and other basic quantities. out that these fundamental fundamental evolutionary other basic quantities. It It turns turns out that these evolutionary processes are are sometimes population structure. structure. processes sometimes strongly strongly affected affected by by even even aa weak weak population
7.2 7.2
GENE FREQUENCY CHANGE CHANGE IN IN METAPOPULATIONS METAPOPULATIONS GENE FREQUENCY Gene frequency can species by four mechanisms: mechanisms: selection, drift, Gene frequency can change change in in aa species by four selection, drift, introgression species, and introgression from from other other species, and mutation. mutation. This This section section reviews reviews mathmath ematical models structure in two more ematical models that that show show the the effect effect of of population population structure in the the two more important of of these these forces, forces, selection selection and and drift. drift. important
Genetic Genetic Drift Drift Genetic drift drift is is the the change change in in allele allele frequency frequency from from one one generation generation to to the the Genetic next caused caused by by random random sampling sampling of of alleles. alleles. Genetic Genetic drift is nondirectional, nondirection aI, next drift is meaning that that the the average average change change due due to to drift is zero, zero, but but as as the the population population size size meaning drift is gets small, frequency in generation can gets small, the the actual actual change change in in allele allele frequency in any any given given generation can be be relatively large. relatively large.
Effective Effective Population Population Size The smaller smaller the the effective effective population population size, size, the the more more random random effects effects can can The become become important. important. A A key key term term here here is is "effective" "effective" m - the the actual actual amount amount of of genetic drift in in aa population population is is determined determined not not only only by by the the actual actual number number genetic drift
7. 7. SELECTION SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS
11155 55
of individuals also by of individuals in in the the population, population, but but also by other other factors, factors, such such as as the the distri distribution species. The bution of of reproductive reproductive success success in in the the species. The effective effective size, size, Ne, of of aa pop population ulation is is defined defined as as the the size size of of an an ideal ideal population, population, which which would would be be expected expected to to have have the the same same amount amount of of genetic genetic drift drift as as the the population population in in question. question. An An ideal ideal population population is is one one in in which which each each of of the the alleles alleles in in the the offspring offspring generation generation have have an an equal equal and and independent independent chance chance of of having having come come from from each each of of the the parental alleles. ideal population population would parent parental alleles. An An ideal would function function as as though though each each parent allele number of pool, and allele contributes contributes an an equal equal and and large large number of copies copies to to aa gamete gamete pool, and then then offspring offspring would would be be formed formed by by random random draws draws from from this this gamete gamete pool. pool. Real Real populations populations are are not not ideal ideal though though for for several several reasons. reasons. First, First, and and most most importantly, importantly, in in real real populations, populations, each each individual individual is is not not expected expected to to contribute contribute equally equally to to the the next next generation: generation: some some are are very very fit fit and and have have aa high high reproductive reproductive suc success, cess, whereas whereas others others die die before before even even reproducing. reproducing. This This variance variance in in reproductive reproductive success increase the success tends tends to to reduce reduce the the effective effective population population size size and and therefore therefore increase the rate rate of of genetic genetic drift. drift. Second, Second, in in real real populations, populations, new new individuals individuals are are not not necessarily formed available alleles. alleles. For example, with necessarily formed at at random random from from the the available For example, with inbreeding, inbreeding, individuals individuals are are formed formed with with aa higher higher than than random random chance chance of of having having similar similar alleles alleles at at homologous homologous sites. sites. Such Such inbreeding inbreeding tends tends to to decrease decrease Ne because because each each individual individual effectively effectively carries carries fewer fewer copies copies of of alleles. alleles. Finally, Finally, both both variation variation in in reproductive success across generations, reproductive success and and nonrandom nonrandom mating mating can can be be inherited inherited across generations, and and the the correlations correlations in in reproductive reproductive success success which which result result can can also also affect affect Ne• Ne. In In structured structured populations, populations, these these three three factors factors are are even even more more important. important. When places, they When organisms organisms live live in in different different places, they are are likely likely to to experience experience different different conditions, variance in conditions, and and therefore therefore there there is is likely likely to to be be greater greater variance in reproductive reproductive success than than in in aa single single well-mixed population. Population structure causes causes success well-mixed population. Population structure kind of of inbreeding inbreeding because because locally locally mating mating individuals individuals are are likely likely to to be be related. related. aa kind Finally, Finally, if if local local conditions conditions are are correlated correlated positively positively from from one one generation generation to to the the next, next, variance variance in in reproductive reproductive success success will will also also be be correlated correlated among among parents parents and offspring, offspring, assuming assuming limited limited migration. migration. and The The effective effective size size of of structured structured populations populations has has been been well well reviewed reviewed by by Wang Wang and 1 999). and Caballero Caballero ((1999). The The Island Island Model Model
Describing Describing the the effective effective size size of of subdivided subdivided populations populations has has aa long long history, history, beginning in 11939. 939. In this paper, the effec beginning with with Sewall Sewall Wright Wright in In this paper, Wright Wright derives derives the effective species subdivided an island island model, be tive population population size size of of aa species subdivided by by an model, finding finding it it to to be
N e,Isiand Model Ne,Island Model = --
Nd Nd - ST
' 1 - FFST ' 1
-
(7. 1) (7.1)
where where N N is is the the number number ooff individuals individuals in in aa deme, deme, d d is is the the number number of of demes, demes, and and FST FST is is given, given, for for large large d d at at equilibrium, equilibrium, by by
FFsT, ST,Isiand Island Modei Model
== -~
11
44Nm Nm ++ l1"'
((7.2) 7.2)
Here, Here, m m is is the the migration migration rate rate among among demes. demes. In In the the island island model, model, each each deme deme contributes contributes aa proportion proportion m m of of its its individuals individuals to to aa migrant migrant pool pool and and then then receives receives the the same same number number of of migrants migrants chosen chosen randomly randomly from from that that migrant migrant
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 56 56
pool. It It is is important important to to note note that that these these are are not not random random proportions, proportions, but but that that pool. each deme deme gives gives and and receives receives exactly exactly Nm Nm individuals individuals to to and and from from the the migrant migrant each pool each each generation, generation, and and each each deme deme consists consists of of exactly exactly N N individuals. individuals. As As aa pool result, me contributes result, each each de deme contributes exactly exactly equally equally to to the the next next generation. generation. This This seemingly seemingly innocuous innocuous assumption assumption turns turns out out to to have have fairly fairly important important effects effects on on interpreting results results obtained obtained from from the the island island model. model. interpreting If each each deme deme contributes contributes exactly exactly equally equally to to the the next next generation, generation, then then there there If is no no variance variance in in reproductive reproductive success success among among demes. demes. We We know know from from classical classical is population population genetics genetics that that aa lower lower variance variance in in reproductive reproductive success success means means higher higher and in in fact fact this this is is the the case case with with structured structured populations as well. well. Look Look again again Ne, and populations as at 1 ) . Give at Eq. Eq. (7. (7.1). Give that that FST FsT is is aa quantity quantity that that ranges ranges between between 00 and and 11,, the the Ne for an an island island model model is is always always something something greater greater than than Nd, in in other other words words for greater than than the the total total number number of of individuals individuals in in the the metapopulation metapopulation as as aa whole. whole. greater This is is because because of of the the assumption assumption that that there there is is no no variance variance among among demes demes in in This reproductive success. success. reproductive
Model Assumptions Relaxing Island Model A more more general general model model of of the the effective effective size size of of structured structured populations populations has has A been derived derived (Whitlock (Whitlock and and Barton, Barton, 11997). The general general form form of of the the equation equation been 997). The for Ne in in aa species species that that has has reached reached demographic demographic equilibrium equilibrium is is given given by by for
Ne me = =
Ndd N
FsT i)·) + 22 2: 2: Nw4-( I I lNiW2(Ndl -FsT, ~.~ wwNN·p wiwI jNiINjPiIJNdI II·· + ~ 2:
--���--��------------���
il
Nd
,1
i1
j!
Nd
(7.3) (7.3)
shown to function of the local population sizes the relarela where where Ne m e is is shown to be be aa function of the local population sizes (Ni (mi), ) , the the FST predicted over over aa set tive contributions contributions of of each FsT predicted set of of demes demes tive each deme deme (wi), (Wi), the and the the correlation correlation among with properties such with demographic demographic properties such as as deme deme ii (FST), (FsT.i), and among which is is defined similar to to FsT, FST, but but instead using demes demes of of allelic allelic identity identity (Pi;, (Pii, which defined similar instead using covariance of of pairs demes).. This This equation few assumptions covariance pairs of of demes) equation makes makes few assumptions about about the of the the spatial spatial subdivision subdivision among among populations, allowing for for variable the nature nature of populations, allowing variable migration rates over different different population pairs, including including isolation migration rates over population pairs, isolation by by disdis tance, local changes changes in size, including local extinction, extinction, and tance, local in population population size, including local and new new population formation via via colonization or fission. population formation colonization or fission. While general, is aa bit for intuitive intuitive use. use. To aid in in explainexplain While general, Eq. Eq. (7.3 (7.3)) is bit unwieldy unwieldy for To aid ing few key ing aa few key features features of of this this result, result, let let us us use use aa simplified simplified version version of of this this equaequa tion few more tion that that makes makes aa few more assumptions. assumptions. If If all all demes demes have have the the same same size size as as each other, other, but but contribute contribute unequally unequally to to the the next next generation via differential differential each generation via migration, write V V as as the the variance variance among among demes demes in in the the expected expected migration, then then we we can can write reproductive success success of of individuals individuals from that deme deme (i.e., (i.e., V V == Var[wi]). Var [wil ). The The reproductive from that effective effective population population size size is is then then
_ Ne Ne = -
Nd
V ) ( 1l -( 1 ++ V)( (1
Nd
FST) + 2NFsTVdfi 2NFsTVd/( dd - 1) 1) FST)
(7.4) (7.4)
-
(Whitlock and Barton, (Whitlock and Barton, 1997). 1 997). Let us us examine examine two two extremes extremes using using Eq. Eq. (7.4). (7.4). If, If, as as in in the the traditional traditional island island Let model, model, the the variance variance among among demes demes in in reproductive reproductive success success is is zero, zero, then then Eq. Eq. (7.4) (7.4)
7. SELECTION SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS 7.
11557 7
reduces to to Eq. Eq. (7.1) ( 7. 1 ) [This [This is is true true not not only only for for the the island island model, model, but but for for any any reduces model for for which which each each deme deme is is equal equal in in size size and and contributes contributes exactly exactly equally equally to to model all other other demes, demes, provided provided that that all all demes demes are are ultimately ultimately reachable reachable by by each each deme deme all via migration migration (Nagylaki, (Nagylaki, 1982). 1 982). This This includes includes classic classic stepping stepping stone stone models.] models.] via At the the other other extreme extreme though, though, let let us us imagine imagine that that one one deme deme is is extremely extremely At successful and and produces produces all all of of the the offspring offspring that that fill fill all all dd of of the the demes. demes. In In this this successful case, we we would would intuitively intuitively predict predict that that the the effective effective population population size size of of the the case, whole system system should should be be the the same same as as the the size size of of the the single single successful successful deme, deme, and and whole Eq. (7.4), (7.4), with with appropriate appropriate modification, modification, shows shows us us that that this this is is in in fact fact the the case. case. Eq. (In (In this this extreme, extreme, the the FST would would be be zero zero and and the the variance variance among among demes demes of of 1 . ) This This extreme extreme example example tells tells us us allelic reproductive reproductive success success would would be be dd- 1.) allelic that Ne Ne can can be be much much smaller smaller than than the the census census size size with with population population structure. structure. that truth obviously obviously lies lies somewhere somewhere in in the the middle. middle. It It turns turns out out that that the the The The truth Ne isis boundary between between whether whether population population structure structure increases increases or or decreases decreases Ne boundary approximately whether whether or or not not demes demes have have greater greater or or less less variance variance in in reprorepro approximately ductive success success than than would would be be expected by aa Poisson Poisson distribution. In other other ductive expected by distribution. In words, if demic demic structure structure acts to increase increase variance variance in in reproductive reproductive success words, if acts to success relative to that that expected by chance, chance, N~ Ne would would be be reduced. reduced. Only Only if if the the effects effects relative to expected by of population population structure structure are are to to reduce reduce the the variance variance among among demes demes in in reproducreproduc of tive success success to to less less than than random random would would N~ Ne be be increased. increased. This This is is perhaps perhaps biobio tive logically yet this is the requirement for for the from the the simple logically unlikely, unlikely, yet this is the requirement the results results from simple island to hold qualitatively. In In real real species, is likely likely to to be be island model model to hold qualitatively. species, the the opposite opposite is true: demes are likely to to have have different resources, and true: different different demes are likely different amounts amounts of of resources, and different demes are to experience experience different different levels levels of of other other ecological fac different demes are likely likely to ecological factors that that might might affect success, such such as as levels levels of of parasitism, parasitism, disease, predation, tors affect success, disease, predation, weather weather fluctuations, fluctuations, and and other other catastrophes. catastrophes. Realistic Realistic ecology ecology implies implies higher higher than than random random variance variance among among demes demes in in reproductive reproductive success, success, and and therefore therefore the the effective effective size size of of aa subdivided subdivided species species is is likely likely to to be be reduced, reduced, perhaps perhaps substan substantially. tially. The The island island model model is is not not aa good good descriptor descriptor of of typical typical population population struc structure, Whitlock and 999). ture, for for this this and and many many other other reasons reasons (see (see Whitlock and McCauley, McCauley, 11999). -
Extinction n d Colonization Extinction aand Colonization
It It will will be be useful useful to to consider consider aa couple couple of of specific specific cases cases that that go go beyond beyond the the simple island model. One aspect of population structure that has attracted simple island model. One aspect of population structure that has attracted some some attention attention is is the the possibility possibility of of local local extinction extinction and and recolonization recolonization (Slatkin, (Slatkin, 11977; 977; Maruyama 980; Whitlock 997). The Maruyama and and Kimura, Kimura, 11980; Whitlock and and Barton, Barton, 11997). The mod models considered in these papers are similar: the basic structure is like an island els considered in these papers are similar: the basic structure is like an island model, model, except except that that each each deme deme has has some some chance chance per per generation generation of of going going extinct extinct independently of its genotype frequencies. An equal number of new independently of its genotype frequencies. An equal number of new demes demes are are colonized, colonized, either either in in the the same same places places recently recently vacated vacated by by the the extinction extinction events events or or in in other other vacant vacant sites, sites, by by aa small small number number of of individuals. individuals. As As aa major, major, unreal unrealistic istic simplification, simplification, each each new new deme deme then then immediately immediately grows grows back back to to N N individuals, individuals, like like all all other other demes. demes. With With local local extinction extinction and and recolonization, recolonization, population population structure structure contributes contributes in in an an obvious obvious way way to to the the variance variance in in reproductive reproductive success success among among demes. demes. Even Even though though this this model model is is based based on on the the island island model, model, even even aa small small rate rate of of extinc extinction tion is is enough enough to to cause cause the the effective effective population population size size of of the the species species to to be be reduced reduced rather rather than than increased. increased. The The main main reason reason is is perhaps perhaps obvious: obvious: with with extinction extinction and and recolonization, recolonization, some some demes demes have have zero zero reproductive reproductive success, success,
1158 58
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK 11.2 .2 1
0.8 NNe e 0.8 Nd 0.6 0.6 Nd ': -
0.4 0.4 0.2 0.2
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Fig. 7.1 7.1 The The effective effective size size (displayed (displayed as as aa proportion proportion of of the the census census size) size) of of aa metapopulation metapopulation with local local extinction extinction and and colonization. colonization. Here Here each each deme deme has has 1100 individuals, and and each each new new popu popuwith 00 individuals, lation lation is is founded founded by by four four individuals. individuals. These These colonists colonists have have aa probability probability ~ that that they they come come from from the /2 in the same same source source population, population, with with &= = 0 0 in in the the dotted dotted line, line, ~ = - 11/2 in the the dashed dashed line, line, and and = = 11 in in the the solid solid line. line. The The migration migration rate rate was was 0.01 0.01 in in all all examples. examples. As As the the local local extinction extinction rate rate increases, the the effective effective population population size size is is reduced reduced greatly. greatly. increases,
whereas manage to survive and whereas others others - - those those that that manage to survive and send send colonists colonists to to start start new new demes demes m have have aa reproductive reproductive jackpot. jackpot. Thus Thus there there is is aa great great deal deal of of vari variance among among demes demes in in reproductive reproductive success, success, which which causes causes the the effective effective size size to to ance This reduction reduction can be extreme extreme (Fig. (Fig. 7. 7.1). reduced. This can be 1 ). be reduced. Sources and Sinks Sources and Sinks
In In some some species, species, some some populations populations have have large large amounts amounts of of resources, resources, whereas whereas others that they replace themselves themselves without without migration migration others have have so so few few that they cannot cannot replace (Pulliam, 1988; Dias, 996; Holt and Gaines, 1 992). These so-called "sources" "sources" (Pulliam, 1988; Dias, 11996; Holt and Gaines, 1992). These so-called and population dynamics to be be different different from and "sinks," "sinks," respectively, respectively, cause cause the the population dynamics to from the demes do contribute equally migrant pool, pool, and and the island island model: model: demes do not not contribute equally to to the the migrant therefore there there is is variance in reproductive reproductive success. of patches patches of of therefore variance in success. If If the the quality quality of is even even more more resource is is correlated correlated positively positively over over time, time, then then the the effect on Ne is resource effect on extreme. extreme. The effects of source-sink source-sink structure and correlation correlation over over time time in in patch patch suitsuit The effects of structure and ability ability can can be be best best seen seen by by another another extreme extreme example. example. Imagine Imagine that that aa fraction fraction of other 80% of demes, demes, say say 20%, 20%, reside reside in in productive productive source source patches, patches, and and the the other 80% of of demes are are what what Bob Bob Holt Holt has has called called "black-hole" " black-hole" sinks sinks m - that that is, is, these these demes demes demes never never contribute contribute migrants migrants to to other other demes demes and and only only persist persist because because of of migramigra tion from from source source populations. populations. In In this this case, case, itit is is clear clear that that only only alleles alleles in in indiindi tion viduals in in source source populations populations can can contribute contribute to to future future generations generations and and so so the the viduals only only individuals individuals that that matter matter to to the the evolution evolution of of the the species species are are in in the the source source populations. Therefore Therefore the the Ne Ne of of the the species species should should reflect reflect only only the the effective effective populations. of this this species species should should be be only only size of of the the source source populations populations alone. alone. Thus Thus the the Ne of size 20% of of what what itit would would have have been been with with equal equal migration. migration. 20% To be be more more general, general, we we can can apply apply useful useful results results from from Nagylaki Nagylaki (1982), ( 1 982), who who To showed that that the the Ne Ne of of aa system system of of populations populations with with aa constant constant migration migration showed matrix could could be be described described with with the the left left eigenvector eigenvector of of that that matrix. matrix. (This (This matrix assumes aa few few technical technical details, details, such such that that all all demes demes are are ultimately ultimately reachable reachable assumes by migration migration from from all all other other demes, demes, even even ifif itit takes takes multiple multiple steps.) steps. ) Consider Consider aa by are case case where where migration migration is is via via aa migrant migrant pool pool so so all all emigrants emigrants from from all all demes demes are
7. 7.
SELECTION IN METAPOPULATIONS SELECTIONAND AND DRIFT DRIFT IN METAPOPULATIONS
1 59 59
1
0.8 0.8 N Nee N Ntot tot
0.6 0.6
0.4 0.4 0.2 0.2 0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
1
Relative Relative contribution contribution of of sink sink populations populations Fig. 7.2 7.2 The The effective effective size size of of aa species species in in which which 20% 20% of of the the demes demes are are sources sources and and the the rest rest axis varies varies the the contribution the sink are The x axis are sinks. sinks. The contribution of of the sink populations, populations, expressed expressed as as aa fraction fraction of of the Here each each 1100 00 demes 00 individuals, and each each receives the contribution contribution of of the the sources. sources. Here demes have have 1100 individuals, and receives five five immigrants immigrants per per generation generation sampled sampled from from the the migrant migrant pool. pool. As As the the contribution contribution of of sinks sinks reaches same as model with reaches zero, zero, the the effective effective size size of of the the system system is is the the same as an an island island model with only only the the 20 20 source source populations. populations.
mixed mixed together together and and then then moved moved on on to to recipient recipient demes demes at at random random with with respect respect to to where where they they originate. originate. Source Source demes demes contribute contribute aa large large number number to to the the migrant migrant pool, pool, whereas whereas "sink" "sink" demes demes contribute contribute aa fraction fraction of of that that number. number. For For simplicity, simplicity, each each deme deme receives receives aa constant constant number number of of immigrants immigrants from from the the migrant ensures that migrant pool. pool. This This ensures that the the FST Fsy among among sources sources and and among among sinks sinks are are approximately equal. Figure approximately equal. Figure 7.2 7.2 shows shows the the effective effective size size of of these these systems systems as as aa function function of of the the relative relative contribution contribution to to the the migrant migrant pool pool by by sinks. sinks. [To [To make make the 7.2, Nagylaki's 1 982) results the calculations calculations in in Fig. Fig. 7.2, Nagylaki's ((1982) results were were used, used, accounting accounting here. for for the the fact fact that that Nagylaki's Nagylaki's definition definition of of Ne Ne differs differs from from the the usage usage here. Nagylaki calculates the variance Nagylaki calculates the Ne that that would would give give the the same same amount amount of of variance within balance; in within aa deme deme at at mutation-migration-drift mutation-migration-drift balance; in other other work work including including in in this this chapter chapter Ne predicts predicts the the amount amount of of variance variance predicted predicted by by the the average average The second of these two two quantities allele frequency of the species as a whole. whole. The can can be be found found from from the the first first by by dividing dividing by by I-FsT• l-EsT. Details Details are are given given in in Whitlock Whitlock (2003).] (2003).1 Note Note that that with with this this form form of of source-sink source-sink structure, structure, the the effective effective size size of of the the populations when when the sinks species is just the effective size of just the source populations not contribute contribute to the future, future, and it reduces to the island model results when when do not "sinks" "sinks" contribute contribute equally equally to to sources. sources.
Selection Selection With With good good reason, reason, the the study study of of selection selection in in subdivided subdivided populations populations has, has, in in the the past, past, focused focused on on the the effects effects of of spatially spatially heterogeneous heterogeneous selection selection (e.g., (e.g., Felsenstein, 976) . A Felsenstein, 11976). A great great deal deal of of important important and and interesting interesting evolutionary evolutionary biol biology ogy results results from from variation variation in in selection selection over over space, space, but but population population structure, structure, perhaps surprisingly, perhaps surprisingly, has has aa lot lot of of interesting interesting effects effects even even on on uniform uniform selection. selection. Arguably, Arguably, most most loci loci have have approximately approximately similar similar selection selection in in different different demes, demes, even even though though the the more more obvious obvious and and more more polymorphic polymorphic cases cases may may reflect reflect spa spatially tially divergent divergent selection. selection. This This chapter chapter focuses focuses on on this this special special case case in in which which genotypes have same relative genotypes have the the same relative fitness fitness in in each each population population of of the the species. species.
1160 60
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
When When selection selection is is uniform uniform across across populations, populations, it it becomes becomes possible possible to to follow follow the population by the state state of of the the meta metapopulation by following following the the mean mean allele allele frequency frequency across across all all local local populations, populations, q. ~. Consider Consider simple simple selection selection between between two two alleles alleles at at the the same same locus, locus, with with the the fitnesses fitnesses of of the the three three genotypes genotypes given given by by 11 :: 11 + + h h ss :: 11 + + s. s. In this case, case, the the change change in in allele allele frequency frequency due due to to selection selection within within each each population population this is is aa third-order third-order function function of of q; q; therefore, therefore, to to understand understand how how the the mean mean allele allele fre frequency quency would would change change by by selection selection requires requires knowing knowing the the expected expected values values of of q, q, 2, and 3 . Fortunately, qq2, and qq3. Fortunately, under under most most circumstances circumstances the the dynamics dynamics of of the the expected expected value value of of qq33 can be be well well enough enough predicted predicted by by an an understanding understanding of of changes changes in in the the which reduces the problem to understanding q~ and and E[ E[q2]. The expected expected first two, which q2] . The value value of of qq22 may may seem seem like like an an exotic exotic quantity quantity to to keep keep track track of, of, but but remember remember that that 2] , and the the variance variance among among demes demes is is derived derived easily easily from from q ~ and and E[q E[q2], and FST Fsy is is derived derived easily from from the the variance variance in in allele allele frequency frequency and and q. ~. Thus, Thus, aa very very good good under undereasily standing of of the the change change in in allele allele frequency frequency across across aa meta metapopulation can be be standing population can obtained obtained by by knowing knowing q ~ and and FST. FST. Moreover, Moreover, as as long long as as the the selection selection coefficient coefficient is is not not much much greater greater than than the the rate rate of of migration migration into into aa deme, deme, the the FST FST predicted predicted from from neutral works extremely neutral theory theory works extremely well well to to predict predict allele allele frequency frequency change change in in struc structured tured populations. populations. These These conclusions conclusions are are derived derived and and discussed discussed in in greater greater detail detail (2002).) . in Whitlock (2002 [One note is these quantities, [One technical technical note is necessary: necessary: when when calculating calculating these quantities, it it is is essential weight each essential to to weight each individual individual equally. equally. The The usual usual calculations calculations of of FST FST weight each Most models weight each local local population population equally, equally, independent independent of of size. size. Most models of of population have assumed and therefore they population structure structure have assumed equal equal deme deme sizes, sizes, and therefore they predict right quantity. not measure predict the the right quantity. Most Most empirical empirical measures measures do do not measure the the appropriate appropriate FST FST exactly. exactly. This This may may be be an an important important issue issue in in some some cases; cases; for example, if smaller demes have higher extinction rates, rates, then then the the subsub for example, if smaller demes have higher extinction FST's would would properly properly be weighted set of the the population population with highest FsT's set of with the the highest be weighted least.] It help to look at at the the equation equation for for the the change mean allele allele frequency frequency It will will help to look change in in mean to selection. From Whitlock (2002),) , we get due to Whitlock (2002 2q ) + q ) ) s( 11 -- r)(FsT FST )(h( 11 -- 2g)+g)) r )(FsT ++ (( 11 -- FST)(h( As-q Asq = == p15 -q q s(
(7.5) (7.5)
where rr is the relatedness relatedness of of two two random random individuals individuals competing competing for for resources. resources. where is the Let us us consider consider the the various various parts parts of of this this equation equation in in turn. turn. First, First, we we see see that that Let the response response to to selection selection is is aa function function of of the the mean mean allele allele frequencies frequencies and and the the the strength of of selection selection io p~ q s. s. These These are are the the classic classic terms terms that that would would appear appear even even strength without population population structure: structure: the the response response to to selection selection is is proportional proportional to to the the without allelic and to to the the strength strength of of selection. selection. allelic variance variance pp qq and Next, we we find find that that the the response response to to selection selection is is proportional proportional to to one one minus minus Next, the the relatedness relatedness of of competing competing individuals. individuals. This This last last phrase phrase deserves deserves some some explanation. Consider Consider aa classic classic dichotomy dichotomy introduced introduced by by Dempster Dempster (1955; ( 1 955; explanation. see also also Christensen, Christensen, 1975) 1 975) between between local local and and global global competition competition for for see resources, i.e., i.e., soft soft versus versus hard hard selection. selection. With With soft soft selection, selection, each each deme deme concon resources, tributes aa number number of of individuals individuals to to the the next next generation generation (whether (whether via via resident resident tributes individuals or or migrants) migrants) independent independent of of the the genotypes genotypes of of the the deme. deme. With With hard hard individuals selection, each each deme deme contributes contributes to to the the next next generation generation in in proportion proportion to to selection, its its mean mean fitness fitness determined determined by by its its genotype genotype distribution. distribution. Under Under soft soft selection, selection, individuals are are competing competing locally locally for for resources, resources, and and therefore therefore there there is is individuals
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
161 161
competition between between relatives. relatives. The The mean mean relatedness relatedness of of individuals individuals from from the the competition same deme deme (without (without inbreeding inbreeding within within demes) demes) is is given given by by rr === 2FsT/(1 2FsT/(1 ++ FsT). FST)' same At the the other other extreme, extreme, under under hard hard selection, selection, there there is is no no local local competition competition for for At resources, and and the the relatedness relatedness of of competing competing individuals individuals is is zero. zero. Putting Putting these these resources, equations into into Eq. Eq. (7.5), (7.5), we we find find that that hard hard selection selection is is always always more more effective effective equations than soft soft selection selection in in changing changing allele allele frequency. frequency. With With local local competition competition for for than resources, if if an an individual individual does does well well because because of of having having aa good good genotype, genotype, it it resources, will, through through competition, competition, reduce reduce the the resources resources available available to to other other individuals individuals will, in the the same same deme. deme. With population structure, these other other local local individuals individuals are are in With population structure, these likely to to share share alleles. alleles. Therefore Therefore the the event event that that would would have have boosted boosted the the numnum likely ber of of copies copies of of this this good good allele allele in in the the next next generation generation (the (the first individual ber first individual doing well) well) is is partially partially counterbalanced counterbalanced by by competition competition against against the the same same doing genotypes. genotypes. Note that that for the relatedness relatedness term, population structure structure tends tends to to Note for the term, increasing increasing population PST results weaken the the response response to to selection. selection. With With soft soft selection, selection, increasing increasing FsT results in in weaken greater relatedness relatedness and lower response response to to selection, selection, all else being greater and therefore therefore aa lower all else being eequal. qual. Finally, we we see the last last term term (FST + ((11 FST)(h ( l -- 2~) 2q) ++ q)) reflec Finally, see in in the (FsT + - FsT)(h(1 ~)) aa reflection the effects increasing homozygosity homozygosity on on the the response response to to selection selection in in tion of of the effects of of increasing structured populations. As As FST increases, so of individ structured populations. FsT increases, so does does the the proportion proportion of individuals that homozygous, even even for for the the same same mean mean allele allele frequency. Greater uals that are are homozygous, frequency. Greater increases the the magnitude magnitude of the response response to to homozygosity, the same homozygosity, for for the same q q,, increases of the selection. increase is is particularly particularly important important if if ~ q isis small small and and the the allele allele is selection. This This increase is at least partially recessive (h < 1/2). 112). In In these these cases, cases, with panmixia, most most at least partially recessive (h < with panmixia, alleles appear and selection therefore cannot alleles appear as as heterozygotes heterozygotes and selection therefore cannot discriminate discriminate increases, most selection is the the recessive recessive alleles. alleles. As As FST FsT increases, most of of the the selection is experienced experienced by by alleles alleles in in the the homozygous homozygous state, state, where where the the alleles alleles have have relatively relatively large large effects. effects. Thus, Thus, in in opposition opposition to to the the effect effect of of relatedness relatedness given given earlier earlier through through its its effects increasing homozygosity, effects on on increasing homozygosity, population population structure structure tends tends to to increase increase the recessive alleles, boost can can be be the response response to to selection. selection. For For nearly nearly recessive alleles, this this boost extremely large. extremely large. This This effect effect of of excess excess homozygosity homozygosity has has been been described described much much earlier earlier with with respect 974). In In respect to to inbreeding inbreeding within within populations populations (Ohta (Ohta and and Cockerham, Cockerham, 11974). fact, between the fact, with with hard hard selection, selection, there there is is no no distinction distinction between the effects effects of of inbreed inbreeding ing due due to to population population structure structure and and that that due due to to local local inbreeding; inbreeding; they they enter enter the the response response to to selection selection equations equations in in exactly exactly the the same same way. way. With With soft soft selection, selection, however, however, the the extra extra effects effects of of competition competition among among relatives relatives change change the the relation relationship ship between between F F and and response response to to selection. selection. The The balance balance between between these these two two effects effects (competition (competition among among relatives relatives and and With hard selection, there is no effect homozygosity) depends on the details. With of rate of of relatedness, relatedness, and and population population structure structure therefore therefore always always increases increases the the rate of response response to to uniform uniform selection. selection. With With soft soft selection, selection, response response to to selection selection can can be be either either increased increased or or decreased decreased depending depending on on the the dominance dominance coefficient coefficient of of the the locus locus under under selection selection and and FST' FsT. The The following following section section shows shows examples examples of of both. both. The The effects effects of of population population structure structure on on even even uniform uniform selection selection are are quite quite complicated. complicated. With With this this selection selection equation equation available, available, aa variety variety of of results results on on basic basic selection selection become become easy easy to to derive. derive. The The next next few few sections sections of of this this chapter chapter show show some some of of these these results. results.
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
1162 62
7.3 7.3
MAINTENANCE ENETIC VARIATION MAINTENANCE OF OF G GENETIC VARIATION IN IN SUBDIVIDED SUBDIVIDED POPULATIONS POPULATIONS One One of of the the oldest oldest questions questions in in population population genetics genetics is is "what "what forces forces are are most most important in in maintaining maintaining genetic genetic variation variation ??"" Population Population subdivision subdivision can can affect affect important the This section the maintenance maintenance of of genetic genetic variation variation in in aa variety variety of of ways. ways. This section reviews reviews aa few these briefly, case of few of of these briefly, focusing focusing on on the the case of spatially spatially uniform uniform selection. selection.
Mutation-Selection M u t a t i o n - S e l e c t i o n Balance Balance Estimates Estimates have have shown shown that that the the genomic genomic rate rate of of mutation mutation to to deleterious deleterious alle alleles les is is reasonably reasonably high, high, ranging ranging from from aa few few per per thousand thousand individuals individuals to to much much greater individual (Lynch greater than than one one per per each each new new individual (Lynch et et aI., al., 1999; 1999; Keightley Keightley and and Eyre-Walker, selection operates Eyre-Walker, 2000) 2000).. Although Although natural natural selection operates to to reduce reduce the the fre frequency of these deleterious quency of these deleterious alleles, alleles, they they are are not not immediately immediately eliminated eliminated com completely. result, some deleterious alleles pletely. As As aa result, some deleterious alleles are are always always segregating segregating in in populations at balance between populations at aa frequency frequency determined determined by by the the balance between mutation mutation and and selection. selection. Some Some have have argued argued that that levels levels of of standing standing genetic genetic variance variance observed observed in in natural natural populations populations could could be be explained explained largely largely by by this this mutation-selection mutation-selection balance. balance. Mutation Mutation is is likely likely not not much much affected affected by by population population structure, structure, but but the the prev previous section showed showed that ious section that the the efficacy efficacy of of selection selection can can be be affected affected greatly greatly by by subdivision. At allele is subdivision. At mutation-selection mutation-selection balance, balance, the the deleterious deleterious allele is likely likely to to be be rare, rare, which which simplifies simplifies Eq. Eq. (7.5) (7.5) to to /lsii As-q == -~ qs( -~s( 11 - r)(Fs r)(FsTT + + ((11 - F FsT)h) ST )h)
(7.6) (7.6)
The allele frequency frequency at at mutation-selection mutation-selection balance balance IS is then then The equilibrium equilibrium allele given by given by J.l. q == q = -s( - s ( 1 1 - r)(Fs r)(FsTT + + ((11 - F FsT)h) ST )h)
^ 2
-
(7.7) (7.7)
(Remember (Remember that that in in the the way way we we have have defined defined fitness fitness in in this this chapter, chapter, aa dele deleterious alleles in terious allele allele has has ss < < 0.) 0.) For For recessive recessive alleles in particular, particular, the the frequency frequency of of deleterious deleterious alleles alleles at at mutation-selection mutation-selection balance balance is is much much reduced reduced with with popu population lation structure structure due due to to the the more more effective effective selection selection against against homozygotes. homozygotes. See See Fig. Fig. 7.3, 7.3, for for some some examples. examples. As As aa result, result, the the amount amount of of variation variation maintained maintained by by mutation mutation selection selection balance balance can can be be reduced reduced greatly greatly in in large large metapopula metapopulations, tions, depending depending on on the the distribution distribution of of dominance dominance coefficients. coefficients. Most Most current current estimates mean dominance dominance coefficient deleterious alleles alleles give estimates of of the the mean coefficient of of mildly mildly deleterious give answers answers around around h h = - 0.1 0.1 (Houle (Houle et et aI., al., 1997; 1997; Garda-Dorado Garcia-Dorado and and Caballero, Caballero, 2000; 2000; Peters Peters et et aI., al., 2003), 2003), so so the the reduction reduction in in variance variance can can be be substantial substantial even even for for relatively relatively small small FS FsTT values. values. The model of genetic mechanism The predominant predominant model of the the genetic mechanism for for inbreeding inbreeding depression depression claims claims that that inbreeding inbreeding depression depression results results from from deleterious deleterious reces recessive mutation-selection balance. sive alleles alleles segregating segregating in in populations populations at at mutation-selection balance. With With population the reduction in mean population structure, structure, the reduction in mean deleterious deleterious allele allele frequency frequency
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
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0.3
FST
Fig. 7 . 3 The equilibrium value of the frequency of a deleterious allele can be changed substantially by population structure. Here solid lines indicate pure soft selection and dashed lines indicate pure hard selection. With very recessive alleles, the equilibrium allele frequency is reduced greatly relative to the case in an undivided population (where C/---- - ~/hs). Parameter values used for these calculations were s -- - 0 . 1 , I~ -- 10 -6, and the three lines correspond to h = 0.4, 0.1, and 0.01 from top to bottom. From Whitlock (2002).
results in in aa potentially potentially large large reduction reduction in in the the amount amount of of inbreeding inbreeding depresdepres results PST values values (see ( see Whitlock, Whitlock, sion predicted predicted for relatively low low FsT sion for aa species, species, even even at at relatively 2002). 2002).
Balancing Selection Balancing Selection Balancing to increase the Balancing selection, selection, by by definition, definition, occurs occurs when when selection selection acts acts to increase the frequencies frequencies of of rare rare alleles. alleles. This This can can happen happen with with overdominance, overdominance, negative negative fre frequency-dependent are favored they are quency-dependent selection selection (where (where rare rare alleles alleles are favored because because they are rare), rare), or or by by spatially spatially heterogeneous heterogeneous selection. selection. Each Each of of these these are are affected affected by by the the spatial spatial population population structure. structure. Overdominance
With With overdominance, overdominance, the the heterozygote heterozygote is is the the most most fit fit genotype. genotype. For For this this sec section tion only, only, let let us us redefine redefine the the fitnesses fitnesses of of the the three three genotype genotype AA, Aa, and and aa as as 1-s 1-s :: 11 :: 1-t, l-t, such such that that the the fitness fitness of of the the two two homozygote homozygote genotypes genotypes is is reduced reduced by by aa factor factor ss or or t.t. With With overdominance overdominance in in aa large large randomly randomly mating mating population, population, there there is is an an intermediate intermediate equilibrium equilibrium allele allele frequency frequency that that stably stably maintains maintains vari variation ation in in the the population population as as aa result result of of the the heterozygote heterozygote being being selected selected for for when whenever ever one one or or the the other other of of the the two two alleles alleles becomes becomes too too rare. rare. In In structured structured populations, populations, the the extra extra homozygosity homozygosity caused caused by by population population structure structure can can change change the the dynamics dynamics of of the the maintenance maintenance of of variance. variance. Nonrandom Nonrandom mating mating causes causes the the marginal marginal fitnesses fitnesses of of the the two two alleles alleles to to be be determined determined more more by by their their homozygous homozygous effects effects and and less less by by their their effects effects in in heterozygotes. heterozygotes. As As aa result, result, if if the the two two homozygotes homozygotes fitnesses fitnesses are are not not equal equal (s (s =1= 4: t), t), then then the the allele allele associated associated with with the the fitter fitter homozygote homozygote will will have have aa higher higher frequency frequency than than expected expected under under random random mating. mating. Mathematically, Mathematically, that that frequency frequency is is given given by by �
ST s - tP tFsT
s
-
q= c)--~ ((ss + + t)( t)(11 - PFST)' ST ) ' -
(7.8) (7.8)
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 64 64
so so long long as as this this value value is is between between zero zero and and one, one, which which it it need need not not be be (Whitlock, (Whitlock, 2002). 2002). If If PST FSTis is large large enough, enough, the the expected expected equilibrium equilibrium leaves leaves the the population population fixed fixed for for the the allele allele with with the the most most fit fit homozygote. homozygote. Thus Thus population population structure structure tends tends to to reduce reduce the the amount amount of of variation variation maintained maintained by by overdominance. overdominance.
Frequency Dependence Dependence Frequency In In some some cases, cases, the the fitness fitness function function changes changes with with the the frequency frequency of of alleles alleles in in the the population population or or species; species; this this is is called called frequency-dependent frequency-dependent selection. selection. If If selec selection displays displays negative negative frequency frequency dependence, dependence, then then alleles alleles are are more more fit fit when when tion rare than when the same allele is common. In this case, selection can act to maintain maintain variation variation in in aa population population because because as as alleles alleles get get rare rare (as (as they they would would from the population) population),, their fitness increases and on the path to being lost from therefore their frequency climbs again. One One of of the the most most studied studied examples examples of of negative negative frequency frequency dependence dependence is is the the self-incompatibility self-incompatibility (51) (SI) alleles alleles common common to to many many species species of of plants. plants. With With 51, SI, pollen parent plant) that shares alleles with the maternal pollen (or, in some cases, cases, its parent plant) that alleles with plant ovules. These plant are are not not allowed allowed to to fertilize fertilize ovules. These processes processes presumably presumably evolved evolved as also prevent as aa mechanism mechanism to to prevent prevent self-fertilization, self-fertilization, but but they they also prevent unrelated unrelated individuals that alleles from alleles at individuals that share share alleles from mating. mating. As As aa result, result, rare rare alleles at the the 51 SI they are able to mate mate with more more other other indi indilocus have higher fitness because they viduals in the population. population. All else being equal, the system always favors new new viduals introduced into into the population, population, but but real species species have limited num numalleles being introduced bers alleles because because of bers of of 51 SI alleles of loss loss due due to to genetic genetic drift. drift. The The smaller smaller the the effective effective population size, the fewer 51 SI alleles maintained at equilibrium. population With population structure, alleles might With population structure, one one might might imagine imagine that that different different alleles might be be maintained increasing the the total in maintained in in different different populations, populations, thereby thereby increasing total diversity diversity in the species out that that this this is is true true for species with very low low the species as as aa whole. whole. It It turns turns out for species with very migration rates rates between but with realistic, intermediate intermediate levels levels of migra migration between demes, demes, but with realistic, of migration the total number number of of 51 SI alleles maintained than would would be tion the total maintained is is slightly lower than be expected 998; Schierup et ai., 2000; Muirhead, Muirhead, expected with with panmixia panmixia (Schierup, (Schierup, 11998; Schierup et al., 2000; 2001). 200 1).
Heterogeneous Selection Heterogeneous It the 1950s varying selection It has has been known known since at least least the 1 950s that that spatially spatially varying can maintain maintain genetic genetic variation, variation, especially especially if if there there is is soft selection (Levene, (Levene, can soft selection 1954; conditions for for this than was was com1 954; Dempster, Dempster, 1955). 1 955 ). The The conditions this are are narrower narrower than com monly Smith and strong, relamonly thought thought (Maynard (Maynard 5mith and Hoekstra, Hoekstra, 1980), 1980), requiring requiring strong, rela tively symmetric symmetric selection. selection. Felsenstein ( 1 976) and and Hedrick Hedrick (1986; ( 1 986; Hedrick Hedrick et et tively Felsenstein (1976) al., ai., 1976) 1 976) reviewed the the theory theory and and empirical evidence for for and and against the the maintenance of of genetic genetic variance variance by by heterogeneous heterogeneous selection. selection. maintenance A form of populations in A different different form of heterogeneous heterogeneous selection selection can can emerge emerge in in populations in which which there there is is already already aa lot lot of of genetic genetic differentiation differentiation among among populations. populations. In these these cases, cases, epistatic epistatic interactions loci can can cause cause different different alleles alleles to to In interactions between between loci be when the describing the be favored favored locally locally even even when the underlying underlying function function describing the relationrelation ship fitness and genotype is Chapters 99 and ship between between fitness and genotype is uniform uniform across across space space (see (see Chapters and 11). 1 1 ). This This sort sort of of heterogeneous heterogeneous selection selection depends depends on on there there being being selectively selectively and epistatically epistatic ally different different alleles alleles in in different local populations, populations, which which becomes becomes and different local important restricted gene important only only under under extremely extremely restricted gene flow flow or or extreme extreme drift. drift.
7. 7.
SELECTION IN METAPOPULATIONS SELECTIONAND AND DRIFT DRIFT IN METAPOPULATIONS
1165 65
One One special special case case of of epistasis epistasis that that may may be be quite quite common common is is that that generated generated on on approximately additively additively interacting interacting alleles that form form aa phenotype phenotype under under stabil stabilapproximately alleles that izing selection. Stabilizing izing selection. Stabilizing selection selection causes causes the the fitness fitness effects effects of of alleles alleles to to vary vary depending all other depending on on whether whether the the sum sum of of the the effects effects of of all other alleles alleles in in the the individual individual add add up up to to aa value value above above or or below below the the optimum optimum for for the the trait; trait; hence hence with with stabiliz stabilizing ing selection, selection, aa population population near near its its optimum optimum will will have have mainly mainly epistatic epistatic variance variance for 995). Barton for fitness fitness associated associated with with that that trait trait (Whitlock (Whitlock et et aI., al., 11995). Barton and and Whitlock Whitlock ((1997) 1 997) have stabilizing selection have shown shown that that with with uniform uniform stabilizing selection and and low low migration, migration, the the amount amount of of genetic genetic variance variance for for aa trait trait that that can can be be maintained maintained can can be be increased increased substantially epistasis. However, likely to substantially as as aa result result of of this this epistasis. However, this this is is only only likely to be be import important ant in in species species with with very very high high values values of of P FST, in the the range range of of P FST > -0.2. -0.2. ST > ST, in
7.4 7.4
ADAPTATION IN SUBDIVIDED ADAPTATION IN SUBDIVIDED POPULATIONS POPULATIONS Population Population structure structure can can affect affect the the pace pace of of adaptive adaptive evolution. evolution. We We have have already already discussed discussed the the conditions conditions under under which which the the response response to to selection selection is is increased increased or or decreased decreased with with population population structure. structure. The The subdivision subdivision also also allows allows novel 1 ), novel patterns patterns of of adaptation, adaptation, such such as as local local adaptation adaptation (see (see Barton, Barton, 200 2001), shifting 1 93 1 ), but Coyne et ai. ((1999) 1 999) and and shifting balance balance evolution evolution [Wright [Wright ((1931), but see see Coyne et al. Whitlock and Phillips Phillips (2000)], (2000)], and and more more rapid evolution with with epistatic epistatic inter interWhitlock and rapid evolution actions Bryant et 986; Goodnight, 9 8 8; see actions ((Bryant et aI., al., 11986; Goodnight, 11988; see Chapter Chapter 9). 9). More More funda fundamentally though, pace of mentally though, population population structure structure strongly strongly affects affects the the pace of evolution evolution even alleles that even for for those those alleles that are are uniformly uniformly selected selected without without any any complicating complicating interactions loci. This interactions with with other other loci. This section section reviews reviews the the effects effects of of population population structure structure on on the the probability probability of of fixation fixation of of new new mutations. mutations.
Probability Probability of of Fixation Fixation One One of of the the most most remarkable remarkable results results in in population population genetics genetics has has to to be be Haldane's 1927) result Haldane's ((1927) result that that aa new new beneficial beneficial allele allele with with heterozygous heterozygous benefit benefit of of hs hs has has only only about about 2hs 2hs chance chance of of ultimate ultimate fixation. fixation. Haldane Haldane assumed assumed that that the ideal (i.e., equaled its the species species in in question question was was ideal (i.e., its its census census size size equaled its effective effective size) size) and undivided. undivided. Even Even in in an an infinite infinite population, population, if if aa new new allele allele is is introduced introduced as as and only only aa single single copy, copy, the the fate fate of of that that allele allele is is partially partially determined determined by by stochastic stochastic changes changes in in the the numbers numbers of of copies copies of of the the allele allele left left in in each each generation. generation. It It turns turns out allele as (as aa rare out that that by by introducing introducing an an allele as aa single single copy copy (as rare mutation mutation would would likely be likely do), do), even even alleles alleles with with moderate moderate selective selective advantage advantage are are more more likely likely to to be lost population than 1 964; see also Crow lost stochastically stochastically from from the the population than fix. fix. Kimura Kimura ((1964; see also Crow and 970) modified and Kimura, Kimura, 11970) modified this this result result to to allow allow for for nonideal nonideal populations populations and and allowed allowed arbitrary arbitrary dominance dominance for for deleterious deleterious alleles alleles as as well. well. He He found found that that the allele is the probability probability of of fixation fixation of of aa beneficial beneficial allele is given given approximately approximately by by 2hsNiN, 2hsNe/N, where where N N is is the the census census size size of of the the population. population. In 11970, Maruyama achieved achieved the the first first results results on on the the probability probability of of fixation fixation In 970, Maruyama in populations. He in subdivided subdivided populations. He showed showed that that in in an an island island model, model, the the probabil probability ity of of fixation fixation for for an an additively additively acting acting allele allele was was simply simply s. s. (For (For additive additive alle alleles, les, h h = = 112, 1/2, so so this this result result is is equivalent equivalent to to the the 2hs 2hs of of Haldane.) Haldane.) Maruyama Maruyama ((1974) 1 974) and 9 8 1 ; Nagylaki, and others others (Slatkin, (Slatkin, 11981; Nagylaki, 1982) 1982) extended extended this this result result to to deal deal with with any any model model such such that that each each deme deme contributes contributes exactly exactly equally equally to to the the next next
MICHAEL MICHAEL C C.. WHITLOCK WHITLOCK
1166 66
generation; the generation; the probability probability of of fixation fixation with with population population structure structure with with this this restriction restriction remained remained s. s. This This was was viewed viewed by by some some as as an an invariant invariant result result of of popu population lation structure; structure; the the claim claim was was made made that that population population structure structure therefore therefore did did not beneficial alleles. not affect affect the the probability probability of of fixation fixation of of beneficial alleles. However, However, this this con conclusion was because other models of pos clusion was premature premature because other models of population population structure structure are are possible ((and and even model) and sible even more more reasonable reasonable than than the the island island model) and because because the the effects dominance were effects of of dominance were not not properly properly accounted accounted for. for. The The first first demonstra demonstration extinction and tion that that this this was was not not true true was was aa model model of of extinction and two two specific specific types types 1 993). In of Barton ((1993). of recolonization recolonization by by Barton In these these cases, cases, the the probability probability of of fixation fixation was reduced by relative to panmictic case. was much much reduced by population population structure structure relative to the the panmictic case. The The probability probability of of fixation fixation in in aa more more general general model model of of structured structured populations populations has has been been found found (Whitlock, (Whitlock, 2003). 2003). Based Based on on Kimura's Kimura's diffusion diffusion equations, equations, this this work work shows shows that that the the probability probability of of fixation fixation can can be be derived derived from from the the equations equations for response to earlier in for drift drift and and response to selection selection presented presented earlier in this this chapter. chapter. Moreover, Moreover, as as long long as as the the strength strength of of selection selection is is lower lower than than the the typical typical immigration immigration rate, rate, the the expected for for neutral neutral loci loci can can be be used used in in these these equations, equations, which which expands expands their their FFsT ST expected 112, the usefulness usefulness greatly. greatly. For For dominance dominance coefficients coefficients differing differing from from 1/2, the equations equations cannot obtained with cannot be be solved solved directly, directly, but but the the answers answers can can be be obtained with numerical numerical inte integration. space, this gration. In In the the interests interests of of space, this chapter chapter will will not not review review the the mathematics mathematics of of the focus on the general general equations, equations, but but will will focus on the the additive additive case, case, as as well well as as an an approxi approximation mation that that works works very very well well for for beneficial beneficial alleles alleles even even with with arbitrary arbitrary dominance. dominance. More details can More details can be be found found in in Whitlock Whitlock (2003). (2003). For For additive additive alleles, alleles, such such that that h h = = 112, 1/2, the the probability probability of of fixation fixation in in struc structured given by tured populations populations is is given by exp[ 2s( 1 11 - exp[-2s(1 - F FsT)Neq] ST )Neq] 2s( 1 -- FFsT)Ne] u[q]= 11 - exp[ exp[-2s(1 ST )Ne]
u [q ]
-
-
(7.9) (7.9)
-
for selection and for soft soft selection and
u[q]
11 --
exp[ -2s( 1 + exp[-2s(1 + F FsT)Neq] ST )Neq]
-2s( 1 ++ FFsT)Ne] u[q]= 11 - exp[ exp[-2s(1 ST )Ne]
(7.10) (7.10)
-
for for hard hard selection, selection, where where q is is the the initial initial allele allele frequency frequency of of the the allele allele in in the the metapopulation. metapopulation. If If the the population population starts starts with with aa single single copy copy of of the the new new allele, allele, the total of the then then q q = = 11/2Ntot, where Ntot mto t is is the total size size of the metapopulation. metapopulation. These These I2Ntot, where equations look fearsome, similar to equations for equations look fearsome, but but in in fact fact they they are are quite quite similar to the the equations for 1 964). There the the panmictic panmictic case case derived derived by by Kimura Kimura ((1964). There are are two two differences. differences. First, First, subdivided population, the the Ne Ne here here is is the the effective effective size size of of aa subdivided population, given given by by Eq. Eq. (7.3). (7.3). Second, Second, the the strength strength of of selection selection ss is is now now modified modified by by aa term term involving involving F FsT, ST, which of selection which reflects reflects the the change change in in the the efficacy efficacy of selection from from population population structure. structure. For can write For beneficial beneficial alleles, alleles, we we can write aa simple simple equation equation for for the the probability probability of of fixation fixation of of aa new new mutant, mutant, even even with with arbitrary arbitrary dominance: dominance:
1 - FST )h )NeINtot. )(FST ++ ((1-FsT)h)Ne/Ntot. -~ 2s( 2s(11 - rr)(FsT uu ==
(7.11) (7. 11)
Here Here it it iiss possible possible to to see see that that this this result result builds builds directly directly on on Kimura's. Kimura's. As As F FsT ST goes zero, this this approaches goes to to zero, approaches the the 2hsN)N 2hsNe/N given given earlier. earlier.
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
1 67 161 A
0.0004 ,.c 0.0004
�
0 O .m ,.i-, cO x ;;;:;
9 0.0003 .---x 0.0003 '0 O
� 0.0002 0.0002
:c 43 co 03 .c
0.0001 se 0.0001 a..
13..
0.02 0.02
0.04 0.04
0.06 0.08 0.08 0.06
0.1 0.1
Extinction rate rate Extinction
B
0.0002 0.0002 0.0001 8 0.00018 0.0001 6 0.00016 0.00014 0.00014 0.000 1 2 0.00012 0.0001 0 0.00010 0.00008 0.00008 0.00006 0.00006 0.00004 0.00004 0.00002 0.00002 0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
Migration rate rate Migration
Fig. 77.4 Examples of of the the fixation fixation probabilities probabilities of of nearly nearly recessive recessive beneficial beneficial alleles alleles (h (h = = 0.01) 0.01 ) Fig. . 4 Examples with soft selection. (A) Extinction and and recolonization. recolonization. In In this this example, example, the the migration migration rate rate between between with soft selection. (A) Extinction populations was 0.05, 0.05, colonization colonization occurred occurred by by four four individuals individuals with with aa probability probability of of common common oriori populations was gin /2, ss == 0.002, and there there were were 1100 00 demes demes with with 100 1 00 diploid diploid individuals each. (Each (Each point point gin of of 11/2, 0.002, and individuals each. represents results 1 07 simulations, so the standard error on the the left left represents results from from 107 simulations, so the standard error ranges ranges from from 6.9 6.9 x • 1100-6 -6 on to 0--66 on on the the right.) right.) As As the the extinction extinction rate the effective effective population population size size of of the to 3.9 3.9 xx 110 rate increases, increases, the the metapopulation decreases, decreases, and and therefore so does probability of (8) A A one-dimen metapopulation therefore so does the the probability of fixation. fixation. (B) one-dimenincreases as as the the sional model. With stepping-stone model, (and therefore sional stepping-stone stepping-stone model. With aa stepping-stone model, FST FST(and therefore Ne) increases migration rate drops so migration. This parmigration so the probability of fixation also also increases with with lower migration. This is par ticularly true true with with recessive recessive alleles, which are are expressed in the the homozygous homozygous state with the the ticularly alleles, which expressed often often in state with concomitant selection. (There are 100 1 00 demes demes with with 100 1 00 diploid individ concomitant increase increase in in the the efficacy efficacy of of selection. (There are diploid individand dots dots represent 1 06 simulations simulations each.) each.) uals each, uals each, ss = = 0.0002 0.0002 and represent 106
These have been been tested simulation in in aa wide wide variety variety of of models models of These results results have tested by by simulation of population population structure, structure, including including the the island island model, model, extinction-recolonization, extinction-recolonization, stepping-stone models, and source-sink models. They work remarkably well (see (see Figs Figs 7.4 7.4 and and 7.5). 7.5). The probability of fixation of beneficial alleles tends to to be much reduced with population structure. with population structure. This This is mainly a result of the fact that the effective
cc
"-= 03 � x 0 O
._x '0 0 ;;;:;
g ~ 9 :c
03 co .c
e a..
O a..
0.002 0.002 0.00175 0.00175 0.001 5 0.0015 0.00125 0.00125 0.001 0.001 0.00075 0.00075 0.0005 0.0005 0.00025 0.00025
•
0.2 0.2
0.6 0.8 0.4 0.4 0.6 0.8 Relative Relative contribution contribution of of sinks sinks
1
Fig. 00 demes, Fig. 7.5 7 . 5 The The probability probability of of fixation fixation in in aa source-sink source-sink model. model. Here Here there there are are 1100 demes, 20 20 of of which 00 individuals, which are are "sources" "sources" and and the the rest rest are are "sinks". "sinks". Each Each deme deme has has 1100 individuals, and and the the immigra immigration tion rate rate to to the the sources sources is is 0.2, 0.2, whereas whereas it it is is 0.25 0.25 in in sinks. sinks. Demes Demes exchange exchange migrants migrants by by aa modi modified fied island island model, model, where where each each sink's sink's contribution contribution to to the the migrant migrant pool pool is is aa fraction fraction of of that that of of each each source. source. As As this this asymmetry asymmetry increases, increases, the the effective effective population population size size is is reduced reduced and and the the probabil probability /2, and ity of of fixation fixation of of beneficial beneficial alleles alleles drops. drops. For For these these examples, examples, ss = = 0.002 0.002 and and hh = = 11/2, and dots dots represent 07 simulations. represent results results of of 1107 simulations.
11 68 68
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
population size models of population size is is reduced reduced in in most most models of population population structure. structure. The The prob probability of loci, especially especially for ability of fixation fixation can can be be increased increased for for some some loci, for nearly nearly reces recessive alleles that sive alleles that can can be be expressed expressed more more strongly strongly in in structured structured populations populations because because of of increased increased homozygosity. homozygosity. Let us the island the island is an Let us return return to to the island model. model. As As mentioned mentioned earlier, earlier, the island model model is an extreme extreme description description of of population population structure structure because because it it allows allows no no variance variance among populations in among populations in reproductive reproductive success. success. For For additive additive alleles, alleles, Maruyama Maruyama and and successors successors found found the the probability probability of of fixation fixation to to be be simply simply ss in in an an island island model, model, the population. The the same same as as in in an an unstructured unstructured population. The more more general general model model predicts predicts that that the the probability probability of of fixation fixation should should be be ss ((11 - FST) FST) NefNtot Ne/Ntot (because (because the the island island model model in in its its basic basic form form as as used used by by Maruyama Maruyama is is also also aa soft soft selection selection model). model). Remember Remember that that the the island island model model has has the the unusual unusual property property of of having having aa larger larger Ne Ne than the probability of fix than census census size: size: Ne Ne= = Nto!(l Ntot/(1 - FST)' FST ). Putting Putting this this Ne N e into into the probability of fixation equation simply s. consistent; what ation equation simplifies simplifies it it to to simply s. The The results results are are consistent; what is is more more important island model important is is that that the the island model is is unrealistic unrealistic and and extreme. extreme. Most Most real real species species and so most will have lower will will have have Ne Ne < < Ntot, mtot, and so most will have lower probabilities probabilities of of fixation fixation of of bene beneficial ficial alleles alleles than than predicted predicted by by Maruyama's Maruyama's formula. formula. Probabilities Probabilities of of fixation fixation are are not population subdivision. not invariant invariant with with respect respect to to population subdivision. Relaxing Relaxing the the assumption assumption of of uniform uniform selection selection has has been been investigated investigated using using the island model (Barton, 11987; 9 8 7; Tachida the island model by by aa variety variety of of authors authors (Barton, Tachida and and lizuka, Iizuka, 11991; 99 1 ; Gavrilets Gavrilets and and Gibson, Gibson, 2002). 2002). Population Population structure structure tends tends to to increase increase the the probability probability of of fixation fixation relative relative to to that that expected expected by by the the mean mean fitness fitness of of the the alleles across demes. known what alleles across demes. It It is is not not yet yet known what effect effect heterogeneous heterogeneous selection selection would have have with with aa more more realistic realistic model model of of subdivision. subdivision. would Population Population structure structure also also substantially substantially affects affects the the time time taken taken for for fixation fixation of of new alleles (Whitlock, (Whitlock, 2003). new alleles 2003). 7.S 7.5
GENETIC GENETIC LOAD LOAD IN IN SUBDIVIDED SUBDIVIDED POPULATIONS POPULATIONS Genetic Genetic load load is is the the reduction reduction in in the the mean mean fitness fitness of of aa population population relative relative to to an an optimal optimal genotype genotype caused caused by by some some particular particular factor, factor, such such as as deleterious deleterious mutation, mutation, genetic genetic drift, drift, and and segregation segregation (Crow, (Crow, 1993). 1993). Load Load is is sometimes sometimes strongly strongly affected affected by by population population structure, structure, as as reviewed reviewed in in this this section. section.
Mutation M u t a t i o n Load Load Mutation Mutation load load is is the the reduction reduction in in mean mean fitness fitness caused caused by by recurrent recurrent delete deleterious mutations mutations in in aa population. Mutation load load is is usually usually calculated calculated at at muta mutarious population. Mutation tion-selection balance: is, it mean reduction reduction in tion-selection balance: that that is, it is is the the mean in fitness fitness associated associated with allele frequency with an an allele frequency predicted predicted by by the the equilibrium equilibrium between between mutation mutation and and selection. In panmictic populations, associated with allele that selection. In panmictic populations, the the load load associated with an an allele that is is not not completely completely recessive recessive is is L L = = 2J.L 2~ (where (where J.L ~ is is the the mutation mutation rate rate from from wild wild type deleterious allele; type to to deleterious allele; remarkably, remarkably, this this is is not not aa function function of of the the strength strength of of selection against selection against the the deleterious deleterious allele). allele). With With population population structure, structure, load load equations equations become become more more complicated complicated (Whitlock, 2002) 2002):: (Whitlock, L -FsT )+FsT)sq L= ~ --( 2(2h( h ( I1--FsT)+FST)S-q
(7. 12) (7.12)
11 69 69
7. AND DRIFT IN METAPOPULATIONS 7. SELECTION SELECTION AND DRIFT IN METAPOPULATIONS
"0 "0
.3
11.2 .2
ttl t~
0 .._1
9 (J) > ._>
�
1
(j) ~ 0.8 nII:
0.6 0.6
~
.
.
-"
~
. ----
--- _
-...- n
0.01 hh=O.01 =
- -
........
-
-
-
- - - - - - - -
0.05 0.05
0.1 0.1
Fig. F i g . 77.6 .6
0. 15 0.15 FFST ST
0.2 0.2
- - _ _ _
-
-
-- - -
0.25 0.25
0.3 0.3
The mutation in aa metapopulation the load in an The mutation load load in metapopulation relative relative to to the load at at aa similar similar locus locus in an undivided undivided population population (-2fL). (-21~). For For the the values values of of FST FSTlikely likely to to be be found found within within species species and and relatively relatively small values values of small of the the dominance dominance coefficient coefficient h, the the mutation mutation load load can can be be reduced reduced substantially substantially in in aa subdivided subdivided population. population. Solid Solid lines lines show show pure pure soft soft selection, selection, whereas whereas dashed dashed lines lines correspond correspond 0. 1 , fL 0--6, 6, and to example are to pure pure hard hard selection. selection. Parameters Parameters for for this this example are s = = -0.1, I~ = = 110 and the the three three pairs pairs 0.01 from of of curves curves correspond correspond to to h = = 0.4, 0.4, 0.1 0.1,, and and 0.01 from top top to to bottom. bottom.
where 7.7). Note where the the value value of of q ~ is is given given by by Eq. Eq. ((7.7). Note that that ss will will cancel cancel out out when when this substitution for q is made, but load remains a function of the dominance this substitution for ~ is made, but load remains a function of the dominance coefficient, case. Figure 7.6 shows coefficient, unlike unlike the the panmictic panmictic case. Figure 7.6 shows the the change change in in load load as as aa function function of of population population subdivision. subdivision. Load Load is is always always reduced reduced with with hard hard selection, selection, but but with with soft soft selection, selection, load load is is increased increased for for high high values values of of FST FST and and near additivity. With nearly recessive alleles, the reduction in load near additivity. With nearly recessive alleles, the reduction in load can can be be nearly %. nearly 5500 %.
Segregation Segregation Load Load Segregation Segregation load load is is the the reduction reduction in in fitness fitness caused caused by by the the inability inability of of aa pop population ulation to to be be composed composed entirely entirely of of heterozygotes heterozygotes even even when when these these genotypes genotypes are are the the most most fit. fit. As As such, such, segregation segregation load load requires requires overdominance. overdominance. With With population structure, there are even fewer heterozygotes population structure, there are even fewer heterozygotes in in aa species species than than under under Hardy-Weinberg Hardy-Weinberg conditions conditions so so the the segregation segregation load load would would be be more more pronounced. pronounced. Using Using the the same same notation notation as as in in the the overdominance overdominance section section given given earlier, earlier, the the segregation segregation load load is is expected expected to to be be L L = =
((11 + + FS FsT)st T )st , ss ++t t '
--
--
((7.13) 7. 1 3 )
which which reduces reduces to to the the segregation segregation load load in in aa panmictic panmictic population population when when FST FST = = 00 ((Crow, Crow, 11958). 958). Therefore, load is Therefore, the the segregation segregation load is ((11 + + FST) FST) times times as as great great in in aa subdivided population one, as expected by subdivided population as as in in an an undivided undivided one, as expected by the the increased increased number number of of homozygotes. homozygotes.
Drift Drift Load Load Drift Drift load load is is the the reduction reduction in in fitness fitness caused caused by by drift drift changing changing allele allele fre frequencies quencies away away from from those those favored favored by by selection. selection. An An extreme extreme form form of of drift drift load load results results from from fixation fixation of of deleterious deleterious alleles alleles by by drift. drift. Drift Drift load load has has received received aa lot lot
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MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
of attention attention in in the the last last several several years years because because of of the the possible possible mutational mutational meltmelt of down of of small small endangered endangered populations populations (Lande, ( Lande, 1994; 1 994; Lynch Lynch et et al., aI., 1995a,b). 1 995a,b). down rate that that deleterious deleterious alleles alleles accumulate accumulate in in aa species species is is aa function function of of the the The The rate efficacy of of selection selection and and of of the the effective effective population population size; size; the the smaller smaller these these two two efficacy values are are the the faster faster drift drift load load will will accumulate. accumulate. We We have have seen seen that that selection selection is is values often more more effective effective in in structured structured populations populations (although (although not not always), always), but but more more often importantly, the the effective effective population population size size tends tends to to be be reduced reduced by by structure. structure. importantly, Because the the latter latter of of these these two two effects effects turns turns out out numerically numerically to to be be more more imporimpor Because tant, in in most most cases, cases, population population structure structure increases increases the the rate rate of of accumulation accumulation of of tant, deleterious alleles alleles (Higgins (Higgins and and Lynch, Lynch, 2001; 200 1 ; Whitlock, Whitlock, 2003). 2003 ). This This is is most most deleterious pronounced in in cases cases with with large large variance variance in in reproductive reproductive success success among among demes, demes, pronounced such as as with with extinction extinction and and recolonization recolonization or or source-sink source-sink models. models. Figure Figure 7.7 7.7 such shows that that the change in in the probability of of fixation fixation of of deleterious deleterious alleles alleles can can shows the change the probability be reasonably reasonably large large (two(two- to to three fold), although although perhaps perhaps in in most most cases cases the the be three fold), change is less less than than aa doubling. doubling. change is
Migration Load Load Migration If the the local local population in aa deme deme is is well well adapted to local local conditions conditions and and if if If population in adapted to migrants to to this this population population come come from populations adapted to other condi migrants from populations adapted to other conditions, then alleles that into the the population population by by migration are likely likely to to tions, then the the alleles that come come into migration are be poorly adapted local conditions. mean fitness be poorly adapted to to local conditions. The The reduction reduction in in mean fitness that that Migration load load increases increases with with increasing increasing dif results is is called called migration results migration load. load. Migration differences the selection coefficients among populations and with migration migration ferences in in the selection coefficients among populations and with most important important type of rate. In some species, migration migration load is likely to to be the the most genetic load. Migration load in determining limits of genetic load. Migration load may may be be key key in determining the the range range limits of species species because because migration migration from from the the species species center center may may prohibit prohibit further further local local adaptation 963; Kirkpatrick 997). adaptation at at the the margins margins (Mayr, (Mayr, 11963; Kirkpatrick and and Barton, Barton, 11997).
B B
A 6• 10 -6 6x1 0-6 5x 10-6 5x1 0-6
tc: 0
'i 0.00003 "~ 0.00003 >< .O m
;._~0.00002 0.00002 o
:0
e
ct! .0 .13
J
4x 10-6 4x1 0-6 3x 10-6 3x1 0-6
._x ""
-6 2xl 00-6 2x1 -6 11xx10-6 10
0.00001
O
c.. 13_
0.02 0.02
0.04
0.06 0 . 0 6 0.08 0.08
Extinction Extinction rate rate
0.1 0.1
0.1 0.1
0.2
0.3 0.3
0.4
0.5
rate Migration rate
Fig. Fig. 7.7 7 . 7 The The probability probability of of fixation fixation of of deleterious deleterious alleles alleles with with (A) (A) extinction extinction and and colonization colonization or or (8) (B) aa one-dimensional one-dimensional stepping stepping stone stone model. model. (A) (A) The The three three lines lines plot, plot, from from bottom bottom to to top, top, the the predicted 1 , and predicted probability probability of of fixation fixation for for alleles alleles with with dominance dominance coefficients coefficients of of 0.5, 0.5, 0. 0.1, and 0.01 0.01,, respectively. 07 replicates respectively. The The symbols symbols mark mark simulation simulation results results over over aa minimum minimum of of 1107 replicates each, each, with with the the three three dominance dominance coefficients coefficients represented represented by by triangles, triangles, squares, squares, and and crosses, crosses, respectively. respectively. Other Other 1 , 1100 00 demes 00 diplOid parameters parameters used used for for these these examples were were 5s = = -0.0002, -0.0002, m m= = 0. 0.1, demes of of 1100 diploid indi individuals viduals each, each, and and colonization colonization by by four four individuals individuals with with aa probability probability of of common common origin origin equal equal to to 11/2. /2. The The probability probability of of fixation fixation isis increased increased substantially substantially by by the the reduction reduction in in Ne N e that that accompanies accompanies extinction 00 extinction dynamics. dynamics. (8) (B) The The parameters parameters in in these these examples examples were were hh = = 0.01 0.01,, 5s = = -0.0002 -0.0002 with with 1100 demes 00 diplOid 08 simulations. demes of of 1100 diploid individuals. individuals. Points Points represent represent the the results results of of 1108 simulations.
7. SELECTION SELECTION AND DRIFT DRIFT IN METAPOPULATIONS METAPOPULATIONS 7.
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Local Genetic Genetic Load Load and and the the Consequences Consequences of of Migration Migration Local In In subdivided subdivided populations, populations, weakly weakly deleterious deleterious alleles alleles can can rise rise by by drift drift to to keephigh frequencies within local populations, even if selection is effective at keep frequency low throughout the species. Crow (1948) (1948) proposed ing their overall frequency that this could be the mechanism for the commonly observed pattern of heter heterthat osis, osis, the the increase increase in in fitness fitness often often observed observed in in hybrids hybrids between between different different popu populations. lations. We We examined examined this this hypothesis hypothesis using using Wright's Wright's distribution distribution of of allele allele frequencies frequencies for for the the island island model model (Whitlock (Whitlock et et ai., al., 2000; 2000; Ives Ives and and Whitlock, Whitlock, 2002) 2002) and and found found that that Crow's Crow's hypothesis hypothesis was was extremely extremely credible. credible. We We referred referred to the reduction reduction in mean fitness caused by this local increase in the frequency local drift drift load load and showed that that reasonably large values of deleterious alleles local of of heterosis heterosis were were consistent consistent with with what what is is known known about about mutation mutation rates rates and and population population structure. structure. These These results results have have been been extended extended by by Morgan Morgan (2002) (2002) and and Gl~min (2003 (2003).). Morgan Morgan (2002) showed showed that that Glemin
(
)
Whybrid ( ((11- - hs hs)2)nVamong )2 nVamong Whybrid == 1 S Wlocal 1 -- S -
(7.14) (7.14)
where demes in where Vamo Vamong the variance variance among among demes in allele allele frequency frequency as as defined defined and and ng is the number of loci. With this we can write a prediction prediction for for the heterosis nn is the number in terms of FST Fsy and q: ~:
heterosis. =. Whybrid WhybYid heterOSIs - . 1l . ---
-Wlocal Wlocal
=
p ((1 I ( l -m hs hs )2 )21)nFSTQ nF~Tqp -1 l -s .
.
1- s
1
(7.15) (7.15)
If the the meta population itself large and and aatt equilibrium, then p == 1 1 If metapopulation itself iiss relatively relatively large equilibrium, then p ~and q ~ is approximately q ~ from and from Eq. (7.7). (7.7). Heterosis interesting biological consequence. If offspring formed formed by by Heterosis has has an an interesting biological consequence. If offspring crosses between between demes have selective advantage, then crosses demes have selective advantage, then the the offspring offspring of of migrants migrants increased fitness (Ingvarsson (Ingvarsson and and Whitlock, Whitlock, 2000; 2000; Morgan, Morgan, 2002). will have increased 2002). Thus of migration actual Thus the genetic effects of migration will will be increased increased relative relative to to the the actual migration rate for aa neutral neutral locus locus is is observed observed migration migration rate. rate. The The effective effective migration rate for approximately approximately erosislf, mee -eheter~ = mm ehet m
(7.16) (7.16)
where 7r is the harmonic harmonic mean mean recombination recombination rate rate between between the the neutral neutral locus locus where is the and and all all selected selected loci loci (Ingvarsson (Ingvarsson and and Whitlock, Whitlock, 2000). 2000). For For low low values values of of FST, FST, the magnification magnification of of the the effective rate of of migration migration can be severalfold. severalfold. This can can be counterbalanced counterbalanced or or reversed by sufficient sufficient local local adaptation adaptation or or strong differ be reversed by strong differences among among populations populations in in epistatic epistatic interactions. interactions. ences
Load Load in in Subdivided Subdivided Populations, Populations, a a Summary Summary Several types types of of load load are are affected by population population structure. structure. Mutation Mutation load load tends tends Several affected by to decline decline at at equilibrium equilibrium with with structure, structure, and and migration migration load load is is lowered lowered with with to lower migration migration rates, rates, whereas whereas drift drift load, load, segregation segregation load, load, and and local local drift drift load load lower tend cumulative, the tend to to increase. increase. Because Because these these different different genetic genetic loads loads are are cumulative, the mean mean
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MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
fitness of of the the population population with with three three different different types types of of genetic genetic load load isis approxiapproxi fitness mately (1 ( 1 -- L1) L1) (1 ( 1 - L2) L2 ) (1 ( 1 -- LL3 3) .) ' IfIf the the loads loads are are small small (they (they are are not not in in general general mately expected to to be) be) then then the the overall overall load load is is approximated approximated by by the the sum sum over over the the types types expected of load. load. Whether Whether population population structure structure increases increases or or decreases decreases mean mean fitness fitness on on of average depends depends on on aa large large number number of of circumstances. circumstances. If If habitat habitat conditions conditions vary vary average strongly, then then population population structure structure allows allows local local adaptation adaptation (in (in other other words, words, strongly, reducing migration migration load) load) and and this this effect effect can can be be paramount. paramount. However, However, if if migramigra reducing tion rates rates become become too too small small and and local local population population size size is is low, low, then then local local drift drift load load tion will become become very very important important and and essentially essentially the the population population will will suffer suffer from from will inbreeding depression. depression. Species-level Species-level drift drift load load could could become become important important ifif there there is is inbreeding lot of of variance variance among among demes demes in in reproductive reproductive success success and and if if the the total total census census size size aa lot of the the species species was was small small (so (so that that the the effective effective size size was was low), low), but but is is likely likely not not very very of important if if the the effective effective size size of of the the species species is is over over about about 10,000. 10,000. Mutation Mutation load load important may be be reduced reduced by by population population structure structure (at (at equilibrium), equilibrium), but but not not by by more more than than may half. In In some some species, species, for for example, example, those those in in which which the the genomic genomic deleterious deleterious aa half. mutation rate rate is is high, high, this this could could be be aa major major effect; effect; but but for for species species with with lower lower mutation mutation rates, rates, this this could could be be aa trivial trivial effect. effect. The The balance balance of of the the effects of these these mutation effects of processes will will depend depend on on the the specifics specifics of of the the species. species. processes -
7.6 7.6
CONCLUSIONS AND INCONCLUSIONS INCONCLUSIONS CONCLUSIONS AND The course of of evolution evolution is is changed changed quantitatively and qualitatively The course quantitatively and qualitatively by by the the subdivision of space. All the population population genetic genetic processes processes subdivision of populations populations over over space. All of of the that populations are substantially, and that act act in in unstructured unstructured populations are affected, affected, sometimes sometimes substantially, and some possible with populations. This some kinds kinds of of evolution evolution are are only only possible with structured structured populations. This chapter chapter focused focused on on the the former: former: quantitative quantitative changes changes in in evolutionary evolutionary rates rates from from population population subdivision. subdivision. Even Even with with uniform uniform selection, selection, the the rate rate of of genetic genetic drift drift and the are changed and the response response to to selection selection are changed substantially. substantially. For this chapter the probabil For some some of of the the quantities quantities described described in in this chapter (e.g., (e.g., Ne, the probability ity of of fixation fixation of of beneficial beneficial alleles), alleles), results results have have already already been been found found for for aa special special case case of of population population structure: structure: the the island island model. model. The The island island model model is is the the oldest oldest in in population population genetics, genetics, and and it it is is rightfully rightfully the the first first to to turn turn to to when when considering considering new new problems problems because because of of its its simplicity. simplicity. Unfortunately, Unfortunately, the the very very simplicity that that make make it it appealing appealing also also makes makes it it an an aberration. aberration. The The island island simplicity model model assumes assumes that that all all demes demes are are equal; equal; not not only only do do all all demes demes have have the the same same population population size size and and migration migration and and immigration immigration rates, rates, but but more more importantly, importantly, it it implicitly assumes assumes that that all all demes demes contribute contribute exactly exactly equally equally to to the the next next gener generimplicitly ation. ation. Clearly Clearly these these conditions conditions do do not not apply apply to to most most (or (or even even all) all) natural natural popu populations, lations, but but this this would would not not matter matter if if these these assumptions assumptions had had no no effect effect on on our our evolutionary evolutionary predictions. predictions. Unfortunately, Unfortunately, this this assumption assumption of of equal equal reproductive reproductive success success has has aa qualitative qualitative effect effect on on our our predictions, predictions, especially especially for for questions questions that that involve involve effective effective size. size. In In this this subtle subtle but but key key respect, respect, the the island island model model is is an an extreme extreme model, model, and and some some of of the the predictions predictions made made from from the the island island model model are are extreme extreme as as aa result. result. Fortunately, Fortunately, it it is is possible possible to to derive derive theory theory that that predicts predicts the the necessary necessary param parameters eters for for other other models models of of population population structure. structure. The The last last couple couple of of decades decades have have seen seen aa lot lot of of development development of of models, models, including including isolation isolation by by distance, distance, local local extinction, extinction, population population size size change, change, variable variable migration migration rates, rates, and and asymmetric asymmetric
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
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migration. Even Even more more fortunately, fortunately, the the results results described described in in this this chapter chapter show show migration. that, at at least least for for weak weak selection, selection, most most of of the the effects effects of of population population structure structure can can that, be described described in in aa few few summary summary statistics, statistics, especially especially FsT PST and and N Ne• This is is be e. This extremely useful useful because because we we know know aa lot lot about about how how FsT PST isis changed changed by by various various extremely demographic processes processes and and we we have have the the theory theory to to predict predict the the effective effective size size for for demographic broad class class of of models. models. FsT PST in in particular particular has has been been very very well well studied, studied, with with aa broad many empirical empirical studies studies devoted devoted to to measuring measuring itit in in aa wide wide variety variety of of species species and and many large number number of of theoretical theoretical models. models. These These include include extinction extinction and and recolonrecolon aa large ization (Wade (Wade and and McCauley, McCauley, 1988; 1988; Whitlock Whitlock and and McCauley, McCauley, 1990), 1 990), populapopula ization tion fission fission and and fusion fusion (Whitlock, (Whitlock, 1994), 1 994), source-sink source-sink models models (Gaggiotti, (Gaggiotti, tion 1 996), and and stepping-stone stepping-stone models models (Kimura (Kimura and and Weiss, Weiss, 1964). 1 964). In In all all of of these these 1996), cases, FsT PST differs differs significantly significantly from from that that predicted predicted by by the the island island model, model, and and in in cases, most the the effective effective population population size size is is also also substantially substantially different different (and (and usually usually most much less less than than the the census census size). size). Moreover, Moreover, itit is is usually usually straightforward straightforward to to much calculate FsT PST even even for for aa novel novel system. system. calculate As an an aside, aside, the the reason reason that that FsT PST has has been been measured measured empirically empirically so so often often has has As little to do do with its importance to predict the effects effects of of population population structure structure on on little to with its importance to predict the selection or or drift. drift. FsT PST has has been been measured measured usually of the the false false hope hope that that selection usually because because of it could could be be used used to to estimate estimate the the number number of of migrants migrants coming coming into into aa population population it per generation (Whitlock and McCauley, 1999). 1 999). It It is is fortunate fortunate then that this this per generation (Whitlock and McCauley, then that effort has not been wasted, important not not to to throw effort has not been wasted, and and it it is is important throw the the evolutionary evolutionary baby the estimator bathwater. PST baby out out with with the estimator bathwater. FsT is is an an excellent excellent descriptor descriptor of of the the nature of population be calculated in genetic nature of population structure structure and and should should be calculated in genetic studies studies of of metapopulations. Unfortunately, Unfortunately, the be said properties as metapopulations. the same same cannot cannot be said for for its its properties as an an estimator estimator of of dispersal. dispersal. There There are are many many unresolved unresolved questions questions on on evolution evolution on on space. space. We We have have made made some some progress progress in in understanding understanding the the effects effects of of population population structure structure on on response response to to uniform uniform selection, selection, but but we we have have not not yet yet made made similar similar progress progress with with the the heterogeneous case. All heterogeneous selection selection case. All of of the the results results considered considered here here deal deal with with dis discrete crete populations populations in in which which organisms organisms are are grouped grouped into into demes demes with with the the space space between between them them empty. empty. Most Most of of the the questions questions presented presented here here have have not not solved solved for for the the spatial spatial case case in in which which individuals individuals are are spread spread continuously continuously over over space, space, aa much much more more challenging challenging topic. topic. These These results results all all assume assume weak weak selection, selection, yet yet some some of of the the most most interesting interesting cases cases involve involve selection selection coefficients coefficients stronger stronger than than migration migration rates. rates. We also also need need many many more more empirical empirical studies studies on on these these topics. topics. This This chapter chapter We has has not not reviewed reviewed the the empirical empirical literature literature at at all, all, but but most most of of the the theory theory pre presented sented here here remains remains untested untested experimentally. experimentally. Furthermore, Furthermore, we we need need better better measures measures of of some some key key parameters. parameters. The The dominance dominance coefficient coefficient has has aa ten tendency dency to to cancel cancel out out of of panmictic panmictic calculations, calculations, but but this this is is not not true true for for evolu evolution tion in in structured structured populations; populations; we we have have very very few few estimates estimates of of the the distribution distribution of of dominance dominance coefficients. coefficients. We We desperately desperately need need more more empirical empirical studies studies of of the the effective effective size size of of structured structured populations. populations. We We also also need need to to develop develop individual-weighted individual-weighted estimators estimators of of PST, FsT, as as has has been been shown shown to to be be required required by by this this theory. theory. The The subdivision subdivision of of aa species species over over space space can can affect affect its its evolution evolution strongly strongly and and in aa variety variety of of ways. ways. Because Because most most species species in in nature nature are are subdivided subdivided over over space, space, in itit behooves behooves us us to to understand understand this this nearly nearly ubiquitous ubiquitous feature feature of of the the natural natural world. world.
sdfsdf
META PO PU PULATIONS M ETAPO LATI O N S AND COALESCENT CO A LES CENT AND TH EORY EO RY John Wakeley Wakeley John
8.1 8.1
INTRODUCTION INTRODUCTION Coalescent Coalescent theory, theory, or or the the study study of of gene gene genealogies, genealogies, provides provides the the framework framework for rapidly moving for empirical empirical molecular molecular population population genetics. genetics. It It is is aa rapidly moving field field that that at at once long history once draws draws upon upon the the long history of of population population genetics genetics theory theory and and responds responds to to the the latest latest advances advances in in biotechnology. biotechnology. The The essence essence of of the the coalescent coalescent is is that that it it models models the the genealogical genealogical history history of of aa sample sample of of genetic genetic data data and, and, via via that that history, history, makes makes predictions predictions about about patterns patterns of of variation variation that that might might be be observed observed development of the coalescent among members of the sample. During the development approach 970s and 980s, there approach between between the the early early 11970s and the the early early 11980s, there was was aa switch switch in in viewpoint from the prospective view view taken by classical classical population population genetics to aa new one that begins with a sample and looks backward in time (Ewens, 11990). 990). The The immense immense practical practical benefit benefit of of this this was was that that it it was was no no longer longer neces necessary sary to to describe describe the the properties properties of of an an entire entire population population and and then then imagine imagine sam sampling piing from from it it in in order order to to make make predictions predictions about about aa sample sample of of genetic genetic data: data: only only the the direct direct ancestors ancestors of of the the sample sample mattered. mattered. The The aim aim of of this this chapter chapter is is to to describe describe the the basic basic features features of of coalescence coalescence in in unstructured unstructured populations, populations, to to dis discuss cuss how how this this forms forms aa basis basis for for inference inference about about population population history, history, and and then then to to discuss discuss the the ways ways in in which which metapopulation metapopulation structure structure changes changes these these basic basic fea features tures and and what, what, in in turn, turn, the the prospects prospects are are for for historical historical inference inference in in metapopu metapopulations. In taking taking the the coalescent coalescent approach, this this chapter chapter complements complements those of of
Ecology, Ecology,Genetics, Genetics,and and Evolution Evolution of of Metapopulations Metapopulations
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Copyright Copyright 2004, 2004, Elsevier, Elsevier,Inc. Inc. 0-12-323448-4 0-12-323448-4
JOHN JOHN WAKELEY WAKELEY
1176 76
Chapters 9, which consider classical, Chapters 77 and and 9, which consider classical, forward-time forward-time dynamics dynamics of of genetic genetic variation in variation in aa metapopulation metapopulation and and the the genetics genetics of of quantitative quantitative traits traits in in aa meta population, respectively. metapopulation, respectively.
8.2 8.2
COALESCENCE COALESCENCE IN IN PANMICTIC PANMICTIC POPULATIONS POPULATIONS Although Although the the seeds seeds of of genealogical genealogical thinking thinking and and coalescence coalescence date date back back at at least 1 972) and Karlin and 1 972) on least to to the the work work of of Ewens Ewens ((1972) and Karlin and McGregor McGregor ((1972) on the the sampling theory neutral alleles alleles under alleles model sampling theory of of selectively selectively neutral under the the infinite infinite alleles model of of mutation, obvious in 1 975) on mutation, and and are are obvious in the the work work of of Watterson Watterson ((1975) on the the number number of of segregating mutation without segregating sites sites in in aa sample sample under under the the infinite infinite sites sites model model of of mutation without recombination, 980s that familiar ancestral recombination, it it was was not not until until the the early early 11980s that the the familiar ancestral process known process known as as the the coalescent coalescent was was firmly firmly established. established. Almost Almost simultaneously, simultaneously, Kingman 1 982a,b,c) proved the convergence of the the ancestral Kingman ((1982a,b,c) proved the convergence of ancestral process process for for aa sample to process, which called the sample to this this simpler, simpler, pure pure death death process, which he he called the n-coalescent, n-coalescent, whereas 1 983a) and 1 983) explored whereas Hudson Hudson ((1983a) and Tajima Tajima ((1983) explored many many properties properties of of gene gene genealogies that biologists. The genealogies that are are of of direct direct interest interest and and use use to to biologists. The mathematically mathematically similar just before similar theory theory of of lines lines of of descent descent was was introduced introduced just before this this by by Griffiths Griffiths ((1980). 1 980). Tavare 1 984) reviewed reviewed these these early early mathematical Tavar~ ((1984) mathematical developments, developments, and and Hudson 1 990) and 1 ) reviewed Hudson ((1990) and Nordborg Nordborg (200 (2001) reviewed the the broader broader biological biological scope scope of of coalescent coalescent theory. theory. This section This section explores explores the the properties properties the the standard standard coalescent coalescent process process that that Kingman described and called called the n-coalescent (a (a bit of terminology that never quite biological practitioners). quite caught caught on, on, at at least least among among more more biological practitioners). Note Note that that ter terminology example, ancestral minology is is used used loosely loosely in in general general here: here: for for example, ancestral process, process, coales coalescent cent process, process, and and genealogical genealogical process process are are used used interchangeably, interchangeably, without without reference to standard coalescent reference to any any particular particular model model of of aa population. population. The The standard coalescent involves involves aa number number of of assumptions assumptions in in addition addition to to the the assumption assumption that that the the popu population panmictic, i.e., lation is is well well mixed, mixed, or or panmictic, i.e., mating mating randomly randomly across across the the entire entire range talking about about diploid also assumed range if if we we are are talking diploid species. species. It It is is also assumed that that variation variation is selectively neutral, neutral, that is selectively that the the effective effective size size of of the the population population has has not not changed changed that there is no recombination recombination within the locus under under study. over time, time, and that The choice model is is rather and typically The choice of of aa mutation mutation model rather flexible flexible and typically depends depends on on the the type of data that that is will be type of data is available available or or will be gathered. gathered. Deviations Deviations from from each each of of these these assumptions sections, although assumptions will will be be considered considered here here and and subsequent subsequent sections, although the the main focus chapter is main focus of of this this chapter is to to describe describe the the effects effects of of metapopulation metapopulation struc structure gene genealogies discuss the ture on on gene genealogies and and the the coalescent coalescent process process and and to to discuss the impli implications cations of of this this for for inference. inference.
The Structure of The Structure of Gene Gene Genealogies Genealogies For For aa wide wide variety variety of of population population models, models, which which differ differ in in terms terms of of important important biological distribution of of offspring number among biological properties, properties, such such as as the the distribution offspring number among members population and members of of the the population and whether whether generations generations are are overlapping overlapping or or dis discrete, 1 9 82a,b,c) proved crete, Kingman Kingman ((1982a,b,c) proved that that the the ancestral ancestral process process for for aa sample sample of of finite finite size size nn converges converges to to the the coalescent coalescent as as the the population population size size tends tends to to infin infinity. happen to ity. In In this this limit, limit, all all of of the the myriad myriad possible possible events events that that could could happen to the the sample looking looking back back in single generation reduce to either all all items sample in time time aa single generation reduce to two: two: either items
8. 8.
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have distinct parents parents or share aa common common ancestor. have distinct or two two members members of of the the sample sample share ancestor. The the popula The other other possibilities, possibilities, whose whose probabilities probabilities become become negligible negligible as as the population tion size size goes goes to to infinity, infinity, are are those those in in which which more more than than one one of of these these common common ancestor ancestor events events happens happens in in aa single single generation. generation. For For example, example, in in aa small small popu population, lation, two two pairs pairs of of samples samples may may have have common common ancestors ancestors or or more more than than two two members single common common ancestor members of of the the sample sample may may share share aa single ancestor in in the the immedi immediately ately previous previous generation, generation, and and the the probability probability of of this this cannot cannot be be neglected. neglected. In In aa large history of large population, population, the the genealogical genealogical history of aa sample sample is is simplified simplified greatly greatly because genealogies are because such such events events are are extremely extremely unlikely. unlikely. The The resulting resulting genealogies are easy easy to describe describe in in words words and and to to model model mathematically. mathematically. to This only for currently sampled items, but This simple simple process process holds holds not not only for the the currently sampled items, but also also for for the the lineages lineages ancestral ancestral to to them them that that existed existed at at some some time time in in the the past. past. Thus, Thus, the the genealogy genealogy of of aa sample sample under under the the standard standard coalescent coalescent is is simply simply aa series series of of com common mon ancestor ancestor events events between between pairs pairs of of lineages, lineages, by by which which the the sample sample of of n n items, items, or or lineages, lineages, can can be be traced traced back back to to aa single single common common ancestor. ancestor. An An example example geneal genealogy .l. ogy is is shown shown in in Fig. Fig. 88.1. Times Times ttoo Common Common Ancestry Ancestry
The The history history of of aa sample sample of of n n items items includes includes exactly exactly n n - 1i coalescent coalescent intervals. intervals. These These are are the the times times in in the the history history of of the the sample sample during during which which there there were were n, n, n - 11,,. .. .., . , 3, 3, 2 2 lineages lineages ancestral ancestral to to the the sample. sample. In In Fig. Fig. 8.1, 8.1, Tj Ti is is used used to to denote denote the lineages. For the time time during during which which there there were were ii ancestral ancestral lineages. For aa broad broad class class of of models models of of aa population population ~ the the "exchangeable" "exchangeable" models models of of Cannings Cannings ((1974) 1 9 7 4 )~ Kingman Kingman showed showed that that these these times times are are independent independent and and distributed distributed exponentially: exponentially: -
fTi(ti) = (~)e-(i~) ti
((8.1) 8.1)
when when time time is is measured measured iinn units units of of G/u G/or 22 generations, generations, where where G G iiss the the total total num number of of copies of each each genetic genetic locus locus in in the the population, population, and and u (r22 is is the the variance variance in in ber copies of offspring offspring number. number. Under Under the the commonly commonly used used Wright-Fisher Wright-Fisher model model (Fisher, (Fisher, 11930; 930; Wright, 93 1 ) of of aa diploid, Wright, 11931) diploid, monoecious monoecious organism, organism, G G is is equal equal to to 2N, 2N,
AA Fig. 8.1
BB
CG
DD
EE
An An example example genealogy genealogy of of five five items items under under the the standard standard coalescent. coalescent.
JOHN WAKELEY WAKELEY iOHN
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where N is equal equal to to 1. 1 . Strictly Strictly speaking, speaking, N isis the the number number of of individuals individuals and and 00".22 is where the coalescent coalescent is is aa model model for for aa haploid haploid population. population. However, However, itit holds holds exactly exactly the as stated stated for for the the diploid diploid Wright-Fisher Wright-Fisher model, model, due due to to the the assumptions assumptions of of ranran as dom mating dom mating and and monoecy, monoecy, and and itit holds holds when when there there are are two two sexes sexes if if N N is replaced with with the the appropriate appropriate effective effective population population size size (M6hle, (Mahle, 1998b). 1 998b). Thus, Thus, replaced on average, average, genealogies genealogies will will look look something something like like the the example example in in Fig. Fig. 8.1 8 . 1 in in on which T2 T2 >> T3 T3 >> T4 T4 > > Ts, Ts, although although the the expectation expectation is is even more skewed skewed than than which even more what is is shown shown in in the the Fig. Fig. 8.1, 8 . 1 , as as E[Ti] 2/[i(i -- 1)]. I}]. The The variances variances of of these these what E [ T;] == 2/[i(i times are are quite quite large large as as well. well. Nordborg Nordborg (2001) (200 1 ) displayed displayed several several realizations realizations times of the the coalescent coalescent process process to to illustrate illustrate this. this. of and TTji are are independent independent if if ii ~-:t j,j, only Eq. Eq. (8.1) ( 8 . 1 ) and and the the fact that T; Using fact that Ti and Using only many useful useful analytical analytical results results can can be be obtained, obtained, including the distribution distribution of of many including the ( TMRCA) and and the the the time time to to the the most most recent common ancestor of the the sample sample (TMRCA) the recent common ancestor of (TTotal), i.e., i.e., the the sum sum of of the the distribution of the the total total length of the the genealogy genealogy (Tvotat), distribution of length of lengths of of all all the the branches in the the genealogy. genealogy. Expressions these two prob lengths branches in Expressions for for these two probability not particularly particularly illuminating, illuminating, and are not not ability functions functions are are complicated, complicated, not and are reproduced here (see Tavare, 984). From From these distributions, or or directly from reproduced here (see Tavar6, 11984). these distributions, directly from Eq. ((8.1), 8 . 1 ), one can obtain the familiar the expected expected values values of of Eq. one can obtain the familiar expressions expressions for for the these quantities: these quantities:
E[TM E[TMRCA] RCA]
( - �),
= 2(11 _ 1 ) , = 2
-111 �1
E [ TTotal] = 22/~1= 2,; 7" --;-. E[rTotal] ;= 1 t
=
((8.2) 8.2) (8.3) (8.3)
As As seen seen later, later, the the second second of of these these determines determines the the expected expected number number of of poly polymorphic, morphic, or or segregating, segregating, sites sites at at aa locus locus when when mutations mutations occur occur according according to to the 975). the infinite infinite sites sites model model (Watterson, (Watterson, 11975). In addition, addition, it it is is not not too too difficult difficult to to derive derive the the expected expected total total length length of of In branches branches in in the the history history that that have have ii descendents descendents in in the the sample: sample:
E['ri]
2 =
_, l
(8.4) (8.4)
((Fu, Fu, 11995), 995), which which for for 11 ::5 -< ii ::5 --- nn - 1 are the individual individual terms in the expected expected sum 8 .3)]. In . 1 , the sum of of all all branch branch length length [Eq. [Eq. ((8.3)]. In Fig. Fig. 88.1, the branch branch above above the the asterisk, asterisk, up up to to the the root root of of the the tree, tree, has has three three descendents. descendents. It It is is the the only only branch branch in in that that particular particular genealogy genealogy that that can can contribute contribute to to 'T3, $3, whereas whereas all all the the other other branches branches contribute contribute to to either either 'T$1l or or 'T$2, and none none in in that that tree tree can can contribute contribute to to 'T4. $4. Under Under 2 , and the represents the opportunity for the infinite infinite sites sites model model of of mutation, mutation, 'T; Ti represents the opportunity for the the cre creation ation of of aa polymorphic polymorphic site site at at which which the the ancestral ancestral base base is is in in nn - ii copies copies and and the copies in 8.4) is the mutant mutant base base in in ii copies in the the sample. sample. Thus, Thus, Eq. Eq. ((8.4) is important important in in mak making ing predictions predictions about about base base frequencies frequencies at at polymorphic polymorphic sites. sites. Branching Pattern of Genealogies
Under Under the the standard standard coalescent coalescent model, model, every every pair pair of of ancestral ancestral lineages lineages has has an an equal equal chance chance of of being being the the pair pair that that coalesces coalesces at at each each common common ancestor ancestor event. event. In In fact, fact, the the simple simple ancestral ancestral process process and and the the rate rate factor factor (�) (~) in in specific specific follow follow
8. 8.
METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY METAPOPULATIONS
11 779 9
from the the fact fact that that each each pair pair of of lineages lineages in in the the coalescent coalescent limit limit coalesces coalesces with with from rate 11 independently independently of of all all other other pairs. pairs. Thus, Thus, every every possible possible random-joining random-joining rate tree is is equally equally likely likely under under the the coalescent. coalescent. In In addition, addition, the the structure structure of of the the tree genealogy and and the the coalescent coalescent times times are are independent independent of of one one another. another. The The many many genealogy useful results results of of the the coalescent, coalescent, some some of of which which are are discussed discussed in in Section Section 8.3, 8 .3 , folfol useful low from from these these facts facts together together with with the the specific specific distributions distributions of of coalescence coalescence low times described described earlier. earlier. In In fact, we have have already already seen seen one one such such result, result, Eq. Eq. (8.4), ( 8 .4), times fact, we in which which the the derivation derivation depends depends on on the the random-joining random-joining structure structure of of genealgeneal in ogies (see (see Fu, Fu, 1995). 1 995). ogies Genealogies and Recombination Genealogies and Recombination
If all all variation variation is is selectively selectively neutral, neutral, and and the other assumptions assumptions of of the stand If the other the standard model model are are true, true, then then the the marginal marginal distribution distribution of of the the genealogy genealogy at at any any ard E [ TMRCA] , nucleotide site site is is given given by by the the coalescent. coalescent. Thus, Thus, quantities such as as E[TMRcA], nucleotide quantities such that do do not not depend depend on on the the joint joint distribution distribution of of the the histories histories and E['ri] E[Tj] that E [ TTotatl, and E[TTotal], at multiple sites in turn turn depend depend on the rate rate of of recombination. recombination. The The joint at multiple sites do do not not in on the joint ancestral process at at two or more critically on recombination. ancestral process two or more sites sites depends depends critically on recombination. Therfore, that do the joint joint histories sites are are affected by Therfore, quantities quantities that do depend depend on on the histories of of sites affected by recombination. For example, the the variances of T MRCA, TTotal, TTotal, and at aa locus locus can can recombination. For example, variances of TMRCA, and TTji at be functions of of the the covariance in coalescence pairs of of sites be expressed expressed as as functions covariance in coalescence times times at at pairs sites in the the sequence, functions of the recombination recombination rate between in sequence, which which in in turn turn are are functions of the rate between the 983b; Hudson Kaplan, 1985). 1 985). McVean McVean (2002) (2002) provided provided the sites sites (Hudson, (Hudson, 11983b; Hudson and and Kaplan, aa simple simple genealogical genealogical derivation correlation in in coalescence coalescence time pair derivation of of the the correlation time for for aa pair of sites. of sites. Hudson ((1983b) 1 983b) and and others, others, including including Kaplan Kaplan and Hudson (1985), ( 1 985), Hudson and Hudson Griffiths 1 996), and 1 999), have Griffiths and and Marjoram Marjoram ((1996), and Wiuf Wiuf and and Hein Hein ((1999), have studied studied the the coalescent process locus. If coalescent process at at aa multisite multisite genetic genetic locus. If there there is is no no recombination, recombination, then then the the entire entire locus locus follows follows one one genealogy. genealogy. Recombination Recombination events, events, viewed viewed backward cause the backward in in time, time, cause the ancestral ancestral segments segments on on either either side side of of the the recombin recombination ation breakpoint breakpoint to to be be separated separated onto onto two two different different copies copies of of the the chromosome. chromosome. Genealogies Genealogies under under recombination recombination become become complicated complicated webs, webs, as as ancestral ancestral sites sites travel travel together together for for periods periods of of time time on on the the same same chromosome chromosome and and are are split split up up by by recombination, recombination, possibly possibly coming coming back back together together later later in in coalescent coalescent events. events. However, However, the the genealogy genealogy of of each each site site individually individually remains remains aa simple simple random randomjoining joining tree, tree, with with marginal marginal distribution distribution described described by by the the standard standard coalescent. coalescent. If If the the recombination recombination rate rate is is very very high, high, then then the the genealogy genealogy of of every every site site is is inde independent pendent of of the the genealogy genealogy of of every every other other site. site. The The effect effect on on the the covariances covariances in in coalescence coalescence times times at at pairs pairs of of sites sites in in the the sequence sequence is is predicted predicted from from these these con considerations. siderations. It It approaches approaches zero zero as as the the recombination recombination rate rate becomes becomes large large and and the the sites' sites' genealogies genealogies become become independent, independent, and and it it grows grows as as the the recombination recombination rate rate decreases. decreases. Extensions Extensions to to the the Coalescent Coalescent
The The basic basic coalescent coalescent technology technology of of modeling modeling the the genealogical genealogical process process for for aa sample been extended sample of of genetic genetic data data has has been extended in in many many different different directions. directions. Examples 1991 ), who Examples include include Slatkin Slatkin and and Hudson Hudson ((1991), who considered considered changes changes in in popu population 1 990), who lation size size over over time; time; Notohara Notohara ((1990), who gave gave aa general general mathematical mathematical model model of of coalescence coalescence in in aa geographically geographically structured structured population; population; Kaplan Kaplan et et al. al. ((1988), 1988), who 1 997), who who modeled modeled strong strong selection, selection, and and Krone Krone and and Neuhauser Neuhauser ((1997), who
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JOHN IOHN WAKELEY WAKELEY
described weak selection. described aa framework framework for for the the coalescent coalescent with with weak selection. Theoretical Theoretical works phenomena in isolation, but works tend tend to to treat treat these these phenomena in isolation, but it it will will often often be be neces necessary sary to to include include several several factors factors when when interpreting interpreting data. data. An An example example is is Kaplan Kaplan et al. ((1991), 19 9 1 ) , who balancing selection, et al. who used used aa model model that that includes includes balancing selection, recombin recombination, subdivision to ation, and and geographic geographic subdivision to explain explain the the decrease decrease in in levels levels of of poly polymorphism morphism with with distance distance from from aa selected selected site site in in samples samples of of the the Adh Adh gene gene in in Drosophila Drosophila melanogaster melanogaster (Kreitman, (Kreitman, 1983). 1983). In In addition, addition, the the standard standard coalescent coalescent has has been been obtained obtained under under aa variety variety of of circumstances, circumstances, such such as as the the case case of of aa two-sex two-sex diploid diploid population population mentioned mentioned earl earlier, in in which which it it is is not not obvious obvious at at first first that that such such aa simple simple model model should should hold hold ier, (M6hle, 998c). These coalescent derive (M6hle, 11998c). These "robustness" "robustness" results results for for the the coalescent derive from from aa lemma of M6hle ((1998) 1998) on chains with lemma of M6hle on the the convergence convergence of of discrete discrete Markov Markov chains with two two timescales timescales to to simpler, simpler, continuous continuous time time processes. processes. For For example, example, in in the the case case of of two two sexes, sexes, lineages lineages switch switch back back and and forth forth between between males males and and females females much much faster is obtained, only with faster than than they they coalesce, coalesce, and and the the standard standard coalescent coalescent is obtained, only with aa rescaled size that number of rescaled effective effective population population size that is is aa function function of of the the number of males males and and females population females in in the the population. population. The The result result for for genealogies genealogies in in aa meta metapopulation described in in Section Section 8.3 8.3 is is based based on on this this kind kind of of separation separation of of timescales. timescales. described
Mutation and Patterns Genetic Variation M u t a t i o n and Patterns of of Genetic Variation in in a a Sample Sample An An aligned aligned set set of of DNA DNA sequences sequences sampled sampled from from aa population population is is aa poten potentially about the tially rich rich source source of of information information about the history history and and current current demography demography of of the the population. population. The The coalescent, coalescent, together together with with aa model model of of mutation, mutation, can can be be used used to to make make predictions predictions about about levels levels and and patterns patterns of of variation variation in in aa sample. sample. The The most most frequently frequently used used mutation mutation model model for for DNA DNA sequences sequences is is the the infinite infinite sites model, which assumes assumes that that every every mutation mutation happens at aa previously previously unmu unmusites model, which happens at tated site. Thus, tared site. Thus, the the infinite infinite sites sites model model is is appropriate appropriate when when the the per-site per-site muta mutation rate is is low. low. Recombination Recombination can, of course, be an an important important factor factor in in tion rate can, of course, be determining patterns patterns of variously assumed assumed determining of genetic genetic variation, variation, and and workers workers have have variously no all (Watterson, 975), independent no recombination recombination at at all (Watterson, 11975), independent assortment assortment among among all all sites 969), or intermediate level sites (Kimura, (Kimura, 11969), or any any intermediate level of of recombination recombination (Hudson, (Hudson, 11983b). 983b). As As noted noted earlier, earlier, the the importance importance of of modeling modeling recombination recombination will will depend on on how how data data are are analyzed. analyzed. depend Undoubtedly about mutation Undoubtedly the the most most important important assumption assumption about mutation in in the the stand standard coalescent is neutral. Genetic ard coalescent is that that all all variation variation is is selectively selectively neutral. Genetic similarities similarities and to past past and present and differences differences among among sampled sampled sequences sequences are are aa view view to and present demography, demography, such such as as metapopulation metapopulation structure, structure, rather rather than than directly directly the the sub subject of natural selection, noted earlier earlier it it is is possible the ject of natural selection, although although as as noted possible to to extend extend the coalescent to to include include selection selection at at aa site under study. study. Due Due coalescent site linked linked to to the the locus locus under to measured under to the the way way in in which which time time is is measured under the the coalescent, coalescent, the the appropriate appropriate mutation mutation parameter, parameter, e, 0, is is similarly similarly scaled. scaled. Under Under the the Wright-Fisher Wright-Fisher model, model, e0 = = 4Nu, 4Nu, where where u u is is the the rate rate of of neutral neutral mutation mutation per per locus locus copy copy per per genera generation. Thus tion. Thus e0 is is equal equal to to twice twice the the average average number number of of mutations mutations introduced introduced into into the the population population each each generation. generation. The The extra extra factor factor of of two two is is due due to to the the histori historical cal importance importance of of the the notion notion of of heterozygosity, heterozygosity, the the expected expected value value of of which which in in randomly mating mating population population is is equal equal to to e. 0. aa randomly Because Because the the mutation mutation rate rate per per generation generation is is very very low, low, mutation mutation is is modeled modeled accurately accurately as as aa Poisson Poisson process process along along the the branches branches of of genealogy. genealogy. Specifically, Specifically,
METAPOPULATIONS AND COALESCENT COALESCENT THEORY THEORY S AND 8. METAPOPULATION
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the the number number of of mutations mutations on on aa lineage, lineage, lineages, lineages, or or entire entire genealogy genealogy of of given given Ot/2. Mutations under length tt follows the Poisson distribution with parameter etl2. neutrality, neutrality, by by definition, definition, do do not not affect affect the the reproductive reproductive rates rates of of individuals. individuals. Thus, Thus, the the genealogical genealogical process process and and the the mutation mutation process process can can be be treated treated separ separately. ately. This This allows allows predictions predictions to to be be made made easily easily under under the the coalescent coalescent about about many many measurable measurable aspects aspects of of DNA DNA sequence sequence polymorphism. polymorphism. The The reason reason for for generating generating such such predictions predictions is is of of course course twofold: twofold: it it builds builds our our understanding understanding how the forces that that maintain variation work, and the predictions can be of how about populations. populations. used for making inferences about Predictions about a b o u t Full Data Data Patterns Patterns Predictions
recomIt is possible under the coalescent with infinite sites mutation and no recom bination bination to to analytically analytically compute compute the the probability probability of of observing observing any any possible possible data data set, set, i.e., i.e., the the full full sample sample of of DNA DNA sequences, sequences, using using aa recursive recursive equation equation ((Griffiths Griffiths and 9 95 ) . The and Tavan\ Tavar~, 11995). The method method can can be be extended extended to to more more general general models Bahlo and models that, that, for for instance, instance, include include geographic geographic structure structure ((Bahlo and Griffiths, Griffiths, 2000). This analytic method method is infeasible except except for for small samples though though 2000). because the number of equations that must be because the number of equations that must be solved solved simultaneously simultaneously becomes becomes astronomically astronomically large large for for complex complex data data sets sets of of many many sequences. sequences. However, However, as as approach can be turned turned into a Monte Monte Carlo Section 8.3 describes, this general approach method of inference. method A A second second issue issue with with this this recursive recursive method method on on full full data data patterns patterns is is that that it it not lend itself to investigations of how how the forces that that produce produce and and main maindoes not act to shape patterns patterns of genetic variation. This This is a general con contain variation act cern cern rather rather than than aa problem problem with with this this particular particular analytic analytic method. method. While While there there obviously aa wealth wealth of of information information in in aa data of DNA DNA sequences, sequences, it it is is is obviously data set set of poorly known which aspects aspects of contain the the bulk bulk of of information information about about poorly known which of data data contain each factor of might have have been been important important in the history history of each factor of evolution evolution that that might in the of the the particular particular species species under under study. study. From From the the theoretical theoretical perspective, perspective, another another aspect aspect this problem is that of this that some of of the the parameters parameters in a complicated historical historical model might might be nonidentifiable nonidentifiable (Beaumont ( Beaumont et al., aI., 2003). 2003 ). There There is essentially just in population in which measure has has been been shown just one one result result in population genetics genetics in which aa measure shown to to contain of the information contain all of information about about a population population parameter. parameter. In the case of of allelic or or haplotypic haplotypic data data from an unstructured, allelic from an unstructured, constant-sized constant-sized population population in in which all variation variation is selectively neutral, numwhich neutral, Ewens Ewens (1972) ( 1972 ) showed showed that that the the num ber for 0, e, i.e., that that the the frequencies frequencies of of the the alleles ber of of alleles is a sufficient sufficient statistic for contain additional information. contain no no additional information. Identifying patterns in in data data that that correspond correspond to to particular particular phenomena phenomena and and Identifying patterns making statements, statements, even even approximate approximate ones, ones, about about the the sufficiency of statistics statistics making sufficiency of will likely be a major major focus focus of of research research in in the the future future given the the current current trends trends in inference inference discussed discussed in Section 8.3. 8 .3. Work Work under under the the coalescent has has focused focused on from the affect gross on how how various various deviations deviations from the standard standard model model affect gross summaries summaries of of the data data such such as as the the expected expected base base frequencies frequencies at at polymorphic polymorphic nucleotide nucleotide sites, sites, the which which form form the the basis basis of of the the "neutrality" "neutrality" tests tests of of Tajima Tajima (1989) ( 19 8 9 ) and and Fu Fu and and Li Li ( 1993). In In the the context context of of subdivided subdivided populations, populations, the the majority majority of of effort has (1993). effort has gone to to studies of Wright's Wright's (1951) ( 1 95 1 ) FF statistics, statistics, most most notably notably the the fixation fixation index index gone studies of FsT, FST, even even though though the the significance significance of of FsT FST in in most most situations situations is is unclear unclear (Whitlock (Whitlock and and McCauley, McCauley, 1999). 1 999). In In order order to to untangle untangle the the complex complex current current and and historical historical demography of of populations, populations, for for instance, instance, those those exhibiting exhibiting metapopulation metapopulation demography
JOHN JOHN WAKELEY WAKELEY
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dynamics DNA sequence be necessary dynamics using using summary summary statistics statistics of of DNA sequence data, data, it it will will be necessary at at aa minimum minimum to to expand expand the the battery battery of of such such measures measures to to include include at at least least as as many many measures measures as as the the number number of of parameters parameters affecting affecting the the population. population. Predictions Predictions about about Summary Summary Measures
The The measures measures of of DNA DNA sequence sequence polymorphism polymorphism that that have have received received the the most most attention attention in in theoretical theoretical studies studies and and the the most most use use in in empirical empirical work work are are the the number of total number sites S, the total number of of polymorphic polymorphic (or (or segregating) segregating) sites the average average number of pairwise pairwise differences differences 'IT, IT, and and the the number number of of polymorphic polymorphic nucleotide nucleotide sites, sites, 'TJi, ~qi, at at which frequent base base is size n. The which the the least least frequent is in in ii copies copies in in the the sample sample of of size The reason reason for for focusing focusing on on these these is is partly partly historical. historical. For For example, example, the the significance significance of of 'IT comes from can be used to comes from the the fact fact that that it it can be used to estimate estimate the the heterozygosity heterozygosity of of aa diploid 993). However, diploid population population (Tajima, (Tajima, 11993). However, concerns concerns about about the the efficiency efficiency of of inferences inferences made made from from sequence sequence data data have have also also been been important. important. For For example, example, 1994) studied possible estimators Fu Fu ((1994) studied the the properties properties of of various various possible estimators of of e0 using using lin linpopulations, the ear ear combinations combinations of of the the 'TJ ms. Again, for for subdivided subdivided populations, the focus focus has has i ' Again, been been on on FST, FsT, which which can can be be seen seen as as aa simple simple extension extension of of average average pairwise pairwise differences population ((Slatkin, Slatkin, 1991; differences to to aa structured structured population 1991; Wilkinson-Herbots, Wilkinson-Herbots, 11998). 998). Because Because mutational mutational and and genealogical genealogical processes processes can can be be treated treated separately separately under under neutrality, neutrality, predictions predictions about about these these and and other other summary summary measures measures can can be be made made by by conditioning conditioning on on the the genealogy genealogy or or on on some some relevant relevant aspect aspect of of the the geneal genealogy. ogy. Conditional Conditional on on the the genealogy, genealogy, the the number number of of mutations mutations in in the the history history of of the the sample sample is, is, again, again, Poisson Poisson distributed, distributed, and and under under the the infinite infinite sites sites model model each each mutation produces produces aa polymorphic polymorphic site. site. Thus, Thus, from from the the probability probability density density func funcmutation the tion total length tion for for the the total length of of the the genealogy, genealogy, Ttotal> Ttotat, the the probability probability function function for for the can be be obtained /(t)dt. Beyond number number of of segregating segregating sites, sites, S, can obtained as as foP{S .f~P{S = klt}hot klt}fTto,at(t)dt. Beyond a this, obtain analytical this, it it is is difficult difficult to to obtain analytical expressions expressions for for the the probability probability functions functions for measures of sequence polymorphism. for measures of DNA DNA sequence polymorphism. However, variances, and However, the the derivation derivation of of expected expected values, values, variances, and covariances covariances of of these these measures measures is is straightforward. straightforward. From From Eqs. Eqs. (8.3) (8.3) and and (8.4), (8.4), and and considering considering the Poisson nature process, we the Poisson nature of of the the mutation mutation process, we have have
(Watterson, (Watterson, 1975) 1975) and and
�1 1 E[S] S] ==0n~ e 2.J11. -;-. E[ i = 111 1 i=
(8.5) (8.5)
0
(8.6) (8.6)
E[Zi] = _, z
(Tajima, 989; Fu 993), where (Tajima, 11989; Fu and and Li, Li, 11993), where Zi Z i is is the the number number of of polymorphic polymorphic copies in sites base is sites at at which which the the mutant mutant base is in in ii copies in the the sample. sample. Typically, Typically, because is not the mutant is the the ancestral because it it is not known known which which is is the mutant and and which which is ancestral base, we base, we have have
E['TJ E[xliil] = =
e (lI + +
_1_. n - I
)
i) 11 + 8~)i,n-i i, n - i +
(8.7) (8.7)
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
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for spectrum, that patterns Zi for the the "folded" "folded" site site frequency frequency spectrum, that is, is, when when the the patterns Z i and and denominator is one Z Z nn--i i are are indistinguishable. indistinguishable. The The Oi 8i,,n-i n_ i term t e r m in in the the denominator is equal equal to to one if if ii = = n n - ii and and zero zero otherwise. otherwise. It It is is needed needed in in order order to to avoid avoid counting counting Zi Zi twice twice variances and covariances of in case where in the the case where ii = = n n - ii = = nl2. n/2. The The variances and covariances of the the 'TJi Tli can can also (Fu, 1995). also be be obtained obtained (Fu, 1995). The The expected expected value value of of the the average average number number of of pairwise pairwise differences, differences, which which can can be be expressed expressed as as aa simple simple linear linear combination combination of of the the mutant mutant base base frequencies, frequencies, 7T ~r = 'i.7:/i En=-(i ((nn -- i) i) Zil\�), Zi/(~), is is equal equal to to e0 (Tajima, (Tajima, 11983), 983), and and this this is is of of course course identical identical to to the the expected expected value value of of S when when n n = = 2. 2. Tajima 1 983) also Tajima ((1983) also obtained obtained the the variance variance of of 7T. ~r. These These and and other other analytical analytical results results have have been been important important in in building building an an understanding understanding about about the the ancestral ancestral process for for aa sample sample and and in in making making inferences inferences about about populations, both when when process populations, both estimating population parameters estimating population parameters and and when when testing testing the the assumptions assumptions of of the the standard standard coalescent coalescent model. model. =
Making nferences Using Making IInferences Using the the Coalescent Coalescent Because Because of of its its close close connection connection to to samples samples of of genetic genetic data, data, the the coalescent coalescent approach approach provides provides aa natural natural framework framework for for inference inference about about the the structure structure and and history history of of populations populations (see (see Stephens, Stephens, 2001). 2001). Inferences Inferences can can of of course course be be made made using the the classical classical forward-time forward-time approach approach to to population population genetics, genetics, but but in in this this case case using it it becomes becomes aa two-step two-step procedure. procedure. First First the the properties properties of of the the entire entire population population are are considered and and then then the the process process of of sampling sampling from from the the population population is is modeled modeled and and considered the the properties properties of of such such samples samples determined. determined. In In some some cases, cases, the the classical classical approach approach may may be be preferable. preferable. For For instance, instance, much much of of the the ease ease and and computational computational efficiency efficiency of of the the coalescent coalescent evaporate evaporate when when weak weak selection selection acts acts on on variation variation (Krone (Krone and and Neuhauser, Neuhauser, 1997; 1997; Neuhauser Neuhauser and and Krone, Krone, 1997). 1997). However, However, the the convenience convenience and and efficiency efficiency of of the the coalescent coalescent approach approach under under neutrality, neutrality, which which stems stems from from the the fact fact that sample can can be reference to that the the genealogy genealogy of of aa sample be modeled modeled without without reference to the the rest rest of of the coalescent aa very the population, population, make make the the coalescent very powerful powerful inferential inferential tool. tool. Analytical Methods Methods
Where Where analytical analytical results results are are available, available, such such as as those those presented presented in in Section Section 8.3, 8.3, corresponding corresponding inferences inferences can can be be made. made. For For example, example, the the analytical analytical expression expression for for the the probability probability of of observing observing S segregating segregating sites sites in in aa sample sample of of size size n n can can be be used to to make make maximum maximum likelihood likelihood estimates estimates of of e0 under under the the assumption assumption of of no no used intralocus intralocus recombination recombination and and infinite infinite sites sites mutation. mutation. However, However, most most analytical analytical methods methods of of inference inference use use the the method method of of moments, moments, i.e., i.e., to to equate equate the the observed observed value of value of aa measure measure of of sequence sequence polymorphism polymorphism with with its its analytical analytical expectation expectation then then to to solve solve for for the the parameter parameter of of interest. interest. This This has has led led to to aa multitude multitude of of esti estie, of population based mators mators of of the the fundamental fundamental parameter, parameter, 0, of the the population based on on S (Watterson, 983), and 'TJl or (Watterson, 1975), 1975), 7T ~r (Tajima, (Tajima, 11983), and ~ql or other other combinations combinations of of the the 'TJi ~qi ((Fu Fu and and Li, Li, 1993; 1993; Fu, Fu, 1994). 1994). Among Among these, these, 7T ~r has has aa rather rather undesirable undesirable statisti statistical property: cal property: it it is is inconsistent. inconsistent. That That is, is, its its variance variance does does not not decrease decrease to to zero zero as as the 995). the sample sample size size tends tends to to infinity infinity (Tajima, (Tajima, 1983; 1983; Donnelly Donnelly and and Tavare, Tavar6, 11995). Therefore, Therefore, estimates estimates based based on on the the number number of of segregating segregating sites, sites, S, or or on on linear linear combinations combinations of of the the site site frequencies, frequencies, 'TJi, xli, are are preferable preferable to to those those made made using using pairwise pairwise differences. differences. These moment-based estimators unbiased and These moment-based estimators are are unbiased and easy easy to to implement. implement. They They also have applicable regardless also have the the advantage advantage of of being being applicable regardless of of the the recombination recombination
WAKELEY JOHN WAKELEY
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rate rate because because the the relevant relevant expected expected values values do do not not depend depend on on the the rate rate of of recomrecom bination. bination. In In fact, fact, their their accuracy accuracy will will increase increase with with the the rate rate of of recombination, recombination, as the the sites sites in in the the sequence sequence become become more more and and more more independent. independent. Of Of course, course, itit as is is not not satisfactory satisfactory to to make make only only point point estimates estimates of of parameters, parameters, and and recombinrecombin ation must must be be considered considered if if any any statement statement about about the the error error of of these these momentmoment ation based estimates estimates is is to to be be made. made. Obtaining Obtaining analytical analytical results results about about the the variances variances based of S, S, ~r, 1T, and and xli 'll i for for arbitrary arbitrary levels levels of of recombination recombination is is not not trivial. trivial. In In fact, fact, itit is is of only only for for ~r 1T that that such such results results are are available available (Hudson, (Hudson, 1987; 1987; Pluzhnikov Pluzhnikov and and Donnelly, 1996). 1 996). In In addition, addition, the the variance variance of of an an estimator estimator is is only only aa useful useful piece piece Donnelly, of information when its its errors are distributed normally or or at at least least when when its its disdis of information when errors are distributed normally tribution is is known known and and is is symmetric, which is is almost almost never never the the case case for meas tribution symmetric, which for measures of of sequence sequence polymorphism. polymorphism. ures Computational Methods Methods Computational
It is is straightforward straightforward and and extremely efficient to to simulate simulate genealogical histor It extremely efficient genealogical histories 1 9 83b, 11990) 990) because because it it is is not not necessary necessary to to simulate the entire entire ies (Hudson, (Hudson, 1983b, simulate the population, just just the sample history. ease of the population, the sample history. The The ease of simulations, simulations, together together with with the desire more about about how how the the errors errors of parameter estimates estimates are are distrib desire to to know know more of parameter distributed, has led led to explosion of computational coalescent coalescent methods methods of uted, has to aa recent recent explosion of computational of the field ment, it it does does appear focus inference. While While the field is is still still in in develope developement, appear that that the the focus inference. has somewhat over over time. time. The first of these methods methods used used Monte Monte Carlo Carlo has shifted shifted somewhat The first of these integration to to compute compute the the likelihood likelihood of full data data set set under under the the coalescent coalescent of aa full integration model. were used over genealogies genealogies by by simulating model. Simulations Simulations were used to to "integrate" "integrate" over simulating aa large number number of of them them and averaging the the results. results. The marks are are in in large and averaging The quotation quotation marks recognition of the of genealogies, continuously distributed recognition of the complexity complexity of genealogies, having having continuously distributed branch lengths lengths and and discrete discrete tree tree structures. This is is an an impossible task if if branch structures. This impossible task genealogies coalescent without genealogies are are simulated simulated using using the the standard standard coalescent without reference reference to to data because the genealogies make data because the overwhelming overwhelming majority majority of of genealogies make aa negligible negligible con contribution to the likelihood the sample sample size or the number number of poly polytribution likelihood unless the morphic morphic sites sites is is small. small. Two Two different different solutions solutions to to this this problem problem were were proposed. proposed. One One was was to to use use the the recursive equations recursive equations for for the the probability probability of of data data under under the the infinite infinite sites sites model, model, discussed .3, to discussed in in Section Section 88.3, to define define an an ancestral ancestral Markov Markov chain chain conditional conditional on on data data and and to to sample sample genealogies genealogies from from this this rather rather than than from from the the "uncondi "unconditional" coalescent process process (Griffiths 994a,b). The tional" coalescent (Griffiths and and Tavare, Tavar~, 11994a,b). The probability probability of of data data is is then then the the average average value value of of aa function function computed computed for for each each simulated simulated path, path, i.e., i.e., genealogy, genealogy, through through this this Markov Markov chain. chain. Because Because only only genealogies genealogies that that are are minimally minimally compatible compatible with with data data under under the the infinite infinite sites sites model model are are generated, generated, the likelihood can the likelihood can be be estimated estimated with with relative relative ease. ease. This This method method has has been been extended extended to to cover cover geographically geographically structured structured populations, populations, both both with with migration migration (Nath 996) and (Nath and and Griffiths, Griffiths, 11996) and without without (Nielsen, (Nielsen, 1998), 1998), and and loci loci that that undergo undergo recombination (Griffiths (Griffiths and and Marjoram, Marjoram, 11996). In the the case case of of recombination, recombination, recombination 996). In the the straightforward straightforward application application of of this this approach approach is is still still quite quite inefficient, inefficient, and and aa that makes better use of importance importance sampling has been more optimal scheme that proposed proposed (Fearnhead (Fearnhead and and Donnelly, Donnelly, 2001 2001).). The other other solution solution to to the the problem problem of of the the enormity enormity of of the the space space of of all all pos posThe sible sible genealogies genealogies was was to to use use aa Markov Markov chain chain Monte Monte Carlo Carlo (MCMC) (MCMC) method method to to focus focus on on genealogies genealogies that that do do contribute contribute substantially substantially to to the the likelihood likelihood (Kuhner (Kuhner et aI., al., 11995). The chain chain is is run run with with aa starting starting genealogy, genealogy, and and each each subsequent subsequent et 995). The
AND COALESCENT 8. 8. METAPOPULATIONS METAPOPULATIONS AND COALESCENT THEORY THEORY
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step involves the proposal of genealogy and acceptance accord step involves the proposal of aa new new genealogy and then then its its acceptance according proposed and ing to to aa probability probability that that depends depends on on how how new new genealogies genealogies are are proposed and on on the the relative relative contributions contributions of of the the current current and and the the new new genealogies genealogies to to the the likeli likelihood. hood. This This is is an an application application of of Metropolis-Hastings Metropolis-Hastings sampling. sampling. If If the the chain chain is is run run long long enough, enough, then then sampling sampling genealogies genealogies from from it it is is equivalent equivalent to to sampling sampling them likelihood of them with with respect respect to to their their relative relative contribution contribution to to the the likelihood of data. data. Mutation Mutation models models other other than than infinite infinite sites sites can can be be incorporated incorporated easily, easily, which which is is an an advantage advantage of of this this approach approach over over the the one one described described earlier. earlier. This This method method has has also also been been extended extended beyond beyond the the standard standard coalescent coalescent to to include include subdivison, subdivison, both both with 999) and 1) with (Beerli (Beerli and and Felsenstein, Felsenstein, 11999) and without without (Nielsen (Nielsen and and Wakeley, Wakeley, 200 2001) migration, migration, and and to to include include recombination recombination (Kuhner (Kuhner et et ai., al., 2000). 2000). By By definition, definition, these these full-data full-data likelihood likelihood methods methods extract extract the the greatest greatest pos possible sible information information from from data. data. However, However, despite despite being being made made feasible feasible by by focus focusing genealogies relevant ing on on genealogies relevant to to data, data, they they are are highly highly computationally computationally intensive, intensive, sometimes sometimes prohibitively prohibitively so. so. Further, Further, it it is is unclear unclear whether whether all all of of this this computa computation tion is is justified justified in in relation relation to to the the questions questions of of statistical statistical sufficiency sufficiency discussed discussed in in Section Section 8.3. 8.3. For For example, example, it it would would be be aa waste waste of of time time to to design design aa full-data full-data method method of of estimating estimating e0 if if the the data data were were allele allele counts counts under under the the infinite infinite alleles alleles model, model, as as all all the the information information about about e0 is is contained contained in in the the number number of of alleles, alleles, not not the the frequencies. frequencies. While While little little is is known known about about the the axes axes of of information information content content in in samples samples of of DNA DNA sequences, sequences, it it cannot cannot be be expected expected that that all all of of the the many many facets facets of contribute equally of polymorphism polymorphism will will contribute equally to to inferences inferences about about particular particular param parameters. eters. Work Work is is clearly clearly needed needed in in this this area, area, both both to to build build our our knowledge knowledge and and intuition intuition and and to to aid aid in in the the development development of of better better computational computational techniques techniques of of inference. inference. Partially Partially in in response response to to these these concerns concerns about about information information content, content, but but mostly mostly due due to to interest interest in in computational computational feasibility, feasibility, there there is is aa growing growing trend trend to to design measures of poly design computational computational methods methods of of inference inference using using summary summary measures of polymorphism 1997), morphism rather rather than than full full data. data. These These methods methods date date back back to to Fu Fu and and Li Li ((1997), Tavare 1 997), and 1 99 8 ) , and Tavar~ et et al. al. ((1997), and Weiss Weiss and and von von Haeseler Haeseler ((1998), and aa more more recent recent example is example is Beaumont Beaumont et et al. al. (2003) (2003).. They They use use simulated simulated genealogies genealogies to to compute compute the probability of the probability of observing observing aa set set of of summary summary measures measures that that are are identical identical to to or or sufficiently sufficiently close close to to the the values values observed observed in in data. data. The The advantage advantage of of this this approach is much larger randomly generated generated genealogies approach is that that aa much larger fraction fraction of of randomly genealogies have have aa chance chance of of producing producing the the observed observed data data summaries summaries than than the the fraction fraction that that contribute contribute significantly significantly to to the the likelihood likelihood of of the the full full data. data. Another Another recent recent trend trend in in inference inference is is the the growing growing popularity popularity of of the the Bayesian Bayesian approach aI., 2003 approach (see (see Beaumont Beaumont et et al., 2003).). The The difference difference between between the the likelihood likelihood and and Bayesian Bayesian approaches approaches is is less less in in the the mechanics mechanics of of the the computational computational methods methods than in than in the the interpretation interpretation of of the the output, output, i.e., i.e., as as aa likelihood likelihood surface surface or or as as aa pos posterior 5 . 1 , Chapter 5 ) . In terior probability probability distribution distribution (see (see also also Box Box 115.1, Chapter 115). In the the former former case, case, the the large large body body of of statistical statistical theory theory on on the the distribution distribution of of likelihood likelihood ratios, ratios, which which holds holds asymptotically asymptotically as as the the sample sample size size tends tends to to infinity, infinity, is is used used to to construct construct confidence confidence intervals intervals and and test test hypotheses. hypotheses. In In the the latter latter case, case, the the credible intervals credible intervals for for parameters parameters given given data data are are drawn drawn so so that that 95%, 95%, or or some some other other chosen chosen percentage, percentage, of of the the posterior posterior distribution distribution lies lies inside inside the the credible credible interval. interval. The The size size of of the the credible credible interval interval can can depend depend strongly strongly on on the the prior prior dis distribution tribution of of the the parameter, parameter, which which may may be be viewed viewed as as aa drawback drawback of of the the Bayesian approach. approach. In Bayesian methods, Bayesian In defense defense of of Bayesian methods, it it is is questionable questionable whether whether
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the available the asymptotic asymptotic theory theory of of likelihoods likelihoods is is valid valid except except when when data data are are available from loci. from aa large large number number of of independent independent loci. "Neutrality" "Neutrality" Tests Tests
The results for produced series series of The analytical analytical results for SS,, '11" ~r,, and and 11 TI1 (or Zl Z1),) , which which produced of 1 (or 8, have also spawned spawned aa series unbiased method of moments estimators unbiased method of moments estimators of of 0, have also series of of statistical coalescent model. model. Tajima's 1 989) D statistical tests tests of of the the standard standard coalescent Tajima's ((1989) D and and the the tests 1 993) are tests of of Fu Fu and and Li Li ((1993) are the the best best known, known, although although aa number number of of others others have proposed (Simonsen 995). All have been been proposed (Simonsen et et aI., al., 11995). All of of these these tests tests are are based based on on the the fact fact that that aa number number different different measures measures of of polymorphism, polymorphism, representing representing different different 8. Under aspects aspects of of data, data, can can all all be be used used to to estimate estimate the the single single parameter parameter 0. Under the the null null model, model, the the expected expected value value of of the the difference difference between between two two such such estimates estimates is is equal zero. Deviations Deviations from equal to to zero. from zero, zero, the the significance significance of of which which are are best best meas measured by by the the simulation scheme of of Simonsen et al. al. ((1995), lead to to rejection rejection of of ured simulation scheme Simonsen et 1 995), lead the the standard standard coalescent. coalescent. Unfortunately, Unfortunately, although although the the standard standard coalescent coalescent involves number of involves aa number of assumptions, assumptions, there there is is aa strong strong tendency tendency to to see see these these tests tests as as tests tests of of selective selective neutrality neutrality only. only. In measure deviations In fact, fact, all all of of these these tests tests simply simply measure deviations from from the the site-frequency site-frequency dis distribution 8.2 displays tribution predicted predicted by by the the coalescent. coalescent. Figure Figure 8.2 displays the the expectation expectation for for this this distribution for aa sample sample of of size size n = = 110, with the of the bars scaled, scaled, by by 0, with the heights heights of the bars distribution for dividing dividing E[Z;] E[Zi] by by E[S] E[S] so so that that they they sum sum to to one. one. The The various various test test statistics statistics detect detect deviations positive or deviations only only in in two two directions d i r e c t i o n s- positive or negative n e g a t i v e- and and if if the the standard standard model is is rejected rejected there there are are aa number number of of possible possible explanations, explanations, which which include include model selection 995). selection and and also also demographic demographic and/or and/or historical historical factors factors (Simonsen (Simonsen et et ai., al., 11995). Some Some statistics statistics assume assume that that information information about about the the ancestral ancestral state state at at each each site site is is known known and and are are thus thus functions functions of of the the mutant mutant base base counts, counts, Zi' Zi. Others Others do do not not make make this this assumption, assumption, instead instead assuming assuming that that the the patterns patterns Zi and and Zn-i Z,-i are are indistin indistinguishable 8.7). The 1989) D, guishable as as in in Eq. Eq. ((8.7). The latter, latter, which which include include Tajima's Tajima's ((1989) D, are are posi positive tive when when the the Zi around around ii = = nl2 n/2 are are inflated inflated relative relative to to Fig. Fig. 8.2 8.2 and and negative negative when when 1I are near either either ii = = 1 I or o r ii = = n n are inflated. inflated. The The statistics statistics that that assume assume the the ances ancesZi near tral tral states states are are known known have have the the potential potential to to detect detect differences differences between between inflated inflated Zi near near ii = = 11 and and inflated inflated Zi Zi near near ii = = n n - 11.. As As shown shown in in Section Section 8.3, 8.3, metapopula metapopulation tion dynamics dynamics can can produce produce aa wide wide variety variety of of site site frequency frequency distributions, distributions, putting putting the the status status of of these these "neutrality" "neutrality" tests tests in in further further jeopardy. jeopardy.
0.4 0.3 E[Z;J -E[SJ
0.2 0.1
2
Fig. 8.2 8.2 The expected expected site frequency distribution under the standard standard coalescent. coalescent.
AND COALESCENT 8. METAPOPULATIONS METAPOPULATIONS AND COALESCENT THEORY THEORY 8.
8.3 8.3
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COALESCENCE IN METAPOPULATIONS COALESCENCE IN METAPOPULATIONS The can be The word word metapopulation, metapopulation, implying implying aa "population "population of of populations," populations," can be applied applied very very broadly broadly to to any any geographically geographically structured structured species, species, particularly particularly ones 968a,b) . ones that that exhibit exhibit local local extinction extinction and and recolonization recolonization (Levins, (Levins, 11968a,b). Hanski Chapter 11)) discussed Hanski and and Gaggiotti Gaggiotti ((Chapter discussed the the current current metapopulation metapopulation con concept, cept, and and other other chapters chapters in in this this book book attest attest to to the the variety variety of of situations situations to to which which the concept has Because population been study the concept has been been applied. applied. Because population geneticists geneticists had had been studying metapopulations for decades before ing metapopulations for decades before the the word word metapopulation metapopulation was was intro introduced, terminology can confusing. In duced, the the terminology can be be confusing. In particular, particular, "population," "population," "subdivided "subdivided population," population, ...."structured structured population," population," and and "total "total population" population" are are often often used used interchangeably interchangeably to to refer refer to to aa metapopulation, metapopulation, and and any any of of "deme" ((Gilmour Gilmour and 939), "subpopulation," "deme" and Gregor, Gregor, 11939), "subpopulation," and and even even "popula "population" tion" are are used used to to refer refer to to geographically geographically local local populations populations within within aa metapopu metapopulation. population and lation. This This section section uses uses meta metapopulation and deme. deme. In In addition, addition, except except for for some work on some brief brief review review of of work on general general models models of of population population structure, structure, this this chapter common meta population notion notion that number of demes chapter adopts adopts the the common metapopulation that the the number of demes is making this assumption implicitly, is not not small. small. Instead Instead of of making this assumption implicitly, the the ancestral ancestral metapopulation number of metapopulation process process described described here here exists exists in in the the limit limit as as the the number of demes demes grows, grows, in in much much the the same same way way that that the the standard standard coalescent coalescent holds holds in in the the limit population size population approaches approaches infinity. limit as as the the population size of of an an unstructured unstructured population infinity. There genetics of There is is aa long long history history of of work work on on the the genetics of structured structured populations, populations, dat dating 1931 ). Much relevant to ing back back at at least least to to Wright Wright ((1931). Much of of this this work work is is relevant to the the discus discussion metapopulation structure, 1 940), who sion of of metapopulation structure, especially especially Wright Wright ((1940), who early early on on saw saw aa major potential recolonization in major potential role role for for extinction extinction and and recolonization in his his shifting shifting balance balance theory 1977) formulated theory of of evolution. evolution. Slatkin Slatkin ((1977) formulated the the basic basic population population genetic genetic model population dynamics commonly used model of of meta metapopulation dynamics that that is is still still commonly used and and identified identified the major possible the major possible effects effects of of extinction extinction and and recolonization recolonization on on genetic genetic variation: variation: ((1) 1 ) that that the the turnover turnover of of demes demes and and recolonization recolonization by by small small numbers numbers of of individ individuals uals can can decrease decrease overall overall levels levels of of variation variation and and (2) (2) that that the the movement movement of of founders population can among founders across across the the meta metapopulation can decrease decrease levels levels of of differentiation differentiation among demes demes in in aa manner manner similar similar to to migration. migration. Pannell Pannell and and Charlesworth Charlesworth (2000) (2000) pro provide (see also 7) which vide an an excellent excellent review review of of these these and and later later works works (see also Chapter Chapter 7) which have have metapopulation structure on well-known summaries of focused on the effects of meta population structure polymorphism 195 1 ) FST' polymorphism such such as as Wright's Wright's ((1951) FsT. approach to the case of population population The formal extension of the coalescent approach structure called the structure occurred occurred only only recently, recently, with with aa general general model model called the structured structured coalescent Nordborg, 11997; 997; Wilkinson-Herbots, 99 8 ) . The coalescent (Notohara, (Notohara, 1990; 1990; Nordborg, Wilkinson-Herbots, 11998). The structured include extinction structured coalescent coalescent does does not not include extinction and and recolonization recolonization of of demes, demes, only only migration migration between between them, them, but but it it could could be be reformulated reformulated to to do do so. so. The The back backme ii that ward ward migration migration rate, rate, mij, mij, is is defined defined to to be be the the fraction fraction of of de deme that is is replaced replaced by by migrants migrants from from deme deme jj each each generation. generation. The The structured structured coalescent coalescent exists exists in in the the limit limit as as the the sizes sizes of of demes demes go go to to infinity infinity but but the the scaled scaled backward backward migration migration rates, weak rates, Mjj Mij = 4Njmj 4Nimij,j, remain remain finite. finite. Thus, Thus, it it assumes assumes that that migration migration is is aa weak force, force, with with aa rate rate roughly roughly comparable comparable to to that that of of genetic genetic driftlcoalescence. drift/coalescence. This This is is not not aa weakness weakness of of the the model. model. If If Mij Mij = = 4Njmij 4Nimij does does not not remain remain finite finite as as the the Ni Ni goes goes to to infinity, infinity, then then migration migration is is aa much much faster faster process process than than drift/coales drift/coalescence, cence, and and the the dynamics dynamics of of the the metapopulation metapopulation converge converge on on those those of of an an unstructured 9 8 0 ) and unstructured population, population, both both forward forward (Nagylaki, (Nagylaki, 11980) and backward backward
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(Notohara, 993) in strong migration limit. In (Notohara, 11993) in time. time. This This is is known known as as the the strong migration limit. In prac practice, tice, the the effects effects of of structure structure are are very very difficult difficult to to detect detect once once the the rates rates Mij Mij are are greater Nordborg and greater than than about about 10. 10. Nordborg and Krone Krone (2002) (2002) studied studied aa structured structured coales coalescent cent in in which which some some of of the the Mij Mq remain remain finite finite whereas whereas others others increase increase without without bound process converges bound and and showed showed that that the the ancestral ancestral process converges to to aa structured structured coa coalescent population that lescent among among the the subunits subunits of of the the meta metapopulation that have have finite finite Mij• Mq. One the major of Wright 19 5 1 ) was was to focus attention One of of the major influences influences of Wright ((1951) to focus attention on on PST FsT as The connection as aa summary summary measure measure of of metapopulation metapopulation structure. structure. The connection to to coales coalescent 1 991 ), who cent theory theory was was made made by by Slatkin Slatkin ((1991), who showed showed that, that, in in the the limit limit of of small small mutation mutation rate, rate, inbreeding inbreeding coefficients coefficients such such as as PST FsTcan can be be expressed expressed in in terms terms of of expected expected pairwise pairwise coalescence coalescence times. times. Under Under the the infinite infinite sites sites mutation mutation model, model, the the expected expected values values of of pairwise pairwise differences differences are are linear linear functions functions of of e0 so so that that tak taking ing ratios ratios of of observed observed pairwise pairwise differences differences within within and and between between demes demes provides provides way of of estimating estimating migration migration parameters. For example, example, under under the the island island aa way parameters. For model metapopulation structure 93 1 ), in model of of metapopulation structure (Wright, (Wright, 11931), in which which every every deme deme exchanges me at exchanges migrants migrants with with every every other other de deme at the the same same rate rate and and all all demes demes are are of of the the same same size, size, expectations expectations of of the the average average number number of of pairwise pairwise differences differences within within and and between between demes, demes, 1Tw ~rw and and 1Tb, %, respectively, respectively, are are given given by by E[rrw]=0
and
E['n'b] = 0(1 + 1 ) ,
(8.8)
population. The where where e0 is is the the scaled scaled mutation mutation parameter parameter for for the the entire entire meta metapopulation. The sole sole migration migration parameter parameter of of the the island island model model can can be be estimated estimated as as = M=
--'"w=---_ __1T'rrw ,IT b
--
(8.9) (8.9)
,rr w
although although this this moment-based moment-based estimator estimator is is certainly certainly not not unbiased. unbiased. However, However, the the island model, many organisms, organisms, is island model, which which is is aa particularly particularly unrealistic unrealistic model model for for many is the the only only model model for for which which there there is is aa simple simple connection connection between between PST, Fsr, or or average average pairwise population (Whitlock pairwise differences, differences, and and the the parameters parameters of of the the meta metapopulation (Whitlock and and McCauley, 999). McCauley, 11999). Theoretical populations has Theoretical work work on on meta metapopulations has focused focused on on pairwise pairwise coalescence coalescence times times or or pairwise pairwise differences differences due due to to their their connection connection with with PST Fsr and and their their utility utility in estimating migration migration rates, rates, but small part in estimating but also also in in no no small part due due to to the the fact fact that that ana analytical lytical results results for for larger larger samples samples under under the the structured structured coalescent coalescent are are difficult difficult to to obtain. obtain. This This is is unfortunate unfortunate because, because, as as noted noted earlier, earlier, estimates estimates made made from from pair pairwise 983; wise differences differences have have relatively relatively poor poor statistical statistical properties properties (Tajima, (Tajima, 11983; Donnelly 995). In Donnelly and and Tavare, Tavar(}, 11995). In hindsight, hindsight, the the historical historical focus focus on on PST FST and, and, relatedly, relatedly, on on pairwise pairwise sequence sequence comparisons comparisons within within and and between between demes demes may may have have drawn drawn attention attention away away from from the the true true goal goal of of such such work, work, which which is is to to understand understand the the dynamics dynamics of of metapopulations metapopulations and and how how these these shape shape the the pat patterns terns of of genetic genetic polymorphism. polymorphism. To To some some degree, degree, this this is is an an unfair unfair statement. statement. The Eqs. ((8.8) 8 . 8 ) and example, can The profound profound importance importance and and utility utility of of Eqs. and (8.9), (8.9), for for example, cannot not be be questioned. questioned. At At the the same same time, time, it it is is clear clear that that these these simple simple measures measures of of polymorphism polymorphism are are not not sufficient sufficient to to untangle untangle the the complicated complicated demography demography of of metapopulations (Pannell (Pannell and 999; Pannell, Pannell, 2003 metapopulations and Charlesworth, Charlesworth, 11999; 2003).).
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
89 11 89
The study study of of genealogies genealogies of of samples from aa metapopulation metapopulation will will help The samples from help ideniden tify likely to con tify the the patterns patterns in in DNA DNA sequences sequences or or other other genetic genetic data data that that are are likely to contain substantial about the tain substantial information information about the dynamics dynamics of of the the metapopulation. metapopulation. Again, Again, the the focus focus here here is is on on the the case case of of aa large large number number of of demes. demes. With With the the additional additional assumption assumption that that the the migrants migrants and/or and/or founders founders that that arrive arrive at at aa deme deme could could have have originated number of originated in in any any one one of of aa large large number of source source demes, demes, it it is is possible possible to to describe these histories, hold for describe the the general general features features of of these histories, which which are are expected expected to to hold for many metapopulation necesmany different different types of meta population structures. Beyond this it will will be neces sary sary to to make make more more detailed detailed assumptions assumptions about about the the structure structure of of the the metapopu metapopulation lation in in order order then then to to explore explore specific specific effects effects on on patterns patterns of of polymorphism. polymorphism. Sampling for for population genetic studies studies is is typically typically not not done done at at random random Sampling population genetic across metapopulation. Instead, number of across the the geographic geographic range range of of aa metapopulation. Instead, aa number of sam samples ples are are taken taken from from aa number number of of different different locations, locations, with with the the geographic geographic dis distances tances between between samples samples from from the the same same location location being being smaller smaller than than those those between locations. This between samples samples from from different different locations. This is is true true even even for for species species with with apparently and it apparently continuous continuous ranges ranges and it is is forced forced by by local local abundance abundance in in species species composed composed of of more more discrete discrete demes. demes. This This is is aa logical logical approach approach to to the the study study of of geographic geographic structure, structure, but but one one that that is is also also conditioned conditioned by by long-standing long-standing notions notions about 8 .9), which, about the the importance importance of of FST FsT and and results results such such as as Eq. Eq. ((8.9), which, again, again, come come directly directly from from the the island island model model of of aa metapopulation. metapopulation. A A consideration consideration of of sam sample genealogies populations may ple genealogies in in meta metapopulations may also also aid aid in in the the design design of of better better sam sampling piing strategies strategies for for studying studying geographic geographic structure. structure. Another Another noteworthy noteworthy aspect aspect of population composed of samples samples from from aa meta metapopulation composed of of aa large large number number of of demes demes or or distributed locations will distributed over over aa very very broad broad range range is is that that many many demes demes or or locations will not not be be sampled sampled at at all. all. Now Now consider consider the the locations locations of of the the lineages lineages ancestral ancestral to to the the sample sample at at some some time time in in the the past. past. Unlike Unlike the the locations locations of of the the present-day present-day samples, samples, which which are are under under experimental experimental control, control, these these will will be be determined determined by by the the history history and and dynamics population and, dynamics of of the the meta metapopulation and, of of course, course, the the depth depth of of time time considered. considered. Recently Recently in in the the past, past, samples samples from from the the same same place place will will tend tend still still to to be be close close together together and and will will be be relatively relatively likely likely to to coalesce. coalesce. Lineages Lineages from from different different loca locations tions are are less less likely likely to to share share aa common common ancestor ancestor recently. recently. In In the the more more distant distant past, lineages past, lineages originally originally sampled sampled from from the the same same location, location, if if they they have have not not coalesced, will coalesced, will have have instead instead moved, moved, by by migration migration and/or and/or extinction/recolo extinction/recolonization, to locations. In nization, to other other locations. In aa metapopulation metapopulation with with aa large large number number of of demes and and in in which which the the number number of of source source demes demes of of migrants migrants and and founders founders is is demes large, lineages are likely to me nor large, these these ancient ancient lineages are not not likely to be be in in the the same same de deme nor are are they they likely likely to to be be in in any any of of the the originally originally sampled sampled demes. demes. Their Their locations locations will will be be the the result of population according result of their their random random movement movement across across the the meta metapopulation according to to the the rates rates of of migration migration and and extinction/recolonization extinction/recolonization between between demes. demes. They They will will tend population that tend to to accumulate accumulate in in the the parts parts of of the the meta metapopulation that contribute contribute greatly greatly to to the the migrant migrant pool pool or or that that send send out out an an abundance abundance of of founders, founders, and and they they will will spend little little time time in in regions regions that that act act as as "sinks" "sinks" instead instead of of "sources" "sources" (Pulliam, (Pulliam, spend 11988). 9 8 8 ) . There There will will be be chances chances for for such such ancient ancient ancestral ancestral lineages lineages to to coalesce, coalesce, mediated mediated by by migration migration and and extinction/recolonization, extinction/recolonization, and and it it may may require require aa lot lot of of wandering wandering of of the the lineages lineages across across the the population population before before the the most most recent recent com common ancestor ancestor of of the the entire entire sample sample is is reached. reached. mon Thus, Thus, for for aa broad broad range range of of specific specific metapopulation metapopulation structures structures that that have have aa large number number of large of demes demes in in common, common, sample sample genealogies genealogies should should exhibit exhibit aa recent recent
JOHN WAKELEY WAKELEY JOHN
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burst of of coalescent coalescent events events among among samples samples taken taken from from the the same same locality locality folfol burst lowed by by aa more more ancient ancient historical historical process process for for the the remaining remaining ancestral ancestral linlin lowed eages. These These have have been been called called the the scattering scattering phase phase and and the the collecting collecting phase phase eages. (Wakeley, 1 999), and and details details of of them them depend depend on on the the details details of of the the dynamics dynamics of of (Wakeley, 1999), the metapopulation. metapopulation. The The next next section section describes an idealized idealized model model of of aa the describes an metapopulation in in which which this this behavior behavior emerges emerges in in the the limit limit as as the the number number of of metapopulation demes goes goes to to infinity infinity and and for which simple simple precise precise descriptions descriptions of of the the scat demes for which scattering and collecting phases are are possible. possible. A A simulation of Pannell Pannell (2003) (2003 ) tering and collecting phases simulation study study of showed, among among other other things, things, that that this this twofold twofold structure structure of of genealogies genealogies is is realreal showed, ized in in aa special special case case of of this model even even if if the the number number of of demes demes is is not not terribly terribly ized this model large. In In addition, addition, Ray Ray et et al. al. (2003) (2003 ) found found this this behavior behavior in in aa model model very very difdif large. ferent from from the the one one described described later. later. Ray Ray et et al. al. (2003) (2003) simulated simulated aa metapopulametapopula ferent tion that that expands expands from from aa single single deme deme over over aa two-dimensional two-dimensional grid and showed tion grid and showed that genetic genetic signatures signatures of of this expansion in in aa sample sample from single deme deme are are that this expansion from aa single strong only if if the the backward backward migration/colonization migration/colonization rate rate of of the the deme deme is is high. high. strong only If the escape the the recent If the migration migration rate rate is is low, low, then then few few lineages lineages will will escape recent burst burst of of scattering-phase be scattering-phase coalescent coalescent events, events, the the effective effective number number of of lineages lineages will will be small, and power to the expansion small, and the the power to detect detect the expansion will will be be low. low.
The Effect on Genealogies The Effect of of Metapopulation M e t a p o p u l a t i o n Structure Structure on Genealogies The same as as the model in The model model considered considered here here is is essentially essentially the the same the model in Wakeley Wakeley and although it it is is unnecessary unnecessary to to assume individual and Aliacar Aliacar (2001 (2001),), although assume that that the the individual demes large (Lessard (Lessard and Wakeley, 2003). 2003). The model assumes assumes that that there there demes are are large and Wakeley, The model are which resembles resembles the the metapopulation metapopulation described described by by are K K "regions," "regions," each each of of which Slatkin 1 977). Each Slatkin ((1977). Each region region may may have have different different values values of of all all parameters, parameters, and and among among regions regions there there is is some some explicit explicit geographic geographic structure. structure. For For ease ease of of discus discussion, case where sion, consider consider the the case where the the demes demes are are large large so so that that the the relevant relevant scaled scaled parameters parameters for for each each such such region, region, or or class class of of demes, demes, are are the the migration migration rate rate Mi Mi = = 4Nimi, 4 N i m i , the the extinction/recolonization extinction/recolonization rate rate Ei Ei = = 4Niei, 4miei, and and the the propagule propagule size size ki' ki, which which is is the the number number of of founders founders of of the the deme deme each each time time it it is is recolonized recolonized after after going going extinct. extinct. Note Note that that ki ki is is the the number number of of founding founding gametes gametes rather rather than than diploid 1 977), and diploid individuals, individuals, as as in in Slatkin Slatkin ((1977), and that that the the parameters parameters Mi M i and and Ei E i dif differ those in 1 ), in fer by by aa factor factor of of two two from from those in Wakeley Wakeley and and Aliacar Aliacar (200 (2001), in keeping keeping with with the the scaled scaled migration migration rate rate M M used used earlier earlier in in this this section. section. The The index index ii of of these 8.3 depicts these parameters parameters ranges ranges from from one one to to K. K. Figure Figure 8.3 depicts one one example example of of such such aa metapopulation. metapopulation. are D D demes demes total total in in the the metapopulation, metapopulation, and and aa fraction fraction [3i ~i of o f these these There are are class ii are in in class class or or region region i,i, where where k ]~iK=l[3i Thus there there are are D D ~[3ii demes demes of of class �l[3i == 11.. Thus so so that that in in the the limit limit D D� ~ 00 oo considered considered later, later, the the number number of of demes demes in in each each class class also also approaches approaches infinity. infinity. Looking Looking back back in in time, time, when when aa lineage lineage experiences experiences aa migration event event or or an an extinction/recolonization extinction/recolonization event, event, it it has has some some probability probability migration of come from me in of having having come from aa de deme in each each of of the the other other regions regions and and some some probabil probability ity of of coming coming from from aa deme deme in in its its own own region. region. That That is, is, mi mi= = kf=l ~K___1 mij mij and and eiei == kf= eij' and E~(=1leij, and these these probabilities probabilities of of movement, movement, given given that that aa migration migration event event or or an an extinction/recolonization extinction/recolonization event event has has occurred, occurred, are are given given by by m;/mi mij/mi and and ei/ej, eij/ei, respectively. respectively. Every Every deme deme in in aa region region has has an an equal equal chance chance of of being being the the source source deme deme of of aa migrant/colonist migrant/colonist from from that that region. region. The The only only constraints constraints on on the the structure structure of of movement movement are are that that lineages lineages can can get get from from any any of of these these K K
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
1191 91
G t~
Fig. 8.3 One conforms to in the the text, One possible possible metapopulation metapopulation that that conforms to the the model model described described in text, in in this this case case with with K K= = 5 5 and and with with an an arbitrarily arbitrarily chosen chosen structure. structure.
regions regions to to any any other, other, given given enough enough time, time, and and that that there there is is aa nonzero nonzero probabil probabilthat a lineage will remain in the same region. Even very strongly con conity that strained strained patterns patterns of of movement movement among among regions, regions, such such as as the the one-dimensional one-dimensional stepping-stone 964), conform stepping-stone model model (Kimura (Kimura and and Weiss, Weiss, 11964), conform to to this this assumption. assumption. A A surprising surprising result result of of this this model model is is that that in in the the limit limit as as D D goes goes to to infinity, infinity, the the details population are details of of this this aspect aspect of of the the geographic geographic structure structure of of the the meta metapopulation are obliterated, obliterated, similarly similarly to to the the way way in in which which all all structure structure disappears disappears in in the the strong strong migration 980; Notohara, migration limit limit (Nagylaki, (Nagylaki, 11980; Notohara, 1993). 1993). Branching Pattern Pattern of of Genealogies
Under 1 998), Under this this model, model, with with the the aid aid of of the the convergence convergence result result of of Mahle M6hle ((1998), it it is is possible possible to to show show that that the the more more ancient ancient part part of of the the history, history, the the collecting collecting phase, 1 982a,b,c) coalescent phase, converges converges to to Kingman's Kingman's ((1982a,b,c) coalescent as as the the number number of of demes demes goes goes to to infinity, infinity, but but with with aa rate, rate, or or an an effective effective size, size, that that depends depends on on all all the the parameters 1 ). This parameters of of the the model model (Wakeley (Wakeley and and Aliacar, Aliacar, 200 2001). This had had been been found found previously include migration extinction and previously in in models models that that include migration but but not not extinction and recoloni recolonization 998, 11999, 999, 2001 zation (Wakeley, (Wakeley, 11998, 2001).). Simulations Simulations imply imply that that the the predictions predictions of of the the model model are are accurate accurate as as long long as as the the number number of of demes demes is is at at least least three three to to four four times 998; Pannell, deal times the the sample sample size size (Wakeley, (Wakeley, 11998; Pannell, 2003). 2003). Surprisingly, Surprisingly, aa great great deal of dynamics of of the the geographic geographic structure structure and and dynamics of the the metapopulation metapopulation ~ the the details details of unsampled of movement movement among among regions regions and and the the values values of of M, M, E, E, and and kk for for unsampled demes demes ~ is is manifest manifest only only through through the the single single effective effective size size of of the the metapopula metapopulation phase. This tion during during the the collecting collecting phase. This results results from from the the fact fact that that when when the the num number ber of of demes demes is is very very large, large, the the lineages lineages will will migrate migrate so so many many times times as as to to reach reach aa stationary stationary distribution distribution over over deme deme types, types, determined determined by by the the movement movement matri matrices ces for for migration migration and and extinction/recolonization, extinction/recolonization, before before two two of of them them end end up up in in the me and the same same de deme and have have the the chance chance to to coalesce. coalesce. Overall, Overall, then, then, if if time time is is scaled scaled by by the the effective effective size size and and e0 is is defined defined accordingly, accordingly, all all the the detailed detailed results results of of the the standard .2, hold standard coalescent coalescent model, model, including including those those discussed discussed in in Section Section 88.2, hold for for the these the these collecting-phase collecting-phase lineages. lineages.
JOHN JOHN WAKELEY WAKELEY
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In this this model, model, for for any any sample, sample, the the transition transition from from the the scattering scattering phase to In phase to this occurs as each ancestral this coalescent coalescent collecting collecting phase phase occurs as soon soon as as each ancestral lineage lineage is is in in aa separate Thus, the taken singly separate deme. deme. Thus, the history history of of aa sample sample taken singly from from different different demes demes in population of in aa meta metapopulation of this this sort sort is is also also described described by by the the standard standard coalescent. coalescent. The The only only evidence evidence of of the the structure structure in in this this "scattered" "scattered" sample sample will will be be in in the the magnitude magnitude of of e, 0, if, if, for for instance, instance, it it could could be be compared compared to to aa scattered scattered sample sample from population of from another another meta metapopulation of the the same same total total size size but but with with different different details details of of structure structure and and thus thus aa different different effective effective size. size. How How many many lineages lineages then then will will enter enter the the collecting collecting phase phase for for other other kinds kinds of of samples? samples? This This is is determined determined by by the the random random outcome outcome of of coalescent, coalescent, migration, migration, and and extinction/recolonization extinction/recolonization within within demes. demes. Assuming Assuming that that demes demes are are large large in in size, size, in in aa deme deme of of type type ii that that contains contains jj lineages lineages these these will will occur occur with with relative relative rates rates j(j j(j - 11),) , jM;, j M i , and and E;. E i. In In the the limit limit as as D D goes goes to to infinity, infinity, the the number number of of demes demes will will be number of lineages ancestral be much much greater greater than than the the number of lineages ancestral to to the the sample, sample, and and migration migration events events will will send send lineages lineages off off to to demes demes that that do do not not contain contain other other ancestral lineages. demes are size, these rates apply apply only ancestral lineages. If If the the demes are small small in in size, these rates only roughly roughly and multiple migration and it it will will be be possible possible for for multiple migration and and coalescent coalescent events events to to occur occur in in aa single single generation. generation. In In either either case, case, both both single single coalescent coalescent events events and and migration migration events sample closer transition to phase by events both both bring bring the the sample closer to to the the transition to the the collecting collecting phase by decreasing deme by decreasing the the number number of of lineages lineages in in the the deme by one. one. If If an an extinction/recolon extinction/recolonization me will ization event event occurs, occurs, whatever whatever lineages lineages remain remain in in the the de deme will be be related related me size possible that through through the the k; ki founders. founders. Even Even if if the the de deme size is is large, large, it it is is possible that several several common common ancestor ancestor events events will will occur occur in in this this step step because because k; ks may may not not be be large. large. one possible sample of Figure Figure 8.4 8.4 shows shows one possible scattering scattering phase phase for for aa sample of size size n n = = 88 from single deme series of events and from aa single deme in in which which aa series of three three coalescent coalescent events and two two migra migration events are tion events are followed followed by by an an extinction/recolonization extinction/recolonization event event in in which which the the remaining all coalesce. result is n' == 33 lineages lineages that remaining three three lineages lineages all coalesce. The The result is n' that will will enter collecting phase. lineages have enter the the collecting phase. These These three three lineages have different different numbers numbers of of descendents descendents in in the the sample, sample, or or different different "sizes." "sizes." Because Because whatever whatever labels labels we we might might assign assign to to these these collecting-phase collecting-phase lineages lineages are are arbitrary arbitrary ~ they they are are ) for the probability that exchangeable exchangeable ~ we we can can write write P(n'; P ( n ' ; a}, a l , a2, a 2 , . ... , , a an) for the probability that n n' lineages there there are are n' lineages are are the the end end of of the the scattering scattering phase phase and and among among these, these, al al have have one one descendent descendent in in the the sample, sample, a2 a2 have have two two descendents descendents in in the the sample, sample, and and so possible size so on. on. The The possible size configurations configurations are are all all those those that that satisfy satisfy !7= En=l1 iaj iai = = n n and 1 a; = and of of course course !7= Ei=lai = n'. n . -
.
n
.
!
a5 = 1 n' = 3
n =8
Fig. 8.4 phase for Fig. 8.4 A A realization realization of of the the scattering scattering phase for aa sample sample from from aa single single deme. deme. The The gray gray cylin cylinder represents the deme back through lines represent lineages ancestral to the sam through time time and lines sample. The ple. The two two attached attached boxes boxes represent represent the the kk = - 2 2 founders founders of of the the deme. deme.
8. 8.
METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY METAPOPULATIONS
1 93 193
Different details details of of the the dynamics dynamics within within demes demes will will give give different different distribudistribu Different an ) . For For example, example, when when the the deme deme size size is is large large and and there there tions, PP(n'; tions, ( n ' ; aab l , aa22, ,. . 9. .9 , an). is no no extinction/recolonization, extinction/recolonization, the the distribution distribution is is identical identical to to Ewen's Ewen's (1972) ( 1972) is distribution, but but with with infinite infinite alleles alleles mutation mutation replaced replaced by by infinite infinite demes demes distribution, migration; the the number number of of alleles alleles becomes becomes the the number number of of collecting-phase collecting-phase lineline migration; ages (n') ( n ' ) and and the the counts of the the allele allele become become the the number number descendents descendents of of these these ages counts of lineages (ai) ( ai) in in the the sample sample (Wakeley, (Wakeley, 1998, 1 998, 1999). 1 999). However, However, if if the the rate rate of of lineages extinction/recolonization is is high high and and migration migration is is absent, absent, then an ) extinction/recolonization then PP(n'; ( n ' ; aab l , aab 2 , 9. 9. .9 , an) will be be the the result result of of tossing tossing nn balls balls into into kk boxes, boxes, with with ai being being the the number number of of will boxes that that contain contain ii balls balls (Wakeley (Wakeley and and Aliacar, Aliacar, 2001). 200 1 ) . Comparable Comparable levels levels of of boxes migration and extinction/recolonization extinction/recolonization combine combine both both these these effects, effects, and and there there migration and are of of course course many many other other possibilities, possibilities, depending depending on on the the local local dynamics dynamics within within are demes and and the the sizes sizes of of demes. demes. demes D goes goes to to infinity, infinity, the the scattering scattering phase phase occurs inde Finally, in in the the limit limit as as D Finally, occurs independently within each each sampled deme so so that probabilities PP(n'; an ) pendently within sampled deme that the the probabilities ( n ' ; aab s , aa22,, 9. .9 . , an) are simply simply multiplied over demes demes to the overall overall chance chance that that n' n' lineages lineages are multiplied over to obtain obtain the enter the collecting phase, with sample enter the collecting phase, with some some distribution distribution of of sizes, sizes, given given that that aa sample of size been sampled number of In sum, the topotopo of size n n has has been sampled among among some some number of demes. demes. In sum, the logical structure logical structure of of sample sample genealogies genealogies in in aa metapopulation metapopulation will will be be identical identical to to that the standard coalescent, except that now the number number of of (collecting (collecting that in in the standard coalescent, except that now the phase) is aa random random variable variable and and each each tip tip will will have have an an associassoci phase) tips tips of of the the tree tree is ated be greater than one one and and is equal to to the the number number of of ated stochastic stochastic size size that that can can be greater than is equal descendents branch in in the the sample. sample. descendents of of that that branch Times Times to to Common Common Ancestry Ancestry
The process. Its The collecting collecting phase phase is is aa metapopulation-wide metapopulation-wide process. Its effective effective size size is is roughly population, although roughly on on the the order order of of the the total total size size of of the the meta metapopulation, although low low rates can make than this this and some types types of of rates of of migration migration can make it it larger larger than and some extinction/recolonization make it situations it extinction/recolonization can can make it smaller. smaller. In In some some situations it is is import important consider these timescale of instance, in ant to to consider these effects effects on on the the timescale of the the coalescent, coalescent, for for instance, in the populations or species, where the context context of of divergence divergence between between two two meta metapopulations or species, where this this timescale reciprocal monophyly timescale determines determines the the probability probability of of reciprocal monophyly of of samples, samples, among among other other things things (Wakeley, (Wakeley, 2000). 2000). Here, Here, we we have have simply simply defined defined the the param parameter eter 60 for for the the metapopulation-collecting metapopulation-collecting phase, phase, and and the the importance importance of of its its effective effective size size is is mostly mostly in in comparison comparison to to that that of of the the scattering scattering phase. phase. This This more more recent phase, which which occurs within demes, depends on the effective sizes of demes. demes. Thus, Thus, the the effective effective size size of of the the collecting collecting phase phase is is about about D D times times larger larger than than that that of of the the scattering scattering phase. phase. In In the the limit limit as as D D goes goes to to infinity, infinity, the the duration duration of of the the scattering scattering phase phase becomes becomes negligible negligible in in comparison comparison to to that that of of the the col collecting phase. lecting phase. Clearly, Clearly, as as in in the the standard standard coalescent, coalescent, the the genealogy genealogy of of aa sample sample from from aa meta population contains exactly n 1 coalescent events. Under the limiting metapopulation contains exactly n - 1 coalescent events. Under the limiting process populations with process described described earlier, earlier, which which holds holds for for meta metapopulations with aa large large number number of of demes, demes, the the first first nn -- n' n' (scattering-phase) (scattering-phase) coalescent coalescent events events have have negligible negligible branch branch lengths, lengths, whereas whereas the the remaining remaining n' n' - 11 have have branch branch lengths lengths determined determined by the Kingman's coalescent process. The scattering phase becomes by the Kingman's coalescent process. The scattering phase becomes an an instant instantaneous aneous adjustment adjustment of of the the sample sample size size and and structure, structure, which which can can be be used used to to obtain obtain results results for for times times to to common common ancestry ancestry as as well well as as predictions predictions about about the the level level and and pattern pattern of of polymorphism polymorphism in in the the sample. sample. It It is is no no longer longer possible possible to to
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JOHN JOHN WAKELEY WAKELEY
write . 1 because write down down aa formula formula like like 88.1 because these these will will depend depend on on the the realization realization of of n' for n' for the the sample. sample. However, However, by by conditioning conditioning on on the the scattering scattering phase, phase, it it is is pos possible 8.2 through 8.4, for sible to to derive derive equations equations like like 8.2 through 8.4, for the the expected expected values values of of T TMRCA, MRCA , [n' ln] to n' lineages Ttotal, can write Ttota h and and 'Tj. a"i. We We can write P P[n'ln] to denote denote the the probability probability that that n' lineages enter enter the the collecting collecting phase, phase, i.e., i.e., without without regard regard to to their their sizes. sizes. Properties Properties of of T TMRCA and Ttotal Ttot~ldepend depend only only on on this this overall overall number number n', n', and and relatively relatively simple simple MRCA and analytic 998, 200 1). analytic expressions expressions can can be be obtained obtained in in some some cases cases (Wakeley, (Wakeley, 11998, 2001). Quantities Quantities such such aass 'Tj T i depend depend oonn the the sizes sizes ooff the the collecting-phase collecting-phase lineages. lineages. IInn the the context context of of aa metapopulation, metapopulation, these these frequency frequency measures measures should should be be redefined redefined to to represent joint frequencies 999). represent the the joint frequencies among among sampled sampled demes; demes; (e.g., (e.g., see see Wakeley, Wakeley, 11999).
The The Effect Effect of of Metapopulation M e t a p o p u l a t i o n Structure Structure on on Genetic Genetic Variation Variation in a a Sample Sample in The The level level and and pattern pattern of of genetic genetic variation variation in in aa sample sample from from aa metapopula metapopulation tion are are determined determined by by the the recent recent history history of of coalescent, coalescent, migration, migration, and and extinc extinction/recolonization tion/recolonization events events in in the the sampled sampled demes demes and and by by the the more more ancient ancient coalescent coalescent process process that that occurs occurs among among the the remaining remaining lineages. lineages. To To obtain obtain predictions predictions for for polymorphism polymorphism in in the the sample, sample, it it is is only only necessary necessary to to condition condition on phase. In on the the outcome outcome of of the the scattering scattering phase. In some some cases cases this this can can be be done done ana analytically, lytically, whereas whereas in in others others it it will will be be necessary necessary to to use use simulations. simulations. These These simulations are are nearly as straightforward straightforward as as simulations simulations of of the the standard standard coa coasimulations nearly as lescent, phase. First lescent, the the difference difference being being the the addition addition of of the the scattering scattering phase. First the the scattering phase is simulated, and his scattering phase is simulated, and all all branches branches during during this this period period of of the the history remaining lineages lineages are tory have have lengths lengths set set to to zero. zero. The The remaining are then then fed fed into into the the usual instance, as described in 1 990). The usual coalescent coalescent simulation, simulation, for for instance, as described in Hudson Hudson ((1990). The advantage advantage of of this this in in terms terms of of the the efficiency efficiency of of the the simulations simulations is is in in not not having having to to represent represent all all of of aa large large number number of of demes, demes, only only those those from from which which samples samples have taken. In have been been taken. In addition, addition, convergence convergence of of the the collecting collecting phase phase to to Kingman's rep Kingman's coalescent coalescent shows shows that that aa lot lot of of time time could could be be wasted wasted trying trying to to represent the myriad myriad details details of of movement movement of of lineage across the the metapopulation metapopulation resent the lineage across during during the the collecting collecting phase. phase. There There are are two two main main effects effects of of metapopulation metapopulation structure structure on on patterns patterns of of genetic genetic variation, variation, which which are are represented represented conveniently conveniently and and separately separately in in the the scattering phase. First, scattering phase phase and and the the collecting collecting phase. First, of of course, course, overall overall levels levels of of vari variation ation are are determined determined by by the the collecting-phase collecting-phase coalescent coalescent process, process, but but the the connection sample to connection of of the the sample to this this more more ancient ancient history history is is mediated mediated by by the the scat scattering overall levels tering phase. phase. In In particular, particular, the the overall levels of of polymorphism polymorphism in in aa sample sample are are greater n' is n' is n' is greater when when n' is larger larger and and smaller smaller when when n' is smaller. smaller. In In fact, fact, if if n' is equal equal to to one, one, which which is is possible possible only only if if all all samples samples come come from from the the same same deme, deme, there there will variation in surprising, but will be be no no variation in the the sample. sample. This This seems seems surprising, but it it is is an an under understandable consequence number of being very standable consequence of of the the number of demes demes being very large large and and e0 being being finite: finite: the the values values of of e0 for for individual individual demes demes must must be be infinitesimal. infinitesimal. This This is is prob probably ably appropriate appropriate for for low low mutation mutation rate rate data data such such as as DNA DNA sequence sequence data. data. The The alternative, alternative, that that demic demic e0 values values are are not not small, small, dictates dictates that that the the metapopulation metapopulationwide wide e0 approaches approaches infinity infinity as as the the number number of of demes demes grows, grows, predicting predicting an an infi infinite nite number number of of polymorphisms polymorphisms in in the the sample. sample. The The assumption assumption that that demic demic e0 values are small may values are not not small may be be appropriate appropriate for for loci loci with with higher higher mutation mutation rates, rates, such such as as microsatellites. microsatellites. In In this this case, case, the the model model would would predict predict an an infinite infinite number number
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of microsatellite microsatellite mutations mutations during during the the collecting collecting phase, phase, and and itit would would be be approappro of priate to to study sample probabilities probabilities of of identity identity as as in in Vitalis Vitalis and and Couvet Couvet priate study sample (2001 a,b). Under Under the the assumption assumption of of finite finite total-population total-population 00 made made here, here, all all (2001a,b). genetic variation variation is is ancestral ancestral in in the the sense sense that that itit is is not not due due to to mutations mutations that that genetic happened during during the the recent recent (scattering-phase) (scattering-phase) history history of of the the sampled sampled demes. demes. happened Scattered samples, taken taken singly some number number of of demes, demes, will will show show Scattered samples, singly from from some patterns of of polymorphism polymorphism identical identical to to those those in in aa completely completely unstructured unstructured patterns population, the the characteristics characteristics of of which which are are reviewed reviewed in in Section Section 8.2. 8.2. To the population, To the extent that that multiple multiple samples samples are are taken taken from from single single demes, demes, the the ancestral ancestral extent collecting-phase variation variation will will be be partitioned partitioned within within and and among among demes' demes' samsam collecting-phase ples. The The following following discussion discussion essentially essentially assumes assumes aa constant constant value value of of 00 for for the the pies. meta population; changes changes in in parameters parameters are are interpreted interpreted as as different different potential potential metapopulation; properties of of the the sampled sampled demes. demes. properties Full Data Data Patterns Patterns Full
the ancestral for samples samples from metapopula The connection connection of of the ancestral process process for from aa metapopulaThe tion samples from an unstructured unstructured population population means means that that the the tools tools tion to to that that of of samples from an of the standard be adapted use here. here. For For example, example, in of the standard coalescent coalescent can can be adapted for for use in prin principle, it should should be be straightforward straightforward to to use use the the recursive recursive approach approach of ciple, it of Griffiths Griffiths and 1 995), with with the the recognition recognition that that the the number number of migration and and and Tavare Tavar~ ((1995), of migration extinction/recolonization events in in the the history history is is not not fixed. fixed. In In other other words, words, it it extinction/recolonization events will be necessary necessary to to account account for for the the stochastic-scattering stochastic-scattering phase, phase, although although will be in case of scattered sample sample the standard coalescent coalescent methods methods can can be be in the the case of aa scattered the standard used directly. At present, present, this this remains remains one one of of several possible areas areas of of future used directly. At several possible future research. research. Summary Summary Measures Measures
Predictions summary measures, segregating sites Predictions for for summary measures, such such as as the the number number of of segregating sites S, average number pairwise differences site-frequency distri S, the the average number of of pairwise differences 1T, ~r, and and the the site-frequency distribution, can make bution, can be be made made by by modeling modeling the the scattering scattering phase. phase. It It is is possible possible to to make analytical predictions about them conditioning on analytical predictions about them by by conditioning on the the number number of of lineages lineages n phase. In n'' that that remain remain at at the the end end of of the the scattering scattering phase. In general, general, any any process process that that tends tends to to decrease decrease n n',' , such such as as restricted restricted migration migration and and extinction/recolonization extinction/recolonization with with aa small small number number of of founders, founders, will will tend tend to to decrease decrease the the number number of of (ances (ancestral) tral) polymorphisms polymorphisms found found to to be be segregating segregating in in the the sample. sample. Wakeley Wakeley and and Aliacar 1 ) showed Aliacar (200 (2001) showed that that larger larger values values of of M M produce produce larger larger average average values values of of S, S, as as do do larger larger values values of of E E if if the the number number of of founders, founders, k, k, is is large, large, whereas whereas increasing increasing E E when when kk is is equal equal to to one one decreases decreases the the average average value value of of S. S. In In add addition ition to to effects effects on on the the average average value value of of S, S, effects effects on on the the shape shape of of its its distribu distribution tion can can be be investigated investigated analytically analytically or or using using simulations. simulations. There There are are many many possible possible summary summary measures measures of of sequence sequence polymorphism polymorphism in in addition addition to to SS and and 1T, ~r, including including the the site site frequencies, frequencies, Z; Zi or or 11;, ~qi,and and it it is is hoped hoped that that the the study study of of gene gene genealogies genealogies in in aa metapopulation metapopulation will will aid aid in in the the development development of of new new statistics statistics that that capture capture the the essential essential features features of of the the dynamics dynamics of of the population. Figure the meta metapopulation. Figure 8.5 8.5 shows shows computer computer simulation simulation results results for for site site frequencies, frequencies, Z;, Zi, in in aa sample sample from from aa single single deme. deme. In In aa single-deme single-deme sample, sample, these these are are adequate adequate to to describe describe the the frequency frequency spectrum spectrum when when patterns patterns of of link linkage age among among sites sites are are not not aa concern. concern. Of Of course, course, there there is is likely likely to to be be some some extra extra
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M = 0. 1 2
0.2 E[Zi)
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8.5 Expected "unfolded" "unfolded" site frequencies, mutant base Fig. 8.5 frequencies, i.e., expected relative counts of the mutant sites, in a sample of size n = 115 from a single de deme with propagule propagule size size kk = = 2. at polymorphic sites, 5 from me with 2. information about about the the meta metapopulation in the the joint joint frequencies at information population contained contained in frequencies at two or or more more linked linked polymorphic sites. In In samples two polymorphic sites. samples from from multiple multiple demes, demes, there there may be contained in of alleles alleles among among demes. may be information information contained in the the joint joint frequencies frequencies of demes. However, the extent of of this this information among However, the extent information in in both both cases cases (among (among loci loci and and among demes) and the the potential potential to to construct construct sufficient sufficient statistics statistics demes) and sufficient or or nearly nearly sufficient for metapopulation not yet yet been explored. for metapopulation parameters parameters have have not been fully fully explored. demonstrates that that the the counts counts of of mutant mutant alleles in a sample sample from Figure 8.5 demonstrates from single deme contain substantial substantial information aa single deme contain information about about the the three three relevant relevant paramters of this of M, and kk for paramters of this model: model: the the values values of M, E, E, and for the the sampled sampled deme. deme. In In addition, the addition, the overall overall number number of of polymorphic polymorphic sites is is likely likely to to constitute constitute the the bulk of the O. In In Fig. Fig. 8.5, 8.5, site site frequencies frequencies are are bulk of the information information available available about about 0. shown shown as as fractions fractions for for the the total total number number of of polymorphic polymorphic sites. sites. In In the the case case of of low 0 . 1 ), when when extinction extinction and and recolonization recolonization are are weak weak forces, forces, low migration migration (M (M -= 0.1), i.e., i.e., when when E E is small, small, the the site-frequency site-frequency distribution distribution is U U shaped. shaped. This This is simisimi lar lar to to the the case case of of alleles alleles under under positive positive Darwinian Darwinian selection, selection, considered considered by by Fay Fay and Wu (2000) and Wu (2000) and and Kim Kim and and Stephan Stephan (2000), (2000), so so we we can can expect expect our our ability ability to to distinguish distinguish between between positive positive selection selection and and migration migration using using single-deme single-deme samsam ples ples to to be be low. low. At At the the other other extreme extreme for for small small M, M, when when extinction/recolonizaextinction/recoloniza tion stronger force force than migration, the tion is is aa much much stronger than migration, the site-frequency site-frequency distribution distribution has might be has aa mode mode at at the the middle middle frequencies. frequencies. This This is is similar similar to to what what might be expected in from two two demes expected in aa combined combined sample sample from demes in in aa metapopulation metapopulation or or if if balbal ancing operating between between two ancing selection selection were were operating two alleles alleles at at aa locus, locus, so so itit is is surprissurpris ing ing to to find find itit here here for for aa single-deme single-deme sample. sample. The The explanation explanation is is that that large large EE means means that that the the deme deme from from which which the the sample sample was was drawn drawn is is very very likely likely to to have have experienced lineages were were experienced aa recent recent extinction/recolonization extinction/recolonization event. event. The The nn lineages immediately immediately related related through through kk ancestors, ancestors, given given aa mode mode in in the the site site frequency frequency distribution distribution around around the the expected expected number number of of descendents descendents per per ancestor, ancestor, n/k, n/k, which which in in this this case case is is equal equal to to 7.5. 7.5. In 1 0 (left ( left side side of of Fig. Fig. 8.5), 8.5), if if EE is is small, small, the the site site frequenfrequen In the the case case of of M M == 10 cies cies are are close close to to those those predicted predicted for for aa sample sample from from aa panmictic panmictic population population
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shown in in Fig. Fig. 8.2. 8.2. As As the the extinction/recolonization extinction/recolonization rate rate increases, increases, aa mode mode shown again develops develops around around n/k. n/k. However, However, because because migration migration is is strong, strong, the the again panmictic pattern pattern continues continues to to hold hold and and the the interaction interaction of of these these two two patterns patterns panmictic produces an an average average site-frequency site-frequency distribution distribution that that has has three three modes. modes. On On the the produces one hand, hand, this this means means that that there there is is potentially potentially aa great great deal deal of of information information one about the the parameters parameters of of the the model model even even in in samples samples from from aa single single deme. deme. On On about the other other hand, hand, this this has has rather rather dire dire consequences consequences for for tests tests of of the the standard standard the neutral model model discussed discussed in in Section Section 8.2. 8.2. It It looks looks as as if if there there will will always always be be neutral neutral metapopulation metapopulation explanation explanation for any significant deviation in in these these aa neutral for any significant deviation statistics. statistics.
A Framework Framework for for Inference Inference in in Metapopulations Metapopulations A Clearly, there there is is great great potential potential to to adapt the many many useful useful methods methods that have Clearly, adapt the that have been developed developed for the standard standard coalescent coalescent model model to to the the case case of of aa metapopumetapopu been for the lation. means taking taking the the scattering phase into into account, lation. Again, Again, this this simply simply means scattering phase account, and and again this work work is is in in its its infancy. infancy. again this Analytical Analytical Methods Methods
As in the if predictions predictions about about summaries of polypoly As in the standard standard coalescent, coalescent, if summaries of morphism do the joint joint distribution of genealogical genealogical lengths morphism do not not depend depend on on the distribution of lengths at at pairs or or groups sites, then then they they will will be be accurate the rate rate of of pairs groups of of sites, accurate regardless regardless of of the recombination. Thus, unbiased method of moments moments estimators estimators of recombination. Thus, unbiased method of of metapopu metapopu(6, M, M, E, E, k) k) lation parameters could be be devised. devised. Because Because there there are at least least four lation parameters could are at four (0, parameters metapopulation model model used used here, here, so this will require the the parameters in in the the metapopulation so this will require proposal of proposal of some some new new summary summary statistics, statistics, tailored tailored to to samples samples from from metapopu metapopulations. lations. It It is is clear clear that that simple, simple, commonly commonly used used measures, measures, such such aa pairwise pairwise dif differences ferences within within and and between between demes, demes, will will not not suffice suffice (Pannell (Pannell and and Charlesworth, 999; Pannell, Charlesworth, 11999; Pannell, 2003). 2003). One One example example of of the the possibilities possibilities for for inference inference is is the the analytical analytical method method given given in 1 999), which bases inferences in Wakeley Wakeley ((1999), which bases inferences on on the the joint joint distribution distribution of of allele allele frequencies deme sample. frequencies among among demes demes in in aa multi multideme sample. It It was was assumed assumed that that the the demes demes were were not not subject subject to to extinction extinction and and recolonization, recolonization, only only migration, migration, and and predictions predictions like like those those shown shown in in Fig. Fig. 8.5 8.5 formed formed the the basis basis of of aa maximum maximum like likemethod of of inference inference using using data data from from unlinked unlinked loci loci such such as as RFLP RFLP or or lihood method SNP SNP data. data. The The model model also also incorporated incorporated aa change change in in the the effective effective population population size size at at some some time time in in the the past, past, illustrating illustrating the the ease ease with with which which such such complica complications tions can can be be treated treated when when part part of of the the history history of of the the sample sample is is given given by by the the stand standard ard coalescent coalescent process. process. Computational Computational Methods Methods
The The development development of of full full data data methods methods such such as as those those of of Griffiths Griffiths and and Tavare Tavar~ ((1994a,b) 1994a,b) and 1 995) appears and Kuhner Kuhner et et al. al. ((199.5) appears promising promising because because those those methods, methods, developed developed for for unstructured unstructured populations, populations, can can be be applied applied directly directly once once the the scat scattering populations with tering phase phase is is taken taken into into account. account. For For the the case case of of meta metapopulations with large large numbers numbers of of demes, demes, this this will will be be much much more more efficient efficient than than the the current current methods methods ((Beerli Beerli and 999; Bahlo and Felsenstein, Felsenstein, 11999; Bahlo and and Griffiths, Griffiths, 2000), 2000), which which require require the the estimation estimation of of migration migration rates rates between between every every possible possible pair pair of of demes demes and and which which
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so so far far have have assumed assumed that that the the sampled sampled demes demes are are the the only only demes demes in in the the metapopulation. Still, population would metapopulation. Still, aa full full data data method method for for aa meta metapopulation would have have to to deal with complicated data deal with more more complicated data and and aa greater greater number number of of potential potential histories histories of population requires, of the the sample sample than than aa panmictic panmictic population requires, so so aa method method of of this this sort sort is is expected expected to to have have aa greater greater number number of of potential potential drawbacks drawbacks than than panmictic panmictic methods. One One potential potential solution solution to to this this is is the the development development of of computational computational methods methods that that use use summary summary statistics, statistics, again again following following the the work work on on the the panmictic panmictic coa coalescent Fu and 997; Tavare aI., 11997; 997; Weiss 998; lescent ((Fu and Li, Li, 11997; Tavar~ et et al., Weiss and and von von Haeseler, Haeseler, 11998; Beaumont Beaumont et et aI., al., 2003 2003).) . These These should should be be much much more more efficient efficient computation computationally than being able able to ally than full full data data methods methods and and hold hold the the promise promise of of being to focus focus on on data data patterns patterns that that contain contain most most of of the the information information about about the the parameters parameters of of the the model, model, assuming assuming that that such such statistics statistics can can be be developed. developed. The The elucidation elucidation of of complicated .5 is goal. complicated patterns patterns like like those those displayed displayed in in Fig. Fig. 88.5 is aa step step toward toward this this goal. An An example example of of aa summary summary statistic statistic computational computational method method for for the the sort sort of of model model discussed discussed here here is is the the approximate approximate maximum maximum likelihood likelihood method method for for multilocus given in 1 ) in which numbers multilocus data data given in Wakeley Wakeley et et ai. al. (200 (2001) in which numbers of of poly polymorphisms morphisms were were used used to to make make inferences inferences about about 8, 0, the the distribution distribution of of those those polymorph isms among polymorphisms among demes demes was was used used to to make make inferences inferences about about migration migration parameters, parameters, and and the the overall overall frequency frequency of of polymorphisms polymorphisms in in the the sample sample was was used make inferences about aa possible used to to make inferences about possible change change in in effective effective size size at at some some time time in the past.
8.4 8.4
SUMMARY S U M M A R Y AND A N D CONCLUSIONS CONCLUSIONS It It is is important important to to remember remember that that the the results results presented presented in in this this chapter chapter hold hold only populations, within only for for large large meta metapopulations, within which which the the number number of of possible possible source source demes demes of of migrants migrants and and colonists colonists is is large. large. Whether Whether this this is is justified justified or or not not will will depend (see Chapters depend on on the the species species under under study study (see Chapters 1133 through through 23 23).) . If If the the num number small, the coalescent (Notohara, 990; ber of of demes demes is is small, the standard standard structured structured coalescent (Notohara, 11990; Nordborg, 11997; Wilkinson-Herbots, 11998) more appropriate appropriate frame frameNordborg, 997; Wilkinson-Herbots, 99 8 ) is a more work. The Kingman's coalescent work. The finding finding of of Kingman's coalescent as as part part of of the the history history of of any any sample sample from from aa large large metapopulation metapopulation immediately immediately makes makes applicable applicable aa plethora and inferential One interesting plethora of of theoretical theoretical and inferential results results and and methods. methods. One interesting consequence unfortunate, is consequence of of this, this, which which might might be be considered considered unfortunate, is that that many many of population are of the the details details of of the the dynamics dynamics of of the the meta metapopulation are folded folded into into aa sin single gle population population parameter: parameter: the the effective effective size size of of the the collecting collecting phase, phase, or or its its mutation-scaled mutation-scaled equivalent, equivalent, 8. 0. This This means means that that many many phenomena phenomena of of bio biological logical interest interest will will not not produce produce any any observable observable effect effect on on patterns patterns of of genetic genetic polymorphism. However, this is akin to the standard standard coalescent coalescent in which which polymorphism. However, the the distribution distribution of of offspring offspring numbers numbers among among individuals individuals in in the the metapopu metapopulation affects levels and only through lation affects levels and patterns patterns of of polymorphism polymorphism only through the the effect effective ive mutation mutation parameter, parameter, 80.. Outweighing Outweighing this this is is the the fact fact that that by by modeling modeling gene population, we about the gene genealogies genealogies in in aa meta metapopulation, we gain gain intuition intuition about the potential potential of of further further theoretical theoretical study study and and the the design design of of more more optimal optimal methods methods of of inference. inference. Even Even the the little little that that is is currently currently known known about about the the complex complex patterns patterns of variation in samples from pro of genetic genetic variation in samples from aa metapopulation, metapopulation, e.g., e.g., Fig. Fig. 8.5, 8.5, provides vides aa great great deal deal of of hope. hope.
POPULATION M ETAPO PU LATI0 N 9~ META
QUANTITATIVE GENETICS GENETICS:: THE QUANTITATIVE QUANTITATIVE GENETICS G EN ETICS OF POPULATION POPU kATION DIFFERENTIATION D IF FERENTIATI O N Charles Charles J. J. Goodnight Goodnight
9.1 9.1
INTRODUCTION INTRODUCTION The 1 930), although The field field of of quantitative quantitative genetics genetics can can be be traced traced to to Fisher Fisher ((1930), although the biometician" school the roots roots of of this this field field can can be be traced traced further further back back to to the the ""biometician" school of 1 ) . The of evolution, evolution, and and ultimately ultimately to to Darwin Darwin (Provine, (Provine, 200 2001). The original original goals goals of of quantitative genetics included explaining and describing the response to direc directional tional selection selection and and to to providing providing analytical analytical tools tools that that could could be be used used in in the the breeding of livestock. With respect to the goal of providing tools for breeders, quantitative quantitative genetics genetics has has been been stunningly stunningly successful. successful. In In developing developing quantitative quantitative genetics, genetics, Fisher Fisher made made the the assumption assumption that that popu populations lations were were large, large, unstructured, unstructured, and and mated mated randomly. randomly. These These assumptions assumptions are are inappropriate for meta populations, which are, by definition, structured metapopulations, structured and in which which the the individual individual demes demes are are frequently frequently small. small. Fisher's Fisher's methods methods remain remain
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valuable valuable in in that that they they predict predict the the response response to to selection selection within within demes; demes; however, however, this prediction not provide this prediction generally generally will will not provide aa useful useful description description of of the the evolution evolution of population as whole. Of many reasons reasons that of the the meta metapopulation as aa whole. Of the the many that this this may may be be true, true, such multilevel selection, such as as environmental environmental heterogeneity heterogeneity and and multilevel selection, one one of of the the most most interesting interesting is is the the effect effect of of population population structure structure on on the the underlying underlying effects effects of of genes genes on on the the phenotypes. phenotypes. In In particular, particular, when when there there is is epistasis, epistasis, defined defined as as inter interactions loci, the actions among among alleles alleles at at different different loci, the effect effect of of aa gene gene on on the the phenotype phenotype of of an an individual individual is is aa function function not not only only of of the the gene, gene, but but also also of of the the genetic genetic back background ground in in which which it it is is found. found. What What this this means means is is that that even even when when selection selection acts acts on phenotype in manner across all demes, demes, the on the the phenotype in the the same same manner across all the genetic genetic conse consequences quences of of that that selection selection may may be be different different in in different different demes. demes. Thus, Thus, an an allele allele favored favored by by selection selection in in one one deme deme may may be be eliminated eliminated by by the the same same selective selective regime me with regime in in aa second second de deme with aa different different genetic genetic background. background. The The goal goal of of metapopulation metapopulation quantitative quantitative genetics genetics is is to to describe describe the the vari variation ation among among demes demes in in the the phenotypic phenotypic effect effect of of alleles. alleles. Conceptually, Conceptually, it it is is aa question of question of what what is is the the variance variance in in the the phenotypic phenotypic effect effect of of aa particular particular allele allele when when it it is is "inserted" "inserted" into into the the different different demes demes in in aa metapopulation? metapopulation? This This vari variance me mean, ance must must be be corrected corrected for for the the effects effects of of the the overall overall de deme mean, which which will will necessarily allele across necessarily affect affect the the phenotypic phenotypic variance variance in in the the effects effects of of an an allele across aa metapopulation. more correct metapopulation. Thus, Thus, the the more correct quantity quantity is is the the phenotypic phenotypic variance variance in in the the effect effect of of an an allele allele relative relative to to the the effects effects of of other other alleles alleles at at the the same same locus locus measured allelic effects measured in in the the same same demes. demes. If If this this variance variance in in the the relative relative allelic effects is is zero, zero, or or in in an an experimental experimental situation situation small small and and not not significant, significant, then then allelic allelic effects effects measured measured in in one one deme deme are are indicative indicative of of allelic allelic effects effects (relative (relative to to other other alleles measured in deme. However, alleles at at the the same same locus) locus) measured in any any deme. However, if if the the variance variance in in the allelic effects nonzero, then allelic effects the allelic effects is is nonzero, then the the allelic effects measured measured in in one one deme deme are are not not predictive predictive of of the the relative relative allelic allelic effects effects in in other other demes. demes. When When there there is is variance variance in in the the effects effects of of alleles, alleles, phenotypic phenotypic selection selection acting acting uniformly all demes become aa diversifying uniformly in in all demes will will become diversifying selection selection at at the the genic genic level. level. That That is, is, selection selection favoring favoring an an allele allele in in one one deme deme may may lead lead to to aa decrease decrease in in the the frequency leads to frequency of of that that same same allele allele in in aa second second deme. deme. This This potentially potentially leads to aa selective restriction of migration between selective restriction of migration between demes, demes, as as the the offspring offspring of of migrants migrants will be will be of of low low fitness. fitness. This This could could interfere interfere with with ecological ecological and and demographic demographic processes, migrants. In processes, such such as as the the "rescue "rescue effect" effect" of of migrants. In the the extreme, extreme, the the fitness fitness of of the the offspring offspring of of migrants migrants may may be be so so low low that that interdemic interdemic gene gene flow flow is is elim eliminated, inated, effectively effectively turning turning the the different different demes demes into into separate separate species. species. Finally, Finally, it it is alleles need is important important to to note note that that variation variation in in the the effects effects of of alleles need not not be be corre correlated with with variation variation in in deme deme means. means. Thus, Thus, even even if if two two demes demes have have very very simi similated lar lar mean mean phenotypes, phenotypes, they they may may nevertheless nevertheless be be differentiated differentiated for for genic genic effects. effects. Conversely, mean phenotypic Conversely, two two demes demes with with very very different different mean phenotypic values values need need not not be be differentiated differentiated for for genic genic effects. effects. This describes traditional This chapter chapter briefly briefly describes traditional or or "Fisharian" "Fisharian" quantitative quantitative genet genetics some of ics and and uses uses this this as as aa framework framework to to discuss discuss some of the the modifications modifications of of this this theory when applying applying this theory that that are are necessary necessary when this theory theory to to aa metapopulation metapopulation rather than than aa single single panmictic panmictic population. It then then discusses discusses the the interpretation interpretation rather population. It of whereas traditional of metapopulation metapopulation quantitative quantitative genetics. genetics. In In particular, particular, whereas traditional or or Fisherian are naturally Fisherian quantitative quantitative genetics genetics are naturally related related to to measuring measuring the the response response to to selection, selection, metapopulation metapopulation quantitative quantitative genetics genetics is is more more naturally naturally related related to to the populations and, consequence, speciation. the differentiation differentiation of of populations and, as as aa consequence, speciation.
9. 9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVEGENETICS GENETICS
9.2 9.2
201 201
FISHERIAN QUANTITATIVE QUANTITATIVE GENETICS GENETICS FISHERIAN When When he he originally originally developed developed the the field field of of quantitative quantitative genetics, genetics, Fisher Fisher ((1930) 1 930) used used the the assumption assumption that that traits traits were were determined determined by by aa very very large large (in (in the the limit, infinite) number of loci each with a very small (in the limit infinitesimal) effect. effect. Under Under this this assumption, assumption, long-term long-term directional directional selection selection would would lead lead to to aa linear mean phenotype linear change change in in the the mean phenotype with with no no discernible discernible change change in in gene gene fre frequency limit of quency at at any any given given locus. locus. Indeed, Indeed, at at the the limit of an an infinite infinite number number of of loci, loci, with an infinitesimal effect on the phenotype, phenotype, changes in gene frequency each with would would be be infinitesimal infinitesimal as as well. well. More More importantly, importantly, he he assumed assumed that that there there was was no no population population structure structure and and that that populations populations were were very very large. large. Finally, Finally, in in order order work, there must be ran ranfor many of the relationships that Fisher described to work, dom mating (Falconer, 1985). Quantitative Quantitative genetics genetics is is built built on on the the idea idea of of partitioning partitioning the the phenotype phenotype into into components. components. If If the the ith ith individual individual has has aa phenotype phenotype Pi, Pi, then then this this can can be be divided divided into Gi) and into components components due due to to genetics genetics ((Gi) and the the environment environment (Ei): (Ei): Pi Pi = = f.L Ix + + G Gii + + Ei Ei
((9.1) 9.1)
where where f.L tx is is the the mean mean ooff the the population. population. IInn this this partitioning, partitioning, it it is is assumed assumed that that there there are are no no genotype genotype by by environment environment interactions interactions or or correlations. correlations. These These are are incorporated incorporated easily, easily, but but are are not not necessary necessary for for the the topics topics discussed discussed in in this this chap chapter. ter. The The genetic genetic component component can can be be further further divided divided into into components components including including the the breeding breeding value value (additive (additive effects, effects, A), A), aa component component that that can can be be attributed attributed to to interactions interactions between between alleles alleles at at the the same same loci loci (dominance, (dominance, D), D), and and interactions interactions between alleles alleles at loci (epistasis). Epistasis can can be be further further divided into between at different different loci (epistasis). Epistasis divided into components due to the the nature nature of the particular particular interaction. interaction. For For example, two components due to of the example, twolocus interactions interactions can can be be divided divided into into additive additive by by additive additive epistasis epistasis (AXA), (AXA), locus additive by by dominance dominance by by additive additive epistasis epistasis additive dominance epistasis epistasis (AXD), (AXD), dominance ( DXA) , and and dominance dominance by by dominance dominance epistasis epistasis ((DXD) DXD) (Table 9 . 1 ) . Similarly, Similarly, (DXA), (Table 9.1). three locus locus and and higher higher interactions interactions can, can, in in principle, principle, be be added. added. Thus, Thus, the the value value three of the phenotype of the ith individual individual becomes of the phenotype the ith Pi = Ai + Di + + AXAi AXDi + + DXAi DXAi + + DXDi +E Eii (9.2) (9.2) Pi = f.L IX + + Ai + Di AXA~ + + AXDi DXDi + + .. ... . + This partitioning partitioning of of the the phenotype phenotype into into components components is statistically using using This is done done statistically the the regression regression of of phenotype phenotype on on the the genetic variance variance components components (I-Iayman (Hayman and and Mather, 1955; 1 955; Goodnight, Goodnight, 2000a,b). 2000a,b). Understandably, Understandably, the the regression regression model model Mather, can potentially becoming complicated as genetic effects can potentially becoming quite quite complicated as more more of of the the genetic effects are are included. included. It is is important important to to emphasize emphasize that partitioning is is aa statistical statistical partitioning partitioning It that this this partitioning done done by by multiple multiple regression. regression. Further, Further, this this multiple multiple regression regression is is weighted weighted by by the the frequency of frequency of the the different different genotypes. genotypes. As a consequence, consequence, when when gene gene frequencies frequencies change, the the partitioning partitioning will will also also change. change. Thus, Thus, the the breeding breeding value value of of an an indiindi change, vidual is is not not only only aa property property of of the the genes genes that that make make up up that that individual, individual, but but also also vidual of population in which which it is measured measured (Falconer Mackay, 1996). of the the population (Falconer and and Mackay, 1 996). Variation is is necessary necessary if if there there is is to to be be evolution, evolution, and and as as aa result result it it is is the the partiparti Variation tioning of of the the phenotypic phenotypic variance variance that that is is of of interest. interest. Because Because the the phenotype phenotype has has tioning been divided divided into into genetic genetic and and environmental environmental components components using using aa least-squares least-squares been
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CHARLES CHARLESj.J. GOODNIGHT GOODNIGHT
TABLE 9.1 9.1 TABLE
The Eight Genetic Effects Effects Used Used in This ChapterQ Chapter a
Additive A A locus locus Additive
Additive B B locus locus Additive
Dominance A A locus locus Dominance
Dominance B B locus locus Dominance
Additive by by additive additive epistasis epistasis Additive
BjBj B1B1 BjB BIB2z B B2B2 zBz
BjB! BIB1 BIB B1B2 z B B2B2 zBz
BIB BIB I1 BIB BIB z2 B B2B2 zB z
BIBI BIB1 BjB BIB2z B B2B2 zBz
Additive by by ddominance o m i n a n c e epistasis epistasis Additive
BIBI BIB1 BIB BIB22 B2B22 B2B
D ominance b byy additive Dominance additive epistasis epistasis
BIB BIB I1 BI B22 B1B B2B22 B2B
D ominance b byy ddominance o m i n a n c e epistasis Dominance epistasis
a
BIBj1 BIB BIB22 BIB BzB2 B2B2
BjB ! BIB1 B ! B2z BIB B2Bz B2B2
A A2A2 zAz
A 1 Al A1A1
A 1 Az AIA2
11 11 11
00 00 0 0
A 1 Al AIA1
A 1 Az AIA2
11 0 0 -- 11
11 00 -- 11
11 00 -- 11
A 1 Al AIA1
A 1 Az AIA2
A AaAa zAz
- 11 - 11 - 11
A 1 Al AIA1
-- 11 -- 11 -- 11
A A2Aa zAz
1 1 11
- 11 - 11 -- 11
A 1 Az AIA2
A A2A2 zAz
- 11
-- 11 11 - 11
-- 11 11 -- 11
A 1 Al AIA1
A 1 Az AIA2
A A2A2 zAz
11 0 0 - 11
0 0 0 0 0 0
- 11 0 0 1
A 1 Al AIA1
A 1 Az AIA2
A A2A2 zAz
-- 11 11
0 0 0 0 0 0
1 --11
A 1 Al A1A1
A 1 Az AIA2
AzAz A2A2
00 -- 11
--1 1 00 11
00 --11
A1 Al A1A1
A1 Az AIA2
AzAz A2A2
1
11
--11 1 -- 11
11 -- 11 11
a These These effects effects fully fully describe describe any any two-locus two-locus two-allele two-allele genetic genetic effects. effects.
1
- 11
- 11
11
--11 1 -- 11
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METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS 9. METAPOPULATION
regression, regression, these these components components are are statistically statistically independent independent of of each each other. other. As As aa result, result, phenotypic phenotypic variance variance can can be be partitioned partitioned in in exactly exactly the the same same manner manner as as the the individual 930; Hayman 955; Falconer Falconer and individual phenotype phenotype (Fisher, (Fisher, 11930; Hayman and and Mather, Mather, 11955; and Mackay, 996; Goodnight, 998, 2000a,b): Mackay, 11996; Goodnight, 11998, 2000a,b): Vp Vp
VA A+ if- V Vo D + q- V VAX Jr- V VAX D + if- V VDX q- V VDX D + + . . .. . . + if- V VE OXAA + == V oxo AXO E AXAA +
(9.3) (9.3)
As As with with the the partitioning partitioning of of the the phenotype phenotype of of an an individual, individual, the the components components of phenotypic phenotypic variance variance are are statistical statistical properties properties of of aa population. As gene gene fre freof population. As quencies will also quencies change, change, the the partitioning partitioning among among variance variance components components will also change. change. Under Under Fisher's Fisher's assumption assumption of of large large population population size, size, and and many many loci loci each each with with small small effect effect gene gene frequencies frequencies will will not not change change appreciably, appreciably, and and the the variance variance components components will will remain remain approximately approximately constant. constant. The The primary primary utility utility of of quantitative quantitative genetics genetics is is that that it it can can be be used used to to predict predict the breeder's equation" equation" ((Falconer Falconer the response response to to selection selection using using the the standard standard ""breeder's and 996): and Mackay, Mackay, 11996): r-
h2s
(9.4)
where where rr is is the the response response to to selection selection measured measured as as the the difference difference in in mean mean pheno phenotype type between between parents parents and and offspring, offspring, ss is is the the selection selection differential differential measured measured as as the the difference difference in in mean mean phenotype phenotype between between the the selected selected parents parents and and all all parents, parents, and and hh 22 is is the the heritability. heritability. Heritability Heritability is is the the ratio ratio of of the the additive additive genetic genetic variance variance to to the the phenotypic phenotypic variance, variance, -VA. ~ . It It is is also also aa constant constant of of proportionalproportional-
Vp
/1
ity change due ity that that "converts" "converts" change due to to selection selection within within generations generations into into change change between Falconer and 996). between generations generations ((Falconer and Mackay, Mackay, 11996). For For predicting predicting the the response response to to selection, selection, only only two two variance variance components components and and the the selection selection differential differential are are needed. needed. The The two two variance variance components components are are the the additive additive genetic genetic variance variance and and the the phenotypic phenotypic variance. variance. The The phenotypic phenotypic vari variance ance is is simply simply the the variance variance in in the the trait trait observed observed in in the the population. population. Thus, Thus, the the additive additive genetic genetic variance, variance, the the genetic genetic variance variance that that can can contribute contribute to to the the resem resemblance blance between between parents parents and and offspring, offspring, is is important. important. In In addition, addition, it it is is often often use usedominance variance both both because it is often a large portion portion ful to measure the dominance of reasonable to of the the phenotypic phenotypic variance variance and and because because it it is is experimentally experimentally reasonable to measure. 1 930) felt measure. Fisher Fisher ((1930) felt that that the the epistatic epistatic variance variance components components could could be be rele relegated component. This emphasizes that gated to to the the environmental environmental variance variance component. This emphasizes that the the "environmental" "environmental" variance variance is is perhaps perhaps better better referred referred to to as as the the "residual "residual"" vari variance, ance, as as it it is is the the sum sum of of all all of of the the unmeasured unmeasured factors, factors, genetic genetic and and environ environmental, contributing variance. mental, contributing to to the the phenotypic phenotypic variance.
9.3 9.3
GENETIC GENETIC VARIANCE VARIANCE COMPONENTS COMPONENTS IN IN A A TWO-LOCUS TWO-LOCUS TWO-ALLELE TWO-ALLELE SYSTEM SYSTEM Consider Consider aa system system with with an an A A locus locus and and aa B B locus. locus. At At each each locus locus there there are are two two alleles, Al and and A and B1 Bl and and B In aa alleles, A1 A22 alleles alleles at at the the A A locus, locus, and B22 alleles alleles at at the the B B locus. locus. In system possible genotypes. purposes, it system such such as as this this there there are are nine nine possible genotypes. For For heuristic heuristic purposes, it is is
204 204
CHARLES CHARLES ). I. GOODNIGHT
convenient to array these as a three by three matrix (Table 99.1). . 1 ). Such a system equations (Goodnight, (Goodnight, 2000a,b). One can be fully described by nine orthogonal equations of these is the mean genotypic value, leaving eight equations to fully describe the genetic two-locus two-allele two-allele system always be genetic effects. effects. Thus, Thus, aa two-locus system can can always be divided divided into into independent genetic effects. Assuming a gene frequency of 0.5 for both alle alleeight independent les les at at both both loci loci and and aa population population in in two-locus two-locus Hardy-Weinberg-Castle Hardy-Weinberg-Castle equilib equilibrium, the eight genetic effects used in this chapter chapter are given in Table 9.1 9.1.. At a gene frequency orthogonal; however, frequency of of 0.5, 0.5, these these genetic genetic effects effects are are orthogonal; however, if if gene gene frequen frequencies deviate from from 0.5 for either locus, they will no longer be independent. independent. In gen general, eral, aa change change in in gene gene frequency frequency will will tend tend to to shift shift the the forms forms of of genetic genetic variation variation involving more interaction into forms forms involving less less interaction. The The statistical statistical shift shift of of genetic genetic variation variation into into forms forms involving involving less less interaction interaction means that periods of small small population will tend to cause cause dominance means that periods population size will dominance and epistasis to diminish and additive additive genetic variance to increase. Figure 9.1 is a pair of graphs of the additive genetic variance as a function function of the Wright's inbreeding coefficient, single locus effects (additive and and dominance coefficient, P, F, with single dominance effects, 1 ) shown Fig. 9.1a epistasis shown . 1 b. effects, Table Table 9. 9.1) shown in in Fig. 9.1a and and digenic digenic epistasis shown in in Fig. Fig. 99.1b.
Additive
11.00 .00
a a
OJ r () r
.lii 0.50 >� ._
>
0.00 .. -.,Ir----. ,I ----.,0.00 ,... -=--"11 I I 0.75 0.5 0 0.75 1 0.25 o Inbreeding Coefficient (F)
b 11.00 .00
AXA
OJ ()
lii 0.50 ._~
.� > >
0.00 +-"'--....::::'-r----;--:--" o 0
0.25
0.5
0.75
1
Inbreeding Coefficient (F)
Fig. 9.1 Additive genetic variance in a population as a function Fig. 9.1 function of of Wright's Wright's inbreeding coeff coefficient, F. F. Total genetic variance in the outbred population (F (F = 0) is standardized at 11.. (a) Single locus effects, additive effects (Additive) and dominance effects (Dominance), (b) Two locus interactions, additive by additive epistasis epistasis (AXA), additive by dominance and dominance by additive epistasis D), and dominance by dominance epistasis (DXD). epistasis (AX (AXD),
9. 9.
METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
205 205
When When there there are are only only additive additive effects, effects, the the additive additive genetic genetic variance variance declines declines lin linearly early as as aa function function of of F F (Fig. (Fig. 9.1a). 9.1a). For For dominance dominance (Fig. (Fig. 9.1a) 9.1a) and and all all forms forms of of epistasis b), the epistasis (Fig. (Fig. 9.1 9.1b), the additive additive genetic genetic variance variance increases increases as as aa function function of of F F until value at until it it reaches reaches aa maximum maximum value at an an intermediate intermediate level level of of inbreeding inbreeding before before declining. interaction, additive variance declines declining. In In the the absence absence of of gene gene interaction, additive variance declines in in direct direct proportion to to the the heterozygosity heterozygosity (equal (equal to to 1-F, l-F, where where F F is is Wright's Wright's inbreeding inbreeding proportion coefficient). In In the the presence presence of of gene gene interaction, interaction, the the change change in in additive additive genetic genetic coefficient). variance variance due due to to aa small small population population size size will will depend depend on on the the competing competing processes processes of of the the increase increase due due to to the the conversion conversion of of dominance dominance and and epistasis epistasis to to additive additive variance and overall loss variance and the the loss loss due due to to an an overall loss in in genetic genetic variation. variation. If If weighted weighted by by the the genotype genotype frequencies frequencies at at Hardy-Weinberg-Castle Hardy-Weinberg-Castle propor proportions, tions, the the genotypic genotypic values values of of each each genetic genetic effect effect in in Table Table 9.1 9.1 sums sums to to zero. zero. This This allows allows them them to to be be used used as as orthogonal orthogonal contrasts contrasts in in aa linear linear regression regression and, and, as as aa consequence, partitioning any consequence, provides provides aa simple simple method method for for partitioning any set set of of two-locus two-locus two-allele two-allele genotypic genotypic values values into into independent independent genetic genetic effects. effects. To To do do multiple multiple regression, the the nine nine observed observed genotypic genotypic values values are are used used as as dependent dependent variables variables regression, and and the the eight eight genetic genetic effects effects are are used used as as independent independent variables. variables. This This is is an an unusual unusual regression regression because because with with nine nine genotypic genotypic values values there there are are only only eight eight degrees each of degrees of of freedom. freedom. With With each of the the genetic genetic effects effects using using one one degree degree of of free freedom, there Thus, with only nine nine geno dom, there are are no no degrees degrees of of freedom freedom left left for for error. error. Thus, with only genotypic typic values, values, the the genetic genetic variance variance can can be be partitioned partitioned into into genetic genetic variance variance components, components, but but no no estimate estimate of of the the accuracy accuracy of of that that partitioning partitioning is is possible. possible. If, If, however, individuals are phenotypic value value of however, aa set set of of individuals are scored scored for for both both the the phenotypic of the the trait trait and and their their genotype genotype (possibly (possibly using using quantitative quantitative trait trait loci), loci), then then aa regression regression could could be be done done that that would would provide provide an an error error variance. variance. The The genotypic genotypic values values must must be be weighted weighted by by their their genotype genotype frequencies; frequencies; thus, thus, aa regression regression done done at at one one gene gene frequency frequency will will give give aa different different answer answer than than aa regression regression done done at at aa different different gene gene frequency. Again, this is the basis of conversion of a nonadditive nonadditive genetic vari varifrequency. ance (Fig. 9.1 ance into into an an additive additive genetic genetic variance variance (Fig. 9.1).). Finally, Finally, the the regression regression must must be be done done using using sequential sequential or or type type 11 sums sums of of squares. squares. Typical Typical regression regression packages packages use use iterative iterative or or type type 33 sums sums of of squares. squares. While While appropriate appropriate for for standard standard uses uses of of regression regression with with moderate moderate unbalance unbalance (type (type 11 and and type type 33 sums sums of of squares squares give give identical results squares give identical results for for balanced balanced data data sets), sets), type type 33 sums sums of of squares give incorrect incorrect results results for for the the genotypic genotypic regressions regressions that that are are unbalanced unbalanced due due to to changes changes in in gene gene frequency. frequency. An An issue issue with with using using type type 1I sums sums of of squares squares is is that that the the order order in in which which the regression model model will change the the independent independent variables variables are are entered entered into into the the regression will change the independent genetic vari variresults of the regression. In the case of regression on independent ance variables must ance components, components, the the order order in in which which the the variables must be be entered entered is is additive additive locus locus A A and and additive additive locus locus B B (order (order not not important); important); dominance dominance locus locus A A and and dominance dominance locus locus B B (order (order not not important); important); additive additive by by additive additive epistasis; epistasis; addi additive tive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (order (order not not important); important); and and finally finally dominance dominance by by dominance dominance epistasis epistasis (Goodnight, (Goodnight, 2000a). 2000a).
9.4 9.4
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS extend Fisherian Fisherian quantitative quantitative genetics genetics to to aa metapopulation metapopulation setting, setting, To extend assume that population structure consisting of assume that there there is is aa meta metapopulation structure consisting of aa large large set set of of demes demes linked linked by by migration. migration. Define Define the the local local breeding breeding value value of of the the ith ith individual individual
206 206
CHARLES J.I. GOODNIGHT CHARLES GOODNIGHT
Aij•
in in the the jth jth deme deme to to be be Aij. In In aa large large randomly randomly mating mating population, population, the the breeding breeding value mean phenotype individual. In value is is the the mean phenotype of of the the offspring offspring of of an an individual. In aa structured structured population, population, it it is is necessary necessary to to identify identify both both the the individual, individual, and and the the deme deme from from which which its its mates mates are are drawn. drawn. The The author author refers, refers, to to Aij as as the the local local breeding breeding value value to 1930). It to distinguish distinguish it it from from the the breeding breeding value value as as defined defined by by Fisher Fisher ((1930). It differs differs in that an individual individual has single Fisherian Fisherian breeding breeding value, value, which which is is an an in that an has only only aa single average population, whereas average across across the the meta metapopulation, whereas an an individual individual has has aa distinct distinct local local breeding breeding value value in in each each deme deme of of aa metapopulation. metapopulation. The The local local breeding breeding value value is is taken as population mean. mean. Using least-squares taken as aa deviation deviation from from the the meta metapopulation Using least-squares partitioning, partitioning, the the breeding breeding value value can can be be divided divided into into components components due due to to an an individual individual effect effect (Ai), aa deme deme effect effect (Ai), and and an an individual individual by by deme deme interaction interaction (Ai*j) (Wade (Wade and and Goodnight, Goodnight, 1998): 1998):
Aij
(Ai),
(Ai*j)
(Aj),
(9.5) (9.5)
Aij = Ai + Aj + Ai.j Within Within mean mean additive additive genetic genetic variance variance within within demes demes will will be be given given by by
1� (�P· ( ·(A
A*·)2 - A2.))
VA VA = = l~ ~j E, PiJ(Ai"· + t- Ai,j) ·f I f 2 - A2j J ..:,.; -4 If I I
f
Pij
(9.6) (9.6)
where me additive where V VAA is is the the mean mean within within de deme additive genetic genetic variance, variance, Pii is is the the fre frequency quency of of the the ith ith genotype genotype in in the the jth jth deme, deme, and and JJ is is the the number number of of demes. demes. The The effect effect of of population population size size on on this this equation equation is is worth worth discussing. discussing. First, First, when when the the population population is is unstructured, unstructured, Pij, the the frequency frequency of of the the ith ith genotype genotype in in the the jth jth population population is is simply simply the the frequency frequency of of the the ith ith genotype. genotype. In In aa structured structured popu population, will, by lation, genetic genetic drift drift will, by random random chance, chance, cause cause frequencies frequencies of of genotypes genotypes to to change change and and some some genotypes genotypes to to disappear. disappear. It rather than genotype fre It is is often often convenient convenient to to focus focus on on allele allele frequencies frequencies rather than genotype frequencies. quencies. Although Although selection selection acts acts on on entire entire phenotypes, phenotypes, much much of of the the recent recent genetical genetical data data has has taken taken the the form form of of mapping mapping the the chromosomal chromosomal regions regions that that affect affect quantitative quantitative traits, traits, i.e., i.e., quantitative quantitative trait trait loci loci (QTL). (QTL). These These studies, studies, which which are level focus whole phenotype phenotype level are naturally naturally related related to to aa gene gene level focus rather rather than than aa whole level focus, allow aa detailed focus, allow detailed dissection dissection of of two two locus locus interactions interactions that that would would be be diffi difficult or more traditional cult or impossible impossible using using more traditional quantitative quantitative genetic genetic methods. methods. In In sys systems tems without without gene gene interaction interaction and and random random mating, mating, the the frequencies frequencies of of individual individual alleles are adequate for alleles are adequate for describing describing the the genetic genetic variance. variance. Wright's Wright's inbreeding inbreeding coef coefmeasure of of the increase in correlation among among alleles randomly ficient, ficient, F, F, is is aa measure the increase in correlation alleles in in aa randomly mating and thus provides aa summary measure of mating population population and thus provides summary measure of the the expected expected change change in in allele allele frequencies. frequencies. In In an an additive additive system, system, the the additive additive genetic genetic variance variance is increases: is aa function function of of F F and and decreases decreases as as the the inbreeding inbreeding coefficient coefficient increases: n
Pij,
V� VA = = ((11 - ()VA f)VA
((9.7) 9.7)
where where VA V~i is is the the additive additive genetic genetic variance variance in in the the derived derived population. population. It It must must be be emphasized emphasized that that this this formula formula applies applies only only in in the the special special case case of of complete complete addi additivity tivity with with no no dominance dominance or or epistasis. epistasis. The The term term Aio should should remain remain approximately approximately constant constant with with inbreeding, inbreeding, as as it it is is an an average average of of breeding breeding value value across across all all demes. demes. This This will will not not be be true true if if there there
Ai.
9. 9.
207 207
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS
is is inbreeding inbreeding depression depression or or other other factors factors that that cause cause shifts shifts in in breeding breeding values values associated associated solely solely with with increased increased F. F. Ai*/,j, however, however, is is aa function function of of the the inbreeding inbreeding in in the the population population when when there there is is Ai* equal zero. gene gene interaction. interaction. In In particular, particular, in in an an outbred outbred deme deme (F (F = -- 0), 0), Ai* Ai. j will will equal zero. As (F > As the the inbreeding inbreeding coefficient coefficient increases increases (F > 0), 0), demes demes will will become become progressively progressively more more differentiated differentiated and and Ai* Ai. j will will increase. increase. It It is is only only when when there there is is gene gene inter interaction inbreeding that nonzero. Figure action (dominance (dominance or or epistasis) epistasis) and and inbreeding that Ai* Ai. j is is nonzero. Figure 9.2 9.2 is is aa graph graph of of the the variance variance in in local local average average effects effects for for the the different different genetical genetical effects effects as as aa function function of of inbreeding inbreeding coefficient, coefficient, F. F. In In the the absence absence of of gene gene interaction, interaction, Ai* Ai. j is zero. For reason, additive is always always zero. For this this reason, additive effects effects are are not not shown. shown. For For the the additive additive by (and dominance dominance by interaction, the variance in by dominance dominance (and by additive) additive) gene gene interaction, the variance in local local average average effects effects is is shown shown separately separately for for the the "additive" "additive" locus locus (A (A locus locus in (B locus in the the additive additive by by dominance dominance interaction) interaction) and and the the "dominance" "dominance" locus locus (B locus in in the the additive additive by by dominance dominance interaction). interaction). The The "additive" "additive" locus locus refers refers to to the the locus locus with with additive additive genotypic genotypic values values within within the the genotype genotype of of its its interacting interacting pair. pair. For by dominance the BlBl For example, example, for for additive additive by dominance epistasis epistasis within within the B1B1 genotype, genotype, genotypic genotypic values values for for the the A A locus locus are are 11,, 0, 0, and a n d --1 1 for for the the AlAb A1A1, AlA AIA2, and A A2A2 2' and 2A2 genotypes, geno genotypes, respectively. respectively. In In contrast, contrast, the the "dominant" "dominant" locus locus has has dominant dominant genotypic values within typic values within genotypes genotypes of of the the interacting interacting pair. pair. For For example, example, within within the the AlAl locus has genotypic values A1A1 genotype, genotype, the the B B locus has genotypic values of of 11,, --1, 1 , and and 11 for for the the BlBb B1B1, BlB BIB2, and B B2B2 genotypes, respectively. respectively. 2, and 2B2 genotypes, Thus, there occur as demes differentiate. Thus, there are are two two competing competing processes processes that that occur as demes differentiate. The The loss loss of of genetic genetic variation variation is is reflected reflected in in the the Pi Pijj and and has has the the effect effect of of decreas decreasing ing the the additive additive genetic genetic variance, variance, whereas whereas when when there there is is gene gene interaction, interaction, the the
2.00 2.00 AXD AXD (dominance (dominance locus) locus) 11.50 .50 -
Q)u c .00 =~ 11.oo �> m o {:: til
0.50 0.50 Dominance AXA
AXD AXD (additive (additive locus) locus) 0.00 0.00
o
o-
0.25 0.'25
0.5 0'.5
0.75 0.'75
Inbreeding Coefficient Coefficient ((F) F)
i
Fig. 9.2 9 . 2 Variance due to to the allele by deme interaction, interaction, Var(a/kl), Var(oLffkl), as a function function of inbreed inbreeding coefficient, variance in in local ing coefficient, F, F, for for the the different different forms forms of of genetic genetic effects. effects. The The variance local average average effects effects for for additive additive effects effects is is zero zero for for all all values values of of F. F. Dominance Dominance effects effects (Dominance), (Dominance), additive additive by by addi additive D), tive epistasis epistasis (AXA), (AXA), additive additive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (AX (AXD), and dominance dominance epistasis, dominance by dominance dominance epistasis (DXD) are �hown. shown. For additive by dominance additive additive and and dominance dominance loci loci are are listed listed separately. separately.
208 208
CHARLES CHARLES jJ.. GOODNIGHT GOODNIGHT
differentiation differentiation of of populations populations will will tend tend to to cause cause the the Ai*j Ai. j to to increase, increase, which which will will in in turn turn increase increase the the additive additive genetic genetic variance. variance. Thus, Thus, the the question question of of whether whether the genetic variance will increase following aa population the additive additive genetic variance will increase or or decrease decrease following population bottleneck bottleneck depends depends on on the the relative relative magnitude magnitude of of these these two two effects. effects. When When there there are (no dominance are only only additive additive effects effects (no dominance or or epistasis), epistasis), the the Ai*j Ai, j will will equal equal zero zero regardless inbreeding. With regardless of of the the level level of of inbreeding. With gene gene interaction, interaction, Ai*j Ai, j will will generally generally 1 9 8 8 ) showed be be nonzero nonzero and and will will increase increase with with increasing increasing F. F. Goodnight Goodnight ((1988) showed for 1/3, aa one for additive additive by by additive additive epistasis epistasis that that if if the the ratio ratio of of VAANA VAA/VA > > 1/3, onegeneration generation bottleneck bottleneck will will lead lead to to an an increase increase in in the the additive additive genetic genetic variance. variance. Similar Similar results results have have not not been been developed developed for for other other forms forms of of epistasis. epistasis.
9.5 9.5
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS AND AND POPULATION POPULATION DIFFERENTIATION DIFFERENTIATION Some Some of of the the most most important important effects effects of of gene gene interaction interaction appear appear primarily primarily among demes. Typically, discussions of population population differentiation differentiation focus focus on on the the among demes. Typically, discussions of differentiation means of common formula differentiation of of the the means of the the populations. populations. That That is, is, aa common formula for for the demes (in (in an system) is the genetic genetic variance variance among among demes an additive additive system) is (e.g., (e.g., Hedrick, Hedrick, 2000) Vbet
--
8) (9. (9.8)
2f VA
variance between demes. Because Because the where where Vb Vbeett is is the the variance between demes. the additive additive genetic genetic variance variance deme mean is is aa portion portion of of the the phenotypic phenotypic variance, variance, Vb V b eett is is the the variance variance among among deme mean phenotypes. phenotypes. However, However, the the following following discussion discussion shows shows that that it it is is also also necessary necessary to to consider consider the the differentiation differentiation of of genetic genetic effects. effects. It It also also show show that that differenti differentiation me means. ation for for genetic genetic effects effects may may not not be be related related to to differentiation differentiation of of de deme means. The The increase increase in in additive additive genetic genetic variance variance following following bottlenecks bottlenecks in in systems systems with interaction must come from somewhere. The with gene gene interaction must come from somewhere. The additive additive genetic genetic vari variance earlier in ance was was given given earlier in terms terms of of breeding breeding value; value; however, however, in in aa randomly randomly mat mating population, population, the the breeding breeding value of an an individual is the the sum sum of of the the average average ing value of individual is effects underlying alleles. discussion it effects of of the the underlying alleles. In In the the following following discussion it will will be be conveni convenient allele is ent to to focus focus on on average average effects. effects. The The Fisherian Fisherian average average effect effect of of an an allele is defined population mean defined to to be be the the mean mean deviation deviation from from the the population mean of of individuals individuals that that received allele from parent, with received that that allele from one one parent, with the the allele allele received received from from the the other other parent parent having having come come at at random random from from the the population. population. Using Using this this relationship, relationship, the population can the additive additive genetic genetic variance variance in in aa randomly randomly mating mating population can be be shown shown to (Falconer and 996) to be be (Falconer and MacKay MacKay 11996)
VA- 2 ~
~
pkt~l
(9.9) (9.9)
k = loci l = alleles
where kk refers refers to to the the summation summation over over all all loci loci affecting affecting the the trait, trait, II refers refers to to the the where summation all alleles each of loci, Pkl is lth summation over over all alleles at at each of the the kk loci, is the the frequency frequency of of the the/th effect of of the the khh kith allele. allele kth locus, and fXkl allele at at the the kth locus, and ~kl is is the the average average effect allele. Examination Examination of equation reveals increase the of this this equation reveals that that there there are are only only two two ways ways to to increase the additive additive genetic One is number of genetic variance. variance. One is to to increase increase the the effective effective number of alleles. alleles. Although Although perhaps not increasing the perhaps not obvious obvious from from this this form form of of the the equation, equation, increasing the number number of of
9. 9.
209 209
M ETAPOPULATION QUANTITATIVE METAPOPULATION QUANTITATIVE GENETICS GENETICS
has the alleles mean Pkl has alleles and and decreasing decreasing the the mean the effect effect of of increasing increasing the the additive additive genetic genetic variance. However, genetic genetic drift drift leads to an an increase in F F and, and, on on aver aver variance. However, leads to increase in 1 age, will will decrease the effective effective number number of of alleles (defined as as ne ne =~) ((Crow and age, decrease the alleles (defined Crow and
=�)
Kimura, 970). As Kimura, 11970). As aa result, result, the the increase increase in in additive additive genetic genetic variance variance does does not not come possible way come from from this this source. source. The The other other possible way for for the the additive additive genetic genetic vari variance ance to to increase increase is is for for the the average average effects effects to to change. change. This This shift shift in in the the average average effects alleles is me interaction effects of of alleles is the the cause cause of of the the individual individual by by de deme interaction (Ai•j) (Ai,j) and and causes causes the the increase increase in in the the additive additive genetic genetic variance. variance. Thus, popu Thus, when when the the additive additive genetic genetic variance variance increases increases as as aa result result of of small small population lation size, size, this this is is evidence evidence that that there there is is an an individual individual by by deme deme interaction. interaction. What What this this means means is is that that genes genes have have different different effects effects in in different different demes demes in in aa genetically genetically differentiated differentiated metapopulation. metapopulation. The me interaction The gene gene by by de deme interaction requires requires aa bit bit more more discussion. discussion. In In moving moving to to aa metapopulation metapopulation setting, Fisherian setting, it it is is again again necessary necessary to to distinguish distinguish the the Fisherian average average effect effect from from local local average average effects. effects. Local Local average average effects, effects, like like local local breeding breeding values, values, are are similar similar to to average average effects, effects, but but defined defined separately separately for for each each deme. deme. Thus, Thus, the the local local average average effect effect of of an an allele allele measured measured in in aa deme deme is is the the mean metapopulation mean mean deviation deviation from from the the metapopulation mean of of individuals individuals that that have have the the allele allele, and allele in in question, question, with with the the other other allele, and all all alleles alleles at at other other loci loci having having come come at Goodnight, 2000a,b). at random random from from the the deme deme in in question question ((Goodnight, 2000a,b). In In theory, theory, meas measuring uring this this would would require require substituting substituting an an allele allele into into genotypes genotypes drawn drawn from from the the deme question without loci. While deme in in question without modifyi.ng modifying the the alleles alleles at at any any other other loci. While this this ideal can be ideal is is not not possible possible experimentally, experimentally, two two locus locus local local average average effects effects can be reconstructed data. Unfortunately, randomly mating reconstructed from from QTL QTL data. Unfortunately, whereas whereas in in aa randomly mating panmictic population the panmictic population the breeding breeding value value of of an an individual individual is is aa simple simple sum sum of of the the average average effects effects of of the the underlying underlying alleles, alleles, the the local local breeding breeding value value of of an an indi individual is is not not aa simple simple sum sum of of the the underlying underlying local local average average effects. effects. Factors Factors such such vidual as population com as linkage linkage disequilibrium disequilibrium generated generated by by drift drift in in aa structured structured population complicate this this summation. summation. Interestingly, Interestingly, this this is is an an example example of of how how reductionist reductionist plicate methods methods that that work work well well in in randomly randomly mating mating populations populations often often fail fail when when there there is population structure. is population structure. For purposes of For the the purposes of this this discussion, discussion, local local average average effects effects are are useful useful because because they interaction they help help in in describing describing the the behavior behavior of of the the different different forms forms of of gene gene interaction and data. Using and are are useful useful in in interpreting interpreting quantitative quantitative trait trait loci loci data. Using local local average average effects, mean additive metapopu effects, the the mean additive genetic genetic variance variance within within aa single single deme deme in in aa metapopulation lation becomes becomes
VA == 22 2: ~m VA
(
2: 2: jk/CiTk l 2 ~, PPik,~i2kljj = = demes es k k = = loci l o c i II = = alleles alleles -
( 2: ~ik-)2) k
Cijk .
((9.1o) 9.10)
where where jj iiss the the summation summation ooff demes, demes, kk iiss the the summation summation over over loci, loci, and and II iiss the the summation summation over over alleles alleles at at the the kth kth locus. locus. The The frequency frequency of of the the Ith lth allele allele at at the the kth /, Cijkl kth locus locus in in the the jth jth deme deme is is Pjk Pikl, ~ikl is is the the local local average average effect effect of of the the kith khh locus locus mean local in in the the jth jth deme, deme, and and Cij oqk. is the the deme deme mean local average average effect effect at at the the kth kth locus. locus. k . is As As with with the the local local breeding breeding value, value, the the local local average average effect effect can can be be divided divided into into components components due due to to aa deme deme effect, effect, Cij ogk., an allele allele effect, effect, Ci.kl o~.kl,, and and aa deme deme by by allele allele k ., an interaction, interaction, Cij*kl' cxi.kt,with with the the only only difference difference being being that that effects effects are are locus locus (k) (k) specific. specific. Using mean within Using this this formulation formulation the the mean within deme deme additive additive genetic genetic variance variance becomes becomes
2 2 1100
CHARLES ].J. GOODNIGHT GOODNIGHT
VA -- 2
j=
d~em E E Pjkl(~ es k = loci I=alleles
+ OL2k*l)-
OLjk,
(9.11)
It It is is particularly particularly interesting interesting to to examine examine the the among among demes demes variance variance in in local local average average effects. effects. Conceptually, Conceptually, this this is is the the equivalent equivalent of of measuring measuring the the local local average me and average effect effect of of aa single single allele allele in in each each de deme and measuring measuring the the variance variance in in these allele local local average local aver these within within allele average effects. effects. The The (mean) (mean) variance variance in in the the local average age effect effect of of an an allele allele at at the the kth kth locus locus is is
Var Var(( <Xkl) cxkl) == 22 ~, PPj(~jkl~-kI)) 22 j( <Xjkl - <X_kl
11 =- 22 E Pj( <xrk- + + <xr* kl - <X;kl) 11 Var(~jk. Var(oLj. kl) == Var ( <Xj* kl) (<Xjk_)) ++ Var
(9.12) (9.12)
The The variance variance due due to to the the deme deme effect effect on on the the kth kth locus, locus, Var( Var(ogk.), can be be dis dis<Xjk_), can j* ) tinguished allele by tinguished from from the the variance variance due due to to the the allele by deme deme interaction interaction Var( Var(o9.~1 ) by <X kl by examining examining the the among among demes demes variance variance in in the the mean mean local local average average effects: effects:
Var(o~k. V ar( <Xk_)) = 22 E p pj(/ <Xjj,. ,.,.)' 2 k_ -- <X_k_) ! --
= =
1
:t pj( �
_
Y \~ 2
E PJ( ~ PI<Xjkl - <X_kI ~ }
((9.13) 9.13)
The The allele allele bbyy deme deme interaction interaction can can then then bbee obtained obtained bbyy subtraction subtraction ((Goodnight, Goodnight, 2000a). 2000a). jk_ is Variance Variance in in the the <X 09k. is not not particularly particularly interesting. interesting. The The mean mean local local average average effect allele changes effect is is how how much much on on average average the the mean mean allele changes the the phenotype phenotype of of an an individual. If If aa de deme particularly high high frequency frequency of of alleles alleles conferring conferring aa individual. me has has aa particularly large phenotypic value, mean local local average average effect will be large phenotypic value, the the mean effect will be negative; negative; if if the the deme alleles, the local average deme has has aa low low frequency frequency of of these these alleles, the mean mean local average effect effect will will be be positive. positive. Indeed, Indeed, in in aa system system without without gene gene interaction, interaction, <X ogk. directly propor proporjk_ isis directly tional demes. More tional to to the the phenotypic phenotypic variance variance among among demes. More importantly, importantly, this this com component local ponent of of the the local local average average effects effects does does not not drive drive any any differentiation differentiation the the local average local average average effects effects of of alleles alleles because because the the local average effects effects of of all all alleles alleles at at aa locus will example, con locus will be be affected affected by by the the deme deme effect effect in in the the same same way. way. For For example, consider sider aa locus locus affecting affecting body body weight. weight. If, If, when when measured measured in in one one deme, deme, one one allele allele causes causes the the body body weight weight to to be be 33 g g heavier heavier than than aa second second allele, allele, this this 3-g 3-g differ difference which deme deme they they are are measured. The ence will will be be maintained maintained regardless regardless of of in in which measured. The mean mass of jk_; mean mass of individuals individuals will will vary vary as as aa result result of of the the variation variation in in the the <X o9~.; however, however, the the mean mean difference difference in in weight weight between between individuals individuals carrying carrying different different alleles alleles will will be be aa constant. constant. j*kl', that It It is is the the gene gene by by deme deme interaction, interaction, <X o9.kl that is is of of more more interest. interest. As As with with the the individual me interaction, individual by by de deme interaction, this this value value equals equals zero zero in in the the absence absence of of gene gene interactions. interactions. When When there there is is dominance dominance and and epistasis, epistasis, it it will will generally generally be be nonzero nonzero (Fig. (Fig. 9.2). 9.2). Unlike Unlike the the deme deme effect, effect, this this interaction interaction does does shift shift the the local local average average effects effects of of alleles alleles relative relative to to each each other. other. In In the the example example given given earlier, earlier, if if the the differ difference ence in in local local average average effect effect between between aa pair pair of of alleles alleles in in one one deme deme is is 33 g, g, this this will will
2 11 211
9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVEGENETICS GENETICS
not demes. In not be be predictive predictive of of the the difference difference in in other other demes. In aa second second deme deme the the differ difference may may be may be ence be 1I g g or or the the ordering ordering of of local local average average effects effects may be reversed. reversed. To local average To quantify quantify the the extent extent to to which which the the local average effects effects of of alleles alleles vary vary among among demes demes together together due due to to Ujke cxik. compared compared to to the the extent extent to to which which they they vary vary among independently due among demes demes independently due to to Uj*k og,kl,l' Goodnight Goodnight (2000a) (2000a) suggested suggested using using the the intraclass intraclass correlation correlation in in local local average average effects: effects: (9.14) (9.14)
cor(e kl ) = Var(cxk, )
This This intraclass intraclass correlation correlation will will vary vary between between zero zero and and one. one. If If it it is is one, one, this this indicates local average average effects locus are indicates that that the the local effects of of the the alleles alleles at at aa locus are maintaining maintaining their less than their relative relative ranking ranking in in all all demes. demes. If If the the correlation correlation is is less than one, one, it it indi indicates cates that that the the local local average average effects effects are are varying varying among among demes demes relative relative to to each each other. other. The The amount amount it it is is less less than than one one indicates indicates the the extent extent to to which which the the local local average average effects effects of of the the alleles alleles are are varying varying independently independently among among demes. demes. In In Fig. Fig. 9.3, 9.3, the the intraclass intraclass correlation correlation in in local local average average effects effects as as aa function function of of the the inbreeding inbreeding coefficient coefficient is is shown shown for for all all of of the the types types of of genetical genetical effects. effects. The The "additive" 1 ) and "additive" locus locus (A (A locus locus of of additive additive by by dominance dominance interaction, interaction, Table Table 9. 9.1) and the the "dominance" "dominance" locus locus (B (B locus locus of of additive additive by by dominance dominance interaction, interaction, Table Table 9.1 9.1)) o are separately for dominance and dominance by are shown shown separately for additive additive by by dominance and dominance by additive additive epistasis. epistasis. Several Several points points can can be be made made from from Fig. Fig. 9.3. 9.3. First, First, additive additive gene gene action action is is aa special special case case and and the the only only case case in in which which cor(ukl) cor(oLki) is is one. one. This This means means that that regard regardless inbreeding, if allele confers less of of the the level level of of inbreeding, if one one allele confers an an advantage advantage over over aa different different allele deme, it will maintain deme it allele in in one one deme, it will maintain that that advantage advantage regardless regardless of of in in what what deme it
/
1.00
0.75
AXD (Additive locus)
cO
-~ 0.50 L O
o
0.25
~~/./ /
0.00
0
, ~
Dominance D~,,D (Dominlant locus)
0.25 0.5 0.75 Inbreeding Coefficient (F)
1
Fig. Fig. 9.3 9.3 Intraclass Intraclasscorrelation in local local average average effects effects as as a function of inbreeding coefficient for the different forms forms of genetic effects. effects. Additive effects effects (Additive), dominance effects effects (Dominance), (Dominance), additive by additive epistasis epistasis (AXA), (AXA), additive by dominance and dominance by additive epistasis epistasis (AXD), (AXD), and dominance by dominance epistasis epistasis (DXD) are shown. For additive by dominance epistasis, epistasis, additive and dominance loci are listed listed separately. separately. Additive effects effects give a correlation of one, which indicates that the values values of alleles alleles relative to other alleles alleles at the same locus are a constant. All other effects have correlations less less than one (AXA, (AXA, AXD additive locus) or zero (all (all interactions involving dominance). dominance).
21 2122
CHARLES j. GOODNIGHT CHARLES J. GOODNIGHT
is is measured. measured. For For the the case case of of the the "additive" "additive" locus locus of of additive additive by by dominance dominance and and dominance epistasis, the approaches one dominance by by additive additive epistasis, the correlation correlation approaches one as as the the inbreed inbreeding one; however, ing coefficient coefficient (f) (f) approaches approaches one; however, this this is is very very different different from from the the addi additive case where one. Second, all interactions interactions involving tive case where the the correlation correlation is is fixed fixed at at one. Second, all involving dominance, locus of dominance and dominance, i.e., i.e., dominance, dominance, "dominance" "dominance" locus of additive additive by by dominance and dominance by by additive additive epistasis, and dominance dominance by by dominance dominance epistasis, have aa dominance epistasis, and epistasis, have correlation in all inbreeding correlation in local local average average effects effects of of zero zero for for all inbreeding coefficients. coefficients. This This means that local average means that to to the the extent extent that that the the local average effects effects of of alleles alleles vary vary among among demes, Thus, for these pure pure forms gene interaction interaction demes, they they vary vary independently. independently. Thus, for these forms of of gene involving dominance, rank involving dominance, regardless regardless of of the the inbreeding inbreeding coefficient, coefficient, the the relative relative ranking one deme ranking in ing of of alleles alleles in in one deme is is not not predictive predictive of of the the ranking in other other demes. demes. Actual Actual two two locus locus interactions interactions will will involve involve aa mixture mixture of of different different genetical genetical effects. effects. When When interacting interacting quantitative quantitative trait trait loci loci have have been been measured, measured, aa large large frac fraction 995; Goodnight, tion of of the the variation variation is is typically typically additive additive (Cheverud, (Cheverud, 11995; Goodnight, 2000b). 2000b). Thus, Thus, it it is is unlikely unlikely that that in in most most circumstances circumstances the the extreme extreme of of cor(ukl) cor(~t) = = 00 will will often often be be observed observed in in experimental experimental situations. situations. When there there is is dominance dominance or or epistasis, epistasis, the the intraclass correlation in in local When intraclass correlation local average less than average effects effects is is less than one. one. This This indicates indicates that that when when there there is is gene gene inter interaction, populations whereas when action, populations differentiate differentiate for for local local average average effects, effects, whereas when there there are differentiation occurs. are only only additive additive effects, effects, no no differentiation occurs. This This can can be be contrasted contrasted to to the differentiation different forms the differentiation of of population population means means (Fig. (Fig. 9.4) 9.4) for for the the different forms of of pure genetical pure genetical effects. effects. Additive Additive effects effects cannot cannot contribute contribute to to the the differentiation differentiation
4.00
AXA
3.50 3.50 3.00
Q)<.J
.�
Additive
2.50 2.00
>
11.50 .50
"~"'~AXD
11.00 .00
Dominance
0.50 oxo
0.00 0.00
-
0
0.25 0.25
0.5
0.75
Inbreeding Inbreeding Coefficient Coefficient ((F) F)
-
1
demes as Fig. 9.4 9 . 4 The The variance variance in in mean mean phenotype phenotype among among demes as aa function function of of inbreeding inbreeding coeffi coefficient for the different different forms of genetic genetic effects. effects. Additive Additive effects effects (Additive), dominance dominance effects (Dominance), (Dominance), additive by additive epistasis (AXA), additive additive by dominance dominance and dominance dominance by additive epistasis (AXD), and dominance dominance by dominance dominance epistasis (DXD) are shown. Additive Additive effects effects have have aa large large effect effect on on the the phenotypic phenotypic variance variance among among demes, demes, but but no no effect effect in in the the vari variance in local average effects. effects. Conversely, dominance dominance effects and DXD epistasis have little little effect on the the phenotypic the variance in in local on phenotypic variance among among demes, but but a much much larger effect on the average (Fig. 9.2). average effects effects (Fig. 9.2).
2 1133 2
9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
ooff average average effects, effects, but but can can contribute contribute to to the the differentiation differentiation ooff population population means, whereas whereas interactions interactions involving involving dominance dominance have have aa large large effect effect on on the the means, differentiation differentiation of of average average effects, effects, but but little little effect effect on on the the differentiation differentiation of of popu population means. means. lation
9.6 9.6
MIGRATION MIGRATION The The effects effects of of migration migration on on quantitative quantitative genetics genetics parameters parameters in in metapopu metapopulations have not been well studied. Whitlock et al. ( 1 993) used a descent lations have not been well studied. Whitlock et al. (1993) used a descent meas measure ure model model to to show show that that with with additive additive by by additive additive epistasis epistasis the the highest highest level level of of additive additive genetic genetic variance variance is is attained attained at at an an intermediate intermediate level level of of migration. migration. The The relationship relationship between between island island model model migration migration and and inbreeding inbreeding coefficient coefficient can be concerning the migration on on can be used used to to make make some some observations observations concerning the effects effects of of migration metapopulation quantitative genetic Following Hedrick Hedrick (2000) metapopulation quantitative genetic measures. measures. Following (2000) under an an infinite infinite alleles alleles model model in in aa meta metapopulation with an an infinite infinite number number of of under population with demes, migra demes, but but aa small small population population size size within within each each deme deme and and island island model model migra(F or, tion demes, the inbreeding coefficient tion among among demes, the change change in in inbreeding coefficient (F or, in in this this context, context, more more correctly correctly FST) Fsy) is is given given by by
1+(1 )Ft] l m,2 1
F(t+l ) = [~--~
((9.15) 9.15)
where em e population where N N iiss the the within within ddeme population size size and and m m iiss the the migration migration rate. rate. A A setting reasonable value of reasonable approximation approximation for for the the equilibrium equilibrium value of F F is is found found by by setting F(t F(t+l F(t ) and and ignoring ignoring terms terms on on the the order order of of m m 22.• Using Using this this approximation, approximation, + l)) == F(t) it it can can be be shown shown that that (Hedrick, (Hedrick, 2000) 2000) 11
/~
F= 1\
4Nm + + 11 4Nm
----
((9.16) 9. 1 6)
This result that migrants (Nm) not This emphasizes emphasizes the the classical classical result that it it is is the the number number of of migrants (Nm) not determines population the per-capita migration migration rate the per-capita rate (m) (m) that that determines population differentiation. differentiation. Solving Eq. Eq. ((9.16) for Nm N m gives gives Solving 9 . 1 6 ) for Nm = ~ Nm
11 - F F 4F 4F -
--
((9.17) 9 . 1 7)
Figure 9.5 additive genetic genetic variance Figure 9.5 is is aa graph graph of of the the additive variance (relative (relative to to aa variance variance of population) as of one one in in the the panmictic panmictic population) as aa function function of of the the number number of of migrants migrants per per deme .1. deme for for each each of of the the different different pure pure forms forms of of genetical genetical effects effects listed listed in in Table Table 99.1. IInn agreement agreement with with intuition, intuition, when when there there are are only only additive additive effects, effects, the the additive additive genetic number of genetic variance variance increases increases as as aa function function of of the the number of migrants migrants and and is is max maximum dominance to imum when when mixing mixing is is complete. complete. The The conversion conversion of of dominance to additive additive genetic maximal at genetic variance variance is is maximal at aa migration migration rate rate of of slightly slightly over over one one migrant migrant epistasis, the conversion of every every other other generation generation (Nm (Nm = = 0.575). 0.575). For For digenic digenic epistasis, the conversion of epistatic epistatic variance variance to to additive additive variance variance is is maximum maximum at at even even lower lower migration migration rates of of between between one one migrant migrant every every four four generations generations and and one one migrant migrant every every six six rates
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generations . 1 75). Nevertheless, generations (Nm (Nm = 0.25 0 . 2 5 -- 00.175). Nevertheless, for for several several forms forms of of gene gene interaction, interaction, notably notably dominance dominance and and additive additive by by additive additive epistasis, epistasis, the the conver conversion substantial, even sion of of nonadditive nonadditive variance variance to to additive additive variance variance can can be be substantial, even with with migration rates rates as as high high as as three three or or more more individuals individuals per per generation. generation. migration It also interesting It is is also interesting to to examine examine population population differentiation differentiation for for average average effects effects as as aa function function of of migration migration rate. rate. Figure Figure 9.6 9.6 shows shows the the variance variance in in local local average average effects effects as as aa function function of of number number of of migrants migrants per per deme deme for for dominance dominance and and each each of of the the different different forms forms of of epistasis epistasis (for (for additive additive effects, effects, this this value value is is zero zero for for all all migration migration rates) rates).. For For the the additive additive by by dominance dominance (and (and dominance dominance by by additive) additive) interaction, interaction, the the "additive" "additive" locus locus (A (A locus locus of of additive additive by by domin dominance 1 ) and B locus ance interaction, interaction, Table Table 9. 9.1) and the the "dominant" "dominant" locus locus ((B locus of of additive additive by 1 ) are by dominance dominance interaction, interaction, Table Table 9. 9.1) are plotted plotted separately. separately. From From Fig. Fig. 9.6 9.6 it it is is apparent apparent that that the the differentiation differentiation of of local local average average effects effects is is very very sensitive sensitive to to the the migration migration rate. rate. For For all all effects effects except except the the additive additive by by dominance dominance additive additive the differentiation differentiation of of local local average average effects effects declines declines with with migration migration rate, rate, locus, the and and with with the the exceptions exceptions of of dominance dominance and and additive additive by by additive additive epistasis, epistasis, the the differentiation differentiation for for average average effects effects is is very very low low with with as as little little as as one one migrant migrant per per generation. generation.
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Fig. Fig. 9.6 9 . 6 Variance Variancedue to the allele by deme interaction, Var(a!'k/), Var(otr, kl), as a function function of number number of migrants per per deme per generation. The variance in in local average for additive effects is average effects for zero for for all values values of Nm. Nm. Dominance Dominance effects (Dominance), (Dominance), additive by additive epistasis (AXA), additive additive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (AXD), (AXD), and and dominance dominance by by domi dominance nance epistasis epistasis (DXD) (DXD) are are shown. shown. For For additive additive by by dominance dominance epistasis, epistasis, additive additive and and dominant dominant loci are listed separately. The variance in local average effects decreases decreases rapidly rapidly as a function function of the is very very small is less the number number of of migrants migrants and and is small unless unless the the number number of of migrants migrants per per deme deme is less than than one per generation.
It It is is important important to to note, note, however, however, that that these these calculations calculations assume assume that that there there is is no selection acting. potentially lower no selection acting. Selection Selection can can potentially lower the the effective effective migration migration rate. rate. Consider the Consider the situation situation of of aa metapopulation metapopulation with with aa migration migration rate rate of of three three indi individuals epistasis. In viduals per per generation generation and and considerable considerable additive additive by by additive additive epistasis. In this this case, the variance would would be elevated by conversion of epi case, the additive additive genetic genetic variance be elevated by the the conversion of epistasis variance (Fig. stasis to to additive additive genetic genetic variance (Fig. 9.5) 9.5) and and the the response response to to selection selection would would reflect reflect this. this. In In addition, addition, there there would would be be some some small small differentiation differentiation of of average average effects effects due due to to the the additive additive by by additive additive epistasis epistasis (Fig. (Fig. 9.6). 9.6). This This indi indicates cates that that to to some some degree, degree, directional directional selection selection would would be be driving driving aa differentia differentiation tion of of average average effects effects as as it it acted acted on on the the gene gene interaction. interaction. This This potentially potentially leads leads to Goodnight, 2003) to aa lowering lowering in in fitness fitness of of the the offspring offspring of of migrants migrants ((Goodnight, 2003),, which which in potential exists pos in turn turn would would lower lower the the effective effective migration migration rate. rate. The The potential exists for for aa posinteraction among among selection, migration, itive feedback to develop. Thus, the interaction drift may generate much more population population differentiation than predicted predicted and drift
216 216
CHARLES I.] . GOODNIGHT CHARLES
by the the apparent apparent migration migration rate rate alone. alone. This process has has not not been been explored explored and and by This process deserves more more attention. attention. deserves
9.7 9.7
EMPIRICAL EXAMPLE: EXAMPLE: THE THE METAPOPULATION M ETAPOPULATION QUANTITATIVE QUANTITATIVE EMPIRICAL GENETICS OF OF A A TEOSINTE-MAIZE TEOSINTE-MAIZE CROSS CROSS GENETICS Doebley et et al. a!. (1995) ( 1 995) identified identified quantitative quantitative trait loci for several traits traits in in aa Doebley trait loci for several wide cross cross between between teosinte teosinte (Zea (Zea mays mays ssp. ssp. parviglumis) parvig/umis) and and cultivated cultivated maize maize wide Zea mays mays ssp. ssp. mays). mays). The The details details of of the the mapping mapping procedure procedure are are ( "corn," Zea ("corn," described in in Doebley Doebley et et al. a!. (1995; ( 1 995; see see also also Doebley Doebley and and Stec, Stec, 1993). 1 993). Several Several described pairs of of QTL QTL regions regions were were shown shown to to interact interact epistatically. epistatically. One One pair pair of of epistaepista pairs tically interacting markers is is BV302 BV302 and and UMC107, UMC1 07, which which are are located located on on difdif tically interacting markers ferent chromosomes. chromosomes. One One trait trait these these loci loci affect affect is is "PEDS," "PEDS," the the percentage percentage of of ferent cupules with with the the pedicellate pedicellate (maize (maize like) like) spikelet spikelet (for (for aa more more complete complete descripdescrip cupules tion of trait, see see Doebley et a!., 1 995). tion of this this trait, Doebley et al., 1995). The PEDS for the different different two two locus locus genotypes genotypes are The genotypic genotypic values values of of PEDS for the are shown in Table 9.2. The values are are simply mean phenotypes of shown in Table 9.2. The genotypic genotypic values simply the the mean phenotypes of those individuals genotype of interest. Using Using the the regression regression procedure procedure those individuals with with the the genotype of interest. described into components components for for this this described earlier, earlier, the the partitioning partitioning of of genetic genetic variance variance into trait is three gene 9.3. This partitioning trait is shown shown for for three gene frequencies frequencies in in Table Table 9.3. This partitioning includes total genetic variance and percentage contributions contributions due includes total variance and due to to additive effects, epistasis. At three gene gene frequencies frequencies effects, dominance dominance effects, effects, and and digenic digenic epistasis. At all all three the mainly attributable attributable to to additive locus domdom the genetical genetical effects effects are are mainly additive and and single single locus inance effects. However, in in all epistasis accounts accounts for for aa substansubstan inance effects. However, all cases, cases, digenic digenic epistasis tial between 20.9 tial proportion proportion ((between 20.9 and and 39.4%) 39.4%) of of the the total total genetic genetic variance. variance. This This is variance due is reflected reflected in in Fig. Fig. 9.7, 9.7, which which is is aa graph graph of of additive additive genetic genetic variance due to to the the BV302 BV302 QTL QTL as as aa function function of of the the frequencies frequencies of of the the BV302 BV302 QTL QTL and and the the UMC1 07 QTL. 07 locus, UMC107 QTL. Within Within each each gene gene frequency frequency of of the the UMC1 UMC107 locus, the the addi additive variance of locus varies varies in manner consistent tive variance of the the BV302 BV302 locus in aa manner consistent with with aa standard standard locus 996); however, shape locus with with dominance dominance (e.g., (e.g., Falconer Falconer and and Mackay, Mackay, 11996); however, the the shape
TABLE TABLE 9.2 9 . 2 Genotypic Genotypic Values Values for PEDS, PEDS, the Percentage of Cupules with with the Pedicellate Pedicellate (Maize-like) Spikelet for aa Cross between Teosinte (T) (T) and Maize (M)Q (M) a UMCI07 UMC107 BV302
T/T TIT
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Note Note that that crosses crosses were were done done with with teosinte teosinte as as the the seed seed parent. parent. The The teosinte teosinte cytoplasm cytoplasm and and random random sampling sampling of of genes genes at at other other loci loci affecting affecting this this trait trait are are likely likely responsible responsible for for the the low low proportion proportion of of maize-like maize-like cupules cupules even even when when both both loci loci are are from from maize maize (data (data from from Doebley Doebley et et a!., al., 1995). 199.5).
9. 9.
221 1 17
METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS METAPOPULATION
TABLE 9.3 9.3 Decomposition Decomposition into Genetic Variance Components Components for for a Pair of of Interacting Interacting TABLE Loci (UMC107 (UMCl 07 and BV302) Affecting Affecting the the Percentage Percentage of of Cupules Lacking the the Pedicellate (Maize-like) Spikelet.
Effect E ffect Additive Additive (UMC107) (UMC107) Additive Additive
(BV302) (BV302) Dominance Dominance (UMC107) (UMC107) Dominance Dominance
(BV302) (By302) AXA AXA AXD AXD DXA DXA DXD DXD Total genetic Total genetic vanance variance (percentage (percentage due to due to epistasis) epistasis)
aa
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00.25 .25 = 00.25 Freq(UMC1 F r e q ( U M C 1 0 707teo.) teo.) = .25
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00..55 FFreq(UMC107teo.) r e q ( U M C 1 0 7 t e o . ) -= 00.5 .5
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00.75 .75 = 00.75 Freq(UMC1 F r e q ( U M C 1 0 707teo.) teo.) = .75
3.7209 (38.6%) (38.6%) 3.7209
0.6328 (21.6%) (21.6%) 0.6328
0.0258 (8.9%) (8.9%) 0.0258
2.7719 2.7719 (28.7%) (28.7%)
0.55 13 (18.8%) ( 1 8.8%) 0.5513
0.0697 (24.1%) (24.1 %) 0.0697
0.8016 (8.3%) (8.3%) 0.8016
0.4556 (15.5%) (15.5%) 0.4556
0.0665 (23.0%) (23.0% ) 0.0665
0.3433 (3.5%) (3.5%) 0.3433
0.1406 (4.8%) (4.8%) 0.1406
0.0137 (4.7%) (4.7%) 0.0137
1.4400 (14.9%) (14.9%) 1.4400 0.3 165 (3.3%) (3.3%) 0.3165 0.2109 0.2109 (2.2%) (2.2%) 0.0445 0.0445 (0.5%) (0.5%) 9.6501 9.6501 (20.9%) (20.9%)
0.4556 (15.5%) 0.4556 (15.5%) 0.3613 (12.3%) (12.3%) 0.3613 0.1953 (6.7%) 0.1953 (6.7%) 0.1406 (4.8%) 0.1406 (4.8%) 2.9331 %) 2.9331 (39.3 (39.3%)
0.0089 0.0089 0.0476 0.0476 0.0130 0.0130 0.0445 0.0445 0.2897 0.2897
(3.1 %) (3.1%) ( 1 6.4%) (16.4%) (4.5%) (4.5%) (15.4%) (15.4%) (39.4%) (39.4%)
Numbers iinn parentheses parentheses are percentage of of the the total total genetic genetic variance variance due the component. component. Total Total genetic genetic Numbers are the the percentage due ttoo the variance and percentage due to digenic digenic epistasis epistasis are also listed. listed. variance and percentage due to are also
of changes dramatically gene frequency frequency at 07 QTL of the the curve curve changes dramatically as as the the gene at the the UMCI UMC107 QTL changes. graph of total variance digenic epistasis changes. Figure Figure 9.8 9.8 is is aa graph of the the total variance in in digenic epistasis in in the the population. Note that greatest at gene fre population. Note that epistatic epistatic variance variance is is greatest at intermediate intermediate gene frequencies. and its quencies. Fixation Fixation of of either either locus locus leads leads to to aa loss loss of of epistasis epistasis and its conversion conversion to to additive additive and and dominance dominance variance variance at at the the locus locus that that is is still still segregating. segregating. When metapopulation undergoes When aa metapopulation undergoes genetic genetic drift, drift, gene gene frequencies frequencies at at both both loci loci in in the the different different demes demes will will change change randomly. randomly. The The expected expected distribution distribution of of gene gene frequencies frequencies can can be be described described using using aa Markov Markov chain chain and and Wright's Wright's inbreed inbreeding ing coefficient coefficient (F). (F). Figure Figure 9.9 9.9 is is aa graph graph of of the the effect effect of of drift drift (measured (measured by by F) F) on on mean mean additive additive genetic genetic variance variance and and the the variance variance in in local local average average effects effects (corrected Also shown (corrected for for demic demic effects) effects) for for the the BV302 BV302 QTL. QTL. Also shown for for comparison comparison is is the the variance variance in in deme deme means. means. Not Not that that due due to to epistatic epistatic interactions interactions with with UMCI07, maximum value UMC107, the the additive additive genetic genetic variance variance has has aa maximum value at at an an inter intermediate value of mediate value of F. F. The The early early increase increase in in additive additive genetic genetic variance variance is is due due to to aa conversion conversion of of epistatic epistatic variance variance to to additive additive variance. variance. At At higher higher values values of of F F the the effects effects of of fixation fixation within within demes demes at at both both loci loci overwhelm overwhelm the the conversion conversion of of epi epistasis stasis to to additive additive variance variance and and the the overall overall additive additive genetic genetic variance variance declines declines until until it it reaches reaches zero zero when when F F equals equals one. one. Along Along with with this this increase increase in in additive additive genetic local average genetic variance variance there there is is also also an an increase increase in in the the variance variance in in the the local average effects effects of of the the BV302 BV302 alleles. alleles. In In the the absence absence of of gene gene interaction, interaction, this this variance variance in local local average average effects effects would would be be zero. zero. The The variance variance in in local local average average effects effects in indicates population would indicates that that BV302 BV302 alleles alleles in in aa teosinte-maize teosinte-maize meta metapopulation would have have different different effects effects on on the the PEDS PEDS phenotype phenotype in in different different demes. demes.
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The locus (corrected deme mean) The local local average average effects effects for for the the BV302 BV302 locus (corrected for for deme mean) are are shown Fig. 9.10 shown in in Fig. 9.10 as as aa function function of of the the gene gene frequencies frequencies at at the the two two loci. loci. In In this this par particular case, maize ticular case, maize genes genes at at this this locus locus always always code code for for aa more more maize-like maize-like pheno phenotype; type; thus, thus, the the interaction interaction is is expressed expressed as as aa change change in in scale scale rather rather than than aa reversal reversal in in sign. sign. This This makes makes sense sense given given that that this this interaction interaction has has aa substantial substantial additive additive
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Inbreeding Inbreeding Coefficient Coefficient (F) (F) Fig. Fig. 9.9 9 . 9 Among Among deme deme phenotypic phenotypic variance variance (Deme (Deme Mean), Mean), variance variance in in local local average average effects effects for the BV302 for the BV302 locus locus corrected corrected for for demic demic effects effects (BV302), (BV302), and and mean mean within within deme deme additive additive genetic genetic variance variance (Va) (Va) as as aa function function of of an an inbreeding inbreeding coefficient. coefficient. The The additive additive genetic genetic variance variance shown shown is is the 07 locus. Maximum additive the sum sum of of the the effects effects for for the the BV302 BV302 locus locus and and the the UMC1 UMC107 locus. Maximum additive genetic genetic variance variance occurs occurs at at an an intermediate intermediate inbreeding inbreeding coefficient, coefficient, indicating indicating that that aa conversion conversion of of non nonadditive additive variance variance into into additive additive variance variance is is occurring. occurring. This This is is reflected reflected in in the the variance variance among among demes, demes, which which is is much much greater greater than than expected expected from from an an additive additive model. model. A A maximum maximum phenotypic phenotypic variance among demes under an model. It variance among demes of of 4.74 4.74 is is possible possible under an additive additive model. It is is also also reflected reflected in in the the vari variance which would ance in in local local average average effects effects for for the the BV302 BV302 locus, locus, which would be be zero zero in in an an additive additive model. model.
component component at at all all gene gene frequencies frequencies (Table (Table 9.3). 9.3). Nevertheless, Nevertheless, it it is is quite quite apparent apparent that 07 locus that the the UMC1 UMC107 locus has has aa very very large large effect effect on on the the average average effects effects of of BV302 BV302 alleles. alleles. The The consequences consequences of of this this can can be be seen seen by by considering considering the the difference difference between between the BV302 alleles alleles in the two two BV302 in demes demes with with different different frequencies frequencies of of the the UMCI07 UMC107 alleles. alleles. In In those those demes demes with with low low frequencies frequencies of of the the UMC107 UMC107 teosinte teosinte allele allele (and (and therefore therefore high high frequencies frequencies of of the the maize maize allele), allele), differences differences between between the the two two BV302 BV302 alleles alleles are are pronounced. pronounced. Selection Selection acting acting on on PEDS PEDS would would effectively effectively distinguish distinguish between between the the two two alleles. alleles. However, However, in in demes demes with with high high frequencies frequencies of of the the UMCI07, UMC107, tesinte tesinte allele allele selection selection would would be be much much less less effective, effective, as as the the difference difference between between the the alleles alleles is is much much smaller. smaller. As As an an aside, aside, the the teosinte teosinte ancestor ancestor of of corn corn may may have have had had genetical genetical effects effects at at the the BV302 BV302 locus locus similar similar to to the the back back corner corner in in Fig. Fig. 9.10. 9.10. For For this this set set of of genotype genotype frequencies, frequencies, there there is is almost almost no no difference difference between between the the two two alleles, alleles, and and the BV302 "maize" been nearly the BV302 "maize" gene gene would would have have been nearly neutral neutral with with respect respect to to the the PEDS PEDS phenotype. phenotype. As As the the domestication domestication of of teosinte teosinte progressed progressed and and it it became became more more maize maize like, like, the the frequency frequency of of the the UMC107 UMC107 maize maize allele allele would would have have presumably presumably increased. increased. This BV302 This in in turn turn would would have have acted acted to to magnify magnify the the differences differences between between the the BV302 alleles. alleles. Thus, Thus, this this is is an an interesting interesting case case where where selection selection converts converts formerly formerly neutral neutral variation variation into into large large differences differences that that can can respond respond to to selection. selection. In In this this case, case, selection, selection, rather rather than than using using up up additive additive genetic genetic variance, variance, generates generates new new additive additive genetic genetic vari variance. ance. These These results results also also suggest suggest why why in in the the past past there there was was considerable considerable debate debate over over the the origins origins of of maize maize (Beadle, (Beadle, 1980). 1980). Genes Genes coding coding for for aa "maize" "maize" phenotype phenotype are are nearly nearly neutral neutral in in aa teosinte teosinte genetic genetic background, background, and and the the pathway pathway for for selecting selecting maize maize from from teosinte teosinte is is not not clear. clear.
220 220
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9.8 9.8
GENE GENE INTERACTION I N T E R A C T I O N AND A N D SPECIATION SPECIATION Although Although there there is is aa substantial substantial literature literature on on specIatIOn speciation (reviewed (reviewed in in Templeton, 9 8 1 ; Turelli 1; Kondrashov, Templeton, 11981; Turelli et et ai., al., 2001; 2001; Schluter, Schluter, 200 2001; Kondrashov, 2001; 2001; Wu, 1 ), aa general Wu, 200 2001), general understanding understanding of of the the relationship relationship between between quantitative quantitative genetics speciation has elusive (Hedrick, genetics or or population population genetics genetics and and speciation has been been elusive (Hedrick, 2000) possible explanation explanation for 2000).. One One possible for this this is is that that although although the the importance importance of of gene interaction 963; interaction in speciation speciation is acknowledged acknowledged (Muller, 1939; Mayr, 11963; Templeton, 9 8 1 ; Futuyma, 986; Orr, 11995), 995), the majority Templeton, 11981; Futuyma, 11986; majority of models that that can be be applied applied to to the the study study of of speciation speciation assume assume only only additive additive gene gene action. action. For For example, in his review article, Wu Wu (20 (2001) "speciation genes," example, 0 1 ) focused on "speciation which which he he identified identified as as genes genes responsible responsible for for differential differential adaptation adaptation primarily primarily to ecological or environment. The to the the ecological or sexual sexual environment. The effect effect of of gene gene interaction interaction on on the the shift in the local average effects of alleles is a potential potential genetical mechanism mechanism for for speciation speciation that that does does not not depend depend on on shifts shifts in in the the ecological ecological or or sexual sexual envir environment. This is not 1 939; Orr, 11995) 995) model model onment. not a new idea; for for example, example, Muller's ((1939;
9. 9.
METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
221 221
provided provided aa qualitative qualitative model model of of how how speciation speciation would would occur occur through through the the accu accumulation mulation of of mutations mutations that that caused caused reproductive reproductive incompatibility incompatibility between between two two populations. However, lack aa mechanism populations. However, these these models models lack mechanism other other than than random random mutation mutation for for the the accumulation accumulation of of incompatible incompatible mutations. mutations. The The model model dis discussed possible mechanism cussed in in this this chapter chapter suggests suggests that that aa possible mechanism for for speciation speciation is is that that genetic genetic drift drift generates generates small small differences differences between between demes demes and and directional directional selec selection tion works works to to magnify magnify these these differences differences to to the the point point that that the the two two populations populations become become reproductively reproductively isolated. isolated. Importantly, Importantly, directional directional selection selection may may be be uniform uniform at at the the phenotypic phenotypic level, level, but but nevertheless nevertheless diversifying diversifying at at the the genetic genetic level. That because of level. That is, is, although although selection selection may may favor favor the the same same phenotype phenotype because of the the different alleles would different genetic genetic backgrounds, backgrounds, different different alleles would be be favored favored in in the the two two populations. populations. To To see see how how genetic genetic drift drift coupled coupled with with directional directional selection selection can can lead lead to to spe speciation, .1; ciation, consider consider the the example example of of dominance dominance by by additive additive epistasis epistasis (Table (Table 99.1; Fig. . 1 1 a; Goodnight, Fig. 99.11a; Goodnight, 2000b). 2000b). With With dominance dominance bbyy additive additive epistasis, epistasis, the the A A locus is locus is overdominant, overdominant, neutral, neutral, or or underdominant, underdominant, depending depending on on the the genotype genotype at allele being at the the B B locus, locus, and and the the B B locus locus is is additive additive with with the the favored favored allele being depend dependent locus. ent on on the the genotype genotype at at the the A A locus. Consider population segregating locus, but Consider aa meta metapopulation segregating for for the the A A locus, but fixed fixed for for the the Bz B2 allele. . 1 1 b. In allele. This This situation situation corresponds corresponds to to the the bottom bottom row row in in Fig. Fig. 99.11b. In this this situ situation, ation, the the A A locus locus is is exhibiting exhibiting simple simple overdominance overdominance with with no no apparent apparent epis epistasis. tasis. Stabilizing Stabilizing selection selection will will tend tend to to drive drive the the gene gene frequency frequency at at the the A A locus locus to 0.5 for to approximately approximately 0.5 for both both alleles. alleles. Note, Note, however, however, that that because because demes demes within populations are within the the meta metapopulations are finite finite and and likely likely small, small, there there will will be be deviations deviations from this equilibrium gene frequency. from introduced into a deme mutation or by migration If a B1 Bl allele is introduced deme either by mutation it it will will be be neutral neutral provided provided that that the the gene gene frequency frequency at at the the A A locus locus is is exactly exactly 0.5 0.5 (Fig. 99.11c, middle column). column). Any Any deviations deviations from from aa gene gene frequency frequency of of 0.5 0.5 at at (Fig. . 1 1c, middle the locus will result in Bl allele . 1 1 c, the A A locus will result in directional directional selection selection favoring favoring the the B1 allele (Fig. (Fig. 99.11c, left resulting increase frequency of left and and right right columns). columns). The The resulting increase in in the the frequency of the the Bl B1 allele allele will will weaken weaken the the strength strength of of stabilizing stabilizing selection selection on on the the A A locus, locus, resulting resulting in in neutrality (Fig. 9. 1 1 b, mid neutrality and and ultimately ultimately disruptive disruptive selection selection at at the the A A locus locus (Fig. 9.11b, middle dle and and top top rows). rows). This This is is aa positive positive feedback feedback system, system, wherein wherein genetic genetic drift drift at at the the A A locus locus in weak weak directional directional selection selection at at the the B B locus. locus. This This directional directional selection selection results in on on the the B B locus locus has has the the effect effect of of weakening weakening the the strength strength of of stabilizing stabilizing selection selection at at the the A A locus, locus, which which in in turn, turn, when when coupled coupled with with genetic genetic drift, drift, will will increase increase the the strength locus. Once strength of of directional directional selection selection on on the the B B locus. Once the the frequency frequency of of the the Bl B1 allele allele exceeds exceeds 0.5, 0.5, the the A A locus locus will will experience experience disruptive disruptive selection selection and and be be fixed fixed quickly Al or quickly for for either either the the A1 or the the Az A2 allele. allele. If population some If this this process process occurs occurs in in several several demes demes within within the the meta metapopulation some of of the the demes demes will, will, by by random random chance, chance, become become fixed fixed for for the the Al A1 allele, allele, whereas whereas others others will will become become fixed fixed for for the the Az A2 allele. allele. If If the the strength strength of of destabilizing destabilizing selec selection tion is is strong strong enough, enough, this this could could be be sufficient sufficient to to cause cause reproductive reproductive isolation isolation and, and, as as aa consequence, consequence, speciation. speciation. Prior allele, there Prior to to the the introduction introduction of of aa Bl B1 allele, there would would be be no no reason reason to to con consider sider the the A A locus locus part part of of an an epistatic epistatic interaction, interaction, nor nor any any reason reason to to consider consider it it as Indeed, the as aa candidate candidate for for aa locus locus that that could could drive drive speciation. speciation. Indeed, the A A locus locus would would appear appear to to be be aa simple simple overdominant overdominant locus locus maintained maintained by by stabilizing stabilizing
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CHARLES J.J. GOODNIGHT CHARLES GOODNIGHT a
A, A, A, A2
A2 A2
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0
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Fig. 9.1 9.111 The The potential potential role role of of dominance dominance by by additive additive epistasis epistasis in in speciation. speciation. (a) (a) Genotypic Genotypic values for dominance epistasis. Gray arrows indicate the change values dominance by additive epistasis. change from from stabilizing selection selection to to disruptive disruptive selection selection at at the the A A locus locus that that occurs occurs as as the the frequency frequency of of the the B, B1 allele allele changes from from zero to to one. (b) Genotypic values for for the three three A locus genotypes when when the fre frequency of the B1 B, allele is 0, 0.5, and and 11.. When When the B, B1 allele is rare, there is stabilizing selection at common, there locus. at the the A A locus, locus, whereas whereas when when the the B, B1 allele allele is is common, there is is disruptive disruptive selection selection at at the the A A locus. (c) Genotypic values for the three BB locus genotypes when the frequency frequency of the A, A1 allele is 0, 0.5, and 0.5, the 0, 0.5, and 11.. At At aa frequency frequency at at the the A A locus locus of of 0.5, the BB locus locus is is neutral. neutral. However, However, if if frequency frequency at locus drifts locus will will be selection favoring at the the A A locus drifts from from 0.5, 0.5, then then the the BB locus be under under directional directional selection favoring the the B, allele. When B1 allele. When the B, B1 allele is rare, genetic drift at the A locus will interact interact with with directional selec selection at the BB locus, eventually leading to fixation and either the A, fixation of the B, B1 allele and A1 or the A2 allele. allele. Redrawn Redrawn from from Goodnight Goodnight (2000a). (2000a).
selection. population selection. Similarly, Similarly, at at the the end end of of the the divergence divergence process, process, the the meta metapopulation will Bl allele, allele, and will be be fixed fixed for for the the B1 and again again the the A A locus locus will will appear appear to to be be aa simple simple underdominant underdominant locus locus with with no no evidence evidence that that it it is is part part of of an an epistatic epistatic inter interaction. just by examining the action. Thus, Thus, just by examining the end end points points of of this this process, process, there there would would be be little little indication indication that that an an epistatically epistatically interacting interacting B B locus locus was was involved involved in in the the speciation speciation process. process. Finally, Finally, note note that that this this process process involves involves an an interaction interaction between between the the random random process process of of genetic genetic drift drift and and the the deterministic deterministic process process of of directional directional selection. selection. Random Bl allele allele is Random drift drift begins begins the the process process (the (the B1 is neutral neutral at at aa gene gene frequency frequency of of 0.5 0.5 at at the the A A locus) locus),, and and directional directional selection selection enhances enhances the the power power of of drift drift and greatly accelerates one of and greatly accelerates the the drive drive to to the the fixation fixation of of one of the the two two alleles alleles at at the the A A locus. locus.
METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS 9. METAPOPULATION
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SUMMARY S U M M A R Y AND A N D CONCLUSIONS CONCLUSIONS Fisher Fisher developed developed quantitative quantitative genetics genetics to to address address specific specific questions questions about about evolution within a single population. population. Indeed, Indeed, quantitative quantitative genetics in its original formulation solely with describing the process of adaptation formulation was concerned concerned solely adaptation to directional selection selection acting within populations. population populations. The goals of meta metapopulation quantitative quantitative genetics genetics are are different different from from those those of of Fisherian Fisherian quantitative quantitative genetics. genetics. Metapopulation Metapopulation quantitative quantitative genetics primarily primarily provides provides measures of popu population differentiation differentiation rather rather than than focusing on the response response to selection that that is central central to to Fisherian Fisherian quantitative quantitative genetics. genetics. One One of of the the central central observations observations of of meta population quantitative metapopulation quantitative genetics genetics is is that that populations populations can can be be differentiated differentiated both both for for population population means means and and for for local local average average effects. effects. The The first first is is well well known known from from studies studies of of genetic genetic drift drift in in additive additive systems. systems. The The second, second, while while it it has 963), had has been been observed observed qualitatively qualitatively (e.g., (e.g., Dempster, Dempster, 11963), had not not been been quanti quantified populations for local average fied in in the the past. past. The The differentiation differentiation of of populations for local average effects effects is is of of particular particular interest interest because because it it is is aa measure measure of of what what alleles alleles do do in in different different demes. This form form of population population differentiation, differentiation, unlike differentiation differentiation of popu population lation means, means, is is directly directly related related to to reproductive reproductive isolation isolation and and speciation. speciation. However, However, it it need need not not be be related related to to the the differentiation differentiation of of population population means. means. For For example, additive additive effects cause differentiation differentiation of population population means but no dif difno dif ferentiation average effects, fixation (E ferentiation of of local local average effects, whereas whereas at at fixation there is is no dif(F == 11)) there ferentiation ferentiation of population population means means due to dominance, dominance, but but the populations populations are strongly differentiated differentiated for local average effects. Population Population genetics and Fisherian genetics have been remarkably successful for Fisherian quantitative quantitative genetics have been remarkably successful for developing developing our populations, but our understanding understanding of of evolution evolution within within populations, but interestingly, interestingly, these these dis disciplines have also been remarkably remarkably unsuccessful unsuccessful at developing our our under understanding population changes lead to speciation standing of how how these within within population speciation and and evolution evolution above the species level. By providing providing new measures of population population differentiation, population quantitive differentiation, meta metapopulation quantitive genetics genetics may may shed shed light light on on this this important important subject.
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Part IV Evolutionary Dynamics in Metapopulations
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LIFE LI FE H ISTORY EVO LUTION IN EVOLUTION IN META PO PUL ATIONS M ETAPO PU LATI ONS Ophelie Oph~lie Ronce and Isabelle IsabeUe Olivieri
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INTRODUCTION INTRODUCTION Life history theory deals with with the the evolution of those those traits that that shape shape an organism's age schedules of birth and death Calow, 1998). Many death ((Calow, Many biological traits potentially potentially affect the patterns patterns of reproduction reproduction and mortality throughout throughout traits the life cycle. Life history traits therefore therefore constitute constitute a loosely defined set of morphological, morphological, developmental, developmental, or behavioral characteristics, characteristics, including, including, for instance, body size, growth patterns, patterns, size and age at maturity, reproductive reproductive effort, mating success, number, size, and sex of offspring, and rate of senes senes990s, life history evolution cence. Despite this diversity of traits, traits, by the early 11990s, had grown successfully into a very productive productive field organized around around a few central central questions questions with a very strong unifying theoretical theoretical background, background, grounded Stearns, grounded in both both optimization optimization principles and quantitative quantitative genetics genetics ((Stearns, 11992; 992; Roff, 11992). 992). Then, the realm of most studies of life history theory was that that of a single, large, undisturbed undisturbed and spatially homogeneous homogeneous population population 992; Kawecki, 1993). Through (see, however, Kawecki and Stearns, 11992; Through several 1 997) illustrated examples, Olivieri and Gouyon ((1997) illustrated how how disequilibrium and the populations might significantly affect spatial structure structure characteristic characteristic of meta metapopulations the evolution evolution of life history traits, a phenomenon phenomenon they called "the metapopu metapopulation effect." Five years later, despite increasing awareness awareness of the importance importance of meta population structure metapopulation structure and dynamics for the demography, genetics, and and
enetics, and Ecology, Ecology, G Genetics, and Evolution Evolution of of Metapopulations Metapopulations
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conservation of many species, little is understood understood about how much much these char characteristics acteristics have shaped basic life histories. The present chapter chapter reviews empir empirtheoretical studies studies published since since 11997 ical and theoretical 997 that have addressed the metapopulation context. We We first comment comment evolution of life history traits in a metapopulation generally about the development of this field of research since 11997. 997. First, studies of life history evolution in a meta population context remain metapopulation rare see rare as as compared compared to to the the blooming blooming of of metapopulation metapopulation demographic demographic studies studies ((see Chapters ) . Using Chapters 44 and and 55). Using aa literature literature search search engine engine (lSI (ISI Web Web of of Science Science v04), v04), combined key words words "metapopulation" "metapopulation" and ""life a search with the combined life history evolution" 997 compared evolution" gave gave only only 11 reference reference matching matching the the query query since since 11997 compared to, to, respectively, respectively, 962 962 and and 297 297 when when searching searching with with the the key key words words "metapopula "metapopula"life-history evolution" evolution" alone. This result reflects, not so much the tion" and "life-history scientific scientific production production on on the the subject, subject, but but rather rather the the fact fact that that life life history history evolu evolution in a metapopulation metapopulation context context is poorly poorly identified identified as a distinct distinct field of research. research. We We restrict restrict our our review review to to empirical empirical or or theoretical theoretical studies studies considering considering intraspecific intraspecific variation variation in life history traits traits in a landscape characterized characterized by spa spatial structure, structure, local extinction, extinction, and dispersal among among patches patches of habitat. habitat. Many Many studies in aa studies outside outside this this range range are are related related to to the the field field of of life life history history evolution evolution in metapopulation. instance, many models models of subdivided subdivided populations populations deal deal meta population. For instance, with with the evolutionary evolutionary consequences consequences of of spatial structure structure but do not not take into into account extinction-recolonization extinction-recolonization dynamics dynamics [see, e.g., Gandon Gandon ((1999) account 1 999) and Pen for models models of dispersal and and reproductive reproductive effort evolution] evolution].. (2000), respectively, for studies when we feel that that they point toward toward relevant and We refer to such studies unexplored unexplored aspects of life history evolution evolution in a metapopulation. metapopulation. Finally, life history evolution related to some evolution in a metapopulation metapopulation context context is closely related aspects of dynamics [see Tilman et al., of community community dynamics [see e.g., Tilman aI., (1997) ( 1 997) or or the discusdiscus sion about trade-offs in Chapter about colonization-competition colonization-competition trade-offs Chapter 6]. Testing predicpredic tions tions about about how how metapopulation meta population dynamics dynamics affect affect selection on life histories histories might actually actually be be achieved achieved more more easily easily by documenting documenting changes changes in in specific might specific composition within studying genetic differences composition within a community community rather rather than than by studying differences within a species. To To limit the the scope within scope of of the the present present chapter, chapter, we do do not not incorpoincorpo rate rate community-based community-based studies studies in our our review review but but we we invite invite the reader reader to to keep keep in mind when reading mind the the connection connection when reading Chapter Chapter 6. Second, the field field is largely dominated dominated by theory, with with very very little little empirical empirical Second, the is largely by theory, research due to to obvious obvious practical practical difficulties. difficulties. Most Most empirical empirical evidence of a research evidence of metapopulation effect effect on on life life history history evolution evolution in in natural natural systems systems is is indirect. indirect. metapopulation Metapopulation theory theory predictions predictions have have been been tested tested by comparing comparing mean mean phephe Metapopulation notypes among among populations populations that that have have been been founded founded for for different different times times (Cody (Cody notypes and and Overton, Overton, 1996; 1 996; Piquot Piquot et et al., aI., 1998; 1 998; Hill et et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., 2002) 2002) or the mean or by comparing comparing the mean phenotypes phenotypes among among landscapes landscapes with with different different degrees of of fragmentation fragmentation (Thomas (Thomas et et al., aI., 1998; 1 998; Hill Hill et et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., degrees 2002). Whether Whether those those phenotypic phenotypic differences differences are ultimately due evolution 2002). are ultimately due to to evolutionary investigated (but ary change change and and not not to to environmental environmental effects effects is still still too too rarely rarely investigated ( but et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., 2002). 2002). Artificial Artificial see Thomas Thomas et et al., aI., 1998; 1998; Hill et metapopulations controlled conditions metapopulations of of short-lived short-lived organisms organisms in controlled conditions provide provide a fascinating opportunity opportunity to to witness witness evolutionary evolutionary change change and and test test metapopulametapopula fascinating tion theory predictions predictions more more accurately accurately (Warren, (Warren, 1996; 1 996; Buckling Buckling et et al., aI., 2000), 2000), tion theory but but such such projects, projects, though though growing growing in in numbers, numbers, are are still still in in the the process process of of develdevel opment opment (Lavigne (Lavigne et et al., aI., 2001). 200 1 ) . How How much much artificial artificial metapopulations metapopulations inform inform
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us about about the the relevance relevance of of metapopulation metapopulation theory theory for for life life history history evolution evolution in in us the real world world is is also also open open to to question. question. The The increasing increasing imbalance imbalance between between the real theoretical production production and and data data collection collection is somehow somehow worrying worrying for for the the develdevel theoretical opment of of the the discipline. discipline. Despite Despite our our efforts efforts to to integrate integrate relevant relevant empirical empirical opment examples, the the present present review, review, with with its its strong strong focus focus on on theory, theory, reflects reflects this this bias. bias. examples, The third point concerns the way way the part of the field The third point concerns the the theoretical theoretical part of the field has has develdevel history theory theory specific specific to to metapopulations metapopulations has has bloomed bloomed essentially oped. Life history around questions questions related related to to dispersal dispersal evolution evolution (we (we counted counted more more than than 40 40 around theoretical papers on dispersal dispersal evolution evolution published 1 997). Because theoretical papers on published since since 1997). Because aa whole whole chapter of of the the present present volume is devoted devoted to to dispersal dispersal (Chapter ( Chapter 13, 1 3, see see also also chapter volume is Chapter 16), 1 6), we we will not here review exhaustively exhaustively models of dispersal dispersal evolution. evolution. Chapter will not here review models of Instead, we we focus focus on on those those studies studies that that help help us us to to understand how the the evoevo Instead, understand better better how lution of dispersal contributes to to an an organism's organism's general history strategy. strategy. In In lution of dispersal contributes general life life history particular, this is illustrated illustrated with studies of of variation variation in in dispersal dispersal strategies strategies with with particular, this is with studies age and and the the interaction interaction of of dispersal dispersal with with other other life life history history characters. characters. age Comparatively, metapopulation context, more clas Comparatively, the the evolution, evolution, in in aa metapopulation context, of of other other more classical life Roy, 11999), 999), age sical life history history traits traits such such as as life life span span (Kirchner (Kirchner and and Roy, age at at matur maturity et al., aI., 2000), reproductive effort 1 997; ity (de (de Jong Jong et 2000), or or reproductive effort (Ronce (Ronce and and Olivieri, Olivieri, 1997; Ronce et aI., 2000c; 2000c; Crowley Crowley and 2002) has Ronce et al., and McLetchie, McLetchie, 2002) has received received little little attention attention to In particular, senescence patterns to date. date. In particular, the the evolution evolution of of senescence patterns or, or, more more generally, generally, of of age-specific reproductive major subjects history age-specific reproductive strategies, strategies, while while major subjects of of classical classical life life history theory, unexplored theoretical theoretical questions questions in the context theory, are are almost almost unexplored in the context of of aa metapop metapopulation (with of the ulation (with the the exception exception of the evolution evolution of of delayed delayed reproductive reproductive strategies strategies such 1 ). We such as as dormancy dormancy and and diapause; diapause; for for aa review, review, see see Olivieri, Olivieri, 200 2001). We suggest suggest reasons why why these these questions questions might promising investigation investigation areas. reasons might constitute constitute promising areas. Life Life than dispersal also deserve more attention because they may, history traits other than in some instances, than dispersal in some instances, be be easier easier to to measure measure empirically empirically than dispersal and and would would thus thus allow allow more more precise precise tests tests of of the the theory. theory. This This chapter chapter is is organized organized by by looking looking for for common common patterns patterns explaining explaining results obtained obtained in different different specific studies. Founding Founding events and small local results population population are population size size in in aa meta metapopulation are two two causes causes of of genetic genetic resemblance resemblance among among neighbors neighbors exploiting exploiting the the same same local local environment. environment. This This chapter chapter illus illustrates trates how how this this genetic genetic structure structure makes makes life life history history evolution evolution in in aa metapopula metapopulation tion deviate deviate from from that that expected expected in in aa single single large large panmictic panmictic population. population. Changes Changes in in population population age age structure structure and and density density following following disturbance disturbance and and recolonization population. Species recolonization are are major major features features of of life life in in aa meta metapopulation. Species whose whose biology is described described most adequately adequately using the metapopulation metapopulation framework framework also also often often occur occur in in habitats habitats subject subject to to successional successional changes. changes. Such Such variations variations in in selection selection pressures pressures associated associated with with colonization colonization and and succession succession have have deep deep implications implications for for life life history history evolution. evolution.
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RESEMBLANCE RESEMBLANCE BETWEEN BETWEEN NEIGHBORS NEIGHBORS population Fragmentation of the habitat is often associated with small population size size in in remnant remnant patches patches of of habitat. habitat. Patches Patches of of the the now now classic classic example example of of meta population, the Finnish populations of the butterfly Melitaea metapopulation, Melitaea cinxia cinxia in in the land Islands, the A Aland Islands, contain contain at at most most aa few few sib sib families. families. Both Both genetic genetic and and demo demographic graphic stochastic stochastic processes processes take take an an increasing increasing importance importance in in small small populations. populations.
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In In aa system system of of small small and and poorly poorly connected connected populations, populations, genetic genetic drift drift results results in in both both the the loss loss of of genetic genetic diversity diversity within within each each local local population population and and an an increasing increasing variance in allelic frequencies among populations. populations. Similarly, a transient transient reduc reducpopulation size size associated with disturbance disturbance and/or recolonization recolonization by a tion in population few founders founders can leave a significant signature in the genetic composition composition of popu populations long after population population regrowth 998; Ingvarsson and Giles, Giles, regrowth (Ingvarsson, 11998; Extinction-recolonization processes can either attenuate attenuate or accelerate the 11999). 999). Extinction-recolonization effect population differentiation, effect of of drift drift on on population differentiation, depending depending on on the the details details of of recolo recolonization, dispersal, and the length of the period of transient transient growth growth following recolonization 988; Whitlock recolonization (Slatkin, 1977; Wade and McCauley, 11988; Whitlock and McCauley, 990; Whitlock 997; Ingvarsson, 997; Pannell McCauley, 11990; Whitlock and and Barton, Barton, 11997; Ingvarsson, 11997; Pannell and and Charlesworth, 999). How metapopulation dynamics Charlesworth, 11999). How metapopulation dynamics affect affect the the structuring structuring of of genetic populations is genetic diversity diversity within within and and among among populations is reviewed reviewed in in Chapters Chapters 7, 7, 8, 8, and and 99 (see (see also also Pannell Pannell and and Charlesworth, Charlesworth, 2000). 2000). Both Both founder founder effects effects and and sub subsequent sequent genetic genetic drift drift within within local local populations populations have have the the result result that that two two individu individuals interacting in the same patch patch of habitat have a higher probability probability of sharing alleles belonging to alleles than than individuals individuals belonging to different different patches. patches. Such Such aa genetic genetic structure structure holds been holds major major implications implications for for the the evolution evolution of of life life history history traits, traits, which which have have been explored incompletely and are often neglected. This section illustrates those con consequences evolution of sequences through through three three examples examples concerned, concerned, respectively, respectively, with with the the evolution of dispersal, life span, and allocation strategies. dispersal, life span, and sex sex allocation strategies.
Dispersal Dispersal Multiple Multiple Causes for for the the Evolution Evolution of of Dispersal Dispersal
Dispersal metapopulation context Dispersal is is often often considered considered in in aa metapopulation context as as aa risky risky behav behavior, compensated by ior, compensated by the the potential potential benefit benefit of of founding founding aa new new population population in in an an empty empty patch patch of of habitat. habitat. Such Such aa view, view, adopted adopted by by early early students students of of dispersal dispersal evolution 971; Roff, 975) , appeared evolution (van Valen, 1971; Gadgil, 11971; Roll, 11975), appeared to be some somehow 1 977) discovered how too too simple simple after after Hamilton Hamilton and and May May ((1977) discovered that that selection selection should populations even should favor favor frequent frequent dispersal dispersal behavior behavior in in subdivided subdivided populations even in in the the absence 1 986) and absence of of empty empty patches. patches. Further Further theoretical theoretical work work by by Frank Frank ((1986) and Taylor ((1988), 1 98 8 ), in particular, particular, allowed a better understanding understanding and and quantification quantification of dispersal in demographically stable of the the forces forces selecting selecting for for dispersal in demographically stable but but genetically genetically structured structured populations. populations. Dispersal in such a theoretical theoretical context context can be seen as an altruistic act by which an individual individual risks its own fitness to alleviate alleviate kin competition competition within within the the natal natal patch. patch. As As with with any any altruistic altruistic act, act, such such aa behavior behavior is favored favored as long as the individual individual fitness fitness cost endured endured by the disperser is smaller smaller than than the the inclusive inclusive fitness fitness benefit benefit of of its its departure departure for for its its kin. kin. As As resources resources freed freed by by the the departure departure of of an an individual individual are are shared shared among among all all its its neighbors, neighbors, the the inclusive inclusive fitness fitness benefit benefit will will depend depend on on its its relatedness relatedness to to other other residents residents in in the the natal natal patch patch compared compared to to its its relatedness relatedness with with the the occupants occupants of of patch ((Gandon and Rousset, Rousset, 11999). its new patch Gandon and 999). Dispersal Dispersal is is aa complex complex character character with with multiple multiple consequences, consequences, whose whose evolution evolution is 1 ). Kin is affected affected by by multiple multiple causes causes (Clobert (Clobert et et ai., al., 200 2001). Kin competition competition avoidance avoidance and recolonization mutually exclusive selective forces recolonization of empty patches are not mutually acting on dispersal evolution. However, theoretical theoretical studies addressing the evolu evolution of dispersal often consider one force or the other major explanation other as the major explanation
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for dispersal dispersal by by either either neglecting neglecting genetic genetic drift drift (Olivieri (Olivieri et et al., aI., 1995; 1 995; Holt Holt and and for McPeek, 1996; 1 996; Doebeli Doebeli and and Ruxton, Ruxton, 1997; 1 997; Parvinen, Parvinen, 1999; 1 999; Ronce Ronce et et al., aI., 2000b; 2000b; McPeek, Mathias et et al., aI., 2001; 200 1 ; Parvinen, Parvinen, 2002; 2002; Kisdi, Kisdi, 2002) 2002) or or ignoring ignoring complex complex metapopmetapop Mathias ulation dynamics dynamics (Ezoe, 1998; 1 998; Gandon, Gandon, 1999; 1 999; Gandon Gandon and and Rousset, Rousset, 1999; 1 999; ulation Hovestadt et et al., aI., 2001; 2001 ; Leturque Leturque and Rousset, 2002). 2002). Such decisions are often Hovestadt and Rousset, are often linked to to trivial trivial technical technical choices, choices, such such as as modeling modeling population population numbers numbers as as aa linked continuous rather rather than than discrete variable or using deterministic deterministic rather rather than than stosto continuous chastic models models [see [see Ronce Ronce et et al. ai. (2001) (2001 ) for for further discussion on on this this topic]. topic]. chastic further discussion simulation models incorporate the the kin Individual-based evolutionary simulation models necessarily incorporate phenomena associated with drift and and genetic structure, structure, although selection phenomena with drift although this is not always clearly acknowledged acknowledged (see, e.g., Travis and and Dytham, Dytham, 1998). not A Kin Selection Model Model for for the the Evolution Evolution of of Dispersal Dispersal A Kin Selection in a a Metapopulation Metapopulation in
One how kin selection and One might might wonder wonder about about how and local extinctions extinctions interact interact in aa metapopulation importance in in explaining patterns metapopulation and and about about their their relative relative importance explaining patterns of early on, on, Comins Comins et et ai. of dispersal dispersal in in recurrently recurrently disturbed disturbed systems. systems. Quite Quite early al. ((1980) 1980) incorporated incorporated the two forces forces in in the the same model. An An analytical by the two same model. analytical model model by Gandon and Michalakis Michalakis ((1999), 1 999), building building on ai. ((1980), 1 980), Gandon and on the the work work of of Comins Comins et et al. helped clarify this question Table 10.1 10.1 for for the the main main assumptions their helped clarify this question (see (see Table assumptions of of their model) . The evolutionarily stable dispersal rate, the fraction fraction of of indiindi model). The evolutionarily stable (ES) (ES) dispersal rate, i.e., i.e., the their natal natal patch patch before viduals leaving their before reproduction, reproduction, can be expressed as a simple function function of of the the extinction extinction frequency, the extramortality extramortality or or "cost" "cost" assoasso simple frequency, the ciated the average relatedness among born in in the ciated with with dispersal, dispersal, the average relatedness among individuals individuals born the same patch, patch, and the probability of common common origin of immigrants. When immi immigrants grants in in the the same same patch patch have have aa null null probability probability of of common common origin origin (the (the migrant 977), the migrant pool pool model, model, see see Slatkin, Slatkin, 11977), the ES ES dispersal dispersal rate rate increases increases with with higher extinction extinction rates local population higher rates and and higher higher within within local population relatedness. relatedness. This This happens, patches are happens, respectively, respectively, because because more more empty empty patches are available available for for coloniza colonization tion and and because because kin kin competition competition is is more more intense intense for for philopatric philopatric individuals, individuals, as as was was predicted predicted by by previous previous models models that that have have considered considered kin kin competition competition phenomena phenomena or or the the extinction-recolonization extinction-recolonization dynamics dynamics separately. separately. However, However, more complex patterns emerge those two more complex patterns emerge due due to to the the interaction interaction of of those two forces. forces. In particular, for a very low probability of surviving migration, the ES dispersal dispersal rate rate can can increase increase with with increasing increasing dispersal dispersal cost cost (Fig. (Fig. 10.1.A), 10.1.A), whereas whereas previous previous models models predicted predicted that that dispersal dispersal should should always always decrease decrease with with increasing increasing cost but see 980). Gandon 1 999) cost of of dispersal dispersal ((but see Comins Comins et et aI., al., 11980). Gandon and and Michalakis Michalakis ((1999) explained this unexpected pattern by a simple kin selection argument. As the dis dispersal persal mortality mortality increases, increases, aa larger larger fraction fraction of of the the individuals individuals competing competing in in the the same patch are philopatric (because immigration is very low), which increases the the probability probability of of competing competing with with related related individuals individuals in in the the natal natal patch. patch. In In aa sys system 986), the tem with with no no empty empty patches patches (Frank, (Frank, 11986), the lower lower inclusive inclusive fitness fitness of of philopatric philopatric individuals individuals is is compensated compensated by by the the increasing increasing difficulty difficulty of of immigrat immigrating into extant populations. The empty patches created by local extinction, as in Gandon and Michalakis ((1999), 1 999), however, represent an extra benefit for dis dispersers, leading to increasing dispersal rates for very high dispersal costs. Interactions Interactions between kin competition and metapopulation dynamics are also complex because because the average level of relatedness among among individuals born more complex
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Effect on ES dispersal rate
Gandon and Michalakis ( 1 999) Spatially implicit Infinite island model of dispersal Catastrophic local extinction independent of population size All patches recolonized before disturbance Local carrying capacity reached at foundation
Dispersal mortality increases
Increases
None
Decreases, then increases
Patch carrying capacity decreases
Increases
None
Increases
Heino and Hanski (200 1 ) Spatially explicit Distance-dependent dispersal model based on focal butterfly species behavior Local extinction due to demographic stochasticity Not all patches recolonized Local carrying capacity reached at foundation
Dispersal mortality per unit distance increases or distance between patches increases
Increases
Increases
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Increases
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LIFEHISTORY HISTORY EVOLUTION 110. 0. LIFE
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Fig. Fig. 11 0. O.11 Effect of dispersal cost on the evolution evolution of dispersal. (A) ES ES dispersal rate as a func function of of dispersal cost as predicted predicted by a spatially implicit implicit kin selection model model for different different values 6, 32, 64, and 1128; 28; generated of patch carrying capacity: capacity: from from top top to bottom bottom N = = 11,, 2, 4, 8, 116, generated from model by and Michalakis 999), with values; here from the the model by Gandon Gandon and Michalakis (1 (1999), with different different parameter parameter values; here extinc extinc0.2 and fecundity dispersal propensity tion rate e = = 0.2 fecundity n = 5. 5. (S) (B) ES ES dispersal propensity as a function function of dispersal mor mortality per unit unit of distance as predicted by realistic spatially explicit explicit simulations based on the biology of Finnish Melitea of Finnish Melitea butterflies for two two hypothetical hypothetical patch patch networks characterized, respect respectivel y, by large interpatch (sparse) or short short interpatch ively, interpatch distances (sparse) interpatch distances distances (dense). Redrawn from from Heino and Hanski (2001 (2001);); for original original parameters, values see their Fig. Fig. 4A.
in same patch depends on in the the same patch depends on the the intensity intensity of of founder founder effects effects and and subsequent subsequent drift within within local populations. Local population population size, fecundity, extinction rate, dispersal cost, and the dispersal rate itself all affect indirectly the evolution of dispersal through their their effect on relatedness. Consistent with with what what has been observed in subdivided populations with with no extinction, the ES dispersal rate always decreases with increasing local population size (see (see Fig. 110.1A) 0.lA) because genetic genetic drift drift is is then then weaker weaker and and within-local within-local population population relatedness relatedness decreases. decreases. A Gandon and 1 ) showed A sensitivity sensitivity analysis analysis ((Gandon and Michalakis, Michalakis, 200 2001) showed that that population population extinctions may have a higher qualitative or quantitative impact on dispersal evolution than kin competition or other other forces, such as inbreeding avoidance. This populations made This is is especially especially true true of of meta metapopulations made of of local local populations populations of of large large size, with weak founder effects. In situations situations with strong strong genetic spatial struc structure, kin selection phenomena phenomena are likely to play a considerable role in the evolution of patterns of dispersal. Spatially Spatially Explicit Explicit Simulations Simulations for for a Butterfly Butterfly Metapopulation Metapopulation
Most Most of the qualitative conclusions of the simple theoretical study of Gandon and Michalakis ((1999) 1 999) have been reproduced by a much more realistic model of dispersal evolution (Heino and Hanski, 200 1 ) based on the biology of Finnish 2001) metapopulations of Melitaea (see Table 10.1 for a Melitaea cinxia cinxia and Melitaea Melitaea diamina diamina (see comparison of the assumptions and predictions of the two models). Each local population in these butterfly metapopulations is composed of few sib families, and mating occurs mostly within local populations before dispersal (for a detailed level of relatedness among description of the system, see Hanski, 1999). The level individuals born in the same patch is thus likely to be high and kin competition
234 234
ELIE RONCE OPH OPHELIE RONCEAND AND ISABELLE ISABELLEOLIVIERI OLIVIERI
avoidance avoidance may may be be aa potent potent force force in in dispersal dispersal evolution evolution in in those those species, species, together together with the the frequent frequent local local extinctions. extinctions. Both Both the the effect effect of of migration migration mortality (Fig. (Fig. with 110.1B) 0. lB) and and the the patch patch carrying carrying capacity capacity on on the the evolution evolution of of dispersal dispersal propensity (see Table Table 10.1) 10.1) can can be be interpreted interpreted with the the same kin kin selection arguments arguments as as in in the the (see analytical 1999). However, analytical model of of Gandon Gandon and and Michalakis Michalakis ((1999). However, an an alternative alternative explanation 1 ) is that, in their simulations, the explanation provided by by Heino and Hanski (200 (2001) number number of of empty empty patches patches increases increases when when recolonization recolonization becomes becomes more more difficult difficult because because of of higher higher dispersal dispersal cost cost or or when when demographic demographic stochasticity stochasticity increases increases because of aa small local local population size size (see (see Table 10.1). More numerous empty patches represent more opportunity opportunity for colonization colonization and favor the most dispers dispersing genotypes. Contrary to analytical studies, it is difficult to disentangle the respective role of kin selection phenomena and extinction-recolonization dynam dynam(see, however, the attempts to remove the genetic structure ics in simulation results (see, artificially in Heino and Hanski, 200 1 ). Spatially realistic models, designed to fit 2001). the biology of a focal species, species, offer nonetheless the major advantage that they can be compared to data more readily and may help build a more precise test of the theory. The predicted average dispersal propensity for the well-studied M. M. diamina diamina metapopulation was not significantly different from that estimated empirically, which is encouraging even though it does not provide a critical test of the theory. Predictions of the model concerning regional variation in dispersal propensity among different clusters of patches could, in the future, be compared to the observed variation in butterfly behavior among such clusters.
Evolutionary Consequences Consequences of Landscape Landscape Fragmentation Evolutionary Predictions of the previous models under Predictions of the previous models have have deep deep implications implications for for our our understanding of the effect fragmentation on the evolution of life life histories (see effect of fragmentation Table 10.1). 1 0. 1 ). For fragmentation were comcom For a long time the the evolutionary effects of fragmentation pared to syndromes of insularity. A rapid loss of dispersal ability of plants in pared Overton, 1996) happen in island populations populations (Cody and Overton, 1 996) was expected to also happen fragmented habitats habitats where where the increasing cost of of dispersal would would select against fragmented dispersal. Previous models (Comins et al., aI., 1980; 1980; Gandon Gandon and and Michalakis, Michalakis, 1999; 1 999; Heino and Hanski, 2001) 200 1 ) show show that that the expected relationship relationship between disperdisper Heino and Hanski, propensity and and dispersal mortality mortality or or distance between patches patches is more more comcom sal propensity than previously imagined (see (see also Leimar and and Norberg, Norberg, 1997). 1 997). Another Another plex than complicating factor factor is is that that fragmentation fragmentation is is also also associated associated with with habitat habitat loss, loss, complicating and thus thus with with a likely likely reduction reduction in local population population size: size: previous previous models models and showed that that a reduction reduction in local population population size selects for for increased increased dispersal. showed How How do do the the effects effects of of increasing dispersal cost cost and and decreasing decreasing local population population combine in a fragmented fragmented landscape landscape to to act act on on the the evolution evolution of of dispersal? dispersal? A size combine M. diamina diamina metapopulation metapopulation (Heino (Heino and and Hanski, Hanski, simulated habitat habitat change change in in the the M. simulated 200 1 ) led to to complex complex predictions predictions depending depending on on the the type type of of change change (removal (removal of of 2001) whole patches, patches, reduction reduction in patch patch area, area, or or quality). quality). In some some instances, instances, the the effect effect whole of of decreasing decreasing local local population population size size and and increasing increasing distance distance between between patches patches counteract each each other other completely completely so that that we we would would not not expect expect any any evolutionary evolutionary counteract change change in in dispersal dispersal in in the the new new landscape. landscape. Empirical evidence evidence for for aa correlation correlation between between habitat habitat fragmentation fragmentation and and disdis Empirical persal propensity propensity in in natural natural systems systems is ambiguous ambiguous (see (see Table Table 10.1 1 0 . 1 and and also also persal Plebejus argus argus butterflies, butterflies, total total Chapter 20). 20). Thomas Thomas et et al. al. (1998) ( 1 998) found found that that in in Plebejus Chapter
LIFEHISTORY HISTORY EVOLUTION EVOLUTION 110. 0. LIFE
235 235
mass, potentially associated with flight ability, increased in the most frag fragHesperia mented heathlands. Comparing two metapopulations of the butterfly Hesperia 1 999) found that the relative size of the thorax was larger comma, comma, Hill et al. ((1999) sugin the landscape characterized by larger distances between patches, which sug gests that higher fragmentation is associated with higher dispersal. The fact that famsuch patterns were observed in controlled conditions, together with strong fam ily effects, suggests that morphological differences are based genetically and that these characters have indeed evolved (or could evolve in the future) in response to fragmentation. Whether selection on dispersal is the major expla explanation for such patterns patterns is, however, difficult to establish. Hanski et al. (2002) found no obvious relationship between morphological measurements and and esti estimated dispersal propensity for Glanville fritillary butterflies, M. M. cinxia. cinxia. Contrary to Hill et al. ((1999), 1 999), they found no differences in migration propen propensity among meta populations characterized by different degrees of connectivity. metapopulations
Life Span Span Life Evolution of of Reproductive Effort Effort in Genetically Genetically Structured Structured Populations Populations
Does Does the the genetic genetic structure structure generated generated by by drift drift and and founder founder effects effects affect affect the the evolution of life history traits other than dispersal? A model of a subdivided population (Pen, 2000) suggests that that the same processes are likely to alter the evolution of reproductive effort effort in a metapopulation. metapopulation. Pen's (2000) model does not 0.2). Adults not incorporate extinction-recolonization extinction-recolonization dynamics (see (see Table 110.2). are sessile and competitively superior to juveniles. They allocate their resources functions, which generate between reproductive and and maintenance physiological functions, trade-off between fecundity and and survival. In a genetically structured popula a trade-off structured population, surviving surviving adults the recruitment recruitment of of related juveniles in in the the same same tion, adults prevent prevent the related juveniles patch, whereas dispersing juveniles compete with unrelated individuals. A patch, with unrelated allocation to offspring production instead of survival can then then be seen higher allocation offspring production another mechanism mechanism for for kin competition competition avoidance. avoidance. As a result, result, Pen (2000) as another found that that the ES reproductive found reproductive effort increases with the increasing level of of among competitors patch. We conjecture that that results relatedness among competitors in the same patch. obtained that the exisexis obtained by Pen (2000) could be generalized to other other life cycles in that tence of a strong strong genetic structure structure should favor the allocation allocation of resources resources to to tence favor the history stages with with the highest highest dispersal propensity. propensity. Because of of their their indiindi life history rect rect effect on on within-local within-local population population relatedness, a lower lower dispersal of of juveniles, higher dispersal dispersal cost, and and smaller local population population size all select for higher higher for higher reproductive effort effort in his model. model. We reached similar conclusions conclusions (see (see Table Table reproductive We reached 1 0 .2), although although for different reasons, reasons, in a metapopulation metapopulation model model with with 10.2), for entirely different but no no genetic structure structure (Ronce (Ronce and and Olivieri, 1997). 1997). Both Both local extinctions but that increasing fragmentation fragmentation of the the habitat habitat would would lead lead to to the the models suggest that evolution evolution of of a higher higher reproductive effort effort and and shorter shorter life span span for for species in which adults adults disperse less than than juveniles. Such a prediction should should be checked which by examining examining rigorously rigorously the the interactions interactions between between kin kin selection and and extincextinc tion-recolonization dynamics dynamics for for the the evolution evolution of of reproductive reproductive effort effort as tion-recolonization Gandon and and Michalakis Michalakis (1999) ( 1 999) did did for for dispersal. From From an an empirical empirical point point of of Gandon would be interesting interesting to to compare compare patterns patterns of of allocation allocation to to reproductive reproductive view, itit would
Yes
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No No Yes No transient population growth after colonization Yes Transient population growth after colonization Dynamic number of colonizers Yes Transient population growth after colonization Fixed number of colonizers Yes Transient population growth after colonization Dynamic number of colonizers
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110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
237 237
and maintenance maintenance structures in landscapes of different fragmentation fragmentation levels, as was done for for traits related related to dispersal. Evolution Evolution of of Life Life Span In in a Metapopulatlon Metapopulation with with Sterilizing Sterilizing Parasites
With a different set of assumptions assumptions than in Pen (2000), Kirchner and Roy ((1999) 1 999) showed that resemblance between neighbors generated by founder effects could profoundly modify the evolution of life life span in a metapopulation. metapopulation. Contrary Contrary to Pen (2000), they assumed no direct trade-off between fecundity and adult survival (see (see Table 10.2). Their evolutionary scenario involves interactions with sterilizing pathogens. In a large panmictic population, a long life life span car carries an epidemiological cost, as increasing adult survival increases the prevalence of the pathogen in the population. This happens because infected infected hosts remain in the population longer. Despite this cost expressed at the scale of the population, longer. population, longer lived individuals still enjoy a higher fitness than short-lived hosts and alle alleles increasing longevity invade the population population readily. The situation is different in aa meta population with metapopulation with strong strong founder founder effects effects and and little little dispersal dispersal among among extant extant local populations. Then, the variance in gene frequencies among local popula populations. frequencies populations advantages the short-lived genotypes. When short-lived individuals have mainly short-lived neighbors, the local prevalence prevalence of pathogens is low and the average fecundity per individual is much higher than in local populations populations domin dominated by long-lived individuals where levels of infection infection and sterility are both high. Because of the higher offspring production production of local populations populations with with a high frequency of short-lived genotypes, new populations populations in empty patches are founded founded more often by such genotypes. Selection among local populations populations favors a short life span, whereas selection within local populations populations still advantages geno genotypes with a long life life span. The frequency of short-lived individuals in the metapopulation metapopulation depends depends critically critically on on the the rates rates of of dispersal dispersal and and local local extinction. extinction. that have explained the coexistence of Note that, differently from other models that different life history strategies in a metapopulation by a competition-colonization competition-colonization trade-off (Olivieri and Gouyon, 11997; 997; Lehman and Tilman, 11997; 997; Jansen and Mulder, 11999), 999), the trade-off here emerges from the genetic structure of the meta population, as short-lived individuals indeed enjoy a higher fecundity only metapopulation, when they are surrounded by other short-lived individuals. Predictions of the model could be tested by comparing survival rates of uninfected individuals belonging to Silene (see Chapter 119) 9 ) differing in the preva Silene alba alba metapopulations metapopulations (see prevaUstilago violacea violacea or in their connectivity. connectivity. lence of their sterilizing parasites Ustilago Life History between Life Span History Syndromes: Syndromes: Correlation Correlation between and Dispersal Dispersal
Starting from Grime ((1977), 1977), much effort in life history history theory theory and ecology has been devoted to identifying patterns patterns of covariation covariation among species life his history traits traits and relating relating these patterns patterns to characteristics characteristics of their habitat habitat ((Southwood, Southwood, 11988; 988; Taylor et aI., 11990; 990; Silvertown 993; etal., Silvertown and Franco, 11993; Westoby, 11998; 998; Charnov, 2002). This led to to the definition of syndromes, such as Hutchinson, 11951), 95 1 ), the as the the syndrome syndrome of of the the "fugitive "fugitive species" species" ((Hutchinson, the "ruderal "ruderal strategy" ((Grime, Grime, 11977), 977), or the "colonizer syndrome" (Baker and Stebbins, 11965), 965), which which associates associates high dispersal dispersal ability, high fecundity, and short short life
ELIE RONCE OPH RONCE AND AND ISABELLE OPHELIE ISABELLEOLIVIERI OLIVIERI
238 238
span as a set of coadapted coadapted traits traits symptomatic of unstable habitats. habitats. How How does metapopulation synthe genetic structure of meta population affect the evolution of such syn dromes? Based on previous theoretical theoretical studies, one would dromes? Based on the the two two previous studies, one would predict predict aa positive correlation correlation between lifespan and dispersal ability, contrary contrary to the col colonizer syndrome (see 0.2). This happens because higher (see Table 110.2). higher dispersal decreases genetic differentiation differentiation between local populations, which selects for a longer life span. Conversely, what what are the consequences of increased life span for for the evolution of dispersal? Despite the potential consequences for life his history tory evolution, our understanding understanding of how how complex life cycles with with several age classes affect the genetic structure structure in a metapopulation metapopulation is still very incomplete ((but but see Rousset, 11999). 999). Several models of subdivided populations, populations, assuming that that only juveniles disperse, have shown that that increasing adult survival rates led among juveniles born born in the the same patch, favor favorto a higher level of relatedness among 2000; ing the evolution of increasing juvenile dispersal (Taylor and Irwin, 2000; Irwin and and Taylor, 2000; Ronce et aI., al., 2000a) 2000a).. They therefore predict again a positive 0.2). positive association association between between life life span span and and dispersal dispersal (see (see Table Table 110.2). As such studies do do not incorporate incorporate extinction-recolonisation extinction-recolonisation dynamics, it is, however, difficult to conclude about the generality of the syndrome (see Section 110.3 0.3 for a further further discussion of syndromes). Better predictions could also be achieved by letting letting both life span and dispersal evolve jointly in the same metapopulation. metapopulation. However, theoretical studies addressing this question do not take into account the genetic structure structure of the metapopulation metapopulation (see (see Table 110.2 0.2 and 0.3 ). Artificial and Section Section 110.3). Artificial microcosms microcosms using using mutant mutant strains strains of of Caenorhabditis Caenorhabditis elegans elegans with with modified dispersal and life history history traits traits represent an exciting perspective for testing experimentally the role of kin structure and extinction-recolonization extinction-recolonization dynamics in the evolution of life life history syndromes ((Delattre Delattre and Felix, 200 1 ; Friedenberg, 2003). 2001; Variation Variation of of Allocation Allocation Strategies Strategies with with Age Age
Iteroparity (i.e., the existence of several reproductive episodes within an organism's life span) can also lead to the evolution of offspring dispersal strat strategy varying with maternal age (Ronce et aI., 998, 2000a) al., 11998, 2000a),, in part part because relatedness with other other juveniles born in the same patch increases with with the number number of times their parent parent reproduced reproduced in the patch. We conjecture that that vari variation in the intensity of kin competition competition could also alter the evolution of age agespecific reproductive reproductive effort in such subdivided populations, populations, and thus affect How those results generalize to metapopulations, metapopulations, with patterns of senescence. How large disequilibrium in age structure generated by local extinction, is an open question (see (see Section 110.3 0.3 for predictions, predictions, not not based on genetic resemblance, concerning concerning age-specific allocation strategies).
Sex Sex Allocation Allocation Spatial Spatial Structure Structure for for Genes Involved Involved In in Sex Determination Determination
Similarly, Similarly, several several theoretical theoretical studies studies have have investigated investigated how how genetic genetic structure structure and could affect and context-dependent context-dependent fitness fitness could affect the the evolution evolution of of resource resource allocation allocation between metapopulation. Such between sexes sexes in in aa metapopulation. Such studies studies focus focus particularly particularly on on the the case of hermaphrodites coexist case of gynodioecy gynodioecy or or of of androdioecy, androdioecy, in in which which hermaphrodites coexist with, with,
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
239 239
respectively, female or male individuals (Pannell, 1997a; McCauley McCauley and Taylor, 11997; 997; Couvet 998; Pannell, 2000; McCauley et al., 2000). Couvet et al., 11998; 978) and Gynodioecious species, such as Thymus Thymus vulgaris vulgaris (Dommee (Domm6e et al., 11978) Beta al., 1988), and androdioecious species, such as Beta maritima maritima (Boutin et etal., Datisca 990), Mercuria 997b), Datisca glomerata glomerata (Liston et al., 11990), Mercuria annua annua (Pannell, 11997b), Schizopepon 999), all Schizopepon bryoniafolius bryoniafolius (Akimoto (Akimoto et et al., al., 11999), all occur occur in in recurrently recurrently dis disturbed habitats. In gynodioecious species, species, gender turbed habitats. In gynodioecious gender is is often often determined determined by by epista epistatic interactions 98 8; Frank, interactions between nuclear and cytoplasmic factors (Kaul, 11988; 11989; 989; Charlesworth and Laporte, 1998). High levels of genetic differentiation among among patches for both neutral cytoplasmic markers in linkage disequilibrium with cytoplasmic male sterility alleles ((Cuguen with Cuguen et al., 11994; 994; Manicacci et al., 11996; 996; McCauley et al., 2000) and nuclear genes involved in sex determination determination (Manicacci et al., 11997) 997) suggest that that founding events and limited gene flow metapopulations, such as those of leave a strong signature in gynodioecious metapopulations, B. maritima, Sex ratio ratio is, is, moreover, highly vari B. maritima, T. T. vulgaris, vulgaris, or or Silene Silene vulgaris. vulgaris. Sex moreover, highly variable among populations populations in such species (Frank, 1989). Despite these common features features shared shared by by many many well-studied well-studied gynodioecious gynodioecious systems, systems, theoretical theoretical stud studies disagree about the consequences of genetic structure for the evolution of sex ratio population. Some ratio in in aa meta metapopulation. Some models models predict predict that that the the frequency frequency of of females females would be lower lower in a metapopulation metapopulation than than expected in a large panmictic popu population (Pannell, 11997a; 997a; McCauley and Taylor, 11997; 997; McCauley et al., 2000), 2000), whereas others reach the opposite conclusion ((Couvet Couvet et al., 11998). 998). We here suggest which specific assumptions may be responsible for those discrepancies and and show show how how different different models models shed shed light light on on different different mechanisms mechanisms acting acting on on sex ratio evolution in a genetically structured structured metapopulation. A list of the models' main assumptions predictions is 0.3. models' main assumptions and and predictions is given given in in Table Table 110.3. Local Local Sex Ratio Ratio Variation Variation and and Pollen Pollen Limitation Limitation
All All four four models models focus focus on on the the evolutionary evolutionary consequences consequences of of the the recolonization recolonization of disturbed patches by a small number of individuals. Such founder effects generate a large variation in the local sex ratio (see 0.3). How How does the (see Table 110.3). spatial clustering of female and hermaphrodite genotypes affect the evolution of the sex ratio at the meta population scale? In S. metapopulation S. vulgaris, vulgaris, both the pollination pollination rate rate of of females females and and the the number number of of viable viable seeds seeds per per fruit fruit born born by by hermaphro hermaphrodites were shown to increase with the frequency of hermaprodites hermaprodites in the neigh neighborhood 998; McCauley et al., 2000), suggesting that borhood (McCauley and Brock, 11998; that seed production production is indeed limited by the availability of outcross pollen in that that species. Because of increased pollen competition, the siring success of an her hermaphrodite plant is, however, correlated negatively with the local frequency of hermaphrodites 998). Using frequency-dependent hermaphrodites (McCauley and Brock, 11998). male and female fitness components as in S. S. vulgaris, vulgaris, McCauley and collabo collaborators 997) modeled rators (2000; McCauley and Taylor, 11997) modeled the combined combined effects of ratio variation variation in a gynodioecious metapopula metapopulapollen limitation and local sex ratio tion of an annual plant where where seed dispersal is global but pollen flow flow strictly localized. They concluded that that the expected frequency of females in the meta population was panmictic population metapopulation was lower lower than than in in aa large large panmictic population and and declined declined with with increasing increasing variance variance in in local local sex sex ratio. ratio. A A higher higher variance variance in in the the local local sex sex ratio at foundation foundation increases the spatial aggregation of females. Most Most of them are then found in patches with numerous other females and suffer from pollen
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limitation. Similar Similar arguments explain the limitation. arguments have have been been used used to to explain the predicted predicted lower lower fre frequency feminizing parasites quency of of sex sex ratio ratio distorters distorters or or feminizing parasites in in genetically genetically structured structured animal metapopulations animal metapopulations as as compared compared to to panmictic panmictic populations populations (Hatcher (Hatcher et et aI., al., 2000). 2000). Note Note that that this this prediction prediction goes goes in in aa direction direction opposite opposite to to the the classic classic local local mate mate competition competition hypothesis, hypothesis, which which states states that that in in the the presence presence of of aa strong strong spa spatial tial genetic genetic structure, structure, higher higher sperm sperm or or pollen pollen competition competition among among related related indi individuals but see viduals should should favor favor the the evolution evolution of of female-biased female-biased sex sex ratios ratios ((but see de de Jong Jong et McCauley and 1 997), polymorphism et aI., al., 2002). 2002). In In the the model model of of McCauley and Taylor Taylor ((1997), polymorphism could could not not be be maintained maintained at at equilibrium equilibrium at at both both nuclear nuclear and and cytoplasmic cytoplasmic sex sex loci. loci. The The prediction prediction of of lower lower female female frequency, frequency, however, however, held held whether whether the the determinism 997) or purely determinism of of sex sex was was purely purely nuclear nuclear (McCauley (McCauley and and Taylor, Taylor, 11997) or purely cytoplasmic 997; McCauley McCauley et cytoplasmic (McCauley (McCauley and and Taylor, Taylor, 11997; et aI., al., 2000). 2000). Reproductive Reproductive Assurance and and Recolonization Recolonization
The The same same conclusion conclusion of of aa lower lower frequency frequency of of females females in in aa gynodioecious gynodioecious metapopulation Pannell ((1997a) 1 997a) in pollen metapopulation was was reached reached by by Pannell in aa model model with with no no pollen limitation Pannell ((1997a), 1997a), however, limitation (see (see Table Table 10.3). 10.3). Pannell however, assumed assumed that that females females fail fail to to found found new new populations populations in in the the absence absence of of hermaphrodites. hermaphrodites. Successful Successful immigration immigration of of female female genotypes genotypes in in aa patch patch can can only only occur occur after after the the arrival arrival of of hermaphrodite hermaphrodite genotypes. genotypes. As As aa result, result, the the frequency frequency of of hermaphrodite hermaphrodite geno genotypes types in in recently recently founded founded populations populations is is higher higher than than expected expected on on the the simple simple basis pool. Hermaphrodites basis of of their their frequency frequency in in the the migrant migrant pool. Hermaphrodites benefit benefit more more than than females females from from the the relaxed relaxed competitive competitive conditions conditions and and higher higher recruitment recruitment rates local populations. variation in rates that that prevail prevail in in recently recently founded founded local populations. Stochastic Stochastic variation in the composition of population therefore the composition of founders founders in in such such aa meta metapopulation therefore tends tends to to favor favor cosexual hermaphrodites, hermaphrodites, at expense of cosexual at the the expense of unisexuals, unisexuals, such such as as females females in in gyn gynodioecious 997a). odioecious species, species, but but also also males males in in androdioecious androdioecious species species (Pannell, (Pannell, 11997a). Such Such an an argument argument bears bears close close connections connections to to Baker's Baker's law law and and the the reproductive reproductive assurance 99 8 ) . assurance concept concept (Pannell (Pannell and and Barrett, Barrett, 11998). Intragenomic Conflicts Conflicts and and Founder Founder Effects Effects Intragenomic
Assumptions 1 99 8 ) are Assumptions of of Couvet Couvet et et aI. al. ((1998) are very very similar similar to to those those used used by by Pannell ((1997a), 1997a), as 0.3. However, aI. ((1998) 1998) Pannell as can can be be seen seen in in Table Table 110.3. However, Couvet Couvet et et al. reached reached strikingly strikingly different different conclusions. conclusions. For For some some parameter parameter sets, sets, the the pre predicted population is dicted female female frequency frequency in in the the meta metapopulation is lower lower than than expected expected in in aa large most of range, the large panmictic panmictic population, population, but, but, for for most of the the explored explored parameter parameter range, the reverse holds. In can reverse prediction prediction holds. In particular, particular, relatively relatively high high frequency frequency of of females females can be values that allow the be maintained maintained in in aa metapopulation metapopulation for for parameter parameter values that do do not not allow the presence presence of of females females in in aa single single panmictic panmictic population. population. Such Such discrepancies discrepancies are are ultimately models. Pannell ultimately due due to to the the mode mode of of sex sex inheritance inheritance in in the the two two models. Pannell ((1997a) 1997a) assumed 1 99 8 ) assumed pure pure nuclear nuclear control control of of sex, sex, whereas whereas Couvet Couvet et et aI. al. ((1998) considered considered the the case case where where sex sex is is determined determined bbyy both both nuclear nuclear and and cytoplasmic cytoplasmic loci. 1 997), polymorphism loci. Contrary Contrary to to McCauley McCauley and and Taylor Taylor ((1997), polymorphism at at both both types types of parameters. of loci loci was was protected protected in in the the metapopulation metapopulation for for aa large large range range of of parameters. Why Why are are assumptions assumptions about about the the genetic genetic architecture architecture of of sex sex so so important important ?? In aI. ((1998), 1 998), founding In Couvet Couvet et et al. founding events events have have qualitatively qualitatively different different conse consequences models in quences than than envisioned envisioned in in all all previous previous models in this this section. section. Stochastic Stochastic varia variation tion in in the the identity identity of of founders founders not not only only generates generates phenotypic phenotypic correlations correlations among among
ELIE RONCE OPHIr:LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI OPH
242 242
neighbors, neighbors, but but also also results results in in local local variation variation in in the the mode mode of of sex sex inheritance. inheritance. Depending alleles borne Depending on on the the diversity diversity of of cytoplasmic cytoplasmic and and nuclear nuclear alleles borne by by founders founders of can show variation locally, of aa new new population, population, sex sex can show no no heritable heritable variation locally, have have aa strict strict nuclear or nuclear or cytoplasmic cytoplasmic inheritance, inheritance, or or be be determined determined by by variation variation at at both both types types of of loci. loci. In In local local populations populations with with aa strict strict cytoplasmic cytoplasmic inheritance inheritance of of sex, sex, female female frequency frequency is is predicted predicted to to reach reach aa frequency frequency close close to to 100% 100% ultimately, ultimately, as as soon soon as as females 94 1 ). In females produce produce more more seeds seeds than than hermaphrodites hermaphrodites (Lewis, (Lewis, 11941). In contrast, contrast, in in aa population expression, females population in in which which only only nuclear nuclear alleles alleles can can modify modify sex sex expression, females will will be be progressively progressively eliminated eliminated if if they they produce produce fewer fewer than than twice twice the the number number of of seeds seeds of of an an hermaphrodite hermaphrodite and, and, whatever whatever their their fecundity fecundity advantage, advantage, will will never never exceed exceed 50% 941 ). Such 50% of of the the local local population population at at equilibrium equilibrium (Lewis, (Lewis, 11941). Such aa discrepancy discrepancy is is explained explained by by the the fact fact that that cytoplasmic cytoplasmic genes genes are are usually usually transmitted transmitted through through seeds whereas nuclear transmitted through pollen and seeds only, only, whereas nuclear genes genes are are transmitted through both both pollen and seeds. seeds. In al. ((1998), 1 998), the generated by In Couvet Couvet et et al. the initial initial variation variation in in sex sex ratio ratio generated by found founding selection within ing events events is is thus thus exaggerated exaggerated by by further further selection within the the established established local local populations. production, local populations populations with with a populations. Because of their higher seed production, sex highly biased females grow local popu sex ratio ratio highly biased in in favor favor of of females grow faster faster than than other other local populations, lations, consistent consistent with with the the observed observed larger larger size size of of female-dominated female-dominated patches patches in 996) . The in natural natural T. T. vulgaris vulgaris populations populations (Manicacci (Manicacci et et aI., al., 11996). The combination combination of of within-local within-local population population selection selection favoring favoring aa large large frequency frequency of of females females in in local local populations populations with with aa cytoplasmic cytoplasmic inheritance inheritance of of sex sex and and the the more more dynamic dynamic growth populations results increase in growth of of such such local local populations results in in an an overall overall increase in female female abund abundance scale of ance at at the the scale of the the metapopulation. metapopulation. Note Note that that founding founding events events here here affect affect the ratio not the evolution evolution of of the the metapopulation metapopulation sex sex ratio not because because female female fitness fitness is is affected affected by by their their neighbors neighbors phenotype, phenotype, but but because because founding founding events events allow allow cytoplasmic cytoplasmic male male sterility sterility alleles alleles to to escape escape the the control control of of nuclear nuclear genes genes in in some some local local populations. populations. Conclusion
Because Because these these different different studies studies have have described described different different facets facets of of the the evolu evolutionary consequences of combine their tionary consequences of founding founding events, events, it it is is difficult difficult to to combine their mes messages to sages to appreciate appreciate their their relevance relevance for for sex sex ratio ratio evolution evolution in in real real metapopulations. metapopulations. In In gynodioecious gynodioecious metapopulations, metapopulations, observed observed female female fre frequency often much much higher than expected quency is is often higher than expected in in aa single single panmictic panmictic population population ((Couvet Couvet et 990) and et aI., al., 11990) and is is still still higher higher than than predicted predicted by by the the metapopulation metapopulation model 1 998) for model of of Couvet Couvet et et ai. al. ((1998) for the the same same female female fecundity fecundity advantage. advantage. The The observed observed decline decline in in female female abundance abundance with with time time since since foundation foundation observed observed for for T. 990; see see also 997) is T. vulgaris vulgaris (Belhassen (Belhassen et et aI., al., 11990; also Olivieri Olivieri and and Gouyon, Gouyon, 11997) is con consistent ai. ((1998), 1 998), but sistent with with the the predictions predictions of of Couvet Couvet et et al. but not not those those of of Pannell Pannell ((1997a). 1 997a). Further Further theoretical theoretical and and empirical empirical work work is is needed needed to to estimate estimate the the rel relative ative impact impact of of frequency frequency dependent dependent fitness fitness and and variation variation in in sex sex transmission transmission mode populations. mode for for the the evolution evolution of of sex sex ratio ratio within within meta metapopulations.
Genetic Genetic Resemblance: Resemblance: Conclusion Conclusion We We find find it it useful useful to to distinguish distinguish two two types types of of genetic genetic resemblance resemblance among among neighbors: relatedness relatedness for expression of neighbors: for genes genes directly directly affecting affecting the the expression of the the trait trait of of interest and relatedness relatedness for for genes affecting selection on the trait trait only indirectly. interest
1 0. 10.
LIFE HISTORY HISTORY EVOLUTION EVOLUTION LIFE
243 2 43
In the the first case, the the fact fact that that an an individual individual with with aa given given life life history history trait trait is is In first case, more likely likely to to be be surrounded surrounded by by individuals individuals with with the the same same phenotype phenotype will will more modify selection selection on on that that trait trait whenever whenever selection selection is is frequency frequency dependent. dependent. modify Examples reviewed reviewed here here have have shown shown that that frequency-dependent frequency-dependent selection selection can can Examples affect the the evolution evolution of of aa large large variety variety of of life life history history traits. traits. When When aa life life history history affect character has has aa complex complex inheritance inheritance mode, mode, as as sex sex in in some some plants plants and and animals, animals, character variation of the genetic composition of local local populations can result in differdiffer variation of the genetic composition of populations can result in ences in in the the transmission transmission of of such such character, character, with with potentially potentially important important conseconse ences quences for evolution. Whether Whether intragenomic intragenomic conflicts conflicts affect affect the the evolution evolution quences for its evolution. of life characters in in addition to sex sex allocation allocation is, is, however, however, open open to to of life history history characters addition to question (but ( but see see the the imprinting imprinting phenomena phenomena and and parental parental conflicts conflicts about about question maternal investment investment during pregnancy in in mammals; Hurst et et al., aI., 1996). 1 996). We We maternal during pregnancy mammals; Hurst have not not discussed discussed here here the the evolutionary evolutionary consequences consequences of of the the second second type type of of have genetic resemblance, namely namely that genes affecting affecting fitness fitness but but not not genetic resemblance, that concerning concerning genes directly the the expression of the the life trait of of interest. In particular, particular, the the loss loss directly expression of life history history trait interest. In of founding events and subsequent can of genetic genetic diversity diversity associated associated with with founding events and subsequent drift drift can result fixation of have result in in the the local local or or global global fixation of deleterious deleterious mutations, mutations, which which can can have dramatic effects on population viability Saccheri et 998; Nieminen Nieminen et aI., dramatic effects on population viability ((Saccheri et aI., al., 11998; et al., 2001 Higgins and Lynch, 200 1 ) . Several (Fowler and Whitlock, 2001;; Higgins and Lynch, 2001). Several theoretical theoretical (Fowler and Whitlock, 11999; 999; Whitlock Whitlock, 2002; 2002; Couvet, empirical Whitlock et et aI., al., 2000; 2000; Whitlock, Couvet, 2002) 2002) and and empirical aI., 2002; Groom uninger, 2000) 2000) studies have investigated investigated (Haag et al., Groom and and Pre Preuninger, how metapopulation functioning affects how meta population functioning affects mutational mutational load load and and inbreeding inbreeding depression. Very little inbreeding depression depression and and hetero depression. Very little is is known known about about how how inbreeding heterosis the evolution life history in aa metapopulation. metapopulation. sis indirectly indirectly affect affect the evolution of of life history traits traits in Gandon 1 999) studied kin competition competition avoidance avoidance and inbreeding avoid Gandon ((1999) studied how how kin and inbreeding avoidance ance interact interact to to influence influence the the evolution evolution of of dispersal dispersal in in aa subdivided subdivided population population with (see also with no no local local extinction extinction (see also Perrin Perrin and and Mazalov, Mazalov, 2000). 2000). Could Could similarly similarly the the evolution evolution of of higher higher reproductive reproductive effort effort or or age-specific age-specific reproductive reproductive strate strategies inbreeding avoidance gies be be understood understood as as inbreeding avoidance mechanisms? mechanisms?
110.3 0.3
CHANGING CHANGING LIVING LIVING CONDITIONS CONDITIONS The The recognition recognition of of the the changing changing and and ephemeral ephemeral nature nature of of life life is is deeply deeply rooted rooted in in the the metapopulation metapopulation concept, concept, which which acknowledges acknowledges that, that, just just as as indi individuals, populations populations do not persist forever. Evolution of many life history traits, 994), traits, such such as as dispersal, dispersal, dormancy dormancy (Venable (Venable and and Brown, Brown, 1988; 1988; Rees, Rees, 11994), iteroparity 994), clutch iteroparity (Rees, (Rees, 11994), clutch size size (Orzack (Orzack and and Tuljapurkar, Tuljapurkar, 2001 2001),) , or or age age at at maturity 1 ) can maturity (Lytle, (Lytle, 200 2001) can be be understood understood as as adaptations adaptations to to this this fundamental fundamental uncertainty. Bet-hedging strategies diminish the risks of genotype extinction extinction by spreading spreading reproduction reproduction over over several several years years or or several several sites. sites. This This section section focuses focuses on variability in on the the evolutionary evolutionary consequences consequences of of aa different different type type of of variability in aa metapopulation: metapopulation: we we are are interested interested in in changes changes associated associated with with return return to to the the equilibrium condition condition within within disturbed disturbed populations. populations. Founding Founding events events not not only only equilibrium leave leave aa signal signal in in the the genetic genetic composition, composition, but but also also in in the the demographic demographic struc structure ture of of aa population. population. It It may may be be useful useful to to consider consider the the fact fact that that local local popula populations tions have have aa history, history, characterized characterized by by aa transient transient period period following following recolonization, recolonization, where where density, density, age age structure, structure, and and genetic genetic diversity diversity change change with with time, time, and and aa quasistationary quasistationary period period where where those those population population variables variables are are
ELIE RONCE OPHELIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI OPH
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Fig. 110.2 population size size for for hypothetical "fast" "fast" (A and B) 0.2 Temporal and spatial variability in population "slow" (C and D) species. Metapopulations Metapopulations of the two two species are characterized by the same and "slow" persistence time for local populations, equal to 11/e time probability probability of expected persistence / e if e is the per time extinction. extinction. Demes of the fast species go through through a short period of transient growth growth (A: hatched area) but spend most of their lifetime in a quasistationary state; as a result, most of the occupied metapopulation at equilibrium equilibrium are at this stationary density (B). Demes patches (in black) in the metapopulation species spend a large amount amount of time in the transient phase phase (C); such temporal of the slow species amount of spatial variability in density in the metapopulation metapopulation at dynamics translates into a large amount equilibrium (D). Density now now varies not only among among empty (white) and occupied patches (black equilibrium varies not but differs differs among among occupied occupied patches patches depending time since their colonization or hatched), but depending on time their colonization growing, black for for demes at their stationary density). (hatched for for demes still growing,
approximately constant state will be approximately constant (see Fig. 10.2). 1 0.2). Note Note that that this this stationary stationary state reached at different different times times for for the different different parameters parameters (in particular, reached particular, genetic equilibrium will be reached after demographic equilibrium). If local equilibrium reached long after demographic equilibrium). extinctions and and recolonization synchronized perfectly extinctions recolonization events are not not synchronized perfectly among among different patches, such temporal different temporal changes within within populations populations translate translate into into increased spatial heterogeneity increased heterogeneity at at the the scale of of the the metapopulation metapopulation (see (see Fig. 10.2). differ not 10.2). Populations Populations in different different patches patches of of habitat habitat then then differ not only on on the the basis differences in patch basis of of intrinsic intrinsic differences patch quality quality or or stochastic stochastic factors, factors, but but also also because different amounts of elapsed since because different amounts of time time have have elapsed since their their foundation. foundation. Theoretical starting with Theoretical studies, studies, starting with the the classical classical Levins (1969) ( 1 969) metapopulametapopula tion often assume tion model, model, often assume that that the the transient transient period period in population population dynamics dynamics is relatively short as compared compared to to the the mean mean local local population population life span span (see, (see, e.g., relatively short Gandon the biology Gandon and and Michalakis, Michalakis, 1999). 1 999). Such an an assumption assumption fits fits well well the biology of of some studied metapopulations metapopulations (Heino some empirically empirically studied (Heino and and Hanski, Hanski, 2001). 200 1 ) . In that that case, we we can can safely ignore ignore the the heterogeneity heterogeneity among among patches patches recolonized recolonized at at difdif ferent ferent times and and assume assume that that local local populations populations in the the landscape landscape have have all reached some state (Drechsler reached some demographic demographic stationary stationary state (Drechsler and and Wissel, Wissel, 1997). 1 997). One One may, may, however, however, wonder wonder about about the the generality generality of of this this assumption: assumption: do do the the majormajor ity ity of of species species living living in a metapopulation metapopulation spend spend most most of of their their time time in populapopula tions tions at at equilibrium equilibrium or or in in populations populations still still recovering recovering from from the the last last disturbance? disturbance? Do Do metapopulation metapopulation dynamics dynamics select select for for life history history traits traits that that the transient transient period period following following recolonization recolonization to to be be short? short? Or Or is the the very very allow allow the concept concept of of stationary stationary demographic demographic state state a mere mere abstraction, abstraction, a concept concept difficult difficult to to reconcile reconcile with with the the ever ever changing changing demographic demographic conditions conditions that that prevail, prevail, for for instance, instance, in successional successional systems? systems? Answers Answers t0 to those those questions questions are are still unclear. unclear.
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
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Through Through several several theoretical theoretical and and empirical empirical examples, examples, this this section section illustrates illustrates how periods of transient transient dynamics or, more more generally, varying ecological con conhow foundation can affect affect the evolution of life histories in ditions with time since foundation aa metapopulation. general, spatial spatial or temporal heterogeneity metapopulation. In In general, or temporal heterogeneity in in selection selection has three possible possible consequences: consequences: (i) it may may facilitate facilitate the coexistence coexistence of geno genotypes types with with different different life life history history strategies strategies in in the the same same metapopulation, metapopulation, (ii) (ii) it it may lead lead to strategies that may to the the evolution evolution of of strategies that represent represent aa compromise compromise between between conflicting iii) it conflicting selection selection pressures, pressures, or or ((iii) it may may promote promote the the evolution evolution of of plastic plastic genotypes, genotypes, which which can can express express different different life life history history strategies strategies depending depending on on their their location location or or at at different different times times in in their their life. life. Such Such consequences consequences are are illus illustrated trated through through the the example example of of the the evolution evolution of of sex sex allocation allocation (i), (i), age age at at matur maturity (i and dispersal (ii (ii and ity (i), (i), reproductive reproductive effort effort (i and ii), ii), and and dispersal and iii). iii).
Coexistence Coexistence Sex Allocation Allocation and and Inbreeding Inbreeding
Pannell Pannell (2000) (2000) suggested suggested that that aa change change in in inbreeding inbreeding levels levels within within local local populations populations following recolonization recolonization could could allow allow the the invasion invasion of males males in a hermaphrodite metapopulation. metapopulation. Androdioecy Androdioecy has always been a puzzle for hermaphrodite evolutionary also Section 0.2). Because evolutionary biologists biologists (see (see also Section 110.2). Because males males transmit transmit their their genes pollen, they genes only only through through pollen, they must must produce produce and and disperse disperse more more than than twice twice the the quantity quantity of of pollen pollen produced produced by by hermaphrodites hermaphrodites to to be be maintained maintained in in aa population 1 978 ), but see Vassiliadis et aI., population [Charlesworth [Charlesworth and and Charlesworth Charlesworth ((1978), al., (2000) for (2000) for an an exception]. exception]. This This condition condition is is less less difficult difficult to to achieve achieve if if her hermaphrodites do not allocate their resources male and maphrodites not allocate resources fairly between between male and female female functions functions and and invest invest more more in in ovule ovule production. production. In In large large populations, populations, her hermaphrodites are only if self maphrodites are expected expected to to have have aa female-biased female-biased investment investment only if they they self a large fraction hermaphro fraction of their their ovules. Invasion of male phenotypes phenotypes in a hermaphrodite population increasing the level level of population is, however, made made more more difficult difficult by increasing selfing Charlesworth, 11984). 984). These selling in in such such hermaphrodites hermaphrodites ((Charlesworth, These paradoxical paradoxical requirements requirements could could explain explain why why androdioecy androdioecy is is so so rare. rare. Pannell Pannell (2000) (2000) claimed claimed that that metapopulation metapopulation dynamics dynamics can can generate generate the the situation situation with with aa high high female investment investment and and low inva female low selfing selfing rate rate in in hermaphrodites hermaphrodites necessary necessary for for invasion by argument involves involves the selfing rate goes sion by males. males. His His argument the change change in in selling rate that that goes along change in density following along with with the the change in density following recolonization. recolonization. Delayed Delayed selfing selling consists in consists in various various mechanisms mechanisms favoring favoring outcrossing outcrossing in in dense dense populations populations but but allowing pollen. Such allowing self-pollination self-pollination in in the the absence absence of of outcross outcross pollen. Such aa strategy strategy is is thought thought to to be be particularly particularly adaptive adaptive in in metapopulations metapopulations with with recurrent recurrent bottle bottlenecks necks that that severely severely limit limit the the availability availability of of pollen pollen donors donors as as well well as as that that of of pollinators. pollinators. Strong Strong inbreeding inbreeding in in generations generations following following recolonization recolonization tends tends to to favor favor the evolution evolution of increased increased female allocation allocation in hermaphrodites, hermaphrodites, whereas to whereas the the low low selfing selfing rate rate in in denser denser and and older older populations populations allows allows males males to invade in those those populations. populations. Such a scenario is consistent with with the fact fact that known androdioecious both plants that known androdioecious species, species, both plants and and animals, animals, are are frequently frequently found and found in in recurrently recurrently disturbed disturbed habitats, habitats, have have highly highly variable variable sex sex ratios, ratios, and are are known known or or suspected suspected to to vary vary in in selfing selfing rates rates in in aa density-dependent density-dependent way (see the Pannell, 2000 verbal model model of Pannell (2000) way (see the review review in in Pannell, 2000).) . The The verbal of Pannell (2000) does does not not allow allow us us to to conclude conclude rigorously rigorously that that coexistence coexistence of of males males with with
ELIE RONCE OPH OPHI~LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI
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hermaphrodites hermaphrodites in in such such aa metapopulation metapopulation is is stable, stable, but but suggests suggests that that aa change change population dynamics dynamics may cre crein inbreeding levels during transient stages of population ate ate some some evolutionary evolutionary window window favorable favorable to to the the establishment establishment of of different different life life history history strategies. strategies. Age Age at at Reproduction Reproduction
Changes in population period of growth Changes population density during during the transient transient period growth fol following recolonization also affect history strategies lowing recolonization can can also affect the the evolution evolution of of life life history strategies because it recruitment of uveniles born because it affects affects the the probability probability of of successful successful recruitment of jjuveniles born in populations times. The populations established established for for different different times. The evolution evolution of age at repro reproduction duction in the the perennial perennial monocarpic monocarpic Carlina Carlina vulgaris vulgaris provides provides an illustra illustration tion (de (de Jong Jong et et ai., al., 2000). 2000). In In semelparous semelparous organisms, organisms, age age at at reproduction reproduction represents delaying reproduction represents aa compromise compromise between between the the benefits benefits of of delaying reproduction to to attain aa greater greater size size and higher fecundity attain and thus thus higher fecundity and and the the risk risk of of dying dying before before reproduction. reproduction. Single population population models models based on this basic trade-off, trade-off, how however, largely overpredict overpredict the age of first reproduction reproduction observed observed in natural natural populations of populations of C. C. vulgaris. vulgaris. In In the the system system studied studied by by de de Jong Jong et et ai. al. (2000), (2000), the the rare rare thistles thistles grow grow in in ephemeral ephemeral patches patches at at the the edge edge of of willow willow scrub. scrub. Local populations are Local populations are founded founded by by seeds, seeds, grow grow to to reach reach aa peak peak density, density, and and then decline to 99 6 ) . Recruitment then decline to extinction extinction (Klinkhamer (Klinkhamer et et ai., al., 11996). Recruitment of of seeds seeds is is highly highly variable variable among among patches, patches, being being high high in in the the period period of of population population growth growth but but very very low low in in the the period period of of decline. decline. Rosette Rosette survival survival is is not not affected affected by density, but varies by density, but varies widely widely between between patches patches within within aa year. year. Peaks Peaks of of den density are asynchronized asynchronized among among patches patches and and are probably probably related related to vegeta vegetation succession. tion succession. de de Jong Jong et et ai. al. (2000) (2000) incorporated incorporated these these features features in in aa metapopulation metapopulation model model with with local extinctions extinctions and limited dispersal between between patches. patches. The The predicted predicted age closer to age at at maturity maturity for for C. C. vulgaris vulgaris was was then then closer to that that observed observed in in natural natural popu populations than lations than the the age age predicted predicted by by the the single single population population model model for for the the same same rosette instances, they rosette mortality. mortality. In In some some instances, they found found that that two two genotypes genotypes with with dif different reproduction could could coexist coexist in same metapopulation. ferent ages ages at at reproduction in the the same metapopulation. In In the the period of following colonization, plants reproducing early are period of growth growth following colonization, plants reproducing early are favored favored because their progeny benefits from benign competitive because their progeny benefits from more more benign competitive conditions conditions than than progeny born later in Once safe become scarce, progeny born later in denser denser populations. populations. Once safe sites sites have have become scarce, it, however, delay reproduction reproduction to it, however, pays pays to to delay to reach reach aa higher higher fecundity fecundity and and the the late-reproducing genotypes are late-reproducing genotypes are advantaged. advantaged. Reproductive Reproductive Effort
The latter was obtained The latter results results bear bear close close resemblance resemblance to to what what was obtained about about the the evolution evolution of of reproductive reproductive effort effort in in aa metapopulation metapopulation (Ronce (Ronce and and Olivieri, Olivieri, 11997). 997). We We indeed indeed similarly similarly found found situations situations where where two two genotypes genotypes with with dif different population with ferent reproductive reproductive effort effort strategies could coexist coexist in a meta metapopulation with a transient but neither neither in an unstructured unstructured population transient period period of growth, growth, but population nor nor in in aa metapopulation metapopulation where where the the stationary stationary density density would would be be reached reached imme immediately diately upon upon recolonization. recolonization. Within Within each each population, population, genotypes genotypes with with aa higher reproductive are selected for during reproductive effort effort are during the period period of growth growth because they occupy more rapidly, they are are replaced because they occupy space space more rapidly, but but they replaced progressively progressively by by genotypes genotypes with with aa higher higher investment investment in in survival survival as as soon soon as as juvenile juvenile
241 247
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Time since foundation two genotypes with with different different reproductive reproductive efforts in populations Fig. 110.3 0. 3 Frequency of two populations established for for different different times. In white, the genotype genotype with with the larger reproductive reproductive effort (49% (49% reproduction); in black, the genotype genotype with with the lower lower reproductive reproductive of resources allocated to reproduction); (41% to reproduction). The arrow arrow indicates the the time at which the effort (41 % of resources allocated to stationary ceiling density is reached within within local populations. same model populations. Generated from from the same model (1997), with different different parameters as in their Fig. Fig. 9. as in Ronce and Olivieri (1 997), with
recruitment becomes adult survival (Fig. 10.3). 1 0. 3 ) . Such a recruitment becomes severely limited by adult prediction is consistent frequency in recently recently founded founded popupopu prediction consistent with the higher higher frequency lations lations of genotypes investing in sexual sexual reproduction reproduction rather rather than than asexual asexual growth observed in the erectum along along a succesSparganium erectum succes growth as observed the macrophyte macrophyte Sparganium sional gradient (Piquot (Piquot et aI., al., 11998). sional gradient 99 8 ) . competition-colonization trade-off trade-off (see Cases ooff coexistence coexistence mediated mediated bbyy a competition-colonization and Gouyon, Gouyon, 11997) at the Olivieri and 997) often involve conflicting selection pressures pressures at level of the meta metapopulation of each population population and and at at the level of population [see [see Olivieri et al. ai. (1995) ( 1 995) for polymorphism and and Jansen Jansen and and Mulder Mulder (1999) ( 1 999) for for dispersal dispersal polymorphism reproductive present example example (Ronce and Olivieri, 1997), 1 997), reproductive effort]. The present together with that Jong et ai. (2000) (200 0 ) about about age at maturity, suggests that that together with that of of De Jong et al. at maturity, the succession succession of of antagonistic antagonistic selection selection pressures pressures with with time time within within each each local the population may broaden broaden the the range of conditions conditions allowing allowing the the maintenance maintenance of of population may range of protected polymorphism polymorphism in a metapopulation. metapopulation. Empirical Empirical evidence evidence on on sideside protected Uta stansburiana, stansburiana, similarly suggest that that temporal temporal changes changes in blotched lizards, lizards, Uta blotched help maintain maintain polymorphism polymorphism for for clutch clutch size strategies strategies (Sinervo ( Sinervo et et al., aI., density help 2000). 2000) . Orange-throated Orange-throated females producing producing large clutches clutches of of small offspring offspring are favored during the the period period of of population population growth, whereas yellow-throated yellow-throated are favored during growth, whereas females producing fewer offspring producing fewer offspring of of higher higher quality quality are are favored favored at at high high The previous previous theoretical theoretical models models also also lead us to to think think differently densities. The about colonization: colonization: successful colonization is a complex complex process, process, which which about successful colonization not only only arriving arriving in empty empty patches patches of of habitat, habitat, but but also an an efficient efficient implies not strategy of of space space occupation occupation once once arrived arrived [see also also Bolker Bolker and and Pacala Pacala (1999) ( 1 999) strategy on on the the same same topic]. topic] .
ELIE RONCE OPH OPHF_LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI
248 248
Evolutionary Evolutionary Compromises Compromises Length Length of of the the Growth Growth Period: Effect of of Productivity Productivity
Classic life history theory has for long shown that that an increase in produc producthat fecundity increases, leaving survival unchanged, unchanged, should select tivity, such that for a higher reproductive reproductive effort in an exponentially exponentially growing growing population population (Charnov 973). In (Charnov and and Shaffer, Shaffer, 11973). In aa density-regulated density-regulated population population where where juve juveniles establish only in safe micro sites freed by the death of an adult, microsites adult, increasing productivity has, however, however, no effect on the evolution of reproductive reproductive effort (see Kisdi and Meszena, 11995) 995) because the number number of recruited offspring is then limited by adult mortality and not by productivity. In a meta population metapopulation where populations populations go through through a period of transient growth growth after recoloniza recolonization, the evolutionary pattern pattern is strikingly different from from that that predicted predicted in a population. The ES reproductive effort increases but then decreases with single population. 0AB, and similar pattern increasing productivity [Fig. [Fig. 110.4B, pattern in Ronce and Olivieri ((1997) 1 997) and Ronce et aI., al., (2000c), but see Crowley and McLetchie, (2002) (2002),, for a different conclusion]. conclusion]. Such a nonmonotonic nonmonotonic response reflects the conflict in selection pressures in the metapopulation. metapopulation. Within each growing local popula population, increasing productivity selects for increasing reproductive reproductive effort, but the period of growth growth during which higher reproductive reproductive efforts are favored favored also becomes shorter as local populations populations grow faster and and reach their stationary state sooner (Fig. 0 AA). At any point of time, the fraction of local popula (Fig. 110.4A). populations in the landscape landscape still in the transient transient phase of growth declines with increasing productivity. Lower reproductive reproductive efforts are selected for at the scale of the metapopulation. metapopulation. This example illustrates how changes in demographic parameters affect the evolution of life life history traits, not only by changing selection pressures within local populations, populations, but also by changing the very composition of the landscape. Longer periods periods of transient transient growth tend to select for lower dispersal dispersal rates in the metapopulation Olivieri et al., aI., 11995, 995, Ronce et aI., metapopulation ((Olivieri al., 2000b,c; see also Crowley and McLetchie, 2002). In particular, ES dispersal rates rates decrease decrease with 0AB, and also Fig. 6 in Crowley and decreasing productivity (see (see Fig. 110.4B, McLetchie, 2002). The availability of safe sites within recently founded, low lowdensity local populations populations indeed makes the venture of risking death to reach an an empty empty patch patch of of habitat less less worthwhile worthwhile compared compared to to aa situation situation where where almost all occupied patches are fully crowded 0.4A). Ellner and crowded (Fig. (Fig. 110.4A). Schmida ((1981) 1 98 1 ) used a similar argument argument to explain the rarity of long-range dispersal adaptations aI. ((1990) 1 990) adaptations in desert floras. Consistently, Wilson et al. showed that the relative frequency of plant plant species whose seeds are dispersed by by wind wind or or vertebrates vertebrates increased increased along along aa fertility fertility gradient gradient in in Australian Australian forests, forests, whereas the frequency of species with ant-dispersed seeds or no special dis dispersal device decreased. Coupling between between Local Local and Regional Dynamics: Dynamics: Dispersal and Extinctions Extinctions
Dispersal has been perceived as an adaptation adaptation to habitat habitat instability. The fre fre962; Denno, quency of of winged winged species species in in insect communities communities (Southwood, (Southwood, 11962; Denno, 11994) 994) or of winged morphs 994; morphs in populations populations of the same species (Roff, 11994; Denno aI., 11996) 996) decreases with increasing stability of the Denno et al., the habitat. habitat. Most Most
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models predict that ES dispersal rates should increase monotonically monotonically with with an increasing frequency of of local extinction extinction (see, e.g., Gandon Gandon and Michalakis, growth fol fol11999). 999). Such models generally assume that the period of transient growth lowing lowing recolonization recolonization is so short that it can be neglected neglected altogether. By relax relaxing this assumption, two two deterministic models (Ronce et aI., al., 2000b; 2000b; Parvinen et aI., al., 2003) 2003) found found that the ES dispersal rate may vary nonmonotonically nonmonotonically with with the frequency of of catastrophic catastrophic extinctions. extinctions. On the one hand, more frequent frequent dis disturbances turbances create more empty empty patches patches in the landscape and thus more coloniz colonizwhich selects for increasing increasing dispersal ability. On the other ing opportunities, which hand, disturbances also affect local dynamics indirectly by reducing the total metapopulation size and thus the average number of of immigrants arriving in any given patch. As a result, local populations populations start off off smaller and grow grow longer before reaching some equilibrium density. The longer growth growth period and higher frequency of of low-density low-density local populations populations tend to select against
250
ELIE RaNCE AND ISABELLE OPH OPHELIE RONCE AND ISABELLEOLIVIERI OLIVIERI
frequent dispersal. The conflict between the effects of disturbance at the two spatial scales results in a hump-shaped pattern of dispersal evolution (Ronce et aI., al., 2000b). Note that that it is not so much the inclusion of a transient phase of local population growth that that created this pattern, pattern, but the coupling between local and global dynamics in the metapopulation. metapopulation. For instance, Crowley and McLetchie (2002) assumed a fixed number of founders per new population and no subsequent immigration (see (see Table 110.2): 0.2): they found found that ES dispersal rates always increased with increasing frequency of local extinction because increasing the fraction of empty patches in the landscape then had no effect on the local dynamics. It has, however, been questioned whether whether the decreasing part of the hump humpshaped curve relating dispersal to local extinctions could be observed in nature, the most serious objection being that in the presence of demographic stochasticity, metapopulations may not persist sufficiently long in this high disturbance regime to disturbance regime to observe observe the the evolution evolution of of declining declining dispersal dispersal rates rates (Heino and Hanski, 2001 2001;; Poethke et aI., al., 2003 2003).). Our deterministic model (Ronce et al., aI., 2000b) assumes that that the global landscape composition composition affects the local dynamics, but does not incorporate the reverse feedback (but see Parvinen et al., aI., 200 3 ) . This might not describe accurately the more general 2003). situation where environmental fluctuations combine with demographic sto stochasticity (see Heino Heino and 1 ). Poethke chasticity to to lead lead to to extinction extinction (see and Hanski, Hanski, 200 2001). Poethke et et ai. al. (2003 (2003)) showed that, in the latter situation, the pattern pattern of covariation between dispersal rates and extinction rates is much more complex than previously envisioned. This happens because the evolving dispersal rate itself then affects the probability of extinction. In particular, increasing environmental variabil variability ity may may select select for for larger larger dispersal dispersal rates, rates, which which in in turn turn reduces reduces the the local local extinc extinction because of Kodric-Brown, 11977). 977). tion rate rate because of the the rescue rescue effect effect (Brown (Brown and and Kodric-Brown, Dispersal and local extinctions, both dynamic parameters, may then covary positively or negatively along gradients of environmental variability or dis disai. (2003) suggested that the two latter param persal mortality. Poethke et al. parameters would be better predictors predictors of the ES dispersal rates than the frequency of local extinction. Habitat Habitat Templates Templates and and Life History History Syndromes Syndromes
Taking into consideration transient population dynamics deeply affects our understanding of life life history syndromes. In particular, we showed that pre predicted patterns of association between reproductive effort and dispersal were strikingly different in ""fast fast species" where local populations reach their equi equilibrium density immediately upon recolonization and in "slow species" where the length of the growth period varies dynamically with landscape and life his history tory characteristics characteristics (Ronce (Ronce et et aI., al., 2000c). 2000c). First, First, both both changes changes in in dispersal dispersal and and reproductive effort effort can affect population dynamics and the length of the tran transient phase or, more generally, the distribution of population densities in the landscape. Evolution of one trait may thus modify the selection pressures on the other indirectly through changes in demographic dynamics, just as they do through changes in the genetic structure of the metapopulation (see (see Section 110.2). 0.2). Two models (Ronce et aI., al., 2000c; Crowley and McLetchie, 2002), how however, suggest that such evolutionary interactions may be of minor importance
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Fig. 10.5 1 0.5 Covariation Covariation among among dispersal and reproductive reproductive effort effort evolving evolving jointly jointly along along envienvi Fig. ronmental gradients. g radients. (A) Along a gradient of habitat fragmentation fragmentation in a fast fast species ronmental (A) Along of increasing habitat (in black) black) where where ceiling ceiling density density is is reached reached immediately immediately upon upon recolonization recolonization and and in in aa slow slow (in species (in (in white) white) with with variable variable periods periods of transient growth; the dispersal dispersal cost cost varies varies from from 00 to species of transient growth; the to 0.9. (B) Along a gradient of increasing disturbance in a fast species 0.9. species (in black) and in a slow white); the local extinction rate varies from respectively, the fast species (in white); from 0.1 to to 0.9 or 0.6 for, for, respectively, species or the slow species. Parameters and results results for the ES ES life life history history traits traits are are as as in in Fig. Fig. 33 species or the slow species. Parameters and for the and 44 in in Ronce Ronce et and et al. al. (2000c). (2000c).
(see Table 0.2). Second, because both both traits respond in in opposite opposite directions to (see Table 110.2). Second, because traits respond directions to change in the length of the transient transient period, we may often expect negative corcor relations between reproductive reproductive effort and dispersal along environmental environmental gra grarelations dients (see Fig. Fig. 10.5), 1 0.5), contrary to the the classical colonizer syndrome syndrome (for (for further further dients (see contrary to classical colonizer discussion see Ronce Ronce et aI., 2000c). 2000c). Again, Again, aa clearer understanding of of such life discussion see et al., clearer understanding such life history syndromes syndromes may history may be be achieved achieved by by acknowledging acknowledging the the fact fact that that aa change change in landscape structure in the the landscape structure or or quality quality will will affect affect the the distribution distribution of of population population numbers at two two different spatial scales. The fraction of occupied patches patches in the numbers landscape occupied space space within patches can can be landscape and and the the fraction fraction of of occupied within occupied occupied patches be considered considered two two measures of the degree of resource resource monopolization monopolization in the metapopulation, reflecting, metapopulation, reflecting, respectively, respectively, the the intensity intensity of of global global or or local local com competition. Unexpected life history syndromes can emerge because the evolution of responds differently of long-distance long-distance dispersal, dispersal, contrary contrary to to reproductive reproductive effort, effort, responds differently to change in intensity of local scales. to change in the the intensity of competition competition at at the the global global and and local scales. While While decreased competition competition at the global scale favors increased allocation to repro reproduction and dispersal, decreased competition locally selects for more repro reproduction duction but but less less dispersal. dispersal.
Plasticity Plasticity Density-Dependent Life Histories
If If the the benefits benefits of of dispersal dispersal vary vary with with local local density, density, as as suggested suggested by by previous previous theoretical work, it is quite natural to expect the evolution of dispersal strate strategies predictions and gies conditional conditional on on local local density. density. Theoretical Theoretical predictions and empirical empirical evi evidence dence for for density-dependent density-dependent dispersal dispersal strategies strategies are are reviewed reviewed in in detail detail in in Chapter 3 . While attention Chapter 113. attention to density-dependent dispersal in a metapopula metapopulation has certainly increased in recent years (see, e.g., theoretical work by J
252 252
ELIE RaNCE AND ISABELLE OPH OPHI~LIE RONCE AND ISABELLEOLIVIERI OLIVIERI
and Gyllenberg, 200 1 ; Poethke 2001; Poethke and and Hovestadt, Hovestadt, 2002), the same conclusion conclusion not hold hold about about the evolution of other other plastic life history characteristics. characteristics. does not Yet given the conflicting conflicting selection pressures acting on reproductive reproductive effort or maturity in growing and and crowded crowded populations, populations, one might might expect simi simiage at maturity larly decisions may larly that that such such life life history history decisions may depend depend on on the the density density of of neighbors. neighbors. Quintana-Ascencio al. ((1998) 1998) found Quintana-Ascencio et al. found that that the reproductive reproductive effort of trans transplanted planted Hypericum Hypericum cumilcola cumilcola changed changed with with time time since since fire fire and and the the density density of of conspecifics conspecifics in patches patches of Florida Florida rosemary rosemary scrub. Variation Variation in mean life his hiswith population population age observed in natural natural populations populations is often often consistent consistent tories with both succession through both with the succession through time of genotypes with different different allocation expression of plastic allocation strategies (see, (see, e.g., strategies and with the expression Houssard 995). Houssard and and Escarre, Escarr~, 11995). Time n d Individual Time since Foundation Foundation aand Individual Age Age
Many Many population population characteristics characteristics other other than than density might might change with with time since foundation, consequences for the evolution conditional disper foundation, which consequences evolution of conditional dispersal behaviors or generally conditional conditional life sal behaviors or more more generally life history history strategies strategies have have been been lit little explored. explored. Drastic changes in the age structure structure may occur during during the transient phase following recolonization, transient phase following recolonization, especially especially in in those those species species where where the the age structure structure of migrants migrants does not not reflect the stable age structure structure of local popu populations 996). If competitive lations (see, e.g., Joly and Grolet, 11996). competitive abilities or, more gen generally, demographic parameters, such as fecundities demographic parameters, fecundities or survival rates, vary among will generate among age age classes, classes, the the disequilibrium disequilibrium in in age age structure structure will generate aa large large variation in recruitment variation recruitment probabilities probabilities or expected reproductive reproductive success among among populations founded populations founded for for different different times, even if they have the same density. Patterns genetic diversity diversity may change with (Whitlock and Patterns of genetic with population population age (Whitlock McCauley, 11990), 990), affecting the level of within local population population relatedness compared compared to genetic resemblance resemblance to individuals individuals from from other other local populations. populations. One One might might then then expect expect the the evolution evolution of of variable variable kin kin competition competition avoidance avoidance depending population age. We are, however, aware of no study trying to depending on population relate patterns patterns of change through through time in the genetic structure structure of populations populations to expression of to the the expression of altruistic altruistic behaviors. behaviors. The The probability probability of of extinction extinction of of aa local also depend depend on since foundation, foundation, either either local population population may may also on the the time time elapsed elapsed since through change in population through a change population size and demographic demographic stochasticity stochasticity or because of negative or positive temporal temporal autocorrelation autocorrelation in disturbance disturbance events. In many either exploiting many species species either exploiting an an ephemeral ephemeral and and nonrenewable nonrenewable resource resource or or subject to successional replacement, replacement, the very quality of the habitat habitat deteriorates deteriorates with time (Valderde and Silvertown, 1998). This led Olivieri and Gouyon Gouyon ((1997) 1 997) to predict that that ES conditional conditional dispersal in a metapopulation metapopulation should should increase like that increase with with population population age. age. A A marginal marginal value value argument argument like that used used by by Metz and Gyllenberg (200 1 ) and Poethke (2001) Poethke and and Hovestadt Hovestadt (2002) when study studying density-dependent density-dependent dispersal enabled us to predict predict the ES reaction reaction norm norm for dispersal age for dispersal as as aa function function of of local local population population age for aa species species subject subject to to succes successional replacement, replacement, with no constraint sional constraint on the shape of the reaction reaction norm norm (0. Ronce, S. Brachet, 1. Gouyon and (O. Ronce, S. Brachet, I. Olivieri, Olivieri, P.-H. P.-H. Gouyon and J. J. Clobert, Clobert, unpublished unpublished result). Our Our analysis does not not incorporate incorporate kin selection effects. In most most 1 997) of an cases, we verified the general prediction prediction of Olivieri and Gouyon Gouyon ((1997) increasing increasing dispersal dispersal rate with with time since foundation. foundation. In details, the patterns patterns of
253 253
LIFEHISTORY HISTORY EVOLUTION 110. 0. LIFE
variation variation in in optimal optimal dispersal dispersal rates rates can can be be more more complex complex than than envisioned envisioned in in during the population their simple study, with several outbreaks of dispersal during density is lifetime (Fig. 110.6.A). 0.6.A). Such oscillations occur when equilibrium density reached reached before before the the stable stable age age structure: structure: the the peaks peaks match match the the oscillations oscillations in in age age structure 0.6A). A structure during during the the transient transient period period (Fig. (Fig. 110.6A). A consequence consequence of of such such con conditional ditional dispersal dispersal strategy strategy is is that that high high rates rates of of emigration emigration in in old old populations populations accelerate the rate of successional replacement and shorten shorten the life span of the accelerate population. population. This This population population level level pattern pattern interestingly interestingly parallels parallels what what happens happens in the evolution of senescence at the individual scale. These These highly highly theoretical theoretical considerations considerations do do not not help help us us understand understand how how indi individuals viduals may may acquire acquire information information about about the the age age of of their their population population and and adjust adjust candidate cue for population age their dispersal strategy accordingly. A candidate popucould be the age of the individual itself in a long-lived species in which popu lations lations take take aa long long time time before before reaching reaching their their stable stable age age structure. structure. We We indeed indeed showed that in the presence of strong variance in age structure among popula populashowed founded for for different times, juvenile dispersal strategies conditional on tions founded
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Individual age Individual age Fig. . 6 Evolution Fig. 11 00.6 Evolution of of conditional conditional dispersal dispersal strategies strategies conditional conditional on on individual individual age age or or deme deme (A) Variation Variation of of the the mean mean dispersal dispersal rate rate per per local local deme deme as as aa function function of of time time since since foundafounda age. age. (A) tion for plastic tion for plastic ES ES dispersal dispersal strategies. strategies. Large Large dots: dots: ES ES dispersal dispersal strategy strategy conditional conditional on on deme deme age age when information about when individuals individuals have have access to to complete complete information about time time since since foundation. foundation. Small Small dots: mean age of of individuals individuals per per deme deme as a function function of of time time since foundation. foundation. Continuous Continuous line: mean mean dispersal rate per local local deme deme when when dispersal dispersal varies varies with with individual individual age age (see B) and and individuals individuals have have dispersal rate per (see B) no of demographic information about about time time since since foundation. foundation. (B) (B) Variation Variation of demographic parameters parameters as as aa funcfunc no information tion age-specific fecundity. age-specific survival tion of of individual individual age. age. Long Long dashes: dashes: age-specific fecundity. Short Short dashes: dashes: age-specific survival rate. Continuous line: line: age-specific dispersal rate when individuals individuals do do not not have have access to to information information Continuous age-specific dispersal rate when about time time since since foundation foundation (generating (generating the the continuous continuous line line in in A). A). Demes Demes persist persist for for 80 80 yr yr with with aa about probability of of local local extinction extinction of of 0.1 0.1 per per year. After 80 yr, local local demes demes go go extinct extinct and and remain remain probability After 80 uncolonizable until until the the arrival of the the next next disturbance disturbance (which (which occurs occurs with with aa probability probability of of 0.01) 0.01 ) to to uncolonizable arrival of mimic mimic the the effect effect of of successional successional replacement. replacement. All All demes demes have have aa constant constant size size of of 200 200 adults. adults. Only Only juveniles patches or juveniles disperse disperse and and establish establish either either in in empty empty patches or within within occupied occupied patches patches in in microsites microsites freed freed by by the the death death of of adults. adults. Seventy Seventy percent percent of of the the dispersing dispersing juveniles juveniles die during during migration. migration.
OPH ELIE RONCE RONCE AND AND ISABELLE ISABELLE OLIVIERI OLIVIERI OPHF.LIE
254 254
maternal age age could could be be selected selected for for in in aa metapopulation metapopulation (Fig. (Fig. 10.6.B; 1 0.6.B; O. O. Ronce, Ronce, maternal S. Brachet, Brachet, I. 1. Olivieri, Olivieri, P.-H. P.-H. Gouyon Gouyon and and J. J. Clobert, Clobert, unpublished unpublished result). result). A A S. variation in mean mean dispersal dispersal rates rates among among populations populations due due to to the the combination combination of of variation age structure structure and and age-specific age-specific dispersal dispersal rates rates results results in in aa pattern pattern mimmim variable variable age icking the the optimal optimal reaction reaction norm norm that that would would evolve evolve if if individuals individuals had had complete complete icking information about about the the time time of of foundation foundation of of their their population population (Fig. (Fig. 10.6.A). 1 0.6.A). information This example example illustrates illustrates how how metapopulation metapopulation processes processes may alter the the evolution evolution This may alter of age-specific age-specific life history strategies. Similar processes processes might might affect affect the the evoluevolu of life history strategies. Similar tion of of age-specific age-specific reproductive reproductive effort effort or or rates rates of of senescence. senescence. How How much of the the tion much of classic life life history history theory theory would would be be modified modified in in aa metapopulation metapopulation with with aa varivari classic able age age structure? structure? able
Change in in Living Living Conditions: Conditions: Conclusion Conclusion Change Previous authors authors have have found found it it useful useful to to distinguish distinguish between between selection selection Previous pressures the local level to to understand understand the the evolution evolution of of pressures acting acting at at the local and and global global level life history population ((Olivieri Olivieri and 997). We have life history traits traits in in aa meta metapopulation and Gouyon, Gouyon, 11997). We have view by by showing of extended this view showing through through several several examples examples how how aa variation variation of extended this selection within local local populations, populations, ultimately selection pressures pressures within ultimately due due to to the the disequi disequilibrium generated librium generated by by local local extinction, extinction, could could affect affect the the evolution evolution of of life life his history that the tory traits. traits. The The same same theoretical theoretical examples examples suggest suggest that the evolutionary evolutionary transient dynamics may depend between importance of such such transient dynamics may depend on on the the coupling coupling between importance of landscape processes, extinction and recolonization, and local landscape processes, such such as as extinction and recolonization, and local growth. patches of the growth. Immigration Immigration and and recolonization recolonization of of empty empty patches of habitat habitat are are the two connect the dynamics within local population population to two events events that that connect the dynamics within each each local to the the regional state regional state of of the the metapopulation. metapopulation. Although Although the the demographic demographic conse consequences quences of of this this coupling coupling have have started started to to be be explored explored both both theoretically theoretically and and empirically see, e.g., e.g., the empirically ((see, the relationship relationship between between species species abundance abundance and and distri distribution 999), little about its bution as as discussed discussed in in Hanski, Hanski, 11999), little is is known known about its impact impact on on life life history evolution. We have presented a few examples where this coupling counterintuitive patterns patterns of evolution. However, much results in initially counterintuitive achievement population theory realized by achievement in in meta metapopulation theory has has been been realized by uncoupling uncoupling the the local local and and regional regional dynamics, dynamics, relying relying on on the the timescale timescale difference difference between between the the two 997; see two processes processes (Drechsler (Drechsler and and Wissel, Wissel, 11997; see also also Chapter Chapter 4). 4). Theory Theory based based on on such such assumptions assumptions has has often often led led to to remarkably remarkably robust robust conclusions conclusions 2002). Some Some authors authors have have also also argued argued that that this this very very uncoupling uncoupling (Keeling, 2002). might define define what what is is aa "real" "real" or or "classic" "classic" metapopulation metapopulation rather rather than than aa sim simple 997). We ple patchy patchy population population (Harrison (Harrison and and Taylor, Taylor, 11997). We do do not not share share this this point point of of view. view. Rather, Rather, we we think think that that details details of of the the focal focal species species biology biology deter determine mine the the degree degree of of coupling coupling between between processes processes at at different different spatial spatial scales scales and and that that any any feedback feedback between between local local and and global global dynamics dynamics should should be be neglected neglected only only with with great great caution. caution. This This is is especially especially true true of of modeling modeling exercises. exercises. For For instance, instance, Crowley Crowley and and McLetchie McLetchie (2002) (2002) consider consider that that the the average average number number of of founders founders per per new new population population is is independent independent of of the the global global demographic demographic state state of of the the metapopulation, metapopulation, while while we we assume assume that that the the number number of of empty empty patches patches in in the the landscape landscape is is independent independent of of the the distribution distribution of of local local popula population tion sizes sizes (Ronce (Ronce et et aI., al., 2000b,c). 2000b,c). More More realistically, realistically, both both parameters parameters vary vary dynamically dynamically with with the the number number of of migrants migrants in in the the metapopulation: metapopulation: Both Both mod models els therefore therefore miss miss part part of of the the real real picture. picture. Data Data are are sorely sorely needed needed to to assess, assess,
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
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in a diversity diversity of organisms, organisms, the extent extent and and importance importance of the transient transient phase phase of population population dynamics dynamics and the degree degree of coupling coupling between between regional and and local dynamics.
110.4 0.4
GENERAL GENERAL CONCLUSION CONCLUSION
Perspectives Perspectives for for Theoretical Theoretical Studies Studies of of Life Life History History Evolution Evolution in in a a Metapopulation Metapopulation The The two two parts parts of of this this chapter chapter unfortunately unfortunately reflect reflect aa major major division division in in the theoretical population context. Studies oretical studies of life history history evolution in a meta metapopulation that that have investigated the effect of genetic structure on life history evolution have generally ignored the complex population dynamics generated by extinc extinction tion and and recolonization recolonization and and vice vice versa. versa. Such Such aa theoretical theoretical gap gap is is essentially essentially explained by the difficulty to analyze kin selection models with with complex complex demography. Individual-based simulations allow the consideration consideration of both both demographic demographic and genetic stochasticity, but it remains difficult to disentangle processes and to identify the mechanisms leading to some evolutionary evolutionary pat pattern. Without Without a mechanistic understanding understanding of such patterns, patterns, it may be difficult to determine their generality. Two methods have been proposed proposed to integrate genetic drift drift and demographic demographic stochasticity in the same analytical framework framework (Metz and Gyllenberg, 2001; Rousset and Ronce, 2004). 2004). The latter bears more resemblance resemblance to to classic classic kin kin selection selection models, models, expressing expressing selection selection gradients gradients as as functions of probability probability of genetic identity between between pairs of individuals found found in the same or different local populations. populations. The effect of kin selection processes can probabilities of can then then be be assessed assessed readily readily by by setting setting all all probabilities of identity identity to to zero. zero. The The application particular class of models application of such methods methods is still restricted to a particular ((infinite infinite island models with no age structure) and relies on extensive numernumer ical ical computations. Still, they they offer offer the the exciting exciting possibility possibility to to better better identify identify and and disentangle the effects of demography demography and genetic structure structure acting acting on the evo evolution of life history traits in a meta population. Finally, with the exception of metapopulation. models for for dispersal evolution (Heino and Hanski, Hanski, 2001 2001;; Travis and Dytham, Dytham, 11999), 999), the evolution of life history history traits traits in a metapopulation metapopulation context context has essentially been modeled modeled in a spatially implicit way. We therefore therefore know know very little about about how how the details of spatial configuration configuration of patches may affect the previous models' predictions. A better better understanding understanding of this question could be achieved achieved by by aa careful careful comparison comparison of of spatially spatially explicit explicit simulations simulations and and analyt analytical models as we did here with the work of Gandon 1 999) Gandon and Michalakis ((1999) and Heino 1 ). Heino and Hanski Hanski (200 (2001). Most this chapter make predictions Most of of the the theoretical theoretical studies studies reviewed reviewed in in this chapter make predictions about population, about evolutionary evolutionary stable life history history strategies strategies in a meta metapopulation, evolutionary end points, points, without without worrying worrying about about realistic realistic amounts amounts of i.e., evolutionary genetic variation variation maintained maintained in such such metapopulations, metapopulations, which eventually fuels fuels evolutionary evolutionary change. change. Both Both genetic genetic drift drift and and demographic demographic asymmetries asymmetries not not only affect the direction direction of selection, but but also constrain constrain the evolutionary see Chapters 6) . Although response to selection ((see Chapters 9 and and 116). Although an increasing num number ber of of theoretical theoretical studies studies address address questions questions related related to to the the balance balance among among selection, dispersal, and and drift in a metapopulation metapopulation context, context, little is still
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ELIE RONCE OPH AND ISABELLE OPHI~LIE RONCE AND ISABELLEOLIVIERI OLIVIERI
known known about about how how constraints constraints on selection selection affect the evolution evolution of of particular particular Boughton ((1999) provided a nice empirical empirical example in the life history traits. Boughton 1999) provided checkerspot butterfly checkerspot butterfly Euphydryas Euphydryas editha, editha, where where extinction-recolonization extinction-recolonization dynamics dynamics and and patterns patterns of of gene gene flow flow severely severely constrain constrain the the evolution evolution of of host host preference. preference.
Perspectives Perspectives for for Empirical Empirical Studies Studies of of Life Life History History Evolution Evolution in a Metapopulation in a Metapopulation As As stated stated in in the the introduction, introduction, the the present present chapter chapter strongly strongly reflects reflects the the bias bias affecting affecting the the field, field, with with aa scarcity scarcity of of empirical empirical studies. studies. Data Data are are needed needed to to assess assess the the validity validity of of model model assumptions assumptions about about the the genetic genetic structure structure or or the the demography thought to have evolved demography of of species species thought to have evolved in in aa metapopulation metapopulation context. context. Obtaining Obtaining the the latter latter type type of of information information can can be be extremely extremely demanding demanding as as it it often often requires requires long-term long-term surveys surveys at at multiple, multiple, and and sometimes sometimes very very large, large, spatial spatial scales. scales. Such Such data data accumulated accumulated on on aa few few systems systems now now allow allow us us to to ask ask evolu evolutionary (as the evolution tionary questions questions about about such metapopulations metapopulations (as evolution of dispersal bserving evolutionary and and host host preference preference in in M. M. cinxia). cinxia). O Observing evolutionary changes changes at at the the scale scale of population is challenging task. of aa meta metapopulation is aa challenging task. Patterns Patterns of of variation variation in in life life history history traits traits along along environmental environmental or or successional successional gradients gradients offer offer an an alternative alternative for for testing evolutionary evolutionary model model predictions, predictions, but but it requires evaluating evaluating the relative role of role of genetic genetic differences differences and and plasticity plasticity in in the the production production of of such such patterns. patterns. Proving mechanism as in the explained the Proving that that the the same same mechanism as envisioned envisioned in the model model explained the observed represents another observed pattern pattern represents another difficulty difficulty of of this this approach, approach, as as for for instance instance seed size or flight ability ability may be selected for reasons not for reasons not related to dispersal. Manipulation Manipulation of of natural natural systems systems offers offers aa much much more more powerful powerful test test of of evo evolutionary scenarios. For instance, Sinervo et al. (2000) manipulated manipulated offspring lutionary size assess selection selection on along cycles size in in lizards lizards to to assess on this this trait trait along cycles of of low low and and high high den density in lizard populations. heritable variation variation for life populations. There is a large amount amount of heritable history strategies in their system due to the coexistence coexistence of very distinct female reproductive documented existing patterns reproductive tactics. Sinervo et al. (2000) documented patterns of genetic genetic covariation covariation among among different different life life history history characters, characters, but but also also artificially artificially modified phenotypes of selection scenarios. modified the the phenotypes of individuals individuals to to test test selection scenarios. In In the the same same study, measured the component of study, they they measured the frequency-dependent frequency-dependent component of selection selection acting acting Combining such approaches approaches with a spatial perspective on on offspring size. Combining such such selective selective processes processes would would provide provide aa fascinating fascinating opportunity opportunity for for empirical empirical studies population. Parasites studies of of life life history history evolution evolution in in aa meta metapopulation. Parasites exploiting exploiting indi individual hosts or groups groups of hosts have been pointed pointed out out as organisms organisms particu particularly amenable for metapopulation theory larly amenable for testing testing metapopulation theory (Grenfell (Grenfell and and Harwood, Harwood, 11997). 997). Many Many studies studies have have now now documented documented interspecific interspecific or or intraspecific intraspecific vari variation ation in in parasites parasites life life history history traits traits in in relation relation to to the the structure structure or or dynamics dynamics of of their ai., 11997; 997; Morand, 2002; Parker their hosts hosts populations populations (see, (see, e.g., e.g., Sorci Sorci et et al., Morand, 2002; Parker et et aI., al., 2003; 2003; Thomas Thomas et et aI., al., 2002). 2002). An An experimental experimental test test of of theoretical theoretical predictions predictions about populations using about life life history history evolution evolution in in meta metapopulations using host-parasite host-parasite systems systems and and manipulating manipulating the the characteristics characteristics of of the the host host population population (see, (see, e.g., e.g., Koella Koella and 999) should and Agnew, Agnew, 11999) should be be further further encouraged. encouraged. We potential directions We conclude conclude by by indicating indicating potential directions for for empirical empirical investigation investigation of of life life history history evolution evolution in in aa metapopulation, metapopulation, summarizing summarizing questions questions emerging emerging from particular models from our our review. review. First, First, in in order order to to judge judge the the generality generality of of particular models
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predictions, one one would would wish wish to to have have aa better better sense sense of of the the importance importance of of trantran predictions, sient population population dynamics, dynamics, of of the the degree degree of of coupling coupling between between local local and and global global sient dynamics, and and of of the the extent extent of of among-local among-local population population differentiation differentiation for for dynamics, quantitative characters characters in in diverse diverse species. species. As As seen seen in in this this review, review, kin kin selection selection quantitative and demographic demographic disequilibrium disequilibrium can can profoundly profoundly modify modify our our expectations expectations concon and cerning life evolution, but but the the quantitative quantitative importance of such such phephe cerning life history history evolution, importance of nomena in in real real metapopulations metapopulations is is somehow unclear. Second, if details details of of nomena somehow still still unclear. Second, if the population population dynamics and genetic structure matter so much, much, it it casts casts doubts doubts the dynamics and genetic structure matter so on the the potential potential of of broad broad scale scale comparative comparative studies studies to to identify identify aa robust robust pattern on pattern of variation variation in in life life history history syndromes syndromes associated associated with with metapopulation metapopulation of processes (for aa discussion discussion of of such syndromes, see see Poethke et al., aI., 2003; 2003; Ronce Ronce processes (for such syndromes, Poethke et et al., aI., 2000c). 2000c). Focusing Focusing on on well-studied well-studied systems, systems, for are et for which which such such details details are known and where quantitative predictions predictions can can be be achieved, achieved, sounds as aa more more known and where quantitative sounds as promising alternative. In In particular, particular, the the present review points points toward toward two two promising alternative. present review main questions questions deserving deserving more more empirical empirical exploration. exploration. The The first first concerns concerns the the main patterns of traits other than dispersal in relation relation to to patterns of evolution evolution for for life life history history traits other than dispersal in landscape fragmentation. reproductive landscape fragmentation. Candidate Candidate traits traits include, include, for for instance, instance, reproductive effort, maturity, and size. The question is is effort, longevity, longevity, age age at at maturity, and clutch clutch size. The second second question related evolution of plasticity or, life hishis related to to the the evolution of plasticity or, more more generally, generally, of of conditional conditional life tory in aa metapopulation. metapopulation. tory strategies strategies as as adaptations adaptations to to aa variable variable environment environment in Of particular evolution of senescence patterns or age-specific Of particular interest interest is is the the evolution of senescence patterns or age-specific reproductive for species species subject reproductive strategies strategies for subject to to extinction-recolonization extinction-recolonization dynamics dynamics or processes. or successional successional processes.
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11
SELECTION SELECTION IN IN M ETAPO PU LATI O N S:: META PO PULATIONS THE COEVOLUTION COEVOLUTION OF PH ENOTYPE ENOTYPE OF AND CONTEXT AND CONTEXT Michael JJ.. Wade
11.1 1 1 .1
INTRODUCTION INTRODUCTION Most live in metapopulations, interacting interacting locally locally with con Most organisms organisms live in metapopulations, with conspecifics in ways that that affect affect individual fitness and in genetic contexts that vary fitness and contexts that from to another another (Loveless (Loveless and Hamrick, Hamrick, 1984; 1 984; Whitlock, 1992; 1 992; from one deme to 1 996; Hanski Hanski and and Gilpin, 1997; 1 997; Ingvarsson, 1999; 1 999; Wade and Goodnight, Kelly, 1996; and Goodnight, 1998). As As aa result, result, natural natural selection selection in in metapopulations metapopulations always always differs differs from from 1998). that in large, large, randomly randomly mating mating populations, populations, whether whether or or not not there there is is an an added added that in component of higher level selection acting among demes. This difference difference component higher level between metapopulations and that popu between adaptive evolution in metapopulations that in nonsubdivided poputradition lations has been overlooked by evolutionary genetic theory with its tradition of partitioning partitioning phenotypic variation into into dichotomous dichotomous genetic and and environenviron (e.g., Falconer and MacKay, 1996). 1 996). This traditional traditional approach approach mental factors (e.g., founders whenever whenever some of the factors for variation variation in individual founders factors responsible for and environmental environmental at at the same time. time. It It is phenotype or or fitness fitness are both both genetic and phenotype factors, often called "indirect "indirect genetic genetic effects" effects" (IGEs) (IGEs) (Moore (Moore et et al., ai., 1997; 1 997; these factors,
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MICHAEL MICHAEL J.I. WADE WADE
Wolf 998, 11999), 999), that Wolf et et aI., al., 11998, that create create causal causal pathways pathways between between the the genes genes in in one one individual individual and and the the phenotypes phenotypes expressed expressed by by others, others, even even if if unrelated, unrelated, and and that that permit permit the the coevolution coevolution of of phenotype phenotype and and context context that that is is unique unique to to metapopu metapopulations. lations. Even Even individual individual traits traits with with no no additive additive genetic genetic variance variance can can evolve evolve in in metapopulations an IGE) metapopulations when when the the mean mean of of aa trait trait in in the the social social partners partners ((an IGE) evolves (Moore et aI., al., 11998; et aI., al., 200 2001). evolves 998; Agrawal et 1). An An IGE IGE can can bbee the the outcome outcome ooff aann overt overt social social interaction interaction between between individ individuals, uals, as as in in altruism altruism or or mutualism, mutualism, or or aa less less conspicuous conspicuous competitive competitive inter interaction, action, such such as as competition competition for for sunlight sunlight or or nutrients nutrients in in plants. plants. The The evolution evolution of interactions interactions among among individuals individuals typically typically has has been been investigated investigated in in the the context context of of 964a,b), aa concep of optimality optimality and and game-theoretic game-theoretic models models (e.g., (e.g., Hamilton, Hamilton, 11964a,b), conceptual tual context context that that does does not not deal deal very very effectively effectively with with genotype-by-environment genotype-by-environment interactions, interactions, epistasis, epistasis, or or indirect indirect genetic genetic effects effects (Wolf (Wolf and and Wade, Wade, 2001 2001;; Shuster Shuster and and Wade, Wade, 2003). 2003). An An IGE IGE iiss aann environmental environmental source source ooff phenotypic phenotypic variation variation that that evolves evolves itself itself in in response response to to selection selection and and random random genetic genetic drift. drift. Thus, Thus, IGEs IGEs are are fundamental fundamental to to the the coevolution coevolution of of phenotype phenotype and and context. context. Whenever Whenever there there are are IGEs, IGEs, the the phenotype phenotype of of an an individual individual depends depends both both on on its its own own genotype genotype and and on on the the genotypes genotypes of of those those with with which which it it interacts interacts (Griffing, (Griffing, 11967, 967, 11977, 977, 11989; 989; Moore Moore et 998), permitting et aI., al., 11998), permitting coevolution coevolution of of individual individual and 1 . 1 and 1 .2). The and social social context context (see (see Fig. Fig. 111.1 and 111.2). The magnitude magnitude of of an an IGE IGE depends depends on the the sign sign of of its its effect effect on on the the individual's individual's phenotype, the reciprocity reciprocity of of the the on phenotype, the interactions individuals, and interactions among among individuals, and the the genetic genetic correlations correlations with with other other traits traits in the the individual individual and and in in those those with with which which it it interacts. interacts. However, However, like like the the coevo coevoin lution Gomulkiewicz et lution of of traits traits in in different different species species ((Gomulkiewicz et aI., al., 2000; 2000; Wade, Wade, 2003) 2003),, genetic genetic correlations correlations are are not not necessary necessary for for IGEs IGEs ttoo affect affect the the rate rate oorr direction direction of evolutionary response to local selection. Whenever different differently to to the same context, Whenever different individuals individuals respond respond differently the same context, component to to the the evolution evolution of of interactions. interactions. then IGEs IGEs introduce then introduce aa nonlinear nonlinear component In large, randomly mating mixing populations, populations, these these effects IGEs are are of In large, randomly mating and and mixing effects of of IGEs of limited importance limited importance because because all all individuals individuals experience experience essentially essentially the the same same average context context (Wade (Wade and and Goodnight, Goodnight, 1998). 1 998). In In metapopulations, metapopulations, however, however, average nonlinear interactions interactions become become more more important important because because they they can can enhance enhance the the nonlinear among-deme variation variation in in the the contexts contexts experienced experienced by by individuals individuals and and are are among-deme Environment Environment
.~~Nurture �
�
Phenotype Phenotype
Genes Genes "
Standard Decomposition Decomposition of of the the IIndividu Phenotype: Standard n d i v i d u a l al Phenotype: zI= aI+ eI
caused by by variation variation in in aalI or or in in el: el: Variation in in zz,I isis caused Variation V(zl)= V(a,) ++ V(el) V(z,) = V(al) V(e,) Heritability: Heritability: h2 IJ2(zl) V(a,)I V(z,) (z,) = V(al)/V(zl) =
Standard decomposition decomposition of of an an individual individual phenotype phenotype into into genetic genetic (al) (al) and and envirenvir Fig. 111.11 1 1 .1 Standard Fig. onmental onmental (el) (el) components. components.
261 2.61
111. 1 . SELECTION SELECTION IN METAPOPULATIONS METAPOPULATIONS (a)
Genes Genes
\ \
/ /
Environment Environment
Jal Partners' Soc Social Mean Phenotype
Zse \. NN~
ZsP
Environment
/
Abiotic f Environment (b) (b)
./ / Genes j Genes
�
Social Partners ~ Abiotic Environment Genes
______ Individual Individual
Phenotype zJ zl
~
� ~ Phenotype Nurture Nurture -'~
Individual Individual
Phenotype
Genes Decomposition of the Indi Individual vidual Phenotype: z,= zl= a,+ al+ e,+ el+ ('If)(Asp+ (~)(Asp+ Esp) Esm) Variation in z, z / is caused by
V(z/) = V(a,) V(al) + V(e,) V(el) + 22~12 Cov(al, Asm) V[E/]) V(z,) '1f12 Cov(a" Asp) + (~) ('Iff2 ((V[Asm] V[Asp] + V[E,D h2(zl) = { V V(al) Cov(al, Asp) Asp) + ('Iff (~)2V(Asp)}/V(zl) If(z,) (a,) + 2~ 2'1f Cov(a" V(Asp)}!V(z,) Fig. f i g . 11 11 ..2 2 (a)The influence influence of environment environment on an individual's phenotype can be partitioned into abiotic and social social components (IGEs). (IGEs). Importantly, the social social component itself can be decomposed into genetic and environmental components. Because Because the environment environment contains genes, the individual phenotype and the social social context can coevolve. (b) The decomposition decomposition of an individual phenotype into genetic (al), IGE components (al), environmental (el), (el), and IGE components ('lt12ZSP)' (~12Zse). The mean phenotype of the social social partner, Zsp, Zsp, is partitioned partitioned into genetic (Asp) (Asp) and environ environmental mental (Esp) (Esv) components. components.
more more likely likely to to create create higher higher levels levels of of selection selection (e.g., (e.g., Aviles Aviles et et a!., al., 2002). 2002). In In short, short, the the evolutionary evolutionary role role of of IGEs IGEs in in metapopulations metapopulations is is always always greater greater than than it is in large panmictic panmictic populations, populations, which which are lacking in conspicuous conspicuous genetic structure see Fig. 111.4). 1 .4). structure ((see In considering considering the role of of IGEs in metapopulations, metapopulations, this chapter chapter shows that that it 1 ) how evolution is it is is important important to to determine determine ((1) how trait trait evolution is affected affected by by local local con context, 3 ) how text, (2) (2) which which traits traits contribute contribute to to local local context, context, ((3) how trait trait and and context context are are correlated variation in correlated ecologically ecologically and and genetically, genetically, and and (4) (4) whether whether variation in mean mean fit fitness that the ness leads leads to to selection selection among among contexts. contexts. This This chapter chapter emphasizes emphasizes that the traits traits affected 1 ) may affected by by context context ((1) may be be the the same same or or different different from from those those traits traits con contributing tributing to to context context (2) (2) and and that that these these may may or or may may not not be be genetically genetically correl correlated. absence of ated. Furthermore, Furthermore, even even in in the the absence of among-deme among-deme selection selection (4), (4), meta populations permit metapopulations permit the the coevolution coevolution of of phenotype phenotype and and context context where where the the more more traditional traditional evolutionary evolutionary genetic genetic models models do do not not (cf. (cf. discussion discussion in in Wade, Wade, 11996; 996; Wade 998; Goodnight Wade and Goodnight, Goodnight, 11998; Goodnight and and Wade, 2000). 2000). Finally, the chapter coevolution of chapter discusses discusses how how these these considerations considerations might might influence influence the the coevolution of species in metacommunities, metacommunities, where the presence or absence of interacting interacting species variation in species contributes contributes to to aa variation in local local context. context.
262 262
11 11..2 2
MICHAEL J. WADE MICHAEL J. WADE
KINDS K I N D S OF OF CONTEXTS CONTEXTS It It is is fundamental fundamental in in evolutionary evolutionary genetic genetic theory theory that that the the environment environment affects expressed as individual phenotype. affects how how aa nuclear nuclear genotype genotype is is expressed as an an individual phenotype. The The general concept concept of general of "environment" "environment" includes includes intraindividual intraindividual cytoplasmic cytoplasmic organelles, organelles, such such as as mitochondria mitochondria and and chloroplasts, chloroplasts, as as well well as as endosymbionts endosymbionts such epistatically with olbachia and Wolbachia and Buchnera, Buchnera, which which can can interact interact epistatically with the the such as as W host expression of phenotype. Classical Classical host nuclear nuclear genome genome and and affect affect expression of the the host host phenotype. physiological epistasis, epistasis, based physiological based on on interactions interactions between between genes genes in in the the same same genome (see Chapter nonclassical epistasis, genome (see Chapter 9), 9), or or nonclassical epistasis, based based on on interactions interactions between genes in maternal and genomes (Wolf, between genes in the the maternal and zygotic zygotic genomes (Wolf, 2000; 2000; Wade, Wade, 2000), 2000), are are ubiquitous ubiquitous intraindividual intraindividual genetic genetic contexts, contexts, wherein wherein the the genetic genetic back background ground at at one one locus locus modifies modifies the the phenotypic phenotypic expression expression of of another another locus. locus. Variation Variation in in these these heritable heritable contexts contexts can can cause cause aa focal focal gene gene to to change change its its Mendelian identity from Mendelian phenotypic phenotypic identity from dominant, dominant, to to recessive, recessive, to to additive, additive, to to neutral, 1 ; see see also neutral, or or to to overdominant overdominant (Wade, (Wade, 200 2001; also Chapter Chapter 99).) . The The variance variance in in context phe context caused caused by by additive-by-additive additive-by-additive epistasis epistasis alone alone can can change change aa gene's gene's phenotypic major to its effect notypic effect effect from from major to minor minor and and change change the the sign sign of of its effect on on fitness fitness so deme but but aa ""bad" bad" gene in another (Wade, so that that it it is is aa "good" "good" gene gene in in one one deme gene in another (Wade, 2002; 995, 2000) intraindividual genetic genetic 2002; Goodnight, Goodnight, 11995, 2000).. These These effects effects of of the the intraindividual environment always more important in populations, where environment are are always more important in meta metapopulations, where subdivi subdivision leads to large populations, populations, sion leads to among-deme among-deme variance variance in in context, context, than than in in large where recombination diminishes variation experi where panmixia panmixia with with recombination diminishes the the contextual contextual variation experienced single genes genes (Fig. 1 1 .3). Indeed, Indeed, the enduring controversy controversy at enced by by single (Fig. 11.3). the enduring at the the foun foundation of theory between between S. S. Wright A. Fisher Fisher (Coyne dation of evolutionary evolutionary theory Wright and and R. R. A. (Coyne et 997, 2000; Wade and and Goodnight, 998; Goodnight et aI., al., 11997, 2000; Wade Goodnight, 11998; Goodnight and and Wade, Wade, 2000) 2000) can varia can be be seen seen as as aa controversy controversy over over the the evolutionary evolutionary significance significance of of such such variations tions in in genetic genetic context. context. The The patterns patterns of of migration migration within within and and among among demes demes and local population variance in and the the local population size size affect affect the the variance in context. context.
Large, Panmictic Population
Metapopulation
Fig. 11 11 ..33 A large, randomly mating "Fisherian" population (left) and a genetically sub subdivided meta population (right). metapopulation (right). The different size circles in the right panel represent demes of different age (smaller is younger) and different abundance. Demes with dotted borders are local extinctions. Small arrows connecting different demes represent migration, whereas larger arrows indicate colonization events.
263 263
111. 1 . SELECTION IN M ETAPOPULATIONS SELECTION IN METAPOPULATIONS
The The abiotic abiotic environment, environment, external external to to the the individual, individual, is is the the most most commonly commonly considered considered context context in in relation relation to to phenotypic phenotypic variation variation among among individuals. individuals. It It is is the the reference reference context context for for most most discussions discussions of of adaptation adaptation and, and, in in most most evolu evolutionary tionary theory, theory, the the external external environment environment is is treated treated as as temporally temporally and and spatially spatially invariant, invariant, at at least least on on average. average. The The traditional traditional partitioning partitioning of of the the variation variation in in aa phenotype, i ) and phenotype, Zl, zl, into into additive additive genetic, genetic, al (with (with mean mean A A1) and environmental, environmental, eb (Fig. 111.1): 1 . 1 ): el, components components reflects reflects this this viewpoint viewpoint (Fig. zl = al + el.
((11.1) 11.1)
Genotype-by-environment Genotype-by-environment interactions interactions (G (G X x E E)) are are the the most most prevalent prevalent format format for introduction of for the the introduction of variation variation in in environmental environmental context context into into evolutionary evolutionary models 998). In models (Schlichting (Schlichting and and Pigliucci, Pigliucci, 11998). In spatially spatially heterogeneous heterogeneous habitats, habitats, geo geographic graphic variation variation in in parameters parameters of of the the niche niche can can lead lead to to variation variation in in local local select selectMigration among among habitats with G G X x E E ive pressures (Holt and Gaines, Gaines, 11992). 992). Migration constrains constrains evolution evolution because because genotypes genotypes adapted adapted to to one one habitat habitat are are moved moved to to other other habitats habitats where where they they are are less less fit fit by by migration. migration. Habitats Habitats where where abundance abundance is is greatest greatest will will tend tend to to dominate dominate evolution evolution in in those those where where abundance abundance is is lower lower if if migration 992). Under migration is is proportional proportional to to local local abundance abundance (Holt (Holt and and Gaines, Gaines, 11992). Under some some circumstances, circumstances, however, however, 'adaptive 'adaptive plasticity' plasticity' can can evolve evolve as as aa response response to to spatial plays aa central understanding of evo spatial variation. variation. Plasticity Plasticity plays central role role in in our our understanding of the the evolution of those involved involved in adaptive response lution of polyphenisms, polyphenisms, particularly particularly those in an an adaptive response to to temporal and par temporal and spatial spatial heterogeneity heterogeneity of of the the abiotic abiotic environment. environment. The The standard standard parE, CCaxe, titioning titioning of of phenotypic phenotypic variation variation is is expanded expanded to to include include aa term term for for G G X x E, axe, Zl -- a l + e l q- Claxe.
((11.2) 1 1 .2 )
This This approach approach iiss appropriate appropriate for for the the abiotic abiotic environment environment but but not not for for biotic biotic environments. environments. This This formulation formulation permits permits aa genetic genetic response response to to abiotic abiotic environ environmental mental variation, variation, but but does does not not allow allow the the reciprocal reciprocal because because the the environment, environment, although variable, is although variable, is not not evolutionarily evolutionarily dynamic. dynamic. Only Only the the genotype genotype changes changes evolutionarily 1 1 .2 ) cannot evolutionarily if if fitness fitness varies varies with with context context in in this this way. way. Thus, Thus, Eq. Eq. ((11.2) cannot adequately adequately account account for for the the context context provided provided by by conspecifics conspecifics or or by by other other species species in ecological contexts in an an ecological ecological community community because because these these ecological contexts (unlike (unlike el) el) also also have ecologically important have aa genetic genetic component. component. Thus, Thus, many many ecologically important contexts contexts differ differ from they are are evolutionarily and can can coevolve with Zl' from G G X x E E because because they evolutionarily dynamic dynamic and coevolve with zl. Interactions Interactions between between males males and and females, females, between between parents parents and and offspring, offspring, between between age age cohorts, cohorts, between between competitors, competitors, between between predators predators and and prey, prey, or or between between hosts hosts and and pathogens pathogens all all contribute contribute to to individual individual phenotypic phenotypic and and fitness fitness variation animals. Notably, variation in in plants plants and and animals. Notably, these these contexts contexts contain contain genes genes and, and, when me to population, genes when they they vary vary from from de deme to deme deme across across aa meta metapopulation, genes and and con context 994; Wolf text can can coevolve coevolve (Thompson, (Thompson, 11994; Wolf et et aI., al., 2003; 2003; Wade, Wade, 2003) 2003)..
111.3 1 .3
EVOLVING EVOLVING CONTEXTS CONTEXTS In In order order to to investigate investigate evolving evolving contexts, contexts, additional additional terms terms must must be be included included in in the the standard standard formulation formulation given given earlier. earlier. Imagine Imagine aa second second trait, trait, Zz2, which is is 2 , which sensitive sensitive to to the the local local mean mean value, value, Zli, Zli, of of the the first first trait. trait. The The first first trait, trait, Zb Zl, is is the the
264 264
M ICHAEL I.). WADE MICHAEL WADE
"context" "context" and and its its local local mean mean value value influences influences individual individual values values of of phenotype, phenotype, Zz2, in that that deme deme as as follows: follows: 2 , in Z2i = a2i + xP12Zli + e2i.
((11.3) 1 1 .3)
The The second second term, term, 'l'1 xIt12Z1i is the the IGE, IGE, where where the the coefficient, coefficient, '1'1 xlt122 ,, represents represents 2Z1i,, is the magnitude of linear effect the scale scale or or magnitude of the the linear effect of of Z1i Z l i relative relative to to the the direct direct effects effects whose have been set to both lin whose coefficients coefficients have been set to 11.. The The effects effects of of Z1i Z l i ccan a n include include both linear nonlinear components ear and and nonlinear components (d. (cf. Agrawal Agrawal et et ai., al., 2001). 2001). Whenever Whenever it it con contains possible to tains nonlinear nonlinear terms, terms, then then it it is is not not generally generally possible to scale scale the the global global means, means, Z1 Z1 and and Z Z2, to zero zero without without also also aa temporal temporal and and spatial spatial rescaling rescaling of of 2 , to the all other 1 1.3) each the coefficients coefficients of of all other terms terms in in Eq. Eq. ((11.3) each generation. generation. Thus, Thus, the the presence an evolvable context means additive presence of of an evolvable context means that that the the coefficients coefficients of of additive .0), must be reevaluated genetic assumed to genetic effects, effects, such such as as aa2i (here assumed to be be 11.0), must be reevaluated 2i (here from remaining constant from generation generation to to generation generation instead instead of of remaining constant as as is is done done in in standard standard theory. theory. It It is is the the nonzero nonzero values values of of '1'1 XItl2 that permit permit the the environment environment 2 that to evolve (Fig. 1 .2). to co coevolve (Fig. 111.2). Substituting 1 1. 1 )) into 1 1 .3 ), we Substituting Eq. Eq. ((11.1 into Eq. Eq. ((11.3), we find find that that 2~2i =
a2i + xP12(A1i + Eli) + e2i.
((11.4) 1 1.4)
This This is is the the effect effect of of IGEs: IGEs: an an individual's individual's phenotypic phenotypic value, value, ZZ2, depends on on the the 2, depends genetic and value of 1 .2). genetic and phenotypic phenotypic value of its its local, local, social social ccontext o n t e x t 'l'1 xI~12Zli (Fig. 111.2). 2Z1i (Fig. Furthermore, Furthermore, with with genetic genetic subdivision subdivision (FST (FsT > > 0), 0), the the average average social social context context itself will Thus, there itself will vary vary from from deme deme to to deme deme across across aa metapopulation. metapopulation. Thus, there can can be be heritable heritable variation variation in in the the additive additive genetic genetic values values of of social social context context from from deme to is deme to deme, deme, affecting affecting the the evolution evolution of of traits traits sensitive sensitive to to context. context. If If FST FST is Wright's population genetic (aj) isis the Wright's measure measure of of meta metapopulation genetic subdivision subdivision and and V V(ai) the vari variance ance in in aj aj in in aa population population without without subdivision subdivision (i.e., (i.e., FST FST-= 0), 0), then then in in the the absence of absence of epistasis, epistasis, the the average average variance variance within within demes demes is is ((11 - FST) FsT)V(aj) and V(aj) and the the variance variance among among demes demes is is 2FsTV(aj) 2FsTV(ai) (Hartl (HaM and and Clark, Clark, 1997). 1997). ia; X• ai When When there there is is additive-by-additive additive-by-additive epistasis, epistasis, V V(ai aj),l , the the average average addi additive tive genetic genetic variance variance within within demes demes can can be be greater greater than than (( 11 -- FST) FsT)V(ai) V(aj) (Goodnight, 990, 1995, (Goodnight, 1987, 1987, 11990, 1995, 1999) 1999) by by an an amount amount approximately approximately equal equal to to 993). With 4FsT(I1 -- FST) FsT)V(ai aj) (Whitlock (Whitlock et et ai., al., 11993). With nonlinear nonlinear interactions interactions 4FsT( Via; xx aj) among contributing to among genes, genes, the the identity identity of of the the genes genes contributing to the the local local additive additive vari variance me to ance will will vary vary from from de deme to deme deme as as will will the the coefficients coefficients of of aa1l and and aa22 in in Eqs. Eqs. ((11.1)-(11.3) 1 1. 1 )-( 1 1.3) ((Goodnight, Goodnight, 2000). 2000). If If the the IGE IGE term, term, 'l'1 xt~1221i is itself itself partitioned partitioned 2Z1i,, is into linear and into linear and nonlinear nonlinear components, components, then then not not only only will will the the coefficients coefficients change deme to change from from deme to deme deme and and from from generation generation to to generation, generation, but but the the strictly strictly additively , will now have an added epistatic additively determined determined phenotypes, phenotypes, Z1 zl and and Zz2, will now have an added epistatic 2 (i.e., (i.e., a1 al x x aa2) component. 2 ) component.
11 11. .4 4
SELECTION S E L E C T I O N IN IN M M EETAPOPULATIONS TAPOPULATIONS To populations is To see see how how selection selection in in meta metapopulations is different different with with IGEs, IGEs, consider consider selection selection only only within within demes. demes. We We need need to to first first define define the the relative relative fitness fitness of of an an individual individual in in the the ith ith deme, deme,
SELECTION IN IN M METAPOPULATIONS 111. 1 . SELECTION ETAPOPULATIONS
W i -- k + ~liZli q- ~2iZ2i -F B 1 Z l i + B 2 Z 2 i
265 265
((11.5a) 1 1 .5a)
and then then substitute substitute the the genetic genetic values values contributing contributing to to the the phenotypes, phenotypes, and W i -- k +
f31iali + ~32i(a2i + xI~12Ali) + BIAli + B2(A2i + xI~12Ali). (11.5b)
[Note that that in in Eq. Eq. ((11.5 b), only only the the additive additive genetic genetic components components of of fitness fitness are are 1 1 .5 b), [Note expressed expressed and and the the abiotic abiotic environment environment and and epistasis epistasis are are left left out.] out.] Here, Here, kk is is aa constant constant common common to to all all individuals individuals and and it it is is assumed assumed that that the the abiotic abiotic envir enviri, rep 1i and onment onment remains remains constant constant across across generations. generations. The The coefficients, coefficients, 13f31i and 132 f32i, represent the the strength strength of of individual individual selection selection within within the the ith ith deme deme on on the the two two resent phenotypes, phenotypes, Z1i Zli and and Zz2i, and the the coefficients, coefficients, B1 B1 and and B B2, represent the the strength strength 2 , represent 2i, and of interdemic selection acting acting on on the the means, means, Z1 and Z Z2, of each each phenotype. phenotype. Z1 and of interdemic selection 2, of B1 and Setting Setting both both B1 and B B22 equal equal to to zero zero eliminates eliminates interdemic interdemic selection. selection. Note Note that, that, considering considering selection selection only only within within the the ith ith deme, deme, the the term, term, 132 ~32ixi~12A1i, i-qr 1 2A li, would individuals and would not not vary vary among among individuals and thus thus would would not not affect affect the the variance variance in in relative fitness within the deme. The population mean values, Z1 The meta metapopulation mean values, Z1 and and Z Z2, change in in proportion proportion to to the the 2 , change covariance covariance between between the the additive additive genetic genetic components components of of Zzl1 and and Zz22 and and relative relative equals I3j. fitness. fitness. Let Let selection selection within within all all demes demes be be the the same same so so that that I3ji ~3ji equals f3i. Considering 1, taking Considering only only the the genetic genetic effects effects for for trait trait 1, taking the the covariance covariance of of Z1 zl and and Wl, we we have have Wh
LlZ1 ( 13 1 V[a l ] ++ 132 C[a1,a2] ) ( 1 -- FST) AZ1 = --([3iV[a1] ~32C[al,a2])(1 FST) ++ 2FsTV(a 1 2 ++ BB2xI~12)+ 1 1 .6a) tl(B1 ++ 132-qr 2-qru) + 2FsTC(aha 2FsTV(al)(B1 ~2xIv12 2FsTC(al,a2)B2, 2 )B 2, ((11.6a) where genetic covariance between Zl Z1 and within aa nonsubnonsub where C(aha C(al,a2) the genetic covariance between and Zz22 within 2 ) isis the B1 and divided population. In the absence absence of of interdemic ( i.e., B1 divided population. In the interdemic selection selection (i.e., and B B22 equal to zero), this reduces to to AZ1 = ([3IV[a1] + [32C[a1,a2])(1 - FST) + 2FsTV(al)([32~12).
( 1 1 .6b) (11.6b)
The first first two of Eq. are the standard expressions The two terms terms of Eq. ((11.6 1 1 .6 b) b ) are the standard expressions for for the the change change Z1, in subdivided metapopulation metapopulation that that result result from in the mean, Z1, in the mean, in aa genetically genetically subdivided from the average average effects effects of of direct and ((1) 1 ) the direct selection selection within-demes within-demes on on phenotype, phenotype, Zl, Zh and the average effects of of indirect selection within-demes within-demes on on phenotype, Z2 ' (2) average effects indirect selection phenotype, z2. (2) the However, the the third third component component is is new new and and results results from from genetic genetic variation variation in in However, 0). It It not not only only affects the social context context among among demes demes (i.e., (i.e., 2FsTV[al] social affects the 2FsTV[a d >> 0). Z2 but but also also affects affects selection selection on on Zl. As aa result, result, social social context context expression of of z2 expression Z1 ' As affects its own own evolution evolution in in aa metapopulation metapopulation through through its its indirect indirect effect on the the affects its effect on context sensitive sensitive phenotype, phenotype, z2. Z2 ' This This effect effect of of context context is is different from standstand context different from ard indirect indirect selection selection because because it it occurs occurs even even when when the the two two traits traits are are not not genetgenet ard = 0 ) . In standard theory, absent a genetic ically correlated correlated (i.e, ( i.e, with with C[al,a2] aha ically = 0). In standard theory, absent a genetic ] C[ 2 correlation, the the evolution evolution of of one one phenotype phenotype would would be be completely completely independent independent correlation, of and and unconstrained unconstrained by by the the evolution evolution of of the the other. other. Thus, Thus, in in aa metapopulation metapopulation of > 00 and and xI~12 nonzero) and and without correlation with IGEs IGEs (i.e., (i.e., FST with FST > without aa genetic genetic correlation -qr 1 2 isis nonzero) between Zl Z1 and and z2, Z2> individual individual selection selection on on Zl Zl will will always always be be affected affected by by selecselec between even in in the the absence absence of of interdemic interdemic selection. selection. This This is is different different from from tion Z2, even tion on on z2, evolution in in aa large large randomly randomly mating mating and and mixing mixing population population where where aa genetic genetic evolution
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MICHAEL MICHAEL J.I. WADE WADE
correlation correlation is is necessary necessary for for the the evolution evolution of of one one phenotype phenotype to to constrain constrain another. Selection Selection on on Zz2, which is is sensitive sensitive to to the the social social context context provided provided by by Zzl, 2, which b isis more complicated, complicated, even even omitting omitting terms terms in in ('1'1 (~12) more 2 ) 22,, AZ2"= ~lC[al,a2])(1 FST) + + 2FsT{( 2FsT{(f32 132 ++ BB2)V[a2] 2 ) V[a2l [a1,a2l ) ( 1 -- FST) 132V[a2l ++ 131C AZ 2' = ((]32V[a2] ++ Wd131 V[a !l 1 ++ 2FsTC(a xlY'12(~1+ + B1) B1)V[al]} 2FsrC(al,a2){2]32xi'12 tl, W12B2 ++ BB1}, ba2 ) {2 132W1 2 ++ 22xI-t12B2
((11.7a) 1 1.7a)
and, in the absence absence of interdemic selection, and,
AZ C[a ba2l )(1 -- FST) 132 V[a2l 132 V[a2l ++ 131 z~Z2 131C[al,a2])(1 Fsr) + + 2FsT( 2Fsr(f32V[a2] 2 == ((132V[a2] + ba2 ) 132W 1 2 . 213 1 V[ad) + 4FsTC(a + '1'1 ~121BIV[a1])+ 4FsrC(a1,a2)(32~12.
((11.7b) 1 1 .7b)
Clearly, populations, aa variation Clearly, in in meta metapopulations, variation in in genetic genetic context context creates creates new new sources sources of of covariance covariance between between genes genes and and fitness fitness that that affect affect the the response response to to selec selection. l 1 .6b) and 1 1 . 7b) have tion. In In the the absence absence of of interdemic interdemic selection, selection, both both Eqs. Eqs. ((11.6b) and ((11.7b) have 1 2 ' Thus, terms terms with with the the coefficient coefficient 2FsTW 2FsT~12. Thus, in in aa metapopulation metapopulation with with IGEs IGEs will be (i.e., (i.e., FST FST > > 00 and and '~12 nonzero), individual individual selection selection will be always always be be differ differ1'1 2 isis nonzero), ent from what it is randomly mating mating and mixing population, population, even in ent from what it is in in aa large large randomly and mixing even in the the absence absence of of interdemic interdemic selection. selection. This This is is true true as as long long as as Z1 Z1 varies varies among among demes demes (i.e., (i.e., 2FsTV[a1 2FsTV[al]l > > 0), 0), whether whether or or not not there there is is aa genetic genetic correlation correlation between between Z1, and social context, social context, zl, and the the sensitive sensitive trait, trait, Zz2, either within within or or among among demes. demes. 2, either With interdemic selection, Eqs. ((11.6a) 1 1 .6a) and 1 1 .7a), the response to With interdemic selection, as as in in Eqs. and ((11.7a), the response to selection becomes becomes substantially substantially more more complex. Note especially that all all of of the the selection complex. Note especially that interdemic coefficient: there interdemic selection selection terms terms have have FST FST as as aa coefficient: there can can be be no no interdemic interdemic selection in the the absence absence of genetic variation. variation. Note Note also also that that interinter selection in of among-deme among-deme genetic results in in aa response response to to interdemic interdemic selection selection in in demic on either demic selection selection on either trait trait results the trait. This This means means that that whether whether aa trait trait is social context context trait trait or or is is the other other trait. is aa social sensitive to context trait, trait, interdemic interdemic selection one affects the evoevo sensitive to aa social social context selection on on one affects the lution of and does does so independent of lution of the the other other and so independent of genetic genetic correlations. correlations.
111.5 1 .5
THE PERVASIVENESS OF INDIRECT THE PERVASIVENESS OF OF EFFECTS EFFECTS OF INDIRECT GENETIC GENETIC EFFECTS IN METAPOPULATIONS EFFECTS IN METAPOPULATIONS Local is an an ubiquitous ubiquitous example example of an IGE Z1 Local density density dependence dependence is of an IGE (like (like Zl shown earlier) earlier) that that affects affects evolution evolution in in metapopulations. metapopulations. Many Many morphological morphological shown and and behavioral behavioral traits traits exhibit exhibit aa response response to to local local density. density. The The emigration emigration rate rate in Tribolium castaneum, castaneum, isis aa good good example example of of aa behavior behavior in the the flour flour beetle, beetle, Tribolium Z2 given given earlier, earlier, is is influenced influenced both both by by the the genotype genotype of of whose expression, expression, like like z2 whose the local population the individual individual and and by by the the local population density density (Craig, ( Craig, 1982). 1 982). The The tendency tendency to to emigrate emigrate of of an an individual individual beetle beetle is is determined determined both both by by its its own own genotype genotype and and by the the local local density density of of conspecifics conspecifics that that itit experiences experiences during during development development by (Fig. (Fig. 11.4). 1 1 .4). An An individual individual reared reared in in aa high-density high-density environment environment is is more more prone prone to low-density envirto emigrate emigrate than than aa genetically genetically similar similar individual individual reared reared in in aa low-density envir onment, even even when when both both are are tested tested for for emigration emigration tendency tendency at at the the same same interinter onment, mediate density. density. That That is, is, the the past past experience experience of of local local density density as as aa larva larva mediate influences influences the the emigratory emigratory tendency tendency of of the the adult. adult. So, So, like like the the theory theory presented presented earlier, (Z2 ) isis sensitive sensitive to to aa genetically genetically earlier, an an individual's individual's emigration emigration tendency tendency (z2)
2267 61
1 1 . SELECTION SELECTION IN IN METAPOPULATIONS METAPOPULATIONS 11. " emigrators' "emigrators"
" emigrators' "emigrators"
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1 1 .4 The emigratory of aa single single flour flour beetle (Tribolium castaneum) castaneum) is influ Fig. 11.4 emigratory tendency tendency of influenced by its own by the social context context represented represented by by local local density density (right). own genotype genotype (left) and by the social (right). Because is an an IGE IGE and Because local local density density has has aa genetic genetic component, component, itit is and aa potentially potentially coevolving coevolving envir environment. for further discussion.) onment. (See (See text text for further discussion.)
variable context, local In this is no no evidence evidence variable social social context, local density density (Zl) (Z1).' In this species, species, there there is tendency and and fecundity. for a genetic correlation correlation between between emigration emigration tendency fecundity. for 1 982) imposed imposed individual individual and interdemic selection When Craig Craig ((1982) When and interdemic selection for for increased (31 i= increased and and decreased decreased emigration emigration rate rate (B1 (B1 and and ~1 * 00 but but B B22 and and (3 [32 - 0), 0), 2= the the local local density density changed changed as as aa correlated correlated response response to to selection selection on on the the emigra emigratory tory rate. rate. The The regression regression coefficient coefficient of of emigration emigration tendency tendency on on reproductive reproductive capacity both expressed capacity ((both expressed as as percent percent deviation deviation of of the the selected selected treatment treatment means means from mean) was from unselected unselected control control mean) was 0.94 0.94 (p (p < < 0.015) 0.015) and and it it explained explained 86% 86% of of the the variation variation in in the the emigration emigration rate rate among among treatments. treatments. Thus, Thus, the the change change in in local local density, density, the the IGE, IGE, accounted accounted for for aa proportionally proportionally greater greater amount amount of of the the individ response to to selection selection in in an an individual's individual's tendency tendency to to emigrate emigrate than than the the individual's genotype! This ual's own own genotype! This demonstrates demonstrates that that the the coefficient coefficient of of context, context, '1'1 xIq2, is 2, is large large relative relative to to the the coefficient coefficient of of the the direct direct additive additive term. term. Genetic Genetic variation variation among among demes demes in in local local density density exists exists in in flour flour beetles beetles and and rapidly rapidly becomes becomes par partitioned populations (Wade titioned among among demes demes in in laboratory laboratory meta metapopulations (Wade and and McCauley, McCauley, 1980; 1980; Wade, Wade, 1982; 1982; Wade Wade and and Griesemer, Griesemer, 1998). 1998). Indeed, Indeed, the the biology biology of of flour flour beetle beetle metapopulations metapopulations is is even even more more complex complex because because different different genotypes genotypes are are not 980; Wade not only only differentially differentially sensitive sensitive to to density density (Wade (Wade and and McCauley, McCauley, 11980; Wade and and Griesemer, Griesemer, 1998; 1998; Wade, Wade, 2000), 2000), but but also also differentially differentially contribute contribute to to den den968; Wade, sity sity effects effects (Lloyd, (Lloyd, 11968; Wade, 2000). 2000). Plant Plant leaf leaf area area in in the the cress, cress, Arabidopsis Arabidopsis thaliana thaliana (Goodnight, (Goodnight, 1985), 1985), is is another another morphological morphological trait trait whose whose expression expression is is sensitive sensitive to to social social context context and and is is representative representative of of aa myriad myriad of of ubiquitous ubiquitous plant-plant plant-plant competitive competitive inter interactions. actions. In In A. A. thaliana, thaliana, an an individual's individual's genotype genotype influences influences its its leaf leaf area area and and so so do do the the leaf leaf areas areas of of neighboring neighboring plants, plants, whether whether genetically genetically related related to to the the focal focal individual individual or or not. not. Here, Here, the the same same trait, trait, leaf leaf area, area, provides provides both both the the direct direct effect effect and and the the social social context. context. In In the the model model given given earlier, earlier, Zz22 would would be be affected affected by by its its
268 268
MICHAEL J. WADE MICHAEL J. WADE
own own mean, mean, Z Z2i. Importantly, as as aa result result of of competition competition for for sunlight, sunlight, the the direct direct 2i' Importantly, effect effect of of an an individual's individual's own own genotype genotype is is in in conflict conflict with with the the IGE IGE of of its its neigh neighbors, bors, i.e., i.e., the the coefficient coefficient of of the the direct direct additive additive term term is is positive positive while while '1J11 ~122 is is nega negative. result, artificial individual selection increase leaf leaf area tive. As As aa result, artificial individual selection to to increase area produces produces an an average decline in leaf area area (Goodnight, 985) as average decline in leaf (Goodnight, 11985) as it it does does in in many many other other domes domesticated ticated plant plant species. species. Without Without the the social social effect effect on on leaf leaf area, area, this this would would not not occur. occur. Plants Plants are are not not typically typically regarded regarded as as social social organisms, organisms, like like the the hymenoptera, leaf area typically considered hymenoptera, and and leaf area is is not not typically considered aa social social context context trait trait or or an IGE. an IGE. The The results results of of Goodnight's Goodnight's experiments experiments support support the the theoretical theoretical findings findings of of 1967, 1977, Griffing Griffing ((1967, 1977, 1981, 1981, 1989). 1989). He He showed showed that that whenever whenever direct direct and and indi indirect rect "associative" "associative" effects effects are are of of opposite opposite sign, sign, as as they they tend tend to to be be in in competi competitive interactions, individual individual selection alone cannot tive interactions, selection alone cannot maximize maximize the the response response to to selection. His findings findings are are particularly selection. His particularly important important to to plant plant breeders breeders trying trying to to maximize maximize yield. yield. A A large large plant plant has has aa manifestly manifestly greater greater yield yield than than aa small small one, one, but but aa stand stand of of N N large large plants plants does does not not produce produce N N times times the the yield yield of of one one large large plant, total yield plant, but but rather rather aa much much reduced reduced total yield due due to to interplant interplant competition, competition, which plant size. which unfortunately unfortunately intensifies intensifies with with plant size. Griffing Griffing (1967, (1967, 1977, 1977, 1981, 1981, 1989) 1989) illustrated illustrated theoretically theoretically how how and and why why interdemic interdemic selection selection could could achieve achieve maximal yields yields whereas whereas individual individual selection selection alone could not. not. Goodnight Goodnight maximal alone could (1985) (1985) demonstrated demonstrated empirically empirically that that artificial artificial interdemic interdemic selection selection alone alone resulted resulted in in aa greater greater response response to to selection selection for for increased increased leaf leaf area area than than artificial artificial individual selection alone Indeed, Goodnight individual selection alone of of comparable comparable intensity. intensity. Indeed, Goodnight showed, showed, as Griffing Griffing (1967, (1967, 1977, 1977, 1981, 1981, 1989) 1989) predicted, predicted, that that individual individual and and interdemic interdemic as selection selection together together would would interfere interfere with with one one another, another, resulting resulting in in aa total total response response to to combined combined selection selection that that was was less less than than the the sum sum of of the the expected expected responses to alone (see also the responses to each each level level of of selection selection acting acting alone (see also the discussion discussion of of this this interference populations with IGEs interference in in Agrawal Agrawal et et aI., al., 2001). 2001). Evolution Evolution in in meta metapopulations with IGEs is is much much more more complex, complex, with with or or without without interdemic interdemic selection, selection, than than the the uncrit uncrit966) would ical use razor (Williams, (Williams, 11966) ical use of of Occam's Occam's razor would suggest. suggest. Applying Applying Occam's Occam's razor to leaf area, one would would get mech razor to the the adaptive adaptive increase increase in in plant plant leaf area, one get both both the the mechanism (interdemic direction of individual selec anism (interdemic selection) selection) and and the the direction of response response to to individual selection tion wrong wrong without without considering considering IGEs, IGEs, which which are are not not part part of of standard standard evolutionary evolutionary genetic theory. Muir 1 996) capitalized Muir ((1996) capitalized on on similar similar findings findings and and used used artificial artificial intergroup intergroup selection increase egg laying in domestic chickens, chickens, Gallus selection on on sire-families sire-families to to increase egg laying in domestic Gallus long-term individual lines is limited by by gallus. gallus. Egg Egg production production in in long-term individual selection selection lines is limited the hens in group cages, where competitive the practice practice of of maintaining maintaining hens in group cages, where competitive inter interactions debeaking is actions have have such such severe severe affects affects on on mortality mortality and and condition condition that that debeaking is aa common Debeaking is common practice. practice. ((Debeaking is the the removal removal of of most most of of the the beak beak to to mini minimize mize feather feather plucking, plucking, injuries, injuries, and and cannibalism cannibalism among among cage cage mates. mates. It It may may be be done done more more than than once once with with laying laying hens. hens. It It limits limits but but does does not not prevent prevent dele deleterious terious interactions interactions among among birds. birds.)) As As with with plants, plants, aa large large hen hen might might yield yield more more or or larger larger eggs eggs than than aa small small hen, hen, but but aa group group of of N N large large hens hens produces produces less than than N N times times the the yield yield of of aa single single hen hen housed housed alone. alone. That That is, is, social social con conless text text as as well well as as individual individual genotypes genotypes have have profound profound effects effects on on the the number number and and size hens. The behavioral concept size of of eggs eggs yielded yielded by by aa group group of of hens. The behavioral concept of of aa domi dominance nance hierarchy hierarchy or or "pecking "pecking order" order" was was also also developed developed in in studies studies of of this this species. species. However, However, the the crude crude attempts attempts to to take take the the peck peck out out of of the the pecking pecking
111. 1 . SELECTION IN METAPOPULATIONS SELECTION IN METAPOPULATIONS
269 269
order did not not eliminate the order by by debeaking debeaking individual individual birds birds (removing (removing trait, trait, Zl) Zl) did eliminate the negative effect of IGE on egg egg production production as effectively as focusing artificial artificial negative effect of IGE on as effectively as focusing interdemic selection selection on on the the IGE. interdemic IGE. The The response response to to artificial artificial intergroup intergroup selection selection for for increased increased egg egg laying laying in in the the Muir Muir experiments experiments was was spectacular, spectacular, especially especially considering considering that that his his founding founding stock elite breed, breed, derived stock was was an an elite derived by by the the application application of of the the most most effective effective and and efficient individual selection protocols protocols for for over over 50 50 generations. generations. In In only only six six gen genefficient individual selection erations selection, mortality erations of of interdemic interdemic selection, mortality in in group group cages cages declined declined sevenfold, sevenfold, from equal to housed from 68.8 68.8 to to 8.8%, 8.8%, which which is is equal to the the background background mortality mortality of of hens hens housed individually. essentially eliminated individually. The The negative negative effects effects of of social social context context were were essentially eliminated in in six six generations generations of of interdemic interdemic selection. selection. Notably, Notably, this this is is aa result result that that could could not not be be achieved achieved by by debeaking debeaking individual individual birds. birds. Mean Mean egg egg number number per per hen hen 91 to because of increased increased more more than than twofold, twofold, from from 91 to 237 237 eggs, eggs, in in part part because of aa 16% 16% increase increase iinn eggs eggs per per hen hen per per day, day, but but also also iinn part part because because ooff aa doubling doubling ooff hen hen longevity. longevity. In In an an individually individually selected selected control control with with single-hen single-hen cages, cages, the the response response to to selection selection was was much much slower slower and and the the setup setup impossible impossible to to implement implement for for large-scale large-scale egg egg production, production, which which requires requires group group cages. cages. The The competitive competitive interactions selection to point that interactions diminished diminished in in response response to to intergroup intergroup selection to the the point that birds birds with with or or without without beaks beaks had had equivalent equivalent survival survival and and debeaking debeaking was was unneces unnecessary to to obtain the increased productivity. sary obtain the increased productivity. It must be be emphasized It must emphasized that that competitive competitive interactions, interactions, like like those those associated associated with leaf area with leaf area in in A. A. thaliana, thaliana, with with density density in in T. T. castaneum, castaneum, or or with with fighting fighting in in G. G. gallus, gallus, are are aa common common form form of of IGE. IGE. Indeed, Indeed, intraspecific intraspecific competition competition was was the the essential essential concept concept from from Malthus Malthus that that Darwin Darwin realized realized would would ensure ensure aa strug struggle words, "" .. ... . gle for for existence, existence, making making natural natural selection selection inevitable. inevitable. In In Darwin's Darwin's words, it it at at once once struck struck me me that that under under these these circumstances circumstances favorable favorable variations variations would would tend Darwin, 1876, tend to to be be preserved preserved and and unfavorable unfavorable ones ones to to be be destroyed" destroyed" ((Darwin, 1876, pp. 19-121). That pp. 1119-121). That is, is, intraspecific intraspecific competition competition is is an an integral integral component component of of the the conceptual conceptual logic logic of of Darwinian Darwinian evolution evolution by by natural natural selection selection and and one one of of the the most most ubiquitous ubiquitous of of IGEs. IGEs. The The effects effects of of IGEs, IGEs, however, however, are are even even more more pervasive pervasive in in metapopulations metapopulations than than the the three three examples examples would would indicate. indicate. Like Like the the role role of of the the beak beak in in competi competition tion in in G. G. gallus gallus or or leaf leaf area area in in A. A. thaliana, thaliana, the the expression expression of of most most social social traits traits involves beetle, involves one one or or more more morphological morphological traits. traits. For For example, example, in in the the flour flour beetle, T. T. confusum, confusum, egg egg cannibalism cannibalism involves involves the the interaction interaction of of both both larval larval mandible mandible 980). Large size size and and egg egg size size (Teleky, (Teleky, 11980). Large eggs eggs are are safe safe from from predation predation by by small small larvae simply simply by by virtue virtue of of their their size size relative relative to to the the mandible mandible size size of of early early instar instar larvae larvae. Thus, mandible size larvae. Thus, "egg "egg cannibalism" cannibalism" is is an an IGE IGE that that involves involves the the mandible size of of the the prospective prospective cannibal, cannibal, its its genetic genetic propensity propensity toward toward cannibalism, cannibalism, and and the the mean local environment, also mean and and distribution distribution of of egg egg sizes sizes in in the the local environment, which which are are also influenced influenced by by the the genes genes in in the the laying laying mothers mothers as as well well as as genes genes determining determining the the ejaculate ejaculate quality quality of of their their mates. mates. The The totality totality of of the the interaction interaction is is highly highly non nonlinear; populations, conspicuous linear; in in laboratory laboratory meta metapopulations, conspicuous among-deme among-deme variation variation in in the the level level of of egg egg cannibalism cannibalism arises arises even even in in the the absence absence of of interdemic interdemic selection selection (Wade, 978, 1979, (Wade, 11978, 1979, 1980). 1980). Neither Neither egg egg size size nor nor mandible mandible size size would would be be aa likely likely candidate candidate for for an an IGE. IGE. Nevertheless, Nevertheless, the the social social trait trait of of cannibalism cannibalism involves involves both both of of these these morpho morphological logical traits traits and and associates associates them them with with fitness fitness via via egg egg viability viability and and via via the the nutritional nutritional fitness fitness advantages advantages that that accrue accrue to to the the cannibal. cannibal. In In this this way, way, many many
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M ICHAEL J.I. WADE MICHAEL WADE
morphological morphological traits traits become become subject subject to to the the evolutionary evolutionary consequences consequences of of selec selection metapopulations. Whether tion of of IGEs IGEs in in metapopulations. Whether aa trait trait is is aa social social context context trait trait or or is is sensitive sensitive to to aa social social context context trait, trait, interdemic interdemic selection selection on on one one trait trait affects affects the the evolution of independent of evolution of the the other other and and does does so so independent of the the existence existence of of genetic genetic cor correlations. population with is relations. Furthermore, Furthermore, in in aa meta metapopulation with IGEs IGEs (i.e., (i.e., PST FST > > 00 and and 61 0122 is nonzero) nonzero),, individual individual selection selection will will be be always always be be different different from from what what it it is is in in aa large randomly and mixing mixing population, population, even even in in the the absence absence of of interlarge randomly mating mating and inter demic demic selection. selection.
11 11 ..6 6
EFFECTS OF INDIRECT IN METACOMMUNITIES EFFECTS OF I N D I R E C T GENETIC GENETIC EFFECTS EFFECTS IN METACOMMUNITIES Whenever Whenever the the environment environment contains contains genes, genes, context context can can evolve evolve along along with with the response to context. This is true the evolutionary evolutionary response to context. This is true whether whether the the environmental environmental 998), as context context is is other other conspecifics conspecifics (Wolf (Wolf et et aI., al., 11998), as in in the the examples examples given given ear earlier, aI., 2002). lier, or or other other species, species, as as in in ecological ecological communities communities (Wolf (Wolf et et al., 2002). Extending Extending the the definition definition of of IGEs IGEs to to include include other other species species is is aa warranted warranted exten extension earlier definitions sion of of earlier definitions (Wolf (Wolf et et aI., al., 1998). 1998). In In the the words words of of Goodnight Goodnight ((1991, 1 991, p. p. 343), interactions are 343), "Inter-species "Inter-species interactions are different different from from other other genotype genotypeenvironment only can can aa second second species environment interactions interactions because because not not only species be be aa signifi significant component of also an cant component of the the environment, environment, but but it it is is also an evolving evolving entity entity that that can can change deterministic forces change as as aa result result of of deterministic forces such such as as natural natural selection selection and and random random forces forces such such as as genetic genetic drift." drift." Keister 1984) modeled Keister et et ai. al. ((1984) modeled the the interaction interaction and and coevolution coevolution of of two two species species interacting population. They interacting randomly randomly in in aa large large panmictic panmictic population. They showed showed how how the the response response to to selection selection on on aa trait trait in in one one species species is is dependent dependent on on the the mean mean value value of of the the context context provided provided by by the the other other species species and and vice vice versa. versa. More More important important for evolution in populations, however, for evolution in meta metapopulations, however, was was the the finding finding that that random random genetic trans-specific, coevolving genetic drift drift affecting affecting two two trans-specific, coevolving characters characters depended depended on on the size. Thus, the species species with with the the smaller smaller effective effective population population size. Thus, if if two two coevolving coevolving species, species A species, A A and and B, B, interact interact and and species A is is strongly strongly affected affected by by genetic genetic subdi subdivision vision but but species species B B is is not, not, traits traits in in species species B B will will nevertheless nevertheless evolve evolve as as though though species highly genetically Thus, the species B B were were as as highly genetically subdivided subdivided as as species species A. A. Thus, the effects effects of IGEs discussed one species apply to to of IGEs discussed in in the the preceding preceding section section for for traits traits in in one species apply coevolving other species, if less coevolving traits traits in in other species, even even if less genetically genetically subdivided! subdivided! A A metacommunity metacommunity can can be be defined defined by by analogy analogy with with metapopulation metapopulation to to be be aa more less genetically more or or less genetically subdivided subdivided collection collection of of interacting interacting species species (see (see also also Chapter Chapter 6). 6). Because Because of of interspecific interspecific IGEs, IGEs, coevolution coevolution in in aa metacommunity metacommunity will result coevolution is will result in in hot hot spots, spots, where where reciprocal reciprocal coevolution is strong, strong, and and cold cold 1994) geographic mosaic spots, weak, according spots, where where it it is is weak, according to to Thompson's Thompson's ((1994) geographic mosaic hypothesis. kind of hypothesis. Empirical Empirical evidence evidence for for this this kind of variation variation in in the the outcome outcome of of coevolution coevolution has has been been forthcoming forthcoming from from the the recent recent studies studies of of natural natural meta metasnake predator, communities communities of of toxic toxic newts, newts, genus genus Taricha, Taricha, and and their their garter garter snake predator, Thamnophis Thamnophis sirtalis sirtalis (Geffeney (Geffeney et et aI., al., 2002; 2002; Brodie Brodie et et aI., al., 2002) 2002).. The The ecologi ecological cal factors factors that that affect affect the the local local abundance abundance of of one one species species are are experienced experienced by by the variation in the other other species species as as among-deme among-deme variation in context, context, which which affects affects the the response across the response to to evolution evolution of of both both species species across the metacommunity. metacommunity. Just as the the mean can change in response Just as mean of of aa trait trait without without heritable heritable variance variance can change in response to (Moore et 997), in community, to selection selection on on aa contextual contextual trait trait (Moore et aI., al., 11997), in aa meta metacommunity,
271 211
111. 1 . SELECTION SELECTION IN METAPOPULATIONS METAPOPULATIONS
the heritability of a trait in species A might depend on the value of context pro provided by species B. B. This has been referred to as "community "community heritability" heritability" and defined as the among-community fraction fraction of the genetic variance affecting coevolving traits (Goodnight, 9 9 1 ; Goodnight 996). Goodnight (Goodnight, 11991; Goodnight and Craig, 11996). Goodnight ((1991) 19 9 1 ) created 10 10 replicate small communities using laboratory laboratory populations of two castaneum and T. T. confusum, confusum, and allowed these two species of flour flour beetles, T. T. castaneum communities to codifferentiate codifferentiate by random random genetic drift for 1166 generations communities (Fig. 111.5). 1 .5). After that (Fig. that period, he factorially combined the members of each species from from each community community to create 100 100 new two-species communities communities and replicated (Fig. 111.6). 1 .6). In 1 .6, the replicated each each three three times times (Fig. In Fig. Fig. 111.6, the 1100 shaded shaded diagonal diagonal squares 1.5. This squares represent represent the the codrifting codrifting 1100 communities communities of of Fig. Fig. 111.5. This design design is is analogous analogous to to the the standard standard diallele diallele design design used used to to detect detect epistasis epistasis as as aa signifi significant inbred strains cant interaction interaction between between crossed crossed inbred strains (see, (see, e.g., e.g., Wade Wade and and Griesemer, Griesemer, 1998). 1998). In In this this case, case, however, however, the the "epistasis" "epistasis" or or "interspecific "interspecific intermixing intermixing abil ability" is due to genetic interactions between species. For For each community, community, Goodnight Goodnight measured measured four traits, offspring numbers numbers of each species and adult emigratory emigratory rate rate of of each each species. species. In In addition addition to to main main effects effects of of community community of of origin origin for for each each species, species, he he found found significant significant interactions interactions (i.e., (i.e., significant significant inter interspecific intermixing castanuem were much more more intermixing ability). Some strains strains of T. T. castanuem productive productive with with particular particular strains strains of of T. T. confusum confusum and and vice vice versa. versa. This This means means that that community community effects on fitness and emigration emigration cannot cannot be decomposed decomposed into simple additive additive effects of the separate separate species. This has very important important impli implications for populations where for ecological models of meta metapopulations where species' growth growth rates and emigration rates are often assumed to be constant. Importantly, they are Differentiated Communities
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Fig. 111.5 1 .5 A schematic representation of the experimental design used 991 ) used by Goodnight (1 (1991) to create genetically divergent small communities, each each consisting of two species of flour two species 0 communities was established by beetles, Tribolium castaneum castaneum and T. T. confusum. Each Each of the 110 taking groups of 1166 adult beetles beetles from laboratory stock populations of each each species. species. These communities were held at constant size, 6 of each size, 32 beetles beetles (1 (16 each species), species), and were allowed to differentiate by random genetic drift for 116 6 generations. (See text and Fig. 1 .6 for further generations. (See Fig. 111.6 discussion.)
212 272
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7 8 9 10
Fig. 1 .6 The 110 0 differentiated 1 .5 Fig. 111.6 differentiated communities generated generated by the protocol depicted in Fig. Fig. 111.5 were separated separated into into 1100 Tribolium Tribolium castaneum castaneum and 1100 T. T. confusum confusum lineages, lineages, one from from each each com community. combined factorially munity. These These lineages lineages were were combined factorially and replicated replicated (three (three replicate replicate populations populations per cell). 00 experimental cell). For For each each of these these 1100 experimental communities, communities, emigratory emigratory rates rates and population population growth Goodnight tested growth rates rates of both species specieswere were measured. measured. Using UsingANOVA, ANOVA,Goodnight tested for direct direct genetic effects effects of T. T. castaneum castaneum (row (row effect) effect) and/or T. T. confusum confusum (column (column effects) effects) as well well as "intermixing "intermixing ability" ability" (interaction (interaction between between rows rows and columns). columns). (See (See text for a discussion discussion of the findings.) findings.)
not 1 ) direct not constant constant in in two two different different ways: ways: ((1) direct genetic genetic effects, effects, where where the the genetic genetic strain species affects strain of of the the focal focal species affects its its population population growth growth and and emigration emigration rate rate and and (2) (2) the the indirect indirect genetic genetic effect, effect, where where the the genetic genetic strain strain of of the the competing competing species emigration rate species affects affects population population growth growth and and emigration rate of of the the focal focal species. species. These These data data are are the the clearest clearest evidence evidence that that random random genetic genetic drift drift acting acting in in aa rela relatively tively brief brief period period of of time time can can create create among-community among-community heritable heritable variation variation and and that that aa component component of of this this variation variation is is not not observable observable in in single single species species and and is heritable property interaction. This experiment is is aa heritable property of of the the interaction. This experiment is particularly particularly inter interesting developed during esting because because the the heritable heritable variation variation developed during aa known known period period of of community community isolation isolation by by random random drift. drift. The only estimate The only estimate of of community community heritability heritability per per se se comes comes from from studies studies of of competitive ability ability in in metacommunities metacommunities of of the the flour flour beetles, beetles, T. T. castaneum castaneum and and competitive T. Goodnight and and Craig 1 996). They T. confusum, confusum, conducted conducted by by Goodnight Craig ((1996). They established established single-species populations as single-species meta metapopulations as well well as as metacommunities, metacommunities, with with both both species species coexisting together. together. After After aa period period of of subdivision subdivision with with no no artificial artificial selection selection coexisting either bee either within within or or among among demes, demes, they they measured measured the the competitive competitive ability ability of of beetles tles from from each each type type of of subdivided subdivided population population using using aa method method similar similar to to that that of of the 1 948). Goodnight 1 996) the classic classic ecological ecological studies studies of of Park Park ((1948). Goodnight and and Craig Craig ((1996) found that that metacommunity metacommunity structure structure affected affected two two ecological ecological aspects aspects of of com comfound petitive ability: ability: ((1) the heritability heritability for for competitive competitive outcome outcome (i.e., (i.e., the the identity identity of of petitive 1 ) the winning winning species) species) and and (2) (2) the the time time to to extinction extinction of of the the losing losing species. species. In In natural natural communities, communities, the the studies studies of of Brodie Brodie et et al. al. (2002) (2002) and and Geffeney Geffeney et et al. al. (2002) (2002) document document geographic geographic covariation covariation between between prey prey toxicity toxicity in in the the newt, newt, Taricha Taricha granulosa, granulosa, and and predator predator resistance resistance in in the the garter garter snake, snake, Thamnophis Thamnophis sirtalis. sirtalis. This This kind kind of of variation variation across across natural natural metacommunities metacommunities could arise by by aa combination combination of of drift drift and and natural selection. These These studies studies of of could arise natural selection.
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natural 1 996) natural metacommunities metacommunities complement complement those of Goodnight Goodnight and Craig ((1996) with laboratory metacommunities metacommunities and and demonstrate demonstrate the potential effects of trans-specific trans-specific IGEs IGEs on on fundamental fundamental ecological ecological processes, processes, such such as as predation predation competition, which shape ecological communities. This important important aspect and competition, of population evolution of meta metapopulation evolution has has received received relatively relatively little little theoretical theoretical attention attention al., 200 2001; 2003; Wolf et ai., al., 2003) 2003).. ((but but see Agrawal et ai., 1 ; Wade, 2003;
7 11 11 ..7
DISCUSSION DISCUSSION Moore 1 997) emphasized Moore et et ai. al. ((1997) emphasized aa critical critical feature feature of of IGEs, IGEs, "" .. .. . . interacting interacting phenotypes phenotypes differ differ from from other other traits traits because because they they are are determined determined in in part part by by an an environment environment that that can can evolve. evolve."" They They investigated investigated the the effects effects of of IGEs IGEs in in the the absence absence of of population population genetic genetic structure structure and and showed showed that that IGEs IGEs profoundly profoundly affect affect the the expected expected response response to to selection. selection. In In the the model model given given earlier, earlier, selection selection because the mean value, value, Zli is on on Zl zl produces produces aa change change in in the the mean mean of of trait trait Zz22 because the mean Z l i >, is aa context poor nutri context that that affects affects the the phenotypic phenotypic expression expression of of trait trait Zz2. Just as as poor nutri2 . Just tion can induce induce small body body size while good good nutrition nutrition can induce large body size, value of phenotype, Zz2, size, changes changes in in the the mean mean value of Zl produce produce effects effects on on the the phenotype, even 2 , even subdivision of in the absence of genetic variation in variation for for Zz2. The genetic genetic subdivision of metapopu metapopu2 . The lations ((i.e., i.e., PST 0) enhances lations FST> > 0) enhances the the influence influence of of IGEs IGEs on on evolution. evolution. Griffing 1 967, 11977, 977, 1981, Griffing ((1967, 1981, 1989) 1989) showed showed how how one one of of the the most most ecologic ecologically common ally common IGEs, IGEs, intraspecific intraspecific competition, competition, negatively negatively affects affects the the response response to selection, especially to individual individual selection, especially selection selection to to increase increase the the rate rate of of population population increase. increase. His His theoretical theoretical findings findings have have found found empirical empirical support support in in aa number number of of competition have been been of organisms, organisms, and and similar similar effects effects of competition and and density density have detected (cf. review by Goodnight Goodnight and 997). detected in in several several studies studies (cf. review by and Stevens, Stevens, 11997). Because of the genetic metapopulations, the the mean mean Because of IGEs IGEs and and the genetic structure structure of of metapopulations, value of an evolutionarily value of one one trait trait becomes becomes an evolutionarily dynamic dynamic component component of of the the environment Thus, IGEs in metapopulations environment experienced experienced by another another trait. trait. Thus, metapopulations alter local individual reinforce the the view view of alter the the outcome outcome of of local individual selection selection and and reinforce of Goodnight in Goodnight and and Wade Wade (2000), (2000), "Multilevel "Multilevel selection selection is is far far more more common common in nature than previously individual selection nature than previously believed, believed, and and "pure" "pure" individual selection is far far less common" 322). common" (p. 322). In metacommunities, interactions interactions between between species species create create aa novel novel class In metacommunities, class of of trans-specific which the of aa trait trait in in one trans-specific IGEs, in which the mean mean value of one species becomes becomes an evolutionarily evolutionarily dynamic dynamic component component of of the the environment experienced by an environment experienced another trait trait in in another another species. species. Because Because the the species species with with the the smaller smaller effective effective another population size size determines determines how how random random genetic genetic drift drift affects affects the the coevolving coevolving population traits, the the metapopulation meta population genetic genetic structure experienced by by one one species species will will traits, structure experienced affect the evolution evolution of of all all the the other other species species with with which which it it interacts, interacts, whether whether or or affect the not they they have have aa conspicuous conspicuous metapopulation metapopulation structure. structure. Whenever Whenever aa trait trait in in not one species an ecological ecological context context trait another species, then evolution one species is an trait in another species, then evolution in both species influenced by metapopulation metapopulation genetic genetic structure structure both species will be uniquely uniquely influenced in either species. Furthermore, Furthermore, interdemic interdemic selection selection on on one species will affect affect of traits traits in the the other other species with with which which it interacts. interacts. Thus, Thus, the the coevolution coevolution of meta population genetic genetic structure structure of of one one species, species, with with or or without without interdemic interdemic metapopulation selection, will will have have important important consequences consequences for for its its evolution evolution and and for for the the selection, coevolution of of many many other other species in in its metacommunity. metacommunity. coevolution
sdfsdf
12
S PECIATION IN SPECIATION IN META PO PU PULATIONS M FTAPO LATIO N S Sergey Gavrilets
112.1 2. 1
INTRODUCTION INTRODUCTION Analysis of ecological and evolutionary dynamics in meta populations, that metapopulations, that is, is, in in populations populations subdivided subdivided into into aa large large number of of local subpopulations subpopulations that that become become extinct extinct and and are are recolonized from from other other locations, has has been aa focus focus of of numerous experimental and theoretical theoretical studies (e.g., Hastings and Harrison, Harrison, numerous 11994; 994; Harrison 996; Hanski 997; Hanski, Harrison and Hastings, Hastings, 11996; Hanski and Gilpin, 11997; Hanski, 1998; 1998; this Volume). The main interest of most of the previous work work on the evolu evolutionary effects of local extinction and colonization in meta populations has been metapopulations mainly on the levels of genetic variation within within and between local populations, populations, on the fixation probabilities probabilities and fixation times, and on Wright's shifting 940; Levins, 1970; 977, 1981, balance theory (e.g., Wright, 11940; 1970; Slatkin, 11977, 1981, 1978; 1978; Wade, 11978; 978; Lande, 1979, 984, 1985, 1979, 11984, 1985, 1992; 1992; Wade and McCauley, 1988; 1988; Whitlock and McCauley, 11990; 990; Barton, 1993; 1993; Michalakis and Olivieri, 1993; 998; Pannell and 1993; Whitlock et aI., al., 1993; 1993; Le Corre and Kremer, 11998; 999). Charlesworth, Charlesworth, 11999). Several recent studies modeled the joint dynamics of speciation, extinction, and and colonization colonization in in aa spatially spatially explicit explicit framework framework using using phenomenological phenomenological descriptions of speciation. These studies did not consider any underlying genetics and simply postulated that a new species with a certain number of individuals (one or more) emerges with a certain probability out of the ances ancestral species. A major focus of these studies was on explaining the so-called
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species-areas species-areas curves. curves.
A A species-areas species-areas curve curve relates relates the the number number of of species species SS found in in aa region region with with its its area area A. A. These These curves curves are are usually usually described described using using found power-law relationship a power-law ((12.1) 12. 1 )
S = cA b
b
where .15 where cc is is aa constant constant and and b is is an an exponent, exponent, which which typically typically ranges ranges from from 00.15 0.40 (e.g., (e.g., Rosenzweig, Rosenzweig, 11995; Hubbell, 200 2001). ttoo 0.40 995; Hubbell, 1). For 1 996) and 1 996) considered For example, example, Bramson Bramson et et ai. al. ((1996) and Durrett Durrett and and Levin Levin ((1996) considered two-dimensional space space divided divided into into square cells. Each Each cell cell is is occupied occupied by by one one aa two-dimensional square cells. individual, and and each each individual individual is is characterized characterized by by its its "type. "type."" Each Each individual individual individual, has has four four "neighbors" "neighbors" (above, (above, below, below, left, left, and and right) right).. During During each each unit unit time time interval, the the state state of of each each individual individual changes changes to to that that of of aa randomly randomly chosen chosen interval, neighbor neighbor with with aa small small probability probability 3. 8. This This event event represents represents the the death death of of the the individual individual immediately immediately followed followed by by dispersal dispersal of of an an offspring offspring of of one one of of its its neighbors cell. With small probability probability vv the neighbors into into the the vacant vacant cell. With aa small the state state of of each each individual individual changes changes to to aa new new type type not not previously previously present present in in the the system. system. This This event event represents represents the the replacement replacement of of the the individual individual by by its its mutant mutant offspring offspring that belongs to aa new new species. species. Numerical simulations of of this this model model show show that that that belongs to Numerical simulations individuals of of the the same same type type tend tend to to form form clusters in space et aI., al., individuals clusters in space ((Bramson Bramson et Durrett and and Levin, Levin, 11996; Hoelzer, 200 2001). This dynamic dynamic pattern pattern is is 11996; 996; Durrett 996; Hoelzer, 1 ) . This explained all individuals same species explained by by the the simple simple fact fact that that all individuals of of the the same species are are ances ancestors of of aa single single mutant mutant individual individual and and therefore therefore are are more more likely likely to to be be found found tors close to course, the close to each each other other (provided, (provided, of of course, the dispersal dispersal is is limited). limited). The The rate rate of of death and and replacement replacement 38 and and the the rate rate of of speciation speciation vv control control the the properties properties of of death the resulting the resulting stochastic stochastic equilibrium equilibrium via analytical via their their ratio ratio cx = vv/8. /3 Using Using analytical methods, Bramson et ai. ((1996) 1 996) and Levin ((1996) 1 996) showed showed that methods, Bramson et al. and Durrett Durrett and and Levin that these different spatial spatial scales. scales. The The characteristic these properties properties are are different different at at different characteristic linear dimension is linear dimension is
a=
.
= �1Va�
Il = , _
1
((12.2) 12.2)
Vot
A >> 12, 12, the the number number of for squares with area area A For large large areas, areas, that that is, is, for squares of species species For b with found is given by Eq. with b -= 11 and and found Eq. (12.1) ( 1 2 . 1 ) with 11 cc = = ~w - or(In a( ln or) a ) 22 21T
(12.3) ( 12.3)
(Bramson et aI., al., 1996). (Bramson et 1 996). For For smaller smaller areas, areas, that that is, is, for for squares squares with with area area A A << 12, F, the results Levin (1996) that the the species-area the results of of Durrett Durrett and and Levin ( 1 996) suggest suggest that species-area curve curve is is given approximately by given approximately by Eq. Eq. (12.1) ( 12. 1 ) with with b
=
=
b =
a
22 In ln lI ++ ln(2/'rr) In( 2/1T ) 22 In ln II
( 1 2.4) (12.4) b
1 2, the and cc = 1. -4 to the exponent exponent b and 1. For For example, example, as as ~ decreases decreases from from 10 1 0-4 to 10 1 0 -12, decreases decreases from from 0.283 0.283 to to 0.174. 0 . 1 74. -
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The 1 996) and 1 996) The model model studied studied by by Bramson Bramson et et al. al. ((1996) and Durrett Durrett and and Levin Levin ((1996) was actually actually aa simplification simplification of of aa model model proposed proposed and and numerically numerically studied studied by by was Hubbell 6) who � 1 ) of Hubbell (2001 (2001,, Chapter Chapter 6) who allowed allowed for for aa number number ((---1) of individuals individuals at at each "local community" each cell cell (interpreted (interpreted as as aa "local community").) . In In Hubbell's Hubbell's model, model, each each death death in aa local local community community resulted resulted in in replacement replacement from from the the local community or or in local community from neighboring local communities with from one one of of the the neighboring local communities with probabilities probabilities 11 - m m and m, respectively. and m, respectively. Parameter Parameter m m is is interpreted interpreted as as the the probability probability of of migra migration. Hubbell Hubbell (200 1 ) demonstrated tion. (2001) demonstrated the the linearity linearity of of the the species-area species-area curves curves on spatial scales scales and on "intermediate" "intermediate" spatial and showed showed that that the the slope slope of of these these curves curves m. He He also also noticed increases decreasing the migration m. increases with with decreasing the rate rate of of migration noticed that that the the dependence approximately linear linear for dependence of of SS on on A A becomes becomes approximately for areas areas that that are are (much) smaller (much) smaller than than those those predicted predicted by by the the characteristic characteristic distance distance II defined defined by by Eq. ((12.2). 12.2). Assuming dispersal limitations limitations (i.e., Eq. Assuming no no dispersal (i.e., disregarding disregarding the the spatial spatial structure Hubbell (2001 structure of of the the system), system), Hubbell (2001)) numerically numerically compared compared two two versions versions of individuals in of the the model model differing differing with with regard regard to to the the number number of of individuals in the the new new species. In "point speciation species. In the the "point speciation model" model" each each new new species species starts starts with with exactly exactly one individual. In "random fission model" the one individual. In the the "random fission model" the new new species species gets gets aa random random proportion proportion (sampled (sampled from from the the standard standard uniform uniform distribution) distribution) of of individuals individuals of of the the ancestral ancestral species. species. This This latter latter procedure procedure was was supposed supposed to to model model speciation speciation resulting Hubbell noticed in resulting from from aa vicariance vicariance event. event. Hubbell noticed dramatic dramatic differences differences in various various characteristics characteristics of of the the system system between between these these two two speciation speciation scenarios. scenarios. A A variant variant of of the the "random "random fission fission model" model" was was used used by by Barraclough Barraclough and and Vogler Vogler (2000) (2000) in in their their numerical numerical study study of of the the dynamics dynamics of of species species ranges ranges that that did did not not allow allow for for extinction. extinction. Both 1 ) and al. ((1996) 1 996) and Both Hubbell Hubbell (200 (2001) and Bramson Bramson et et al. and Durrett Durrett and and Levin Levin ((1996) 1996) were were primarily primarily concerned concerned with with the the number number of of species species found found within within aa certain be certain area area embedded embedded within within aa much much larger larger area. area. Their Their results results cannot cannot be used used to to evaluate evaluate the the overall overall number number of of species species in in the the system system (i.e., (i.e., the the overall overall diversity). al. ((1998) 1 99 8 ) using diversity). This This latter latter question question was was approached approached by by Allmon Allmon et et al. using numerical numerical simulations simulations of of finite finite two-dimensional two-dimensional square-lattice square-lattice systems systems in in which which each population rather each square square cell cell was was interpreted interpreted as as aa population rather than than an an individual. individual. [Note [Note that that the the same same interpretation interpretation is is applicable applicable to to the the models models of of Hubbell Hubbell (2001 1 996), Durrett 1 996).] Allmon al. ((1998) 1 998) (2001),), Bramson Bramson et et al. al. ((1996), Durrett and and Levin Levin ((1996).] Allmon et et al. allowed allowed for for the the probability probability of of speciation speciation to to be be dependent dependent on on the the number number of of and and the the distance distance to to other other populations populations of of the the same same species. species. The The main main focus focus of Allmon et 1 998) was diversity is is of Allmon et al. al. ((1998) was on on demonstrating demonstrating that that the the overall overall diversity maximized local populations. maximized at at intermediate intermediate rates rates of of extinction extinction of of local populations. Pelletier 1 999) used Pelletier ((1999) used numerical numerical simulations simulations to to study study the the species-area species-area curves curves both embedded within regions and isolated areas. both for for areas areas embedded within much much larger larger regions and for for isolated areas. He He used used aa different different modeling modeling framework framework that that explicitly explicitly treated treated the the dynamics dynamics of population densities dispersal of of local local population densities and and allowed allowed for for density-dependent density-dependent dispersal of individuals. equation. In In his his individuals. The The latter latter process process was was modeled modeled using using aa diffusion diffusion equation. simulations, the speciation at point was be simulations, the probability probability of of speciation at aa given given grid grid point was set set to to be inversely point. This assumption inversely proportional proportional to to the the species species abundance abundance at at this this point. This assumption implied that speciation was Pelletier's implied that speciation was most most probable probable in in small small populations. populations. Pelletier's results of species-area log-log scale results demonstrated demonstrated the the linearity linearity of species-area curves curves on on the the log-log scale [which 1 2 . 1 ) was [which implies implies that that Eq. Eq. ((12.1) was adequate] adequate].. Curves Curves corresponding corresponding to to nested nested subareas subareas had had shallower shallower slopes slopes and and were were positioned positioned higher higher than than curves curves corresponding isolated areas. 1 999) also also demonstrated corresponding to to isolated areas. Pelletier Pelletier ((1999) demonstrated that that in in -
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his model model the number given species species has his number of new species originating originating from a given aa power-law distribution and power-law distribution and the the time time series series of of extinctions extinctions and and originations originations have have aa lIf 1/f power power spectrum spectrum where where ff denotes denotes frequency. frequency. Although these earlier approaches training our intuition Although these approaches are very useful for training intuition about about the the process process of of diversification diversification in in metapopulations metapopulations and and for for providing providing aa basis for additional additional numerical numerical and analytical analytical work, the phenomenological phenomenological treatment treatment of of speciation speciation they they employ employ is is not not satisfactory. satisfactory. Excluding Excluding processes processes such such as polyploidy polyploidy and major major chromosomal chromosomal changes, speciation speciation does not not occur instantaneously. loci are needed for instantaneously. Changes Changes in in at at least least several several loci are needed for the the degree degree of of reproductive reproductive isolation isolation or or morphological morphological change change necessary necessary for for assigning assigning an an individual this happening individual or or aa population population to to aa new new species. species. The The probability probability of of this happening instantaneously example, if 10 genes instantaneously is is extremely extremely small. small. For For example, if 10 genes have have to to be be changed, then then using the standard standard estimates estimates of the mutation mutation rates rates ((10-5-10-6), 10-5-10-6), changed, probability of instantaneous instantaneous speciation speciation is on the order order of of 110-5~ the probability 0 - 5°_10 --6~ 6°. If there are L loci and and changes in any K K of them will result in strong reproduc reproductive isolation (or a significant morphological morphological difference), then the probability probability of speciation k )10 5K. Here, Here, (-~) (k) isis the i.e., speciation is is approximately approximately ((-~)10-5K. the binomial binomial coefficient coefficient ((i.e., the the number number of of combinations combinations of of K K objects objects chosen chosen from from aa set set of of L L objects) objects) and and that the mutation mutation probability probability is 10-5 10 -5 per locus per generation. it is assumed that For example, example, with with L = = 100 100 and and K K= = 10, 10, the probability probability of instantaneous instantaneous specia specia35• Hubbell's approximately 1010 -35. Hubbell's numerical numerical work work shows that that both both the tion is approximately initial number species and their initial initial spatial initial number of individuals individuals of the new species distribution have dramatic distribution dramatic effects on various dynamic dynamic characteristics characteristics of the system. However, system. However, the the phenomenological phenomenological approaches approaches cannot cannot provide provide informa informaappropriate values or ranges of these parameters. parameters. Hubbell Hubbell (2001) (2001) tion on the appropriate and others argued convincingly that that species-area curves must bbee derived from from the rather than than be be the underlying underlying processes processes of of extinction, extinction, speciation, speciation, and and dispersal dispersal rather postulated follow aa certain postulated to to follow certain statistical statistical distribution. distribution. However, However, in in aa similar similar way, way, the the dynamics dynamics of of speciation speciation must must be be derived derived from from the the underlying underlying micro microevolutionary evolutionary processes processes rather rather than than be be postulated postulated to to follow follow aa certain certain statistical statistical distribution (e.g., distribution (e.g., aa "point "point speciation" speciation" model model or or the the "random "random fission" fission" model). model). This This chapter chapter attempts attempts to to expand expand these these previous previous approaches approaches by by developing developing aa model incorporating multiple model incorporating multiple genetic genetic loci loci underlying underlying reproductive reproductive isolation isolation and and species differences. It starts starts by describing an approach approach for for modeling modeling speciation that classical Bateson-Dobzhansky-Muller speciation that is is based based on on the the classical Bateson-Dobzhansky-Muller (BDM) (BDM) model. Then Then this approach approach is incorporated incorporated into the metapopulation metapopulation frame framework. work. The resulting model allows one to study the dynamics of both both the overall diversity diversity and under overall and the the genetic genetic structure structure of of the the system system of of populations populations undergoing radiation. radiation. -
1 22.2 .2
MODELING M O D E L I N G GENETICS GENETICS OF OF REPRODUCTIVE R E P R O D U C T I V E ISOLATION ISOLATION Most Most of the existing approaches approaches for modeling modeling the the genetics of reproductive reproductive isolation utilize the idea first expressed explicitly explicitly by Bateson Bateson (1909), (1909), isolation Dobzhansky 1 937), and 1942) according Dobzhansky ((1937), and Muller Muller ((1942) according to to which which reproductive reproductive isol isolation is a consequence incompatibilities" between consequence of ""incompatibilities" between different different genes and and two-locus two-allele model model traits. This section starts by describing a simple two-locus formalizing idea. Then it briefly discusses the notions formalizing this idea. notions of nearly neutral neutral
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generalizanetworks and holey adaptive landscapes that provide a multilocus generaliza Bateson-Dobzhansky-Muller model. Finally, a specific multilocus tion of the Bateson-Dobzhansky-Muller model is described, which is used later in studying the dynamics of speciation within the meta metapopulation and diversification within population framework.
The Bateson-Dobzhansky-Muller Bateson-Dobzhansky-Muller Model Model The The The BDM BDM model model makes makes aa very very specific specific assumption assumption about about the the genetic genetic architecture of reproductive reproductive isolation, that there are two two alleles at architecture isolation, namely namely that different loci that that are "incompatible." "incompatible." Growing Growing experimental evidence different strongly assumption (Wu 994; Orr, 995; Orr Orr strongly supports supports this this assumption (Wu and and Palopoli, Palopoli, 11994; Orr, 11995; and Orr, Orr, 11996; 2001). alternative alleles at the two two loci under and 996; Wu, 200 1 ). Let the alternative under consideration denoted as A, a and and B, b. The The easiest way way to to illustrate consideration be denoted illustrate the BDM the corresponding corresponding adaptive landscapes shown shown in BDM model model is by means means of the adaptive landscapes Figure 112.1 assumes that that alleles a and and B are incompatible incompatible in the the Fig. 112.1. 2. 1 . Figure 2 . 1 assumes that individuals individuals carrying both both of them them have zero viability (Fig. 112.1a) sense that 2.1a) that females carrying allele a ddoo not not mate mate with with males carrying carrying allele B oorr that occurs if two two populations populations that (Fig. 112.1b). 2 . 1 b ) . Speciation occurs that have initially the same composition end up up at the opposite opposite sides of the ridge ridge of of high high fitness genetic composition There are two two pos posvalues at genetic states with with genotypes AABB and aabb. There for this this to happen. The two two populations start with with the sibilities for happen. The populations can can start the same genotype AAbb then evolve in fixing incompat incompatgenotype AAbb and and then in different different directions directions by by fixing ible alleles (as illustrated illustrated by the arrows in Fig. 12.1a) the arrows 1 2 . 1 a ) or or the the populations populations can start with with the the same genotype genotype AABB (or and then of them them will fix start ( or aabb) aabb) and then one one of fix two new alleles (as illustrated the arrows two the illustrated by the arrows in Fig. 12.1b). 1 2 . 1 b). In this model, model, the populations populations are not not required required to to cross any adaptive adaptive valleys to to evolve reprorepro ductive isolation, as they simply follow of high high fitness ductive isolation, follow a ridge ridge of fitness values. The The evolution along this ridge can can be driven driven by any factors, evolution any of of the evolutionary evolutionary factors, such drift, and or sexual selection such as mutation, mutation, random random genetic drift, and natural natural or ((Gavrilets, Gavrilets, 1999a, 1 999a, 2000). 2000).
--/bb --/8b --/88 (\o\'JQe (a)
(b)
�e ((\a.\e
Fig. Fig. 12.1 1 2.1 Fitness Fitness landscapes landscapes in in the the diploid diploid BDM BOM model. model. (a) (a) Fitness Fitness of of an an individual. individual. (b) (b) Fitness of of a mating mating pair.
280 280
SERGEY SERGEYGAVRILETS GAVRILETS
Holey Holey Adaptive Adaptive Landscapes Landscapes How How common is the genetic architecture implied by the BDM model illustrated in Fig. 112.1? 2. 1 ? Starting with Wright ((1932) 1 932) typical adaptive, land landscapes are usually imagined as very rough surfaces with many different peaks and valleys (see (see Fig. 112.2a). 2.2a). Continuous evolution on such landscapes requires crossing adaptive valleys. Wright's metaphor of rugged adaptive landscapes enforces a belief that that the ridges of high fitness values, which are explicit in the BDM model, are very improbable. However, analyses have shown that that the properties of the three-dimensional geographic landscapes implicit in Wright's metaphor are a rather poor indicator of the properties of adaptive landscapes describing genetic systems with thousands of loci ((Gavrilets Gavrilets and Gravner, 11997; 997; Gavrilets, 11997, 997, 2003) 2003).. The most prominent parts of three-dimensional landscapes are peaks and valleys. In contrast, the most prominent feature of adaptive landscapes of very high dimensionality are extensive nearly nearly neutral neutral networks, that is, connected networks of genotypes with very similar fitnesses networks, that expand throughout 997; throughout the genotype space (Gavrilets and Gravner, 11997; Gavrilets, 11997, 997, 2003; Reidys, 11997; 997; Reidys et aI., 997). Among different al., 11997). nearly neutral networks, those with sufficiently high fitnesses are of particular importance as they allow for continuous evolutionary innovations without any significant loss in fitness. An important notion describing such networks is that of "holey adaptive landscapes." A holey holey adaptive adaptive landscape landscape is defined as an adaptive landscape where relatively infrequent high-fitness genotypes form a contiguous set that that percolates (i.e., expands) throughout throughout the genotype space. An appropriate three-dimensional image of such an adaptive landscape that focuses exclusively on the percolating network of genotypes is a nearly flat surface with many holes representing genotypes that do not belong to the network (see (see Fig. 12.2b). The smoothness of the surface in Fig. Fig. 12.2 reflects close similarity between the fitnesses of the genotypes forming the corresponding nearly neutral network. The "holes" include both lower fitness genotypes (("valleys" "valleys" and "slopes" "slopes")) and very high fitness genotypes (the "tips" of the adaptive peaks). The BDM model considered earlier provides one of the simplest examples of a holey adaptive landscape. Many more examples are known (Gavrilets and Gravner, 11997; 997; Gavrilets, 11997, 997, 2003).
enen Q)
enen Q) .E
.E
;;:::
;;:::
genotype space
genotype space (a)
genotype space
genotype space (b)
Fig. 112.2 2.2 Adaptive landscapes. landscapes. (a) A rugged adaptive landscape. landscape. (b) A holey adaptive
landscape.
281 281
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS A A
Multllocus Multilocus Generalization Generalization of of the the BDM BDM Model Mo d el The The BDM BDM model model was was formulated formulated in in terms terms of of only only two two loci. loci. However, However, existing the genetics of reproductive existing data data on on the genetics of reproductive isolation isolation show show that that typically typically there there are underlying reproductive are many many different different loci loci underlying reproductive isolation, isolation, even even at at very very early early stages of divergence (Wu and Palopoli, 1 994; Naveira and Masida, 1 998; stages of divergence (Wu and Palopoli, 1994; Naveira and Masida, 1998; Wu, 200 1 ) . When there are many loci, rather than studying the dynamics Wu, 2001). When there are many loci, rather than studying the dynamics of of speciation (which is speciation given given aa specific specific genetic genetic architecture architecture (which is generally generally unknown) unknown),, iitt becomes becomes much much more more fruitful fruitful ttoo look look at at the the dynamics dynamics ooff speciation speciation expected expected "on average. " One such approach reduces the complexity land "on average." One such approach reduces the complexity of of adaptive adaptive landscapes underlying reproductive isolation to simpler effects of genetic "incom scapes underlying reproductive isolation to simpler effects of genetic "incompatibilities" 995; patibilities" of of certain certain types types that that arise arise with with certain certain probabilities probabilities (Orr, (Orr, 11995; Orr and Orr, 1 996; Orr and Turelli, 200 1 ) . Orr and Orr, 1996; Orr and Turelli, 2001). Let Let us us assume assume that that there there are are two two populations populations that that have have diverged diverged iinn d d diallelic diallelic comloci potentially affecting reproductive isolation. Let us consider all possible possible com binations of binations of kk alleles alleles with with each each allele allele taken taken from from aa different different locus. locus. For For example, example, BDM model, d d = = 2 (see (see Fig. 12.2). If kk = = 2, there are two two "parental" "parental" in the BDM combinations, aBo With combinations, AB AB and and ab, and and two two "hybrid" "hybrid" combinations, combinations, Ab and and aB. With dd = 55 loci, there are are 38 and 78 78 such such loci, there 38 "hybrid" "hybrid" combinations combinations if if kk = = 33 alleles, alleles, and combinations each "hybrid" can be be combinations if if kk = = 4 4 alleles. alleles. Assume Assume that that each "hybrid" combination combination can incompatible with probability incompatible with probability q q.. Here, Here, "incompatibility" "incompatibility" means means epistatic epistatic inter interaction between between the the alleles alleles potentially potentially resulting resulting in in aa loss loss of of fitness. fitness. Note Note that that in in action the combination aB incompatible the BDM BDM model model it it is is assumed assumed that that combination aB is is definitely definitely incompatible and combination Ab is compatible. In just and that that combination is definitely definitely compatible. In contrast, contrast, in in the the model model just formulated, formulated, the the overall overall number number of of incompatibilities incompatibilities is is aa stochastic stochastic variable. variable. The incompatibilities translate The next next question question is is how how exactly exactly the the incompatibilities translate into into repro reproductive isolation between 1 996), ductive isolation between the the populations. populations. Following Following Orr Orr and and Orr Orr ((1996), that complete reproductive isolation isolation occurs when when C C incom incomlet us assume that patibilities BDM model, patibilities separate separate the the populations. populations. Note Note that that in in the the BDM model, C C= = 11.. that aass genetic distance d d between between the populations populations IInn general, one expects that increases, increases, the the expected expected number number of of incompatibilities, incompatibilities, I, I, increases increases as as well. well. One One wid) that also probability w(d) also expects expects that that the the probability that two two populations populations are are not not reproducti reproductively intuition more vely isolated isolated decreases. decreases. Box Box 12.1 12.1 makes makes this this intuition more precise. precise. Results Results given 2 . 1 are given in in Box Box 112.1 are compatible compatible with with the the general general observation observation that that the the degree degree of parental of reproductive reproductive isolation isolation increases increases with with genetic genetic divergence divergence between between the the parental organisms (Edmands, should also intuitively clear clear that organisms (Edmands, 2002). 2002). It It should also be be intuitively that adaptive adaptive landscapes implied landscapes implied by by this this model model are are "holey" "holey" and, and, thus, thus, allow allow for for extensive extensive genetic genetic divergence divergence in in aa (nearly) (nearly) neutral neutral fashion. fashion. To To predict predict the the dynamics dynamics of of speciation, speciation, it it will will be be assumed assumed that that the the prob probability ability of of compatibility compatibility of of two two populations populations that that have have diverged diverged in in d d loci loci is is given given by by the the threshold threshold function function
wi d) w(d)
=
{�
1 0
for for d d< < K, K, ford>--K for d :2: K
((12.5) 12.5)
((Gavrilets Gavrilets et aI., 11998; 998; Gavrilets, 999a, 2000). etal., Gavrilets, 11999a, 2000). This This function function implies implies that that genotypes genotypes that that are are different different in in less less than than K K loci loci are are perfectly perfectly compatible, compatible, whereas whereas genotypes isolated reproductively. genotypes different different in in K K or or more more loci loci are are isolated reproductively. The The neutral neutral case case
SERGEY SERGEYGAVRILETS GAVRILETS
282 282
BOX 12.1
Properties of . Multllocus 80M Model
Consider two populations that have diverged in d loci. The number of incom patibilities between them is a random variable that follows a Poisson distribution with parameter
(81 ) where ('f) is the binomial coefficient (Walsh, 200 3 ). Parameter I gives the expected number of incompatibilities. The value of I increases very rapidly ("snowballs")
with genetic distance d, approximately as the kth order of genetic distance d. Note that the snowball effect is much more pronounced with larger values of k (Orr,
1 995). Because the number of incompatibilities follows a Poisson distribution, the prob ability that two genotypes (or populations) at genetic distance d are not isolated repro ductively is approximately
w (d) =
�1 Ii 2. exp( - /) "7j" ,.
;=0
[(C. / )
(82)
= --
f(C)
Here f( ' , ' } and r( . } are the incomplete gamma function and gamma function, respectively (Gradshteyn and Ryzhik, 1 994), and I is given by Eq. (B 1 ). The probability
that two genotypes at distance d are isolated reproductively is 1 - w(d). The average K, variance var(K), and the coefficient of variation CV(K) of the number of substitutions required for speciation (i.e., for complete reproductive isolation) can be found in a straightforward manner using Eq. (B2). These values are
K
=
Vf 'Ik
var( K } - vf 21k _
C V( K}
f( C + 1 + 1 1k} "" r( C + 1 )
()
C 'lk
'
Vk
(B3a)
r ( C + 1 + 2Ik } r ( C + 1 } - f ( C + 1 + 1 /k }2 f ( C + 1 }2
k ) f ( C +_1_} f_ + 2/_ 1_ (C -::+ =--=_ -:- -:-�:-:n C + 1 + 1 /k } 2
_
1
""
I_
_
k Vc
'
(83b)
(83c)
where Vk = q (2k - k + 1 )/k! and the approximations assume that both k and C are not too small (> 3 ). Figure 8 1 2.1 shows that as the genetic distance d exceeds the value K, the probability of no complete reproductive isolation undergoes a rapid tran sition from 1 to O. This "threshold effect" is especially strong when many complex incompatibilities are required for complete reproductive isolation. The latter feature is also apparent from the fact that the coefficient of variation CV(K) quickly goes to zero as C or k become large.
283 283
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
0.9 0.8 0.7 ,g ro c
o .!!?
o c
15 �
0.6 0.5
:B 0.4
ro .J:J
e 0.3 c.
0.2 0.1
1 .5 0.5 1 normalized genetic distance, d/K
2
Fig. 812.1 The probability of no reproductive isolation with C = 1 0. Different lines correspond to k = 2 (the shallowest), 4, 6, and 8 (the steepest). Note that on the normalized scale d/K, the probability of no reproductive isolation does not depend on q.
(i.e., (i.e., the the case case of of no no reproductive reproductive isolation) isolation) corresponds corresponds to to K K larger larger than than the the num num12.5) can be viewed as a limiting ber of loci. The function defined by the Eq. ((12.5)can w(d) given by Eq. ((12.20) C are large, 12.20) in Box 12.1 when kk or C case of the function w(d) that that is, is, when when reproductive reproductive isolation isolation is is due due to to many many complex complex incompatibilities. incompatibilities. As As noted previously, the threshold function of reproductive compatibility was utilized to explore various features of the dynamics of speciation in systems of two 999a, 2000) two stable populations using analytical approximations (Gavrilets, 11999a, and and in one- and and two-dimensional stepping-stone systems with stable populations using individual-based simulations (Gavrilets et aI., 998, 2000b). al., 11998,
112.3 2.3
DYNAMICS D Y N A M I C S OF OF DIVERSIFICATION DIVERSIFICATION IN IN DEME-BASED DEME-BASED MODELS MODELS Analyzing the dynamics of genetic diversification in systems with a large number number of interconnected interconnected populations and with individuals characterized by number of genes requires simplifying approximations. approximations. The approxima approximaa large number tions made in this study are discussed explicitly later at appropriate appropriate places. However, there there is one approximation approximation that that has to be clarified right away away to avoid possible confusion. A straightforward straightforward interpretation interpretation of the models to be considered considered next is that that they describe speciation speciation caused caused by by the the spatial spatial spread spread of neutral genes. That is, these models are of mutually mutually incompatible incompatible nearly nearly neutral genes. That "neutral" (d. Hubbell, 200 1 ) in the sense that "neutral" (cf. 2001) that they do not explicitly specify the effects of genetic differences on viability and fertility of individuals ((and and popu populations). However, such effects are implicitly present. The underlying underlying picture is that that of the evolution along an extensive system of ridges in a holey adaptive
SERGEY SERGEYGAVRILETS GAVRILETS
284 284
landscape following as landscape with with reproductive reproductive isolation isolation following as aa by-product by-product of of sufficient sufficient genetic (as in genetic divergence divergence (as in the the model model discussed discussed in in the the previous previous section). section). The The exis existence tence of of "holes" "holes" in in the the adaptive adaptive landscape landscape means means that that the the differences differences in in fitness fitness are are present present explicitly. explicitly. However, However, because because of of (i) (i) the the separation separation of of the the time time scales scales (i.e., (i.e., rapid rapid "adaptive" "adaptive" evolution evolution from from aa "hole" "hole" toward toward aa high-fitness high-fitness ridge ridge fol followed by lowed by slow slow "nearly "nearly neutral" neutral" evolution evolution along along the the ridge), ridge), (ii) (ii) the the assumption assumption that that there there are are always always many many possible possible directions directions (i.e., (i.e., ridges) ridges) for for the the evolution evolution of of populations, populations, and and (iii) (iii) the the fact fact that that the the "choice" "choice" of of aa specific specific direction direction is is to to aa large random, the dynamics of large degree degree random, the dynamics of speciation speciation can can be be treated treated as as effectively effectively neutral. problematic in neutral. This This "nearly "nearly neutral" neutral" approach approach is is problematic in situations situations where where reproductive isolation depends on reproductive isolation depends on ecological ecological factors factors that that vary vary between between the the populations. populations. However, However, this this approach approach appears appears to to be be aa good good approximation approximation when when reproductive reproductive isolation isolation is is controlled controlled genetically genetically and and is is not not affected affected by by external external conditions. conditions. The The end end of of this this section section discusses discusses how how adaptation adaptation is is expected expected to to affect affect the the conclusions conclusions reached reached within within the the nearly nearly neutral neutral framework. framework.
Model Model Description Description Throughout Throughout this this section, section, sexual sexual species species are are considered considered with with nonoverlapping nonoverlapping generations. The generations. The main main motivation motivation of of this this section section is is to to develop develop aa mathematical mathematical model describing describing the the dynamics dynamics of of (adaptive) (adaptive) radiation radiation following following colonization colonization of of model aa new appearance of new environment environment or or appearance of aa "key "key innovation. innovation."" [The [The word word "adaptive" "adaptive" is is put put in in parentheses parentheses because because adaptation adaptation is is treated treated implicitly implicitly rather rather than than explicitly.] populations in explicitly.] The The whole whole set set of of populations in the the system system is is considered considered aa "clade." "clade." Spatial Arrangement Arrangement
A A habitat habitat is is considered considered subdivided subdivided into into aa large large but but finite finite number number n n of of discrete discrete "patches" arranged on a line (in the case of one-dimensional systems) "patches" arranged on a line (in the case of one-dimensional systems) or or on on aa two-dimensional two-dimensional square square lattice. lattice. Each Each patch patch can can support support one one population population of of aa from up to two two (in the one-dimensional species. Each patch can receive colonizers from cases) or (in the cases) neighboring cases) or up up to to four four (in the two-dimensional two-dimensional cases) neighboring patches. patches. The The number number of of neighboring neighboring patches patches will will be be smaller smaller for for patches patches at at the the boundary. boundary. Population State
It It is is assumed assumed that that there there is is aa large large number number L L of of possibly possibly linked linked diallelic diallelic loci loci affecting affecting reproductive reproductive isolation isolation or or other other phenotypic phenotypic traits traits (morphological, (morphological, behavioral, etc.) (genera, families, behavioral, etc.) that that differentiate differentiate species species (genera, families, etc.). etc.). Each Each popula population tion is is characterized characterized by by the the genetic genetic sequence sequence of of its its most most common common genotype. genotype. Note possible sequences Note that that because because L L is is large, large, the the number number of of possible sequences is is enormous. enormous. This variation. This This chapter chapter neglects neglects within-population within-population genetic genetic variation. This implies implies that that the populations is the size size of of local local populations is relatively relatively small small and and that that the the rates rates of of mutation mutation and migration are small small as well. System System State
The The system system state state is is characterized characterized by by the the set set of of states states of of the the populations populations present. present. One One can can image image aa population population as as aa point point in in the the multidimensional multidimensional genotype genotype space. space. The The clade clade will will be be aa cloud cloud of of points points that that changes changes both both its its structure structure and and location location in the the genotype genotype space space as as aa consequence consequence of of ecological ecological and and evolutionary evolutionary processes. processes. in
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
285 285
Fixation Fixation of of Mutations Mutations
At At each each time time step step in in each each population population aa mutation mutation can can be be fixed fixed at at each each locus locus under probability fL. under consideration consideration with with aa very very small small probability ~. The The fixation fixation rate rate per per genotype, genotype, v = fLL, ~L, is is assumed assumed to to be be small small as as well. well. Following Following the the general general framework framework discussed discussed earlier, earlier, it it is is assumed assumed that that mutations mutations are are nearly nearly neutral. neutral. For For neutral neutral mutations, mutations, the the probability probability of of fixation fixation is is equal equal to to the the mutation mutation rate rate equal to probability of (Kimura, (Kimura, 1983). 1983). Thus, Thus, for for haploid haploid species, species, fL ~ is is equal to the the probability of mutation mutation per per allele, allele, whereas whereas for for diploid diploid species, species, fL ~ is is equal equal to to twice twice the the probability probability of of mutation mutation per per allele. allele. Extinction Extinction and and Recolonization Recolonization
At population may At each each time time step, step, each each population may go go extinct extinct with with aa small small probability probability
o. B. Extinction Extinction is is followed followed rapidly rapidly by by colonization colonization from from one one of of the the neighboring neighboring
patches patches chosen chosen randomly. randomly. Alternatively, Alternatively, one one can can think think of of extinction extinction of of aa local local population population as as being being caused caused by by successful successful invasion invasion from from one one of of the the neighboring neighboring demes. A established population population grows demes. A newly newly established grows to to the the equilibrium equilibrium size size rapidly. rapidly. Genetic and Species Genetic Clusters Clusters and
It It will will be be assumed assumed that that genetic genetic differences differences lead lead to to genetic genetic or or phenotypic phenotypic incompatibilities populations, for example, as incompatibilities between between different different populations, for example, as specified specified in in the the previous populations are previous section. section. Different Different populations are assigned assigned to to different different genetic genetic clusters clusters ((species, species, families, based on genetic divergence families, genera, genera, etc.) etc.) based on the the degree degree of of genetic divergence number of characterized characterized by by genetic genetic distance distance d. d. Recall Recall that that d d is is defined defined as as the the number of genes genes that that differ differ between between two two populations. populations. The The maximum maximum divergence divergence allowed allowed within 12.5)]. In within aa cluster cluster is is characterized characterized by by parameter parameter K K [see [see Eq. Eq. ((12.5)]. In the the numerical simulations simulations described technique is numerical described later, later, the the single single linkage linkage clustering clustering technique is used used (e.g., (e.g., Everitt, Everitt, 1993). 1993). This This means means that that two two populations populations separated separated by by aa dis distance tance d d equal equal to to or or larger larger than than the the corresponding corresponding threshold threshold K K may may potentially potentially still intermediate" population still belong belong to to the the same same cluster cluster if if there there is is another another ""intermediate" population ""linking" linking" them them together. together. For For example, example, if if both both the the genetic genetic distance distance d12 d12 between between populations 22 and populations 11 and populations and 22 and and the the genetic genetic distance distance d2 d233 between between populations and 33 are smaller than K, all three 2, and are smaller than K, then then all three populations populations 1, 1, 2, and 33 will will belong belong to to the the same same cluster, cluster, even even if if the the genetic genetic distance distance d1 d133 between between populations populations 11 and and 33 is is equal all populations equal to to or or larger larger than than K. K. According According to to this this definition, definition, all populations forming forming 942, 1963; 997; Irwin 1 ) would aa ring ring species species (e.g., (e.g., Mayr, Mayr, 11942, 1963; Wake, Wake, 11997; Irwin et et aI., al., 200 2001) would belong to compatible with belong to the the same same species. species. Note Note that that the the case case of of K K= -- 11 is is compatible with the the case, each previous work Section 12.1 previous work discussed discussed in in Section 12.1.. In In this this case, each cluster cluster (e.g., (e.g., species) species) is is defined defined by by aa unique unique sequence sequence of of genes genes (at (at the the set set of of loci loci under under con consideration) sideration).. The The case case of of K K=2 2 corresponds corresponds to to the the BDM BDM model. model. Alternatively, Alternatively, if new new species result from from the accumulation accumulation of a number number of genetic (morpho (morphological) differences, logical) differences, then then larger larger values values of of K K are are more more appropriate. appropriate. Genetic Genetic clus clusters ters corresponding corresponding to to different different values values of of K K can can also also be be interpreted interpreted as as describing describing different different levels levels of of taxonomic taxonomic classification. classification. For For example, example, let let us us specify Then, all all populations specify an an increasing increasing sequence sequence Kl K1 < < K2 K2 < < K K33 < ~ .. ... .. . Then, populations at at genetic distance distance less than than K K11 can can be be thought thought of of as as belonging belonging to to the the same same a genetic species, all populations at equal than species, all populations at genetic genetic distances distances that that are are larger larger or or equal than Kl K1 but but are are smaller smaller than than K2 K2 can can be be thought thought of of as as belonging belonging to to different different species species within within the the same same genus, genus, all all populations populations at at genetic genetic distances distances that that are are larger larger or or equal equal
2286 86
SERGEY SERGEY GAVRILETS GAVRILETS
than K2 K2 but but are are smaller smaller than than K3 K3 can can be be thought thought of of as as belonging belonging to to different different than species and and genera genera within within the the same same family, family, etc. etc. species Migration into Occupied Patches Migration
It is is assumed assumed that that migration migration into into occupied occupied demes demes has has no no effect effect on on the the It genetic composition composition of of the the resident resident population population even even if if immigrants immigrants are are coming coming genetic from the the same same species species and and are are genetically genetically compatible compatible and and able able to to mate mate with with from the residents. residents. As As aa working working example, example, aa plant plant metapopulation metapopulation is is envisioned envisioned the where local local demes demes produce produce aa large large number number of of seeds seeds of of which which only only few germin where few germinate. In In this this case, case, migrant migrant seeds seeds will will have have an an extremely extremely small small probability probability of of gerger ate. minating unless unless there there is is an an extinction extinction event event eliminating eliminating all all or or most most resident resident minating plants. In In aa similar similar way, way, if if there there is is frequency-dependent frequency-dependent selection selection against against immiimmi plants. grants, then then again again one one can can neglect neglect effects of migration migration (other (other than than bringing bringing grants, effects of colonizers into into an an empty empty patch). patch). The The assumption assumption of of no no effects of gene gene flow flow is is colonizers effects of justified only only if if the the rates rates of of immigration are very small or or the against justified immigration are very small the selection selection against immigrants is very strong. model's dynamics immigrants is very strong. The The effects effects of of gene gene flow flow on on the the model's dynamics will considered elsewhere (M. Saum Saum and S. Gavrilets, Gavrilets, unpublished unpublished results). results). will be be considered elsewhere (M. and S. Dynamic Scenario
Identifying features of process of of diversificadiversifica Identifying and and understanding understanding dynamic dynamic features of the the process tion following colonization new environment key. As As an an initial initial concon tion following colonization of of aa new environment are are key. dition, it is assumed that all patches are occupied by populations with exactly dition, it is assumed that all patches are occupied by populations with exactly the same "founder" genotype. the same "founder" genotype. This This implicitly implicitly assumes assumes that that the the spread spread of of the the species patches from the point point of of its its initial initial invasion invasion hap species across across the the system system of of patches from the happens on (much) shorter that of mutation. This pens on aa (much) shorter time time scale scale than than that of mutation. This assumption assumption appears to be reasonable. Initial spread is followed by the diversification appears to be reasonable. Initial spread is followed by the diversification phase phase during during which which the the founder founder species species splits splits into into an an increasing increasing number number of of different different clusters. stochastic equilibrium clusters. Eventually Eventually the the system system reaches reaches aa state state of of stochastic equilibrium in in which which the the number number of of clusters clusters and and their their different different characteristics characteristics fluctuate fluctuate around certain certain values. reaching this clade keeps around values. Note Note that that even even after after reaching this state, state, the the clade keeps evolving, evolving, as as different different species species (or (or clusters) clusters) go go extinct extinct and and their their place place is is taken taken by by Parameters and different dynamic characteristics of the new species (clusters). Parameters model to studied are model to be be studied are defined defined in in Box Box 12.2. 12.2. The analytical approximations approximations for for the the diversity diversity S, S, the the average average cluster cluster The R, and and the turnover turnover rate rate T T will will assume assume that that the the system system size is sufficiently range R, large large so so that that the the effects effects of of the the boundaries boundaries are are negligible. negligible. For For the the model model under under consideration, consideration, the the characteristic characteristic linear linear size size is is
/oK - ~/SK Ilcc =
'Y --;12
((12.6) 12.6)
(d. 977a,b, 11979; 979; Bramson 996; Durrett 996). (cf. Sawyer Sawyer 11977a,b, Bramson et et aI., al., 11996; Durrett and and Levin, Levin, 11996). spatial) distances separated by by ((spatial) distances much much larger larger than than Ie Ic demes demes are are expected expected Patches separated to to behave behave largely largely as as independent. independent. Also Also for for systems systems with with the the linear linear dimension dimension Il larger larger than than 10 Ic, the the effects effects of of borders borders will will be be small. small. [In [In one-dimensional one-dimensional systems, systems, Il = - n, n, whereas whereas in in in in two-dimensional two-dimensional systems, systems, Il = Vn.J ~ n . ] This This implies implies that that for for large large systems systems (with (with Il > > lcl, lc), the the diversity diversity SS will will increase increase linearly linearly with with the the number number of of patches patches in in the the system, system, whereas whereas the the range range R R will will not not depend depend on on n. n. Note Note that that for for the the parameter parameter values values used used in in the the simulations, simulations, Ie Ic ranges ranges
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
BOX 1 2.2
281 287
Model Characteristics
Parameters and dynamic characteristics. The following is a list of the parameters of the model considered in the main text: • • • •
•
the dimensionality of the system (one dimensional or two dimensional) the system size n (i.e., the number of patches in the system) the local extinction-recolonization rate B
the fixation rate per genotype v (which is actually the product of the number of loci L and the fixation rate per locus joL) the clustering level K
The effects of these parameters on the following characteristics need to be understood: •
•
•
•
•
•
•
•
•
the average time to the beginning of radiation, t/]l defined as the average waiting time until the first split of an initially uniform population into at least two clusters the average duration of radiation, td, defined as the average waiting time from tb to the time when the number of clusters reaches the (stochastic) equilibrium value for the first time the diversity, 5, defined as the average number of clusters in the clade at the stochastic equilibrium the average cluster range, R, at the stochastic equilibrium, defined as the number of populations that belong to an average cluster
the average pairwise genetic distance, cluster
0"
between the members of the same
the cluster diameter, Dc,max, defined as the maximum genetic distance between the members of the same cluster the turnover rate, T, defined as the number of new clusters emerging per unit of time divided by the standing diversity 5
the clade disparity, 0, defined as the average pairwise distance between all popu lations in the system the average genetic distance of the clade from the founder, d" defined as the average of pairwise distances between all populations and the species founder
The main text described analytical approximations for 5, R, T, and d, and used numerical simulations both to check the validity of these approximations and to understand the dynamics of other characteristics. The following is a list of parameter values used in numerical simulation: • • • • •
•
number of loci L 1 00 fixation rate per locus joL = =
1 0-6, 4
extinction-recolonization rate B = clustering levels K 1, 2, 4, 8, 1 6 =
system sizes:
X 1 0 6, 1 6 X 1 0-6 0.25 x 1 0-2, 1 0-2, 4 x 1 0-2 -
- 82 x 1 , 1 62 x l , 322 x l , 642 X 1 for one-dimensional systems and -8 x 8, 1 6 x 1 6, 32 x 32, 64 x 64 for two-dimensional systems the number of runs for each parameter configuration is 40
288 288
SERGEY SERGEY GAVRILETS GAVRILETS
from .25 (for the smallest rate 0~ = from 11.25 (for the smallest extinction extinction rate = 0.0025, 0.0025, K K= = 11,, and and the the largest largest fixation 16 X 6 ) to 80 (for fixation rate rate fJ. Ix = = 16 x 1100 --6) to 80 (for the the largest largest extinction extinction rate rate 08 = = 0.04, 0.04, K K= = 16, 16, and and the the smallest smallest fixation fixation rate rate fJ. Ix = = 11 00-6). - 6 ) . This This suggests suggests that that in in the the one-dimensional numerical examples, examples, the borders will one-dimensional numerical the effects effects of of borders will be be insig insignificant = 16). nificant except except for for the the smallest smallest system system (64 (64 X x 11)) with with the the largest largest K K ((=16). IInn contrast, contrast, in in the the two-dimensional two-dimensional examples, examples, the the effects effects ooff borders borders will will be be important important even even in in the the largest largest system system (64 (64 X x 64) 64) if if K K is is large. large. Unfortunately, Unfortunately, increasing the system size size is currently impossible impossible because because of computation increasing the system is currently of computation considerations. speed considerations.
Transient Transient Dynamics Dynamics Figure Figure 12.3 12.3 illustrates illustrates the the transient transient dynamics dynamics of of the the number number of of different different clusters clusters as as well well as as the the clade clade disparity disparity D D and and the the average average distance distance from from the the founder founder dr. df. The The dynamics dynamics of of the the two two latter latter measures measures do do not not seem seem to to depend depend on spatial dimensionality. The dynamics the founder on the The dynamics of of the the average average distance distance from from the founder depends depends only only on the fixation fixation rate rate per per locus locus fJ. I~ and and is is approximated approximated by by
dr(t)
L - e = ~-[1
-
2~t]
((12.7) 1 2.7)
((Gavrilets, Gavrilets, 11999b). 999b). That That is, is, df df asymptotically asymptotically approaches approaches the the distance distance equal equal to to one-half possible distance. one-half of of the the maximum maximum possible distance. This This implies implies that that after after aa sufficiently sufficiently long in half long time time the the members members of of the the clade clade will will be be different different from from the the founder founder in half of clade can equally likely of the the genes genes on on average. average. Moreover, Moreover, the the clade can be be equally likely found found in 12.7) can can be in any any part part of of the the genotype genotype space. space. Equation Equation ((12.7) be used used both both to to check the rate of check the constancy constancy of of the the rate of evolution evolution in in time time and and to to estimate estimate its its value value (see Gavrilets, Gavrilets, 11999b). (see 999b).
-���----�--j 200 r-���. . . . . . . .
250
1175 75 200
1150 50 1125 25
1150 50
K=2
1100 00
1100 00 75
K=4
50
50
0
25
0
e + 05 20000 40000 60000 80000 80000 1le+05 generation a) ((a)
O0 ������ o 0
20000 e + 05 2 0 0 0 0 40000 40000 60000 60000 80000 80000 1le+05 generation
(b)
2.3 Dynamics different clustering Fig. 112.3 Dynamics of of diversity diversity 5S at at five five different clustering levels levels K K (marked (marked in in the the figure), figure),
of average distance the founder of clade clade disparity disparity D D (bold (bold line), line), and and average distance from from the founder df df (dashed (dashed line). line). 0 --6, 6, 88 = Parameters: Parameters: J.L I~ = 4 4 x x 110 = 0.01 0.01.. The The statistics statistics are are computed computed every every 250th 250th generation. generation. (a) (a) One-dimensional One-dimensional 32 3222 X x 11 system. system. (b) (b) Two-dimensional Two-dimensional 32 32 x x 32 32 system. system. Notice Notice the the difference in the scale scale of of the the vertical vertical axes. axes. difference in the
112. 2. SPECIATION SPECIATION IN IN METAPOPULATIONS METAPOPULATIONS
289 289
The are similar The initial initial dynamics dynamics of of the the clade clade disparity disparity D D are similar to to that that of of df df with with the increases twice the difference difference that that D D increases twice as as fast fast [i.e., [i.e., the the exponential exponential term term in in an an analog rather than than 2J.1t]. 2~t]. A simple explanation of this analog of Eq. ((12.7) 12.7) has 4~t 4J.1t rather simple explanation that while the disparity is computed on the basis of pairs of evolving fact is that lineages, in lineage in pair (i.e., lineages, in computing computing db d6 one one lineage in each each pair (i.e., the the founder) founder) does does not not change. change. For For aa clade clade with with no no spatial spatial structure structure the the dynamics dynamics of of disparity disparity D D are are understood 999b). Unfortunately, understood (Gavrilets, (Gavrilets, 11999b). Unfortunately, for for spatially spatially explicit explicit systems, systems, nor its dynamics on the intermediate time neither the equilibrium value of D nor scales equilibrium for scales are are known. known. In In larger larger systems, systems, approaching approaching an an equilibrium for D D takes takes aa very very long long time. time. Figure Figure 12.3 12.3 illustrates illustrates the the important important observation observation that that the the diversity diversity SS at at low low taxonomic taxonomic levels levels (i.e., (i.e., at at small small K) K) equilibrates equilibrates faster faster than than the D, whereas the clade clade disparity disparity D, whereas the the equilibration equilibration of of the the diversity diversity at at higher higher taxonomic levels K) can time. In taxonomic levels (i.e., (i.e., at at large large K) can take take aa comparable comparable or or longer longer time. In the the latter case, very latter case, very high high values values of of D D (relative (relative to to its its asymptotic asymptotic equilibrium equilibrium value) value) can can be be observed observed simultaneously simultaneously with with very very low low taxonomic taxonomic diversity. diversity. The The pattern pattern of elevated disparity early in many clades clades has been traditionally of elevated disparity early in the the history history of of many has been traditionally explained explained by by paleontologists paleontologists by by invoking invoking explanations explanations that that postulate postulate temporal temporal changes 980; changes in in the the types types and/or and/or levels levels of of forces forces driving driving divergence divergence (Valentine, (Valentine, 11980; Foote, 992, 11999; 999; Erwin, 994; Wagner, 995; Lupia, 999). However, Foote, 11992, Erwin, 11994; Wagner, 11995; Lupia, 11999). However, both both previous work Gavrilets, 11999b) 999b) and models studied these previous work ((Gavrilets, and models studied here here show show that that these patterns patterns are are perfectly perfectly compatible compatible with with the the null null model model of of time time homogeneous homogeneous diversification. diversification. The The dynamics dynamics of of other other characteristics characteristics depend depend crucially crucially on on the the spatial spatial dimensionality dimensionality of of the the system. system. Note Note that that the the one-dimensional one-dimensional version version of of this this model model was was introduced introduced and and analyzed analyzed in in Gavrilets Gavrilets et et al. al. (2000a). (2000a). Figure Figure 12.4 12.4 illustrates illustrates the the dependence dependence of of the the time time to to the the beginning beginning of of radiation, radiation, tb, t6, and and the the duration duration of of radiation, radiation, td, ta, on on parameters parameters in in more more detail. detail. The The time time to to the the beginning increases ((approximately approximately exponentially) beginning of of radiation radiation increases exponentially) with with K. K. In In bio biological logical terms, terms, higher higher taxonomic taxonomic groups groups arise arise later later in in the the history history of of the the clade. clade. tb t6 decreases decreases with with the the fixation fixation rate rate vv (apparently (apparently as as 1/v). 1/v). Increasing Increasing vv by by aa certain factor results in than the certain factor results in aa smaller smaller increase increase in in tb t6 than the proportional proportional increase increase in in K. K. The The time time tb t6 increases increases weakly weakly with with the the extinction/recolonization extinction/recolonization rate rate 8 and size n. n. At can be inverse of and system system size At K K = 11,, tb t6 can be approximated approximated as as the the inverse of the the expected clade, that is, tb expected number number of of mutations mutations per per clade, that is, t6 = ~ lI(nv). 1/(nv). The The duration duration of of increasing weakly radiation radiation td is is not not very very sensitive sensitive to to K K and and 88 ((increasing weakly with with both both these these parameters) parameters) but but is is much much more more sensitive sensitive to to v. v. It It appears appears that that td ta is is on on the the order order of compatible with of lIv. 1/v. This This feature feature of of the the dynamics dynamics of of radiation radiation is is compatible with that that for for the dynamics where the the time the dynamics of of parapatric parapatric speciation speciation where time interval interval during during which which the the intermediate intermediate forms forms are are present present has has the the order order of of the the reciprocal reciprocal of of the the mutation mutation rate Gavrilets, 2000). simulation also also show, rate ((Gavrilets, 2000). Numerical Numerical simulation show, as as expected, expected, that that small small n) reach systems systems (with (with small small n) reach stochastic stochastic equilibrium equilibrium faster faster than than large large systems systems (with (with large large n). n).
Stochastic Equilibrium: One-Dimensional Systems Systems One One can can use use certain certain approximations approximations to to evaluate evaluate cluster cluster ranges, ranges, diversity, diversity, and turnover et aI., The following and turnover rates rates (Gavrilets (Gavrilets et al., 2000a). 2000a). The following formulas formulas assume assume that that the the fixation fixation rate rate per per genotype genotype is is much much smaller smaller than than the the extinction/colonization extinction/colonization rate v« rate ((v << 8). ~).
290 290
SERGEY SERGEY GAVRILETS GAVRILETS
6
6
5
5
4
4
'(; 3 i2 .5'
o
o
(a)
3>
(b)
6
6
5
5
,.
4
�3
4
�C> 3
.2 2
.2 2
C>
0
0
K
(e)
(d)
K
Fig. 2.4 average duration Fig. 11 2 . 4 The The average average waiting waiting time time to to the the beginning beginning of of radiation radiation and and the the average duration of of radiation. radiation. Left Left column: column: 32 3222 x x 11 system. system. Right Right column: column: 32 32 x x 32 32 system. system. First First row: row: the the aver average time time to average duration tb. Second Second row: row: the the average duration of of radiation radiation td' td. age to the the beginning beginning of of radiation radiation tb. Within (from left Within each each figure figure the the three three sets sets of of bars bars correspond correspond to to ()8 = 0.0025, 0.0025, 0.01 0.01,, and and 0.04 0.04 (from left to over 40 to right). right). Average Average values values over 40 runs. runs.
The The average average range range of of aa cluster cluster can can be be approximated approximated as as R = R =
r:;;8 'V b
~~r8 (( KK - 11) !) ! ', ~ f F (( K - 11/2 / 2 )) K-
((12.8a) 12.8a)
where Gradshteyn and 994). This where f F is is the the gamma gamma function function ((Gradshteyn and Ryzhik, Ryzhik, 11994). This expres expression sion simplifies simplifies to to R R =
2v b 'V/0
((12.8b) 12.8b)
1 2. 12.
291 291
SPECIATION IN IN METAPOPULATIONS METAPOPULATIONS SPECIATION
K== l1aand ffor orK n d t oto
)
7rOK ~ / ~rSK R R == ~/ ~ 2v
(12.8c) ( 12.8c)
for large large K. R is obtained as as the the inverse inverse of the probability probability that that two two populapopula for K. R is obtained of the tions belong belong to to the the same same cluster. cluster. The The aforementioned aforementioned equations assume that that tions equations assume each mutation mutation is is unique. unique. A A correction correction can can be be made made to to account backward each account for for backward mutations. In In this this case, case, instead instead of of K K one one needs needs to to use use mutations.
-
f K) K) - In(1 In( 1 - -~ = K= ,, ln(1 In( 1 - 2) f) L
(12.9) ( 1 2.9)
-
which is the the overall overall expected expected number of mutations which is number of mutations needed needed to to move move at at gen~ic distance distance K K from from aa reference reference genotype. genotype. The The aforementioned aforementioned expression expression genetic for for K, K, which which was found from from Eq. Eq. (12.7), simplifies to was found ( 12.7), simplifies to K K if if the the number number of of loci loci L L is very very large. is large. nlR, leading leading to The is just just SS == n/R, The average average diversity diversity is to
SS ==
1I2) ~ /(h 2 v f(KF ( K - 1/2) n. ~r8 (K( K - l1) !) ! n.
\) ;8
((12.10) 12 . 1 0 )
Figure Figure 12.Sa 12.5a illustrates illustrates the the dependence dependence ooff S S oonn the the parameters parameters ooff the the model model observed observed in in simulations. simulations. Biological Biological intuition intuition tells tells one one that that increasing increasing the the rate increase the rate of of fixation fixation of of new new mutations mutations should should increase the rate rate of of speciation, speciation, thus thus increasing increasing the the number number of of species species in in the the system. system. Decreasing Decreasing the the rate rate of of
1 50
1 50
1 00
1 00
(f)
(f)
50
50
a
a
(a)
(b)
Fig. 112.5 2.5 The The number number of of clusters. clusters. (a) (a) 322 322 X • 11 system. system. (b) (b) 32 32 x • 32 32 system. system. Within Within each each
figure figure the the two two sets sets of of bars bars correspond correspond to to /)6 = 0.01 0.01 (left) (left) and and 0.04 0.04 (right). (right). The The averages averages over over generations generations 200,000 200,000 through through 500,000 500,000 and and over over 40 40 runs. runs.
SERGEY SERGEYGAVRILETS GAVRILETS
292 292
extinction-colonization extinction-colonization should should have have aa similar similar effect effect because because larger larger levels levels of of genetic variation variation will will accumulate accumulate in in the the system. system. Equations Equations ((12.8) and ((12.10) genetic 12.8) and 12. 1 0 ) support support these these intuitions. intuitions. For For example, example, decreasing decreasing 88 bbyy factor factor 44 will will result result iinn an an increase increase in in the the number number of of species species by by factor factor 2. 2. These These results results provide provide aa formal formal justification justification for for the the idea idea that that species species can can accumulate accumulate rapidly rapidly after after colonizing colonizing aa new environment environment if if local populations in in the the novel novel environment environment have reduced new local populations have aa reduced probability 963; Allmon 998; Schluter, probability of of extinction extinction (e.g., (e.g., Mayr, Mayr, 11963; Allmon et et aI., al., 11998; Schluter, 11998, 998, 2000). rate v. 2000). Similar Similar effects effects can can be be achieved achieved by by increasing increasing the the fixation fixation rate v. With With K K= = 1, 1, the the rate rate of of turnover, turnover, T T,, can can be be evaluated evaluated by by dividing dividing the the number number of of new new clusters clusters per per generation, generation, which which is is vn, v n , by by the the standing standing diversity diversity S, S, leading to
TT == ~/Sv � 2n" 'V b;'
((12.11a) 12.1 1a)
The consideration consideration of of the the time time that that it it takes takes for for aa typical typical cluster cluster to to go go extinct extinct The leads to
T T = = � K 1,'
((12.11b) 12.1 1b)
K
for Gavrilets et for large large K K ((Gavrilets et aI., al., 2000a). 2000a). The The turnover turnover rate rate depends depends weakly weakly on on 88 for K. The latter for K K = = 11 and and becomes becomes approximately approximately independent independent of of 88 for for large large K. The latter counterintuitive prediction is counterintuitive prediction is explained explained by by the the fact fact that that the the increase increase in in the the overall extinction extinction rate resulting from overall rate of of species species resulting from an an increase increase in in 88 is is exactly exactly maintained in in the the system. system. balanced by by aa decrease number of species SS maintained balanced decrease in in the the number of species To analytical approximations approximations and and to to get further insights insights into into To check check these these analytical get further the model model dynamics, numerical simulations were performed performed (Gavrilets (Gavrilets et et al., aI., the dynamics, numerical simulations were 2000a). In In most most cases, cases, Eq. ( 12 . 1 0 ) underestimates the average average number number of 2000a). Eq. (12.10) underestimates the of species by by aa couple of percents, percents, whereas whereas Eq. 1 2. 1 1 ) overestimates turn species couple of Eq. ((12.11) overestimates the the turnover rate by by about about 5-10%. 5-10 % . In In the case of of the the smallest smallest mutation mutation rate, rate, the the errors over rate the case errors are slightly higher. Additional were used used to to analyze analyze the the structure structure of of the the clade clade in in the the Additional simulations simulations were genotype space. of Fig. Fig. 12.6 12.6 illustrates illustrates within-cluster within-cluster average average genotype space. The The left left column column of 12.6 shows shows that that De> and cluster diameter diameter DDc,max' pairwise pairwise distance, distance, Dc, and cluster c , m a x. Figure Figure 12.6 although there there is is plenty plenty of of within-cluster within-cluster genetic genetic variation, variation, typical typical members members of of although a cluster Effects of and 88 do do not not seem cluster are are at at distances distances that that are are smaller smaller than than K. Effects of vv and to be be significant. significant. In In fact, Fig. 12.6 1 2.6 and and similar similar results results not not shown shown here here suggest suggest to fact, Fig. K12. that for for one-dimensional one-dimensional systems, systems, roughly roughly DDcc ~= KI4 and DDc,ma that K / 4 and c,ma xx ~= U /2.
Stochastic Equilibrium: Equilibrium : Two-Dimensional Two-Dimensional Systems Approximating the the average average range range of of clusters clusters R R in in the the two-dimensional two-dimensional case case Approximating 1 , then then the the is than in is much much more more difficult difficult than in the the one-dimensional one-dimensional case. case. If If K K == 1, results of of Bramson Bramson et et al. ai. (1996) ( 1 996) on on the the number number of of species/~ species R found found within within results square impeded impeded within within aa much much large large area area [see [see Eqs. Eqs. (12.1) ( 12.1 ) and and (12.3)] ( 12.3)] give give aa square /~ = 2-rr8 11 2 1T8 R = vv [ln(8/v)]2". [ In( 81v ) J A
-
2
( 12. 12a) (12.12a)
293 293
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
" 0
8
8
6
6 " 0
4
4
2
2
o
o
(b)
(a) 15 10
� ';
0
� "
0
5 0
1
5 0
(e)
(d)
Fig. Fig. 112.6 2.6 Within-cluster Within-cluster genetic genetic variation. variation. First column: column" 32 3222 x • 11 system. Second column: column32 32 x • 32 32 system. system. First First row: row: average average pairwise pairwise distance distance Du Dc, Second Second row: row: cluster cluster diameter, diameter, Damax. the three sets of Domax. Within Within each each figure figure the three sets of bars bars correspond correspond to to l)~ = = 0.0025, 0.0025, 0.01 0.01,, and and 0.04 0.04
(from (from left left to to right). right). The The averages averages over over generations generations 200,000 200,000 through through 500,000 500,000 and and over over 40 40 runs. runs.
This R This approximation approximation implies implies that that the the size size of of the the system system is is sufficiently sufficiently large. large./~ overestimates many species occupying nearby areas may still have overestimates R R because because many species occupying nearby areas may still have few few representatives representatives within within the the sampling sampling square. square. One One can can also also estimate estimate the the average average number number of of populations populations R R that that have have the the same chosen patch. same type type as as the the population population from from aa randomly randomly chosen patch. R R is is somewhat somewhat larger R. For larger than than R. For example, example, if if there there are are five five clusters clusters with with 11,, 2, 2, 33,, 44, , and and 55 populations, respectively, respectively, then then R R = = ((11 + + 2 2 + + 33 + + 4 4 + + 55)/5 = 33 but but R R = = ((12 + populations, )/5 = 12 + 2 2 2 + 332 + + 4 42 + + Y)11 52)/155 = = 3.67. 3.67. If K = = 11,, then then range R R can be found found by inte inte222 + grating grating the the probability probability of of identity identity [(d, I(d, v) of of two two genotypes genotypes found found aa certain certain distance (as found 977a) over possible spatial distance apart apart (as found by by Sawyer, Sawyer, 11977a) over all all possible spatial positions. positions. This to This approach approach leads leadsto
294 294
SERGEY SERGEY GAVRILETS GAVRILETS
- 1T� 11 9r8 R R == - -----4v �/4v ) + 21T� 4v In( ln(8/4v) + 2~8
((12.12b) 12.12b)
(see Appendix 12. 1 ) . For For large (see Appendix 12.1). large K, K, --rrSK 1T�K (K,v), R R == Tv -~(U,v),
((12.12c) 1 2.12c)
2v
where where (K, ~(K, v) v) is is aa function function that that depends depends only only weakly weakly on on its its arguments arguments (see (see Appendix 112.1). Appendix 2. 1 ) . Simulations Simulations were were performed performed to to check check the the validity validity of of the the approximations approximations ((12.12) 1 2.12) for R. The for the the average average range range size size R. The fit fit was was within within 30-50%, 30-50%, which which is is satisfactory satisfactory given given aa number number of of simplifying simplifying assumptions assumptions involved involved and and the the small small size size of of systems systems used used in in numerical numerical simulations. simulations. As before, the just 5S = or n / R ) . Both As before, the average average diversity diversity is is just = nlR n/R ((or n/R). Both analytical analytical approximations results show approximations and and numerical numerical results show that that the the diversity diversity in in two twodimensional dimensional systems systems is is (much) (much) lower lower than than in in comparable comparable one-dimensional one-dimensional systems. systems. The The differences differences are are most most apparent apparent when when 5S is is relatively relatively small, small, which which number of happens happens with with large large �8 and and K K and and small small /L. ~. Some Some data data on on the the number of clusters clusters are summarized in Fig. 12.5b. are summarized in Fig. 12.5b. With 1, the With K K= = 1, the turnover turnover rate rate can can be be estimated estimated by by dividing dividing the the number number of of leading to new clusters, vn, new clusters, vn, by by the the standing standing diversity diversity 5, S, leading to �
2~r8 T = LI )]:'n'8"v'~2 {/
((12.13a) 12.13a)
in 12. 12a) and in the the case case of of R R as as given given by by Eq. Eq. ((12.12a) and to to 11 44 In( �/4v ) + 21T� ln(8/4v) + 2~r8
1T� 9r8
T T = = - �-----
((12.13b) 12. 1 3b)
given by Eq. ((12.12b). 12. 12b). in in the the case case of of R R as as given by Eq. To approximate the turnover rate To approximate the turnover rate in in the the case case of of K K> ) 1, 1, an an intuitive intuitive but but not not rigorous approach is to consider the average waiting time until a cluster rigorous approach is to consider the average waiting time until a cluster of of an an average average size size goes goes extinct. extinct. Because Because the the process process is is "neutral," "neutral," the the average average time time to same as average time to extinction extinction starting starting with with aa size size R R should should be be the the same as the the average time t" t* to grow to size R starting with a single population. The latter time t': ' to grow to size R starting with a single population. The latter time t* can can be be approximated approximated by by the the solution solution of of equation equation R R -= -
1T�t* In( t* ) In(t*)
-rrSt* �-
(Kelly, 977; Sawyer, 979; Bramson Bramson and 980). Then (Kelly, 11977; Sawyer, 11979; and Griffeath, Griffeath, 11980). Then the the average average · ), which lifetime average size turnover rate lifetime of of aa cluster cluster of of average size is is 2t" 2t*.. The The turnover rate T T is is 1/(2t': 1/(2t*), which leads being given leads to to T T being given by by aa solution solution of of
vv 11 . K ( K,v) In(1/2T) In( 1/2T} K ~(K,v)
T T = = -
((12.13c) 12. 1 3c)
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
295 295
This suggests that T This suggests that T is is of of the the order order of of vlK v / K [the [the logarithmic logarithmic dependence dependence in in Eq. 12. 13c) is The latter not depend on the Eq. ((12.13c) is less less important]. important]. The latter expression expression does does not depend on the extinction extinction rate, rate, which which is is similar similar to to the the one-dimensional one-dimensional case. case. Simulations Simulations were were also also used used to to analyze analyze the the structure structure of of the the clade clade in in the the genotype genotype space. space. The The right right column column of of Fig. Fig. 12.6 12.6 illustrates illustrates the the average average pairwise pairwise (> and within-cluster within-cluster distance, distance, D De, and cluster cluster diameter diameter Dc D c,ma x. As A s in in the the one onemax' dimensional dimensional case, case, Fig. Fig. 12.6 12.6 shows shows that that although although ther there� is is plenty plenty of of within withincluster variation, typical members of cluster genetic genetic variation, typical members of aa cluster cluster are are at at aa distance distance that that is smaller than is smaller than K. K. Effects Effects of of vv and and 8~ on on Dc D c and and Dc,max D c,max do d o not not seem seem to to be be significant. belong to significant. Typically, Typically, populations populations that that belong to the the same same cluster cluster are are also also spatially spatially contiguous. contiguous. The patches in The number number of of patches in the the system system n n is is also also of of importance. importance. If If n n is is too significant diversification too small, small, significant diversification will will be be prevented. prevented. For For example, example, consider aa case case with with 8~ = = 11 00 � - 22,, f.L ~ = = 11 00�6, - 6 , and and K K = = 16. 16. Then Then numerical numerical consider simulations simulations show show that that small small systems systems with with 88 X x 88 patches patches fail fail to to diversify diversify and and have 16 X 16 systems, have aa single single cluster cluster present. present. In In contrast, contrast, in in 16 x 16 systems, there there are are on ust over on average average jjust over two two clusters, clusters, whereas whereas in in 32 32 X x 32 32 systems systems this this number number goes goes to to eight. eight. There There is is also also aa number number of of additional additional observations observations valid valid for for both both oneone- and and two two dimensional dimensional systems systems that that follow follow from from numerical numerical simulations simulations similar similar to to those simulations reported those described described here here and and individual-based individual-based simulations reported elsewhere elsewhere ((Gavrilets Gavrilets et 998, 2000b). et ai., al., 11998, 2000b). 9 Clusters Clusters differ differ in in their their "life "life spans. spans."" The The number number of of unsuccessful unsuccessful spe speciation ciation events events (i.e., (i.e., the the number number of of clusters clusters that that are are very very short short lived) lived) is is number of num much larger larger than than the the number of "real" "real" speciation speciation events events (i.e., (i.e., the the nummuch ber ber of of clusters clusters that that exist exist for for aa long long time). time). • 9 The The distribution distribution of of species species range range size size is is right right skewed skewed on on the the linear linear scale scale and and becomes becomes left left skewed skewed on on the the log log scale. scale. These These properties properties are are similar similar to to those those of of the the species species range range distributions distributions estimated estimated from from real real data data (e.g., (e.g., Gaston, 996, 11998). 998). Species Gaston, 11996, Species are are more more likely likely to to break break at at the the center center of of their 998, 2000b). their range range (cf. (cf. Gavrilets Gavrilets et et ai., al., 11998, 2000b). • 9 The The larger larger the the species species range, range, the the more more likely likely it it will will break. break. However, However, because there there are are not not many with very very large large range range sizes, the species species because many species species with sizes, the that that contribute contribute the the largest largest number number of of new new species species are are those those with with inter intermediate mediate range range sizes. sizes. •
Effects Effects of of Adaptation Adaptation The The aforementioned aforementioned results results are are based based on on models models treating treating the the dynamics dynamics of of diversification diversification as as aa neutral neutral process. process. The The justification justification of of this this approach approach was was given given at the the beginning of the the previous section. An An important important question is how how these these at beginning of previous section. question is results results will will be be affected affected by by adaptation adaptation that that is is expected expected to to take take place place simultan simultaneously eously with with diversification. diversification. One One simple simple qualitative qualitative approach approach is is to to consider consider the the expected assumed expected effects effects of of adaptation adaptation on on the the parameters parameters vv and and 88 (which (which were were assumed to to be be constant constant in in the the previous previous subsection). subsection). Adaptation Adaptation to to local local conditions conditions can can be be controlled controlled by by loci loci different different from from the loci underlying loci can can overlap the loci underlying reproductive reproductive isolation, isolation, or or the the two two sets sets of of loci overlap partially partially or or completely. completely. Let Let us us first first assume assume that that the the two two sets sets of of loci loci are are
296 296
SERGEY SERGEYGAVRILETS GAVRILETS
completely Then adaptation completely different. different. Then adaptation to to local local conditions conditions is is expected expected to to result result in in decreasing decreasing the the rate rate of of local local extinction extinction 0~ and and is is not not expected expected to to affect affect the the rate rate of of fixation fixation of of new new alleles alleles v in in the the genes genes underlying underlying reproductive reproductive isolation. isolation. A A consequence consequence of of these these changes changes will will be be an an increase increase in in the the equilibrium equilibrium level level of of species species diversity diversity S. S. The The turnover turnover rate rate T T will will not not be be affected affected or or will will decrease decrease somewhat. somewhat. Next Next assume assume that that the the two two sets sets of of loci loci overlap. overlap. Now Now the the process process of of fixation fixation of neutral anymore. early stages of mutant mutant alleles alleles will will not not be be neutral anymore. At At the the early stages of of adaptation adaptation (and diversification), (and diversification), one one expects expects many many possible possible directions directions for for evolution. evolution. However, However, as as the the clade clade as as aa whole whole rises rises higher higher and and gets gets closer closer to to aa ridge ridge of of high high fitness values in become more fitness values in the the adaptive adaptive landscape, landscape, one one expects expects that that it it will will become more and and more more difficult difficult to to find find mutations mutations increasing increasing adaptation adaptation further. further. This This will will result level, expected result in in aa decrease decrease in in the the rate rate of of fixation fixation vv from from aa high high level, expected when when mutations are level, expected mutations are mutations are adaptive, adaptive, to to aa lower lower level, expected when when mutations are nearly nearly neutral. neutral. As As before, before, adaptation adaptation is is expected expected to to result result in in decreasing decreasing the the rate rate of of local local extinction extinction O. 8. The The effects effects of of aa simultaneous simultaneous decline decline in in vv and and 0~ on on the the clade clade diversity diversity will will depend depend on on which which parameter parameter has has experienced experienced aa larger larger change. change. A A larger larger change change in in vv than than in in 0~ will will result result in in aa drop drop in in the the species species diversity diversity S. S. A smaller smaller change change in in v than than in in 08 will will result result in in increasing increasing S. S. In In both both cases, cases, one one A expects expects aa decrease decrease in in the the turnover turnover rate rate T. T. These These conclusions conclusions are are preliminary preliminary and and more more concrete concrete modeling modeling work work is is definitely definitely necessary. necessary.
11 22.4 .4
DISCUSSION D I S C U S S I O N AND A N D CONCLUSIONS CONCLUSIONS The The dynamics dynamics of of speciation speciation and and diversification diversification in in spatially spatially explicit explicit systems systems undergoing local extinction been aa subject subject of undergoing frequent frequent local extinction and and recolonization recolonization have have been of several recent theoretical studies (Bramson 996; Durrett Levin, 11996; 996; several recent theoretical studies (Bramson et et aI., al., 11996; Durrett and and Levin, Allmon 998; Pelletier, 999; Hubbell, 1 ) . In Allmon et et aI., al., 11998; Pelletier, 11999; Hubbell, 200 2001). In describing describing speciation, speciation, these these studies approaches, which postulated that studies used used heuristic heuristic approaches, which postulated that new new species species emerge emerge with with certain probability of certain probabilities probabilities and and at at certain certain population population densities. densities. Both Both the the probability of speciation speciation and and the the number number of of individuals individuals in in the the new new species species have have been been shown shown to to be be very very important important in in controlling controlling various various aspects aspects of of the the diversification diversification process. process. However, However, the the heuristic heuristic nature nature of of the the approaches approaches used used did did not not allow allow one one to to uncover uncover the characteristics and the relationships relationships between between these these characteristics and microevolutionary microevolutionary processes. processes. The major goal The major goal of of this this study study was was to to develop develop more more general general approaches approaches in in which which speciation explicitly rather rather than speciation is is modeled modeled explicitly than heuristically. heuristically. The the The approach approach adapted adapted here here is is based based on on aa multilocus multilocus generalization generalization of of the classic classic two-locus two-locus two-allele two-allele Bateson-Dobzhansky-Muller Bateson-Dobzhansky-Muller model. model. In In the the BDM BDM model, model, reproductive reproductive isolation isolation is is reduced reduced to to aa single single "incompatibility" "incompatibility" of of two two alleles different loci. This incompatibility alleles at at two two different loci. This incompatibility is is manifested manifested in in aa reduced reduced fitness viability (in (in the case of fitness component component such such as as an an individual's individual's viability the case of postmating postmating reproductive reproductive isolation) isolation) or or the the probability probability of of mating mating between between two two parental parental forms (in the the BDM model, forms (in the case case of of premating premating reproductive reproductive isolation). isolation). In In the BDM model, reproductive isolation evolves reproductive isolation evolves as as aa by-product by-product of of genetic genetic divergence, divergence, which which can be random can be driven driven by by any any of of the the evolutionary evolutionary factors, factors, such such as as mutation, mutation, random genetic genetic drift, drift, selection selection for for adaptation adaptation to to local local biotic/abiotic biotic/abiotic environment, environment, and and sexual selection. model provides provides aa way for the the (sub)population sexual selection. The The BDM BDM model way for (sub)population to to avoid avoid any any adaptive adaptive valleys valleys on on its its was was to to aa state state of of (complete) (complete) reproductive reproductive
1 2. 12.
SPECIATION IN IN METAPOPULATIONS METAPOPULATIONS SPECIATION
2297 91
isolation, as as the the (sub)population (sub)population evolves evolves along along aa ridge ridge of of high high fitness fitness values values in in isolation, the corresponding corresponding holey holey adaptive adaptive landscapes. landscapes. Although Although the the genetic genetic archiarchi the tecture implied implied by by the the BDM BDM model, model, which which results results in in aa ridge ridge of of high high fitness fitness tecture genotypes, genotypes, may may appear appear to to be be rather rather specific specific and and uncommon, uncommon, theoretical theoretical studies of of multidimensional multidimensional adaptive adaptive landscapes landscapes strongly strongly suggest suggest that that itit should should studies be widespread widespread (Gavrilets ( Gavrilets and and Gravner, Gravner, 1997; 1 997; Gavrilets, Gavrilets, 1997; 1 997; Reidys, Reidys, 1997; 1 997; be Reidys et et al., aI., 1997). 1 997). These studies have shown that neutral and and nearly nearly neutral neutral Reidys These studies have shown that neutral divergence along along the the corresponding corresponding holey holey adaptive adaptive landscapes landscapes can can lead lead to to divergence strong reproductive reproductive isolation. isolation. strong This chapter chapter developed developed aa model model representing representing aa straightforward straightforward multilocus multilocus This generalization of of the the BDM BDM model model for for the the case case when when complete complete reproductive reproductive generalization ( 2:: 1 ) of of incompatibilities incompatibilities between between sets sets of of kk (---2) ( 2:: 2 ) isolation requires requires aa number number C isolation C (>-1) loci. The The adaptive adaptive landscape landscape underlying underlying this this model model belongs belongs to class of holey loci. to aa class of holey adaptive landscapes. landscapes. The The accumulation accumulation of of reproductive isolation in in the the model model adaptive reproductive isolation is characterized characterized by by the the "threshold "threshold effect": effect": as as genetic genetic distance distance between between the the two two is parental forms certain value, value, the the strength strength of of reproductive reproductive isolation isolation parental forms exceeds exceeds aa certain undergoes aa rapid to high. This transition transition is is especially especially undergoes rapid transition transition from from low low to high. This rapid if if both and kk are are large. large. This has been been used used previously previously to to rapid both C C and This property property has explore various various features speciation in two stable explore features of of the the dynamics dynamics of of speciation in systems systems of of two stable populations Gavrilets, 1999a, 1 999a, 2000), 2000), in one- and two-dimensional steppingstepping populations ((Gavrilets, in oneand two-dimensional stone with stable stable populations Gavrilets, et aI., 1998, 1 998, 2000b), 2000b), and stone systems systems with populations ((Gavrilets, et al., and in one-dimensional aI., 2000a). 2000a). This section in one-dimensional metapopulations metapopulations (Gavrilets (Gavrilets et et al., This section consideres two-dimensional metapopulations, metapopulations, paying paying special special consideres both both oneone- and and two-dimensional attention structure of diversifying clade. clade. Speciation and attention to to the the cluster cluster genetic genetic structure of the the diversifying Speciation and diversification were modeled modeled as continuous process process of mutation accumulaaccumula diversification were as aa continuous of mutation tion tion accompanied accompanied by by the the generation generation of of new new genetic genetic clusters clusters and and contractions contractions or or expansions expansions of of their their ranges. ranges. The The main main motivation motivation was was to to get get aa better better understanding understanding of of the the processes processes following colonization of following colonization of aa new new environment environment or or appearance appearance of of aa new new key key innovation. innovation. This This section section focuses focuses both both on on the the properties properties of of the the transient transient dyna dynamics mics of of diversification diversification and and on on the the characteristics characteristics of of the the long-term long-term stochastic stochastic equilibrium. equilibrium. Currently, Currently, empirical empirical data data on on the the dynamical dynamical features features of of (adaptive) (adaptive) diversification Schluter, 2000; best data diversification are are scarce scarce ((Schluter, 2000; Section Section 3.5), 3.5), with with the the best data coming 980; Foote, 992, 11999; 999; Erwin, coming from from the the fossil fossil record record (e.g., (e.g., Valentine Valentine 11980; Foote, 11992, Erwin, 1994; 995; Lupia, Lupia, 11999). 999). A 1994; Wagner, Wagner, 11995; A number number of of potentially potentially important important gener generalizations alizations have have emerged emerged from from the the analyses analyses described described here. here. i.i. The The waiting waiting time time to to the the beginning beginning of of radiation radiation tb t6 increases increases with with decreasing decreasing the the fixation fixation rate rate per per locus locus f.L ~ and and increasing increasing the the number number of of genetic changes necessary for speciation K. K. The local extinction/recol extinction/recolonization onization rate rate B, 8, the the dimensionality dimensionality of of the the system, system, and and the the number number of of patches patches nn have have much much smaller smaller effects. effects. In In numerical numerical simulations, simulations, the the genwaiting time to the beginning of radiation was on the order of 1103 03 gen erations for K 2) to 05 generations for large erations ((for K= -- 11 and and 2) to 1105 generations ((for large K). K). ii. The duration duration of radiation radiation td td depends mostly on the fixation rate f.L. ~. tb t6 increases increases weakly weakly with with K K and and system system size size nn and and decreases decreases weakly weakly with with B. 8. The The duration duration of of radiation radiation is is longer longer in in one-dimensional one-dimensional systems systems than in two-dimensional two-dimensional systems. In numerical simulations the order t6 ranges from 110 044 to tO 105 10 5 generations. of tb
298 298
SERGEY SERGEYGAVRILETS GAVRILETS
iii. iii. The The transient transient dynamics dynamics of of the the diversity diversity SS (i.e., (i.e., the the number number of of clusters) clusters) and and the the disparity disparity D D (i.e., (i.e., the the average average pairwise pairwise distance distance between between populations populations in in the the clade) clade) are are decoupled decoupled to to aa certain certain degree. degree. At At low low taxonomic taxonomic levels levels (with (with small small K), K), the the diversity diversity increases increases faster faster than than the the disparity, disparity, whereas whereas at at high high taxonomic taxonomic levels levels (with (with large large K), K), the the diversi diversity ty increases increases slower slower than than the the disparity. disparity. This This observation observation explains explains the the difference difference between between the the patterns patterns of of diversification diversification as as observed observed in in the the fossil summarized at fossil record record (which (which are are usually usually summarized at higher higher taxonomic taxonomic levels, 980; Foote, 992, 11999; 999; Erwin, levels, e.g., e.g., Valentine, Valentine, 11980; Foote, 11992, Erwin, 1994; 1994; Wagner, Wagner, 11995; 995; Lupia, 999) and are usually Lupia, 11999) and for for more more recent recent groups groups (which (which are usually summarized at at lower lower taxonomic taxonomic levels, levels, e.g., e.g., Schluter, Schluter, 2000). 2000). summarized IV. iv. The The average average genetic genetic distance distance from from the the species species founder founder increases increases monotonically monotonically at at aa constant constant rate rate controlled controlled by by the the fixation fixation rate. rate. Note Note that 1 999b) used that Gavrilets Gavrilets ((1999b) used this this property property to to develop develop aa method method for for test testthe constancy constancy of of the the rate rate of of evolution evolution and and estimating estimating its its rate rate using using ing the morphological data. v. v. The The clade clade as as aa whole whole keeps keeps changing changing genetically genetically as as it it moves moves along along the the underlying underlying holey holey adaptive adaptive landscape landscape even even after after the the number number of of species species (or (or other other genetic genetic clusters) clusters) has has approached approached an an equilibrium equilibrium level. level. VI. vi. The The general general effects effects of of the the model model parameters parameters on on different different equilibrium equilibrium characteristics population are characteristics of of the the meta metapopulation are mostly mostly as as suggested suggested by by bio biological logical intuition. intuition. For For example, example, diversity diversity increases increases with with mutation mutation rate rate and and decreases decreases both both with with the the local local extinction/recolonization extinction/recolonization rate rate and and the speciation. However, the number number of of genetic genetic differences differences required required for for speciation. However, the the model model predicts predicts counterintuitively counterintuitively that that the the turnover turnover rates rates do do not not depend depend (or weakly depend) on extinction rates and and are controlled mostly mostly by by (or weakly depend) on extinction rates are controlled parameters vv and Intuition is is aa poor poor guidance guidance as far as as the the structure structure parameters and K. K. Intuition as far of genetic clusters clusters in in the the multidimensional multidimensional genotype genotype space space is is of different different genetic concerned. Results of of numerical numerical simulations simulations show that both both the the averaver concerned. Results show that age pairwise distance cluster and the cluster cluster diameter mostly age pairwise distance within within cluster and the diameter are are mostly K and are close close to to its its numerical numerical value. value. controlled by by parameter parameter K controlled and are Vll. Diversification requires that the overall number of of patches patches in in (or spatial vii. Diversification requires that the overall number (or spatial area the system minimum value. value. This This effect may area of) of) the system exceeds exceeds aa certain certain minimum effect may have to the the fact in adaptive radiation of the west Indian have contributed contributed to fact that that in adaptive radiation of the west Indian Anolis speciation occurred occurred only bigger islands, islands, Anolis lizards, lizards, within-island within-island speciation only on on bigger despite that the of spatial heterogeneity does does not not seem seem to to despite the the fact fact that the degree degree of spatial heterogeneity differ between between the the islands islands (Losos, (Losos, 1998). 1 998). In In very large systems, the overover differ very large systems, the all with the all diversity diversity increases increases linearly linearly with the number number of of patches patches (or (or area). area). viii. of spatial Vlll. The The results results presented presented here here show show profound profound effects effects of spatial dimensionaldimensional ity on ity on the the dynamics dynamics of of diversification diversification and and significant significant differences differences between between one-dimensional systems systems (such (such as as describing describing rivers, rivers, shores shores of of lakes lakes and and one-dimensional oceans, and and areas areas at at aa constant in aa mountain mountain range) range) and and twotwo oceans, constant elevation elevation in dimensional systems systems (such (such as as describing describing oceans oceans and and continental continental areas). areas). dimensional a. In general, general, the the characteristics characteristics of two-dimensional systems systems are are (much) (much) a. In of two-dimensional more sensitive to parameter values values than of one-dimensional one-dimensional more sensitive to parameter than those those of by aa factor factor systems. For For example, example, increasing increasing the the local local extinction extinction rate rate 88 by systems. 25 will will typically typically decrease decrease the the species species diversity diversity by by the the same same factor factor in in 25 two-dimensional systems. systems. In In contrast, contrast, in in one-dimensional one-dimensional systems systems the the two-dimensional
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS
299 299
decrease 5. Therefore, decrease will will only only be be by by factor factor 5. Therefore, the the diversity diversity of of one-dimen one-dimensional sional systems systems is is expected expected to to be be more more stable stable over over aa long long period period of of time. time. b. The b. The diversity diversity in in one-dimensional one-dimensional systems systems is is predicted predicted to to be be (much) (much) higher higher than than that that in in two-dimensional two-dimensional systems. systems. For For example, example, let let 100 loci, there there be be L L = = 100 loci, the the mutation mutation rate rate per per locus locus per per generation generation be be = 4 4 X 3< 1100-5, -5, the the local local extinction/recolonization extinction/recolonization rate rate be be 0~ = = 1010 -22 fL = per deme deme per per generation, generation, and and let let there be n n -= 11024 local demes. demes. 024 local per there be Then Then if if the the demes demes are are arranged arranged in in aa 32 32 X x 32 32 square, square, numerical numerical sim simulations 2.6 genetic ulations show show that that there there are, are, on on average, average, 2.6 genetic clusters clusters at at the the clustering clustering level level corresponding corresponding to to K K = = 16. 16. In In contrast, contrast, if if the the demes demes are 322 X are arranged arranged on on aa line line (i.e., (i.e., in in aa 322 3< 1I pattern), pattern), there there are, are, on on aver average, 7.2 such age, 117.2 such clusters. clusters. These These effects effects may may have have contributed contributed to to the the extraordinary extraordinary divergence divergence of of cichlids cichlids in in the the great great lakes lakes of of Africa, Africa, most inhabit the relatively narrow along the most species species of of which which inhabit the relatively narrow band band along the shoreline shoreline (e.g., (e.g., Kornfield Kornfield and and Smith, Smith, 2000). 2000). c. c. Typically, Typically, the the genetic genetic clusters clusters in in one-dimensional one-dimensional systems systems are are denser denser (i.e., (i.e., are are characterized characterized by by smaller smaller values values of of D D and and Dma Dmax) than those those x ) than in in two-dimensional two-dimensional systems. systems. It has has been been argued argued that that species species can can accumulate accumulate rapidly rapidly after after colonizing colonizing ix. It aa new new environment environment if if the the species species in in the the novel novel environment environment have have aa reduced probability happen because because reduced reduced probability of of extinction. extinction. This This could could happen reduced extinction can extinction can extend extend the the lifetime lifetime of of aa lineage, lineage, thus thus increasing increasing its its chance enough genetic genetic changes result in chance to to accumulate accumulate enough changes to to result in reproductive reproductive isolation (Mayr, 963; Allmon, Allmon, 11992; 992; Schluter, 998, 2000) same isolation (Mayr, 11963; Schluter, 11998, 2000).. The The same could also also happen happen after developing aa "key could after developing "key innovation" innovation" decreasing decreasing the the extinction rate. The results presented here quantify these arguments. extinction rate. The results presented here quantify these arguments. As As discussed certain factor discussed earlier, earlier, decreasing decreasing the the extinction extinction rate rate by by aa certain factor in in aa two-dimensional will increase two-dimensional metapopulation metapopulation will increase the the equilibrium equilibrium diver diversity by approximately the same factor. sity by approximately the same factor.
IX.
The based on model and The afotementioned afotementioned conclusions conclusions are are based on aa specific specific model and certain certain cautiousness real cautiousness is is required required when when trying trying to to apply apply them them to to more more general general and and realistic istic situations. situations. A A number number of of directions directions must must be be pursued pursued in in order order to to evaluate evaluate the the generality generality of of the the results results presented presented here. here. Here Here the the spatial spatial arrangement arrangement of of demes demes in in aa two-dimensional two-dimensional system system was was (unrealistically) (unrealistically) symmetric. symmetric. Allowing Allowing for for some some demes demes to to be be unsuitable unsuitable is is expected expected to to increase increase the the possibilities possibilities for for differentiation differentiation and and speciation speciation in in both both one oneand and two-dimensional two-dimensional systems. systems. However, However, if if space space were were continuous continuous rather rather than than discrete, discrete, these these possibilities possibilities would would be be reduced reduced significanly significanly in in two-dimensional two-dimensional systems. systems. To To achieve achieve more more realism, realism, one one needs needs to to account account for for the the effects effects of of migration migration into into occupied patches and Migration is expected to occupied patches and the the resulting resulting gene gene flow. flow. Migration is expected to make make splitting splitting of of the the population population into into different different clusters clusters much much more more difficult. difficult. The The big big question question is is how how the the characteristics characteristics studied studied here here scale scale with with the the migration migration rate. rate. The The common common wisdom wisdom is is that that migration migration rates rates on on the the order order of of one one immigrant immigrant per per population population per per generation generation are are sufficient sufficient to to prevent prevent any any significant significant diver divergence neutral alleles. proof that gence in in neutral alleles. One One can can be be tempted tempted to to interpret interpret this this as as proof that speciation speciation will will not not be be possible possible either. either. However, However, this this interpretation interpretation is is not not
300 300
SERGEY SERGEY GAVRILETS GAVRILETS
necessarily ustified because not account necessarily jjustified because it it does does not account for for the the possibility possibility of of large large fluc fluctuations genetic distances between neighboring tuations in the genetic distances between neighboring populations, populations, which can lead Gavrilets, 2000). lead to to reproductive reproductive isolation isolation ((Gavrilets, 2000). Numerical Numerical individual-based individual-based simulations allowed) show simulations (with (with no no extinction extinction allowed) show that that speciation speciation by by random random drift drift and mutation the order and mutation is is possible possible even even if if migration migration rates rates are are on on the order of of several several immigrants Gavrilets et 998, 2000b). immigrants per per population population per per generation generation ((Gavrilets et aI., al., 11998, 2000b). A A simple approach account for within the simple approach to to account for the the effects effects of of migration migration within the framework framework used locus from used here here is is to to adjust adjust the the probability probability of of mutation mutation in in aa locus from /-L ~ to to
/-LIt,ee ==
p, + q/-L
re.M, m N,
where Nis the where m m is is the the rate rate of of migration migration and and A/'is the number number of of neighboring neighboring populations populations that allele fixed consideration. The that have have the the alternative alternative allele fixed at at the the locus locus under under consideration. The afore aforementioned utilizes the fixation of neutral mentioned expression expression utilizes the fact fact that that the the probability probability of of fixation of aa neutral allele is equal to are brought the allele is equal to its its frequency. frequency. With With migration, migration, new new alleles alleles are brought in in the /-L) and migration migration (at (at rate m N) . In this approxi patch both by mutation (at rate ~) mutation (at mH). approximation, alleles that mation, the the only only role role of of migration migration is is to to bring bring in in new new alleles that are are fixed fixed quickly quickly or example, if population or lost lost by by random random genetic genetic drift. drift. For For example, if initially initially both both the the population under under consideration consideration and and its its four four neighbors neighbors have have allele allele 00 at at the the locus locus under under con consideration, probability that sideration, then then the the probability that an an alternative alternative allele allele 11 is is fixed fixed is is /-L ~ ee- -= /-L. ~" However, However, once once this this has has happened, happened, the the probability probability of of switching switching back back to to allele allele 00 is is /-L~ee = /-L~ ++ 44m. If the the migration migration rate rate is is larger larger than than the the mutation mutation rate rate per per locus, locus, m . If switching back will happen happen much faster. However, because there there are many genes and and many many populations, populations, the the accumulation accumulation of of enough enough genetic genetic differences differences may may even eventually tually take take place, place, resulting resulting in in the the splitting splitting of of the the system system into into different different clusters. clusters. Also, multiple popula Also, it it is is necessary necessary to to consider consider the the effects effects of of allowing allowing for for multiple populations per demes. A simple approach approach for for doing this is to introduce introduce another another threshold (> K), reaching threshold genetic genetic distance, distance, say say Kcomp Kcomp(> reaching which which will will allow allow for for coexistence coexistence in in aa deme. deme. If If the the genetic genetic divergence divergence is is below below the the threshold, threshold, the the competition coexistence. In competition between between different different species species prevents prevents their their coexistence. In this this case case the the expected expected evolutionary evolutionary dynamics dynamics will will consist consist of of aa series series of of parapatric parapatric splits splits followed followed by by range range expansions expansions and and an an increase increase in in the the number number of of populations populations per deme after accumulating accumulating enough enough genetic differences. These two elsewhere (M. These two generalizations generalizations are are discussed discussed elsewhere (M. Saum Saum and and S. S. Gavrilets, Gavrilets, unpublished unpublished results). results). It It is is also also important important to to introduce introduce spatial spatial heterogeneity heterogeneity of of selection selection into into the the modeling modeling framework. framework. This This heterogeneity heterogeneity is expected to affect significantly both both the probabilities probabilities of fixation fixation and the overall dynamics Ohta, 11972; 972; Eldredge, Eldredge, 2003; overall dynamics of of diversification diversification ((Ohta, 2003; Gavrilets Gavrilets and and Gibson, 2002). most importantly, needs to analyze the effects 2002). Finally, and most importantly, one needs of of adaptation adaptation explicitly. explicitly.
APPENDIX 2. 1 APPENDIX 112.1 Derivation 2.1 2b) Derivation of of Eq. Eq. (1 (12.12b) Sawyer's 1977a) Eq. Sawyer's ((1977a) Eq. (3.2) (3.2) describes describes the the probability probability l(r, I(r, v) v) that that two two genes genes found found aa distance distance r apart apart are are the the same same type type in in an an infinite infinite allele allele selectively selectively neutral migration-mutation-random model with mutation rate neutral migration-mutation-random drift drift model with mutation rate v: v:
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS n �)
r
301 301
=
Ko(q(x)(2u)l/2) 2 Ko ( q(x)(2u) 1I2 )
,
I(r,v) ~ In( In(1/2u) 4~[2Ncrlcr2 + Co Co]' ] [ 2NCTICT2 + 1I2u ) + 41T
where . ) is where Ko( K0(') is the the Bessel Bessel function function of of the the second second type type of of order order zero. zero. In In terms terms of of our our model, model, 2N 2N = = 11 (there (there is is aa single single sequence sequence per per colony) colony) and and u u = -- vv (which (which is is the the mutation mutation rate rate per per sequence sequence per per generation). generation). The The probability probability that that aa sequence sequence in in aa given given site site is is substituted substituted by by the the sequence sequence from from aa neighboring neighboring site site is 0/4 (where 08 is is 8/4 (where is the the probability probability of of site site extinction, extinction, and and factor factor 114 1/4 because because there there are 44 neighbors each of can colonize extinct site). are neighbors each of which which can colonize the the extinct site). Therefore, Therefore, Sawyer's Eq. ((5.1)] 5. 1 )] is equal to similar way, Sawyer's (r2 = ml [see [see below below his his Eq. is equal to 0/2 8/2 and, and, in in aa similar way, 0/2. Using Using Sawyer's given below his Eq. Eq. (3.4), CT cr�2 = - m2 = - 8/2. Sawyer's expression expression for for Co Co given below his (3.4), Thus, we can rewrite rewrite (2/0)xI2 + Coo = = ((1/4~r) 1n(8/2). Finally, Finally, q q(x) + (2/0)x (2/8)x �2.. Thus, we can 1141T) In(0/2). (x) = X/(2/8)x C the v) as the equation equation for for I(r, v)as
m2
=
CTT ml
i
=V
i
( Vyj yi) , = = In(ln(1/2v) 1/2v ) + 21T0 + In( 0/2 ) In( 0/4v ) + + 2~r~ +1n(8/2) ln(8/4v) + 21T0 2~r8
II(r,v) ( r, v ) ~
yi
yi
(
)
2Ko(~/4~x21 v x�2 2KO 4 xj + 4 4~x2)
+ y2) 2Ko( 2K o X/y2 +
,
=
where y2 = - 4� 4~ x x i2 ,, y2 = - 4� 4~ x x i2 ·. The The average average range range of of aa cluster cluster is is where R ==n X R n A
II
11 I( r, V)dXldx2 I( r' vv ) dXl dX2, X f f I( r' vn )dx~dx2 == j f I(r,
where n system. Then where n is is the the number number of of sites sites in in the the system. Then
R= =
Ir' Ko (VYT + YO dx1dx2 dY ldY2. [KoKo(V'y~ ( VYT ++ Y�) 2) ++ 21T0 II0 g / 4 v� 2'rrg I y~)dyldy2. :4vv In(l n (0/4v �
0/4V + 21T0 In( ln(g/4vi + 2~rg
e
f;Ko (r)rdr
=
Finally, one Finally, using using polar polar coordinates coordinates rr and and 0 and and the the fact fact that that foKo(r)rdr = 11,, one finds finds that that
I J OO
fro" 0
Ko
( VYT Y�)dYl dY2
221
+ y dyldy2 = =
JfoOO(1r|J /2Ko( rr)rdrdO 1T/2 ) = )rdrde == 11 Xx ((~r/2) = 1T/2, ~r/2, 0 "o0
which Eq. ((12.12b). 12. 12b). which leads leads to to Eq.
Derivation 2. 1 2c) Derivation of of Eq. Eq. (1 (12.12c)
The The expected expected number number R R of of the the populations populations that that belong belong to to the the same same cluster cluster as as the the population population at at patch patch 0 0 is is estimated. estimated. Looking Looking back back in in time, time, demes demes 0 0 and and x x can can be be traced traced to to aa single single founding founding deme deme 'To "r0,x generations ago. ago. The The coalescence coalescence ,x generations x is time time 'To "rO,x is aa random random variable. variable. These These two two demes demes belong belong to to the the same same cluster cluster if if �ve accumulated x generations ago. they they h have accumulated less less than than K mutations mutations since since time time 'To $0,x generations ago. , With With small small vv the the process process of of mutation mutation accumulation accumulation is is approximately approximately Poisson Poisson
K
302 302
SERGEY SERGEY GAVRILETS GAVRILETS
2V%,x.
and the the expected and x is 2VTo,x' The and expected number number of of mutations mutations separating separating demes demes 00 and x is The latter equation equation assumes assumes that fixation results in a genotype genotype that comlatter that each fixation that is com pletely Poisson random pletely new new to to the the system. system. X(A) X(;~) is is used used to to denote denote aa generic generic Poisson random variable with parameter parameter A. ;~. Therefore, Therefore, variable with R populations at /~ = = 2: ~] Pr( Pr(populations at 00 and and x x differ differ by by
xx
than K substitutions less than substitutions
= Tx,o = = 2: 2~2 2: ~2 Pr( Pr(~x,O = tt ))PPr( r ( XX(A) (k) xx tt =
[
� � Pr( Tx,o
=
]
< < K K ))
t ) Pr( X( A )
x
< K),
((12.14a) 1 2. 1 4a) (12.14b) (12. 14b)
((12.14c) 1 2.14c)
where vt and where A~ = - 2 2vt and the the sums sums are are taken taken over over all all demes demes and and over over all all possible possible coa coalescence lescence times. times. can be written The The probability probability Pr(X(A) Pr(X(k) < < K) K)can written as - 1..Ai �1 e-aMi!
Pr(X(;~) < K K)=/__~1 X( A) < ) = Pr( i=
. ,� e� � = t .
f( K' AX)) F(K, f( K) F(K)
((12.15) 12.15)
Gavrilets, 11999a; Gavrilets et aI., al., 2000a). 2000a). (e.g., Gavrilets, 999a; Gavrilets The 1 2. 14c) can be approximated The sum sum in the square square brackets brackets in Eq. ((12.14c) approximated by the the derivative respect to can be derivative of of 2: ~ xx Pr(Tx,o Pr('rx,0 :::; - tt)) with with respect to t.t. The The latter latter sum sum can be approxi approximated mated as as 'ITot -rrgt ((12.16) 12.16) Tx o :::; ~'" Pr( Pr('rx,0 -< tt)) "" ~ 1 In(t) n(t ) � ' x -
for large t (e.g., Kelly, 1977; 979; Bramson 980), 1977; Sawyer, 11979; Bramson and and Griffeath, Griffeath, 11980), leading to to an an approximation approximation leading ~r8 ~Pr(~rX'~ ' = t ) ~ In(t)
((12.17) 12.17)
for for large t. Therefore, Therefore,
R/~ ""~ ~2:l
2vt)
-rr8 f( F(K, 'ITO K, 2vt ) t ) f( In(t) F(K) t> K) tT1 ln(
((12.18a) 12.18a)
'IToK K, 2vt) -rrgK t>~l 11 f( F(K, 2vt)2v 2v "" b t ) f( 2v t ln( In(t) F ( K+ K + I1) )
((12.18b) 12.1 8b)
~rgK "" 'IToK
((12.18c) 12.18c)
�
where where
I
x
f( K, A ) f4 ~ F(K, X) ).dx dA cI) = 4vv f( F(K ln(k/2v K + 1 ) In( AI2v )
=
112. 2. SPECIATION IN METAPOPULATIONS SPECIATION IN METAPOPULATIONS TABLE 2. 1 TABLE 112.1 v v
-4 110 0 -4 4 x x 1100-4 -4 4 1166 x x 1100-4 -4
303 303 Values Valuesooff r K K == 2 2
K K ==4 4
K K ==8 8
K K == 116 6
0.122 0 . 1 22 0.148 0.148 0 .187 0.187
0.113 0 .1 13 0.135 0.135 0.167 0 .167
0.105 0 . 1 05 0.124 0 . 1 24 0 . 1 50 0.150
0.098 0.098 0.114 0 . 1 14 0 .136 0.136
Note Note that that x. X= = 4v 4v corresponds corresponds to to tt = = 2. 2. Table Table 12.1 12.1 was was found found by by evaluating evaluating numerically. arguments is numerically. Table Table 12. 12.11 shows shows that that the the dependence dependence of of 9 on on its its arguments is weak. weak.
sdfsdf
Part V Integration and Applications
sdfsdf
I
@
CAUSES , CAUSES, MECH ANISMS MECHANISMS AND CONSEQUENCES OF DIS PERS AL DISPERSAL Jean Clobert, Rolf Anker Ims, and Francois Fran<;ois Rousset
113.1 3. 1
INTRODUCTION INTRODUCTION The movement (or dispersal) of propagules propagules among among suitable suitable patches patches of habi habiThe tats population dynamics. tats is is an an essential essential ingredient ingredient of of meta metapopulation dynamics. At At the the birth birth of of the the meta population concept, 1 970) only metapopulation concept, Levins Levins ((1970) only considered considered colonization, colonization, (i.e., (i.e., movement movement to to empty empty patches) patches).. However, However, all all patches patches within within aa metapopulation metapopulation are are to to some some extent extent exchanging exchanging individuals individuals due due to to dispersal, dispersal, even even those those which which are are already already occupied. occupied. This This phenomenon phenomenon leads leads to to aa reenforcement reenforcement of of extant extant local 9 77). local populations populations (the (the rescue rescue effect; effect; Brown Brown and and Kodrick-Brown, Kodrick-Brown, 11977). This This chapter chapter focuses focuses on on condition-dependent condition-dependent dispersal dispersal because because we we feel feel it it is is important important to to take take condition condition dependence dependence into into account account to to make make realistic realistic predic predictions tions about about dispersal dispersal evolution evolution and and its its consequences. consequences. In In Levins' Levins' original original model model and and some some subsequent subsequent extensions extensions of of it, it, dispersal dispersal was was mostly mostly considered considered as as aa metapopulation had the same probabil probabilfixed trait (i.e., any individual in the metapopulation ity ity of of dispersing dispersing successfully). successfully). In In more more recent recent developments developments of of metapopulation metapopulation theory, theory, dispersal dispersal has has been been considered considered to to be be function function of of the the density density in in the the patch patch of of departure departure and and other other features features such such as as the the patch patch size size and and the the distance distance between between
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Copyright Copyright 2004, 2004, Elsevier, Elsevier,Inc. Inc. 0-12-323448-4 0-12-323448-4
308 31)8
JEAN JEAN CLOBERT CLOBERTET ET AL. AL.
patches 999b). In patches (Hanski, (Hanski, 11999b). In addition addition to to these these attempts, attempts, dispersal dispersal was was largely largely considered considered to to be be unconditional unconditional of of the the status status of of the the individual, individual, the the potential potential donor, donor, and and recipient recipient patches, patches, as as well well as as the the matrix matrix between between patches. patches. fixed trait in popula populaSimilarly, migration has been mostly considered as a fixed genetic models (Barton et al., 2002; Chapter 8). There are some excep exception genetic tions where dispersal is allowed to vary with some aspects of individual condition, 9 8 1 ; Chesser, condition, such such as as sex sex or or age age (Prout, (Prout, 11981; Chesser, 1991; 1991; Rousset, Rousset, 1999b), 1999b), or or with deme the and Barton, 997; with the the deme the individual individual is is belonging belonging to to (Whitlock (Whitlock and Barton, 11997; Rousset, Rousset, 2004), 2004), but but these these do do not not consider consider the the evolution evolution of of dispersal dispersal and and its its demographic 1 966) for demographic consequences consequences [see [see Maynard Maynard Smith Smith ((1966) for an an exception]. exception]. The The origin origin of of dispersal dispersal as as aa behavioral behavioral trait trait at at the the level level of of individuals individuals and and population level consequences were not fully recognized before a popula populaits population experition ecologist, Charles Krebs et al. demonstrated in 11969, 969, by an elegant experi ment, ment, that that population population dynamic dynamic of of voles voles was was affected affected dramatically dramatically when when individuals individuals were were prevented prevented to to move move freely freely (the (the fence fence or or Krebs Krebs effect) effect).. Krebs Krebs himself component by invoking the himself emphasized emphasized this this phenomenon's phenomenon's behavioral behavioral component by invoking the term 969). Lidicker Lidicker ((1962) 1 962) also also pointed pointed out term spacing spacing behavior behavior (Krebs (Krebs et et al., al., 11969). out early phenomenon by early on on that that dispersal dispersal was was most most probably probably aa complex complex phenomenon by identi identifying saturation dispersal. fying two two types types of of movement: movement: presaturation presaturation and and saturation dispersal. Somewhat 1 980, Somewhat later later the the field field of of behavioral behavioral ecology ecology emerged. emerged. Greenwood Greenwood ((1980, 1983) 1983) used used aa comparative comparative approach approach based based on on what what was was known known about about disper dispersal conclude that sal in in birds birds and and mammals mammals at at this this time time to to conclude that mating mating system, system, resources levels, and resources levels, and inbreeding inbreeding were were the the forces forces shaping shaping sex-specific sex-specific natal natal and and breeding dispersal (see (see also also Greenwood 982). The breeding dispersal Greenwood and and Harvey, Harvey, 11982). The evidence evidence that that individual crowding and individual departure departure from its natal site was dependent dependent on local crowding that individuals individuals were were not not choosing choosing to to settle settle in in aa new new habitat habitat at at random random also also that started to to accumulate accumulate (Lambin et al., 2001 ; Kokko Kokko et et al., al., 2001). 200 1 ). From an evo started (Lambin et al., 2001; From an evolutionary (and (and theoretical) theoretical) viewpoint, viewpoint, many many biotic biotic and and abiotic abiotic factors factors were were lutionary identified as as potential potential causes causes for for dispersal dispersal evolution (reviewed in in Clobert Clobert et et al., identified evolution (reviewed al., 200 1 ) . However, However, up up to recently, there there has no comprehensive 2001). to recently, has been been no comprehensive consider consideration of the evolution of aa state-dependent state-dependent dispersal, dispersal, whereas whereas the available ation of the evolution of the available for such such a theory. empirical evidence was was pleading pleading for In the the last last two two decades, decades, dispersal has been been subject subject to to renewed renewed interest. interest. In dispersal has At least least five books have have been been produced produced on on this this subject subject (Stenseth (Stenseth and Lidicker, At five books and Lidicker, 1992; al., 2001; et al., al., 2001; et aL, al., 1 992; Dingle, Dingle, 1996; 1 9 96; Clobert Clobert et et al., 2001 ; Woiwood Woiwood et 200 1; Bullock Bullock et 2002), and the number number of of papers, papers, especially especially theoretical, theoretical, has has increased increased 2002), and the markedly. Indeed, Indeed, understanding understanding why why and and how how animals animals and and plants plants are are moving moving markedly. has become become of of prime prime importance, importance, especially especially if if we we want want to to predict predict what what will will be be has the the result result of of habitat habitat fragmentation fragmentation and and global global changes. changes. It emerges emerges from from these new bodies bodies of of empirical and theoretical theoretical studies studies that that It these new empirical and previous assumptions assumptions about about dispersal dispersal modeling modeling were were far far too too simple. simple. Then, Then, the the previous question of of how how much much details details on on aa species' species' dispersal dispersal ecology ecology must must be be known known to to question predict predict metapopulation meta population dynamics dynamics and and evolution evolution is is still still largely largely unknown. unknown. For For instance, under under which which circumstances circumstances can can we we consider consider dispersal, dispersal, at at least least practicpractic instance, ally, as as the the kind kind of of random random process process usually usually assumed assumed in in models models of of metapopulametapopula ally, tions? To what extent extent are are dispersal dispersal patterns, patterns, in in term term of of dispersal dispersal distances distances and and tions? To what rates, molded molded by by the the cause cause of of dispersal? dispersal? To what extent extent are are patch patch settlement settlement rates, To what decisions conditional conditional on on causes causes of of departure/emigration? departure/emigration? Are Are the the effects effects of of decisions dispersal on on local local patch-specific patch-specific dispersal dispersal proportional proportional to to the the fraction fraction of of dispersal
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individuals individuals leaving leaving and and arriving? arriving? When When can can the the matrix matrix between between habitat habitat patches patches which which organisms organisms must must disperse disperse through through be be considered considered as as neutral? neutral? Because Because every every movement movement might might be be undertaken undertaken for for aa different different reason reason and and may may modify modify both local conditions departure and arrival, it both the the local conditions of of the the patch patch of of departure and arrival, it is is not not intui intuitively tively obvious obvious to to predict predict its its overall overall impact impact on on the the ecological ecological and and evolutionary evolutionary dynamics dynamics of of the the metapopulation. metapopulation. Indeed, Indeed, the the multiplicity multiplicity of of causes causes and and poten potential tial feedback feedback effects effects demonstrated demonstrated by by some some new new empirical empirical studies studies strongly strongly com complicates plicates the the issue issue of of the the role role of of dispersal dispersal in in metapopulations. metapopulations. To To make make the the topic topic tractable tractable for for this this chapter, chapter, we we will will restrict restrict ourselves ourselves to to consider 1 982) for consider natal natal and and breeding breeding dispersal dispersal [see [see Greenwood Greenwood and and Harvey Harvey ((1982) for rather than definitions]. We use use the term term of dispersal rather than migration for reasons given given in in Clobert Clobert et et al. al. (2001 (2001),) , but but both both terms terms have have been been used used for for describing describing the the same phenomenon throughout book. Although same phenomenon throughout this this book. Although sometimes sometimes important important to to consider population framework, consider in in aa meta metapopulation framework, other other kinds kinds of of movements movements (e.g., (e.g., feeding feeding migrations) migrations) are are not not discussed discussed here. here. Due Due to to our our background background there there will will be be an an inherent inherent zoological zoological bias bias in in our our perspectives perspectives on on dispersal. dispersal. In In order order to to discuss discuss the the consequences consequences of of dispersal dispersal on on metapopulation metapopulation dynamics dynamics and and evolution, evolution, we we will will first first review review the the potential potential causes causes of of dispersal dispersal evolution. evolution. The The many many factors factors proposed proposed to to promote promote the the evolution evolution of of dispersal dispersal ((Comins Comins et 3.1). et aI., al., 1980) 1980) can can be be grouped grouped in in three three categories categories (Fig. (Fig. 113.1). 11.. Habitat-specific Habitat-specific factors. factors. All All abiotic abiotic and and biotic biotic factors factors that that are are not not intrin intrintemsic to the organism itself or conspecific individuals. individuals. Examples are tem perature, parasites, and perature, food, food, predators, predators, parasites, and interspecific interspecific competitors. competitors. 2. Factors choice (i.e., 2. Factors related related to to mate mate choice (i.e., inbreeding, inbreeding, mating mating system) system).. Although Although constituting constituting aa special special case case ooff social social factors factors (see (see later), later), we we will will
Social factors
h,
kin interactions asymmetric competition mating system system cooperation
Global Globalequilibrium equilibrium between costs and benefits �
\\ \\ \\
o
_
\\
� ,
~
Benefits Benefits exceed exceed costs costs
Costs exceed b enefits nefits
Ecological factors
Genetic factors
o0
demographic demographic stochasticity temporal heterogeneity heterogeneity 9 temporal spatial heterogeneity
/ ~
inbreeding depression 0 outbreeding outbreeding depresssion depres7
Fig. 3.1 Multiple Fig. 113;.1 Multiple causes that that act act on dispersal evolution. evolution. To each each cause cause is associated associated some cost cost and benefits, the zero zero on the axes axes symbolizing the point where costs and benefits balance each result from all these selection selection pressures each other. other. The evolution of dispersal dispersal may result pressures act acting together, depending depending on how the movement called called dispersal dispersal can be considered as as one well welldefined trait or uncover distinct behaviors behaviors under the control of different different mechanisms. mechanisms. Modified Modified from Clobert et al. (1 994) and J. J. F. (1994) F. Le Le Galliard (unpublished results).
JEAN JEANCLOBERT CLOBERTET ETAL. AL.
310 3 10
treat treat aspects aspects of of mate mate choice choice separately, separately, as as they they have have been been seen seen as as aa major major force force driving driving the the evolution evolution of of dispersal dispersal and and because because it it may may be be strongly strongly involved in process of populations (Hanski, involved in the the process of local local extinctions extinctions in in meta metapopulations (Hanski, 11999b; 999b; Higgins 1) Higgins and and Lynch, Lynch, 200 2001) 3 . Social Social factors. factors. All All types types of of intraspecific intraspecific interactions interactions fall fall into into this this cat category. egory. In In the the broadest broadest sense, sense, such such interactions interactions may may be be treated treated as as density densitydependent dependent sources sources of of dispersal. dispersal. However, However, it it may may be be useful useful to to distinguish distinguish between competitive interactions place within between competitive interactions taking taking place within and and among among differ different age cycle stages, ent age or or life life cycle stages, sexes, sexes, and and individuals individuals with with different different genetic genetic relationship relationship (e.g., (e.g., kin kin or or nonkin), nonkin), as as they they may may lead lead to to different different dispersal dispersal strategies. strategies. 0
We will also also consider interactions among among these We will consider interactions these factors factors and and the the likely likely mechanisms by which dispersal decisions (state-dependent dispersal) can mechanisms by which dispersal decisions (state-dependent dispersal)can be be achieved. achieved. We We will will then then try try to to characterize characterize them them in in term term of of their their potential potential effects effects on on movement movement patterns patterns and and see see to to what what extent extent such such movements movements are are likely likely to to end end in in successful successful settlement settlement either either reinforcing reinforcing local local populations populations or or creating creating new new ones ones (colonization (colonization of of empty empty patches). patches). However, However, because because of of the the lack lack of of empir empirand a distinct distinct theory, a discussion discussion of the consequences consequences of con conical examples and sidering sidering more more realistic realistic dispersal dispersal scenarios scenarios in in aa metapopulation metapopulation context context would would necessarily necessarily be be very very speculative speculative and and will will not not constitute constitute the the main main body body of of the the chapter. chapter. Indeed, Indeed, there there are are still still large large methodological methodological challenges challenges (reviews (reviews in in Ims Ims and 997; Clobert aI., 200 1; Bullock aI., 2002) with and Yoccoz, Yoccoz, 11997; Clobert et et al., 2001; Bullock et et al., 2002) associated associated with obtaining information about about virtually virtually all all aspects aspects of obtaining reliable reliable empirical empirical information of disper dispersal metapopulations. sal that that are are relevant relevant for for metapopulations. .2 11 33.2
DISPERSAL DISPERSAL AND A N D HABITAT HABITAT SPECIFIC SPECIFIC FACTORS FACTORS
Habitat in Space H a b i t a t Heterogeneity H e t e r o g e n e i t y in Space and and Time Time Habitat Habitat heterogeneity heterogeneity in in space space has has long long been been taught taught not not to to be be sufficient sufficient to to promote 983; Holt, Holt, 11985), 985), especially especially in promote evolution evolution of of dispersal dispersal (Hastings, (Hastings, 11983; in source-sink metapopulations (McPeek (McPeek and 992). When source-sink metapopulations and Holt, Holt, 11992). When dispersal dispersal is is costly, costly, the the inclusion inclusion of of temporal temporal fluctuations fluctuations (of (of which which extinction extinction is is an an extreme extreme case) case) was was found found to to be be necessary necessary to to promote promote the the evolution evolution of of dispersal dispersal (Levin aI., 11984). 984). Indeed, (Levin et et al., Indeed, any any population population limited limited by by the the carrying carrying capacity capacity of of the Caswell, 200 1 ) due the habitat habitat has has an an extinction extinction probability probability equal equal to to one one ((Caswell, 2001) due to to demographic environmental stochasticity. demographic or or environmental stochasticity. In it seems In this this context, context, it seems trivial trivial to to predict predict that that genotypes genotypes that that have have the the cap capacity acity to to produce produce some some dispersing dispersing offspring offspring will will enhance enhance their their fitness fitness compared compared to escape this to those those that that do do not. not. Indeed, Indeed, dispersing dispersing individuals individuals will will escape this local local certi certitude tude of of extinction extinction and and will will colonize colonize other other patches patches or or reenforce reenforce other other popula populations tions (rescue (rescue effect; effect; Brown Brown and and Kodric-Brown, Kodric-Brown, 1977). 1977). Rescue Rescue effects effects are are usually usually viewed viewed as as something something advantageous advantageous at at all all levels levels in in the the hierarchy, hierarchy, from from the the immigrant immigrant individual, individual, the the recipient recipient population, population, and and to to the the metapopulation metapopulation as whole. However, example, dispersing as aa whole. However, this this is is not not universally universally true. true. For For example, dispersing indi individuals viduals may may disrupt disrupt the the social social structure structure of of aa local local population, population, which which can can lead lead to to negative negative population population growth growth (Gundersen (Gundersen et et aI, al, 2002 2002).) . Furthermore, Furthermore,
113. 3. DISPERSAL DISPERSAL
3111 31
connecting two landscapes with different locally adapted populations might drive both populations to extinction (Olivieri and Gouyon, 11997). 997). Dispersal may increase the meta population extinction probability because the spatial metapopulation autocorrelation aI., 11997). 997). autocorrelation of temporal fluctuations increases (Heino et al., metapopulation (Pulliam, 11988), too high dis disMoreover, in a source-sink meta population (Pulliam, 9 8 8 ), a too persal rate from the source to the sink may drive the entire system to extinc extinction (Pulliam, 11996; 996; Gundersen et aI., 1 ) . In this case of source-sink al., 200 2001). dynamics, dispersal should be selected against unless dispersal back to the source is permitted (Watkinson and Sutherland, 11995). 995). In some cases, disper dispersal sal is is found found to to be be counterselected counterselected at at high high extinction extinction probabilities probabilities (Ronce et et aI., al., 2000b; Parvinen et aI., al., 2003) 2003),, although this result has been found to be debat debatable (Heino and Hanski, 200 2001). 1). Spatial heterogeneity bbyy itself has been found to have aann impact oonn evolu evolution Holt, 11992). 992). These be tion of of the the dispersal dispersal rate rate (McPeek (McPeek and and Holt, These findings findings might might be understood in the context of habitat selection theory, using the ideal free dis distribution (IFD) concept derived by Fretwell and Lucas ((1970). 1 970). With a simple (IFD)concept model, Fretwell and Lucas ((1970) 1 970) suggested that that animals were distributing themselves such as to equalize each individual's contribution to the future future gene pool of a population. The IFD concept has been most developed in stud studies of animal behavior, particularly foraging studies (Kennedy and Gray, 11993; 993; Tregenza, 11995). 995). The theory of IFD postulates that that an individual will change foraging habitat only when its realized fitness will be higher by mov moving than by staying at the same place. Application of the IFD concept to the evolution of dispersal was first touched upon by Levin et ai. 1 984) and Holt al. ((1984) Holt ((1985) 1 985) and 1 992). Using and was was then then developed developed by by McPeek McPeek and and Holt Holt ((1992). Using aa simple simple two-patch model with with dispersal independent of density, but potentially dependent on habitat quality, McPeek and Holt ((1992) 1 992) found found [see [see also Lemel et ai. 1 997) for an extension] that when the fecundities at the carrying capac al. ((1997) capacity ity were were equal equal and and carrying carrying capacity capacity differed differed between between habitat habitat patches, patches, the the evolutionary stable habitat-dependent dispersal rate was inversely propor proportional to the carrying capacity. This illustrates the superiority of state-depend state-dependent (here habitat) dispersal strategies over state-independent ones. IFD here develops develops in in the the sense sense that that long-term long-term reproductive reproductive success success will will be be equalized equalized across 1 ; Khaladi aI., 2000; across habitats habitats by by selection selection (Holt (Holt and and Barfield, Barfield, 200 2001; Khaladi et et al., 2000; Lebreton et aI., 999a). Moreover, these IFD-based models al., 2000; Rousset, 11999a). predict predict balanced balanced exchanges exchanges (equal (equal number number of of emigrants emigrants and and immigrants) immigrants) among patches, and these predictions have been found compatible with some empirical results (Doncaster et aI., 997; Diffendorfer, 11998; 998; although see al., 11997; Rousset, 11999a). 999a). However, However, the the theoretical theoretical results results have have been been derived derived based based on on stringent stringent assumptions with respect to the environment (no temporal fluctuation, no con constraints straints on on dispersal), dispersal), population population dynamic dynamic and and structure structure (fixed (fixed point point equilib equilibrium, number of patches), mode of life cycle cycle (timing and number of dispersal event), and behavioral capacities of the organisms (existence of environmental and social cues, perfect knowledge of the landscape), all of which may be severely metapopulations. Some severely violated violated in in the the typical typical settings settings of of metapopulations. Some recent recent explor explorations of more realistic models have indeed demonstrated that, in many situ situations, deviations from IFD distribution are found and that the relationship between habitat-specific dispersal rate and habitat-carrying capacity can be
JJEAN EAN CLOBERT CLOBERTET ET AL. AL.
3 11 22
varied varied (Leturque (Leturque and and Rousset, Rousset, 2002). 2002). In In particular, particular, the the size size of of aa patch patch as as well well as as the the distance distance among among patches patches might might play play an an important important role role in in both both metapopula metapopulation persistence and dispersal (Hanski, 9 9 1 ) . Indeed, tion persistence and the the evolution evolution of of dispersal (Hanski, 11991). Indeed, many many models spatial patterns models predict predict complex complex spatial patterns of of patch patch occupancy occupancy or or abundance abundance aris arising dispersal range ing from from an an interaction interaction between between spatial spatial constraints constraints on on dispersal range (maxi (maximum (see Chapter mum dispersal dispersal distance) distance) and and specific specific local local population population dynamics dynamics (see Chapter 3). 3). The The kind kind of of emergent emergent large-scale large-scale dynamics dynamics resulting resulting from from increased increased fragmenta fragmentation may may then evolution of decrease or tion then feed feed back back on on the the evolution of dispersal dispersal as as to to decrease or increase increase the balance between the dispersal dispersal rate, rate, depending depending on on the the balance between forces forces operating operating at at the the local local (i.e., (i.e., within within the the population) population) versus versus aa more more regional regional (e.g., (e.g., metapopulation) metapopulation) scale. al. ((1995), 1995), for scale. Based Based on on such such considerations, considerations, Olivieri Olivieri et et al. for instance, instance, predicted predicted aa decrease decrease of proportion of of the the proportion of aa dispersal dispersal genotype genotype with with population population age. age.
Individuals' in Space Space and Individuals' Decisions Decisions in and Time Time Most population set Most predictions predictions about about the the evolution evolution of of dispersal dispersal in in aa meta metapopulation setting random or ting rely rely on on the the assumption assumption that that individuals individuals are are moving moving at at random or are are fol following simple rules and and thus thus have lowing simple density-dependent density-dependent rules have limited limited capacities capacities for for making making dispersal dispersal choices. choices. Much Much of of the the empirical empirical information information about about individual individual decisions decisions that that are are con conencountered during the different different stages of dis disditional on spatial heterogeneity encountered persal persal (departure, (departure, transience, transience, and and settlement) settlement) has has been been accumulated accumulated over over the the last last decade. decade. For For example, example, during during transience, transience, individuals individuals are are typically typically not not mov moving ing at at random random with with respect respect to to spatial spatial characteristics characteristics of of their their environment environment [see [see Ims ((1995) for aa review] review].. Much Much experimental experimental evidence evidence shows shows that that habitat habitat cor corIms 1 995) for ridors facilitate movements in several species ridors facilitate fast fast and and straight-lined straight-lined movements in several species (Andreassen 996a; Rosenberg 997; Aars 999; (Andreassen et et al., al., 11996a; Rosenberg et et al., al., 11997; Aars and and lms, Ims, 11999; Haddad, 999b; Tewksbury Haddad, 11999b; Tewksbury et et al., al., 2002). 2002). Many Many ground-dwelling ground-dwelling species species are are following following landscape landscape features features such such as as habitat habitat patch patch boundaries. boundaries. Consequently, Consequently, the between two quite different the distance distance between two patches patches might might be be quite different from from how how aa human human will map. will perceive perceive it it on on aa map. Species-specific Species-specific environmental environmental tropisms tropisms will will interact interact strongly strongly with with the the landscape structure to pattern probably landscape structure to produce produce aa dispersal dispersal pattern probably far far from from that that pro produced 997). Specifically, duced by by aa random random walk walk (Wiens, (Wiens, 11997). Specifically, the the perception perception of of land landscape scape heterogeneity heterogeneity by by an an organism organism will will strongly strongly depend depend on on the the graininess graininess of of the the landscape landscape in in terms terms of of its its mobility mobility and and assessment assessment of of risks/cost risks/cost per per time time unit matrix habitat habitat between habitat unit during during transience. transience. Thus, Thus, aa stretch stretch of of matrix between two two habitat patches will less hostile hostile if needs only only l10O ss for patches will be be perceived perceived as as less if aa species species needs for cross crossit than than if if 11 h h is is needed needed or or if if it it is is devoid devoid of of predators. predators. This This kind kind of of vari variing it ability also, to ability can can also, to some some extent, extent, be be found found between between individuals individuals within within aa species species with with aa significant significant temporal temporal component component due due to to changing changing ambient ambient abiotic abiotic (wind, (wind, humidity, humidity, temperature) temperature) or or biotic biotic conditions conditions (food (food resources, resources, predators) predators) (Wiens, 200 1 ). Especially predation are (Wiens, 2001). Especially high high levels levels of of predation are often often thought thought of of as as the the main obstacle during main obstacle during the the transient transient phase phase in in animals. animals. However, However, little little empirical empirical evidence evidence available available that that can can be be used used to to quantify quantify such such aa cost cost during during the the tran transient 996; Johannesen sient phase phase of of dispersal dispersal (Belichon (B~lichon et et al., al., 11996; Johannesen and and Andreassen, Andreassen, 11998; 998; Woodroffe, Woodroffe, 2000). 2000). The The nature nature of of the the transient transient habitat habitat (i.e., (i.e., the the matrix matrix between between habitat habitat patches) patches) has also aa potentially has also potentially strong strong impact impact on on departure departure and and settlement settlement decisions. decisions.
DISPERSAL 113. 3. DISPERSAL
3313 13
There There are are now now many many studies studies where where the the matrix matrix surrounding surrounding suitable suitable patches patches of of habitats habitats or or the the distance distance separating separating suitable suitable patches patches has has been been found found to to influence influence dispersal propensity. For example, in a study of the common lizard, the exchange exchange rate rate between between two two populations populations separated separated by by aa distance distance less less than than aa home home range range diameter diameter (20 (20 m) m) was was decreased decreased from from 50% 50% to to 00 when when open open habi habitat Clobert et 994). Rather tat was was replaced replaced by by forest forest ((Clobert et ai., al., 11994). Rather small small gaps gaps in in habitat habitat corridors corridors may may be be sufficient sufficient to to impede impede movements movements significantly significantly in in voles voles (Andreassen 996b). Reviews (Andreassen et et ai., al., 11996b). Reviews on on how how spatially spatially explicit explicit landscape landscape fea features, matrix structure, structure, and tures, such such as as patch patch size, size, patch patch edge edge characteristics, characteristics, matrix and inter interpatch dispersal have patch distances, distances, affect affect the the rate rate and and direction direction of of dispersal have shown shown that that is is it it difficult 995; Ims difficult at at the the present present stage stage to to find find valid valid generalizations generalizations (e.g., (e.g., Ims, Ims, 11995; Ims and 997; Wiens, 1 ) . The and Yoccoz, Yoccoz, 11997; Wiens, 200 2001). The existence existence of of dispersal dispersal functions functions valid valid as as We think, however, that a species-specific fixed trait is most probably a myth. We to establish establish dis diswith more relevant empirical information it will be possible to persal functions that are conditional on a spatially explicit landscape landscape feature. empirical studies lot of of empirical studies show show that that departure departure and and settlement settlement decision decision A lot depend on on habitat habitat quality quality in in terms terms of of food food resources, resources, amount amount of of refuges, refuges, preda predadepend tors, ai., 2001 tors, parasites, parasites, and and intraintra- and and interspecific interspecific competitors competitors (Clobert (Clobert et et al., 2001).). Most species studied appear appear to to have a state-dependent state-dependent dispersal response response to Most changes in in habitat quality (particularly (particularly well-documented well-documented examples are aphids, aphids, changes habitat quality examples are MacKay and Wellington, 11977; Massot et aI., al., MacKay 977; Weisser et al., ai., 11999; 999; reptiles, Massot 2002). 2002). The The degree degree of of state state dependence dependence is, is, however, however, likely likely to to vary vary among among the the species, and variation of dispersal potential species, and some some species species might might show show no no variation of their their dispersal potential Roff and Fairbairn Fairbairn (20 (2001) for a review]. In the present present absence of quan quan[see Roff 0 1 ) for titative titative data data available available for for metaanalyses, metaanalyses, we we predict predict that that species species with with the the least least environment-sensitive be highly species ((in in environment-sensitive dispersal dispersal strategies strategies will will be highly specialized specialized species term of habitat requirement) living in in habitats habitats varying varying either either in in aa term of habitat requirement) or or species species living systematic (seasonal or aa random random manner (Ronce et et al., ai., systematic (seasonal or or successional) successional) or manner (Ronce 200 1 ) . In no need need of of information information (when 2001). In such such cases, cases, there there is is either either no (when environment environment is changing systematically) there is (randomly varying varying envirenvir is changing systematically) or or there is no no information information (randomly onments) available predict the the value that the the environment environment onments) available at at any any given given time time to to predict value that will The conditions then met met for for the the evolution will take later later on. on. The conditions are then evolution of a fixed fixed dispersal rate.
113.3 3.3
DISPERSAL, INTERACTIONS, AND AND INBREEDING DISPERSAL, SEX-SPECIFIC SEX-SPECIFIC INTERACTIONS, INBREEDING Sex-biased dispersal dispersal (i.e., (i.e., either either males males or or females females are dispersing in in higher higher Sex-biased are dispersing proportions) proportions) has has been been observed observed in a large number number species. In In addition addition to to their their sex, males males and and females females differ in many many respects respects mainly mainly because because they they are are not not sex, differ in subject to to the the same same selection selection pressures. pressures. Because Because of of sexual sexual selection, selection, the the sexes sexes subject differ in in their their morphology, morphology, physiology, physiology, and and behavior behavior (Gross, ( Gross, 1996). 1 996). In In may may differ particular, females are limited limited by by the the number number of of zygotes zygotes they they can can produce produce and and particular, females are therefore will will maximize maximize offspring offspring quality quality though resource acquisition acquisition and/or and/or therefore though resource mate choice, choice, whereas whereas males males are are more more limited limited by by the the number number of of mates mates to to which which mate they have have access access (Andersson, (Andersson, 1994). 1 994). The The type type of of mating mating system system (monogamy, (monogamy, they polygamy, polyandry) polyandry) will will also also constrain constrain the the way way sexual sexual selection selection will will operate operate polygamy, on each each sex sex and, and, as as aa by-product, by-product, influence influence their their respective respective investment investment into into on resource resource and and mate mate acquisition. acquisition.
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Based on on these arguments, Greenwood Greenwood (1980) ( 1 980) attributed attributed the the widespread widespread sexsex Based these arguments, biased dispersal dispersal observed in mammals mammals and and birds birds to to the the fact fact that that resource-based resource-based biased observed in territoriality was was mainly mainly found found in in one one of of the the sexes sexes (males (males in in birds birds and and females females territoriality mammals). Territorial Territorial sex sex was was the the philopatric philopatric sex, sex, as as nondispersing nondispersing offspring offspring in in mammals). of the the same same sex sex would would have have good good chances chances of of inheriting inheriting its its father/mother father/mother territerri of tory. Advantages for for philopatric philopatric sex sex are are important: important: familiarity familiarity with with the the habitat habitat tory. Advantages and not not much much competition competition to to acquire acquire aa territory. territory. The The opposite opposite sex sex offspring offspring and would also also have have some some advantages advantages in in such such circumstances, circumstances, but but would would have have to to would mate with with relatives relatives and and thus thus pay pay the the cost cost of of inbreeding inbreeding depression. depression. mate Although evidence evidence for for inbreeding inbreeding depression depression is is accumulating accumulating (e.g., (e.g., Saccheri Saccheri Although et al., aI., 1998; 1 998; Ebert Ebert et et al., aI., 2002), 2002), its its reported reported impact impact on on dispersal dispersal has has been been et mainly correlative correlative (but ( but see see Wolff, Wolff, 1992). 1 992). Moreover, Moreover, the the hypothesis hypothesis that that mainly inbreeding avoidance avoidance is is aa major major determinant determinant of of dispersal dispersal has has been been challenged challenged inbreeding on several several grounds. grounds. There There are are other other mechanisms mechanisms to to avoid avoid mating mating with rela on with relatives, such as as kin kin recognition. recognition. Also, Also, moderate may even even be be advanadvan tives, such moderate inbreeding inbreeding may tageous under under certain as to to avoid avoid breaking breaking coadapted tageous certain circumstances, circumstances, such such as coadapted genes complexes and to to purge purge deleterious alleles (although the latter latter may may have have genes complexes and deleterious alleles (although the little impact impact on little on dispersal dispersal evolution). evolution). Inbreeding avoidance avoidance per per se was proven proven theoretically theoretically to able to to promote promote Inbreeding se was to be be able the evolution evolution of dispersal (Motro, (Motro, 1991), 1 99 1 ), but but objections objections have have been been raised raised the of dispersal based on theoretical considerations Gandon (1999), ( 1 999), Perrin Perrin and based on aa set set of of theoretical considerations by by Gandon and 1 999), and and Perrin Goudet (2001) (200 1 ) about about the the conditions pro Malazov Perrin and and Goudet conditions that that proMalazov ((1999), mote dispersal. They demonstrated that mote a sex-biased sex-biased dispersal. They demonstrated that inbreeding inbreeding depression depression is promoting dispersal dispersal in in one (the best best solution solution being being one promoting one sex sex only only (the one sex sex dispers dispersing all, all, the the other other remaining remaining philopatric). philopatric). However, However, in in aa vast vast majority majority of of ing species, some degree, suggesting that species, both both sexes sexes are are dispersing dispersing to to some degree, suggesting that factors factors other other than than inbreeding inbreeding are are important. important. In In addition, addition, inbreeding inbreeding is is not not the the only only force force that can promote 1 ) . Therefore, that can promote sex-biased sex-biased dispersal dispersal (Perrin (Perrin and and Goudet, Goudet, 200 2001). Therefore, the precise role inbreeding in generating dispersal dispersal movement the precise role of of inbreeding in generating movement (including (including inbred individuals more, see inbred individuals having having aa tendency tendency to to disperse disperse more, see Cheptou Cheptou et et aI., al., 200 1 ) is 2001) is still still aa widely widely open open question. question. However, However, it it is is difficult difficult to to imagine imagine that that the the cost cost of of inbreeding inbreeding will will not not be be important to to consider consider in in aa metapopulation metapopulation context, context, particularly particularly at at colonization colonization important or possible that or in in small small and and isolated isolated patches. patches. It It is, is, however, however, possible that situations situations leading leading to to aa potential potential risk risk of of inbreeding inbreeding are are incidentally incidentally avoided avoided by by dispersal dispersal having having evolved evolved for for solving solving other other types types of of individual, individual, especially especially kin-based kin-based interactions. interactions.
113.4 3.4
DISPERSAL DISPERSAL AND AND SOCIAL SOCIAL FACTORS FACTORS
Individual Individual Interactions Interactions that theoreticians demonstrated that that demo demoIt was not until very recently that graphic graphic stochasticity stochasticity will will favor favor aa density-dependent density-dependent dispersal dispersal (Travis (Travis et et aI., al., 11999; 999; Poethke Poethke and and Hovestadt, Hovestadt, 2002; 2002; Cadet Cadet et et a!., al., 2003) 2003).. Interestingly, Interestingly, demo demographic graphic stochasticity stochasticity in in this this case case is is playing playing somehow somehow the the same same role role as as spatially spatially uncorrelated uncorrelated environmental environmental stochasticity. stochasticity. it was was demonstrated demonstrated quite quite early early on on that that dispersal dispersal was was crucial crucial for for Although it population 1 969), and population regulation regulation [the [the enclosure enclosure experiments experiments by by Krebs Krebs et et al. al. ((1969), and
1 3. 13.
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315 3115
Boonstra and and Krebs Krebs (1977), ( 1 977), but but see see Ostfeld Ostfeld (1994)], ( 1 994)], the the mechanisms mechanisms involved involved Boonstra have, until until recently, recently, been been elusive. elusive. For For example, example, to to what what extent extent dispersal dispersal was was have, negatively or or positively positively density density dependent dependent has, has, until until recently, recently, been been controvercontrover negatively sial (Gaines ( Gaines and and McClenaghan, McClenaghan, 1980; 1980; Stenseth, Stenseth, 1983; 1 983; for for aa review, review, see see Ims Ims and and sial Hjermann, inconsistencies are Hjermann, 2001). 200 1 ) . These These inconsistencies are likely to to be be the the result result of of confusion confusion at several several levels: levels: (1) ( 1 ) direct direct versus versus delayed delayed effect of density density and and (2) (2) density density as as at effect of an ultimate ultimate cause cause versus versus aa proximate proximate cue. cue. an Individuals Individuals might might leave a patch patch because because density density is a good good descriptor descriptor of of the the current level level of of intraspecific intraspecific competition. competition. However, However, ideally, ideally, local local crowding crowding has has current to be be compared compared to to the the one one found found in in other other patches. patches. For For instance, instance, as as the the sign sign of of to density is likely likely to to differ differ in in the the emigration emigration (usually (usually positive) positive) and and density dependence dependence is immigration process immigration process (negative), (negative) , respectively respectively (e.g., (e.g., Andreassen Andreassen and and Ires, Ims, 2001), 200 1 ), the resultant resultant dispersal dispersal rate rate will will depend depend on on the the spatial spatial covariance covariance in in population population the density in relation relation to to the the dispersal dispersal range range of of aa given given species species (Ims (lms and and Hjermann, Hjermann, density in 2001 ; see see later). later). Moreover, Moreover, population regulation may may impose impose aa negative negative temtem 2001; population regulation poral autocorrelation in population density poral autocorrelation in population density in in which which case case crowding crowding at at time time t predicts less less crowding crowding at 1 . Thus Thus life life stages stages that that can can endure endure temporal temporal predicts at tt + + 1. crowding may therefore therefore choose in aa patch patch at at high high densities. densities. Depending Depending crowding may choose to to stay stay in on the level level of of temporal in densities, one might might on the temporal and and spatial spatial autocorrelation autocorrelation in densities, one predict very the way density affects predict very different different outcomes outcomes for for the way density affects dispersal. dispersal. Appropriate study designs taking taking such such scaling considerations Appropriate considerations seriously have rarely been used in in studies density-dependent dispersal. dispersal. rarely been used studies of of density-dependent Another may complicate density dependence dependence of dispersal Another aspect aspect that that may complicate the the density of dispersal processes that density not only only determines the potential potential for processes is is that density not determines the for antagonistic antagonistic inter interactions resources, it it may may also proximate cue cue for habi actions in in competition competition for for resources, also act act as as aa proximate for habitat tat quality. quality. The The idea idea that that the the presence presence of of conspecifics conspecifics can can be be used used as as aa cue cue for for habitat has been been proposed many researchers habitat quality quality has proposed by by many researchers (Danchin (Danchin and and Wagner, Wagner, 11997) 997) and 1988, 11991). 99 1 ) . and was was first first demonstrated demonstrated experimentally experimentally by by Stamps Stamps ((1988, Evidence Evidence for for the the fact fact that that the the presence presence ooff conspecifics conspecifics and and their their specific specific charac characteristics teristics (e.g., (e.g., their their reproductive reproductive success) success) may may influence influence departure departure from from and and arrival 1 ) and arrival to to aa patch patch has has since since accumulated accumulated [see [see Stamps Stamps (200 (2001) and Danchin Danchin et et al. al. (200 1 ) for (2001) for reviews]. reviews]. The The use use of of density density of of conspecifics conspecifics as as aa cue cue for for habitat habitat qual quality ity can can therefore therefore explain explain some some of of the the cases cases where where an an inverse inverse relationship relationship between between dispersal (Denno and 995; Lambin dispersal and and density density has has been been found found (Denno and Peterson, Peterson, 11995; Lambin et et aI., al., 2001; 1 ). In 2001; Ims Ims and and Hjermann, Hjermann, 200 2001). In such such cases, cases, the the quality quality component component of of the the cue cue "density" "density" may may overshadow overshadow the the competition competition component. component. In In the the case case of of the the etal., a!', 11996), 996), where individuals left Glanville fritillary butterfly (Kuussaari et patches population density, patches with with aa low low population density, it it was was suggested suggested that that low low density density may may serve serve as as aa cue cue for for low low mating mating probability. probability. It It may may also also be be that that different different individual individual categories categories are are responding responding differently differently to to density, density, depending depending on on their their position position in in the the competitive competitive hierarchy hierarchy (their (their competitive competitive ability) ability) or or their their life life history history characteris characteristics tics (see (see Gundersen Gundersen et et a!., al., 2002). 2002). In In that that case, case, the the realized realized density-dependent density-dependent dis dispersal persal rate rate will will be be conditional conditional on on the the demographic demographic structure structure in in aa given given patch. patch. Although of prime importance to understanding dispersal processes, density dependence dependence is is aa complicated complicated issue issue because because it it may may act act both both as as an an ultimate ultimate cause cause and and as as aa proximate proximate cue. cue. Unfortunately, Unfortunately, there there are are few few experiments experiments explicitly explicitly designed designed to to unravel unravel the the effects effects of of population population density density on on departure departure and and settle settlement ment processes processes (for (for some some exceptions; exceptions; Aars Aars and and Ims, Ims, 2000; 2000; Gaggiotti Gaggiotti et et a!., al., 2002; 2002; Gundersen Gundersen et et a!., al., 2002). 2002).
JJEAN EA N CLOBERT CLOBERTET ET AL. AL.
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Interactions Interactions between between Kin Kin The The nature nature of the the interacting interacting individuals, in particular particular their genetic related relatedness, ness, is is central central to to the the question question of of sociality sociality in in animal animal populations populations and, and, more more specifically, specifically, how how altruistic altruistic behaviors behaviors have have evolved. evolved. Genetically Genetically proximate proximate indi individuals viduals should should tend tend to to congregate congregate spatially spatially by by means means of of offspring offspring philopatry philopatry and/or and/or delayed dispersal as to avoid misdirect misdirect helping and invasion invasion by cheaters cheaters (Packer 997). For (Packer and and Pusey, Pusey, 11997). For example, example, in in the the Seychelles Seychelles warbler warbler (Komdeur (Komdeur et 997), where available territories et al., al., 11997), where most most of of the the good good available territories are are occupied, occupied, off offspring pair holding good territory spring produced produced early early in in the the reproductive reproductive life life of of aa pair holding aa good territory tend parent raise raise other Philopatry then tend to to stay stay to to help help their their parent other offspring. offspring. Philopatry then enhance enhance the the chances chances of of inheriting inheriting aa high-quality high-quality parental parental territory. territory. In In lekking lekking species, species, genetically genetically related related males males tend tend to to concentrate concentrate in in the the same same leks leks (Petrie (Petrie et et al., al., attracted by big groups of males. However, 11999) 999) because females tend to be attracted the dependent. For For example, the intensity intensity of of the the cooperation cooperation is is also also state state dependent. example, Lambin Lambin and 1 998) showed and Yoccoz Yoccoz ((1998) showed that that spatial spatial association association among among kin kin Townsends Townsends voles voles increased (kin cooperation increased as as the the population population increased increased (kin cooperation for for resource resource holding) holding).. The The same same pattern pattern has has been been found found in in red red grouse grouse (Lambin (Lambin eett al., al., 2001 2001).). In In nonsocial nonsocial species, species, where where competitive competitive interactions interactions dominate dominate over over cooper cooperative interactions, interactions, genetically similar individuals individuals are expected expected to avoid avoid situ situcompetition. Hamilton Hamilton and May May ((1977) provided the first theoretical theoretical ations of competition. 1 977) provided support dispersal being being one interactions. They support for for dispersal one way way to to avoid avoid such such interactions. They demon demonstrated homogeneous population, strated that, that, in in aa homogeneous population, kin kin competition competition was was promoting promoting the evolution evolution of dispersal even in the presence presence of a high cost to disperse. Further Further extensions extensions relaxing relaxing some some of of the the assumptions assumptions of of earlier earlier models models (Perrin (Perrin and Goudet, 999; Ronce 998) all con Goudet, 2001 2001;; Gandon Gandon and and Michalakis, Michalakis, 11999; Ronce et al., 11998) converge 1). verge to to the the same same conclusion conclusion (Gandon (Gandon and and Michalakis, Michalakis, 200 2001). Kin competition competition can take several forms, (i.e., between between parents parents and off offspring, offspring of some species spring, among among offspring of opposite opposite or or same same sex). sex). In In some species of of jays, jays, dis dispersers are actively expelled siblings by other siblings from from family groups groups (Strickland, 11991). 99 1 ) . In some rodents, rodents, dispersal increases with with the number number of siblings 992). In siblings in in the the litter litter (Ribble, (Ribble, 11992). In the the common common lizard, lizard, mother-offspring mother-offspring competition (de Fraipont competition leads leads to to the the dispersal dispersal of of female female offspring offspring (de Fraipont et et al., al., 2000). 2000). In In the the latter latter case, case, not not only only the the presence presence of of the the mother mother but but also also her her con condition dition determines determines the likelihood likelihood that that her offspring will disperse (Una (L~na et al., are observational observational and and can can be explained explained 11998). 998). However, most of these cases are by by other other factors, factors, such such that that empirical empirical evidence evidence for for any any form form of of kin kin competition competition promoting promoting dispersal is still scarce in the literature. Only a few experiments experiments demonstrated an effect effect of kin interactions interactions on dispersal (e.g., Lambin, Lambin, have demonstrated L~na et al., 1998). 11994; 9 94; Una Restricted Restricted dispersal dispersal is is not not always always aa prerequisite prerequisite of of the the evolution evolution of of altruism, altruism, not because cycles not only only because cycles of of coevolution coevolution between between the the two two traits traits can can lead lead to to aa temporary between dispersal Galliard and temporary positive positive relation relation between dispersal and and altruism altruism (Le (Le Galliard and 2003b al., 2003a), 2003b et et al., 2003a), but but because because individuals individuals might might have have evolved evolved ways ways to to assess their new social environment environment in term of genetic relatedness relatedness (Hamilton, (Hamilton, 11987). 987). For For example, example, aa colonial colonial ascidia ascidia was was found found to to settle settle in in the the vicinity vicinity of of similar individuals individuals ((Grosberg and Quinn, Quinn, 11986). genetically similar Grosberg and 986). The best evidence of of dispersal dispersal being being caused caused by by competition competition among among or or by by attraction attraction toward toward genet genetically similar individuals individuals comes from from an experiment experiment done done on offspring disper dispersal in the side-blotched side-blotched lizard Uta stansburiana stansburiana (Sinervo et al., 2003). In this
113. 3. DISPERSAL DISPERSAL
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annual male morphs morphs distinct distinct by their throat during repro annual species, species, three three male by their throat color color during reproduction blue, and yellow) coexist duction (orange, (orange, blue, and yellow) coexist in in aa frequency frequency dependent dependent way way analo analogous 996): orange gous to to aa rock-paper-scissor rock-paper-scissor game game (Sinervo (Sinervo and and Lively, Lively, 11996): orange males males are are very very aggressive aggressive toward toward any any other other males males and and easily easily take take over over females females of of blue-throated blue-throated males; males; yellow yellow males, males, which which look look like like females, females, sneak sneak females females of of orange-throated orange-throated males; males; and and blue-throated blue-throated males males successfully successfully avoid avoid being being sneaked sneaked by by yellow-throated yellow-throated males. males. To To avoid avoid being being sneaked sneaked by by yellow-throated yellow-throated males, blue-throated appear to males, blue-throated males males appear to cooperate. cooperate. After After having having randomly randomly dis distributed population, young tributed offspring offspring within within the the population, young orange-throated orange-throated males males were were found based on found 11 yr yr later later to to actively actively avoid avoid each each other other based on their their genetic genetic proximity, proximity, whereas to each whereas genetically genetically proximate proximate blue-throated blue-throated males males were were found found closer closer to each other other than than expected expected by by chance chance alone alone (Sinervo (Sinervo and and Clobert, Clobert, 2003) 2003).. Therefore, Therefore, it decisions can based on it seems seems that that both both departure departure and and settlement settlement decisions can be be based on kin kin and, and, more more generally, generally, on on the the local local genetic genetic structure structure depending depending on on the the cost cost and and bene benefits fits expected expected of of the the interactions interactions within within aa local local population. population. Empirical Empirical evidence evidence is, however, is, however, only only starting starting to to accumulate. accumulate. The The case case of of the the side-blotched side-blotched lizard, lizard, however, however, already already strongly strongly suggests suggests that that individuals individuals have have derived derived direct direct or or indir indirect assess the ect ways ways to to assess the level level of of expected expected kin-based kin-based interactions interactions and and condition condition their their dispersal dispersal behavior behavior to to this this information. information. How interactions between dispersal probabilities opposed to How interactions between kin kin affect affect dispersal probabilities as as opposed to interaction between between unrelated individuals is probably probably a very relevant question question in population setting. in the the typical typical meta metapopulation setting. Relatedness Relatedness in in local local populations populations may may be be expected expected to to depend depend on on patch patch size size and and isolation, isolation, as as well well as as the the time time since since colonization, colonization, and and is is thus thus aa factor factor by by which which spatial spatial structure structure and and demography demography may feed the may feed back back on on the the dispersal dispersal rate rate and and ultimately ultimately on on the the dynamics dynamics of of the meta population. metapopulation.
113.5 3.5
DISPERSAL: DISPERSAL: A A SAME SAME RESPONSE RESPONSE FOR FOR DIFFERENT DIFFERENT FACTORS? FACTORS? For been to For aa long long time, time, the the goal goal of of many many researchers researchers has has been to discover discover the ultim ultimate evolution of realm of ate cause cause of of the the evolution of dispersal. dispersal. In In the the realm of this this effort, effort, many many factors factors has been promote dispersal has been demonstrated demonstrated theoretically theoretically or or empirically empirically to to promote dispersal ((Clobert Clobert et 1 ) . Most et aI., al., 200 2001). Most of of these these studies studies have have been been unifactorial unifactorial in in the the sense sense that time. However, that the the effect effect of of one one factor factor is is considered considered at at aa time. However, the the most most com common situation situation in in nature nature is is that that individuals individuals are are affected affected simultaneously simultaneously by by mul mulmon tiple factors tiple factors that that may may be be involved involved in in the the decision decision of of whether whether an an individual individual should should depart depart from from aa patch patch or or not, not, how how far far it it will will move, move, and and eventually eventually where where it it should should settle. settle. How How such such multiple multiple factors factors interact interact to to shape shape the the overall overall dis dispersal patterns persal patterns is is aa question question of of critical critical importance importance to to our our understanding understanding of of the the evolution evolution of of dispersal dispersal as as well well as as for for predicting predicting transfer transfer rates rates between between patches patches in ecological setting. in aa given given ecological setting. It It is is only only recently recently that that models models of of the the evolution evolution of of dispersal dispersal considered considered the the action action of Mazalov ((1999, 1 999, 2000) of several several factors factors at at the the same same time. time. Perrin Perrin and and Mazalov 2000) considered considered the joint effects the evolution evolution of of sex-biased sex-biased dispersal dispersal under under the the joint effects of of inbreeding inbreeding depres depression, sion, local local mate mate competition, competition, and and local local resource resource competition. competition. They They found found that that dif different way local each sex sex may may result in ferent assumptions assumptions about about the the way local competition competition affects affects each result in sex not and sex biased biased dispersal dispersal or or not and that that inbreeding inbreeding depression depression could could enhance enhance biases biases due 1 ) examined due to to other other factors. factors. Similarly, Similarly, Gandon Gandon and and Michalakis Michalakis (200 (2001) examined the the
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respective respective roles roles of of local local extinction, extinction, kin kin competition, competition, and and inbreeding inbreeding on on the the evolu evolution tion of of dispersal. dispersal. Extinction Extinction seems seems to to be be aa stronger stronger selective selective force force than than the the other other two, interaction among leads sometimes sometimes to counterintuitive results. two, and and the the interaction among forces forces leads to counterintuitive results. For For example, example, the the dispersal dispersal rate rate was was sometimes sometimes found found to to increase increase with with the the cost cost of of dispersal. (see also 1 ) that dispersal. They They also also pointed pointed out out (see also Perrin Perrin and and Goudet, Goudet, 200 2001) that many many fac factors in one one of back into into the tors are are covarying covarying because because variation variation in of them them feeds feeds back the other other (espe (especially cially kin kin interaction interaction and and inbreeding). inbreeding). A A few few other other models models have have attempted attempted to to incorporated ecoevolutionary imposed by local (patch incorporated ecoevolutionary feedbacks feedbacks such such as as those those imposed by local (patchspecific) probability of specific) population population dynamics dynamics on on relatedness, relatedness, probability of inbreeding, inbreeding, and and prob probability of 997; Mathias ability of extinction extinction (Doebeli (Doebeli and and Ruxton, Ruxton, 11997; Mathias et et aI., al., 2001; 2001; Heino Heino and and Hanski, 2001; aI., 2003, Hanski, 2001; Parvinen Parvinen et et al., 2003, Rousset Rousset and and Ronce, Ronce, 2004). 2004). These These modeling modeling exercises exercises have have been been fruitful fruitful in in pointing pointing out out the the complexity complexity of variables describing social environment of the the interactions interactions among among variables describing the the social environment and and in in understanding understanding the the respective respective role role of of each each factor. factor. The The models models are, are, however, however, still still too too simplistic simplistic and and neglect neglect many many important important facets facets of of the the problem. problem. In In addition, addition, in in most most models, models, dispersal dispersal is is not not state state dependent, dependent, whereas whereas results results of of these these mod models els pledge pledge for for the the widespread widespread existence existence of of such such aa state-dependent state-dependent dispersal. dispersal. Indeed, Indeed, variables variables describing describing the the social social environment environment are are both both predictors predictors of of the the future future social social environment environment and and are are influencing influencing each each other other in in aa somehow somehow pre predictable way. autocorrelation in social environment dictable way. Thus Thus temporal temporal autocorrelation in the the social environment then then ground for for the evolution of a sensitivity to cues reflecting it. Finally, sets the ground factors influencing settlement taken into account in factors influencing settlement decisions decisions are are not not taken into account in models, models, whereas whereas empirical empirical research research on on habitat habitat selection selection has has proved proved that that such such decisions decisions are (Stamps, 2001 aI., 200 1). are influential influential (Stamps, 2001;; Danchin Danchin et et al., 2001). Thus Thus dispersal dispersal models models are are still still far far from from offering offering aa good good predictive predictive frame framework respect to interplay between between multiple multiple determinants work with with respect to the the interplay determinants of of dispersal. dispersal. However, However, empirical empirical evidence evidence regarding regarding this this issue issue is is also also poor, poor, and and only only aa few few studies have reported studies have reported interactive interactive effects effects between between determinants determinants of of dispersal. dispersal. Although Although many many of of them them are are correlational, correlational, these these studies studies have have recurrently recurrently found found interactions among factors acting either same time time and interactions among factors acting either at at the the same and location location or or at at different moments in life different locations locations (departure (departure and and settlement) settlement) and and different different moments in the the life of English grain of an an individual. individual. For For example, example, in in the the English grain aphid aphid (Dixon, (Dixon, 1985) 1985) and and in armatus, Bengtsson Bengtsson et 1 994), see in aa collembolan collembolan [Onychiurus [Onychiurus armatus, et aI., al., ((1994), see Ims Ims and and Hjermann 1 ) for Hjermann (200 (2001) for further further examples], examples], dispersal dispersal at at one one location location was was trig triggered between habitat gered by by an an interaction interaction between habitat quality quality and and population population density. density. However, with kin kin and However, in in an an experimental experimental setting, setting, competition competition with and with with unrelated unrelated congeners (Le Galliard congeners has has been been demonstrated demonstrated to to be be additive additive (Le Galliard and and 2003a 2003a et et aI., al., 2003b). In the collared flycatcher, the the decision to leave and and to select a patch patch 2003b). are are both both influenced influenced by by the the average average reproductive reproductive success success in in that that patch, patch, whereas whereas population density 999, population density only only affects affects departure departure not not settlement settlement (Doligez (Doligez et et aI., al., 11999, Interactions often result from factors exerting their influence at differ differ2002). Interactions ent ontogenetic stages (even level of ent ontogenetic stages (even at at the the level of grandmother grandmother in in aphids; aphids; MacKay MacKay and 977). In reciprocal transplant and Wellington, Wellington, 11977). In aa reciprocal transplant experiment experiment on on the the common common lizard, conditions (e.g., lizard, Massot Massot et et al. al. (2002) (2002) found found that that local local conditions (e.g., humidity humidity and and temperature) at temperature) at different different moments moments during during the the embryonic embryonic development development interact interact among conditions at among them them as as well well as as with with the the local local conditions at birth birth to to shape shape the the disper dispersal sal response response of of the the juvenile. juvenile. Although Although the the empirical empirical evidence evidence at at present present is is scarce scarce and and needs needs more more experi experimental Hjermann, 200 1 ) is mental studies, studies, multiple multiple state state dependence dependence (Ims (Ims and and Hjermann, 2001) is
113. 3. DISPERSAL DISPERSAL
3 1 99
expected expected to to be be common common (Massot (Massot and and Clobert, Clobert, 2000) 2000).. Based Based on on our our present present knowledge, knowledge, the the evolution evolution of of dispersal dispersal under under multiple multiple dependence dependence can can be be seen seen in 1 ) aa slow, in two two opposite opposite ways: ways: ((1) slow, progressive progressive building building up up of of factors factors influencing influencing dispersal starting dispersal starting from from aa primitive primitive cause. cause. For For example, example, one one theoretical theoretical per perspective spective is is that that the the minimal minimal model model of of dispersal dispersal evolution evolution is is aa model model with with kin kin competition competition alone alone (Perrin (Perrin and and Goudet, Goudet, 2001; 2001; Gandon Gandon and and Michalakis, Michalakis, 2001 2001;; Leturque Leturque and and Rousset, Rousset, 2002). 2002). Indeed, Indeed, solving solving kin kin competition competition problems problems is is inherent because of inherent to to most, most, if if not not all, all, organisms organisms because of the the obligate obligate spatial spatial cooccur cooccurrenee, least for rence, at at least for aa certain certain amount amount of of time, time, between between parents parents and and offspring offspring or or of of offspring. offspring. All All other other forces forces will will come come as as modifiers modifiers of of this this initial initial situation situation and and therefore therefore should should interact interact with with kin kin competition. competition. (2) (2) At At the the opposite, opposite, an an omnibus omnibus response response to to different different unrelated unrelated problems problems with with their their own own controlling controlling pathways. pathways. One One might might think think that that kin kin interactions, interactions, mate mate searching, searching, intraspecific intraspecific competition, habitat characteristics perceived at competition, and and habitat characteristics are are all all perceived at different different spatio spatiotemporal 992; Ims, 995; Ims 1 ). In temporal scales scales (Krebs, (Krebs, 11992; Ims, 11995; Ims and and Hjermann, Hjermann, 200 2001). In this this situ situation, common response ation, dispersal dispersal is is just just aa common response to to very very different different situations situations and and one one should should expect expect many many additive additive effects effects of of various various factors factors with with only only limited limited interactions among among them. interactions At At this this stage, stage, no no perspective perspective can can be be discarded. discarded. Indeed, Indeed, although although aa state statedependent dependent dispersal dispersal at at the the level level of of single single factor factor seems seems to to be be the the rule rule and and aa fixed fixed dispersal dispersal the the exception, exception, the the few few studies studies looking looking to to the the existence existence of of interactions interactions among Stamps, 2001; among factors factors used used at at departure departure and/or and/or settlement settlement ((Stamps, 2001; Doligez Doligez et 2002; Le et aI., al., 2002; Le Galliard Galliard et et aI., al., 2003a) 2003a) give give contradictory contradictory results. results. The The study study of of the the mechanisms mechanisms and and cues cues involved involved in in departure, departure, transience, transience, and and settlement settlement might might help help build build aa more more precise precise view view of of dispersal dispersal evolution, evolution, as as well well as as its its expected population dynamic. expected consequences consequences on on the the meta metapopulation dynamic.
113.6 3.6
PROXIMATE PROXIMATE CONTROL CONTROL OF OF DISPERSAL DISPERSAL
Genetic Control Genetic Control Evidence Evidence for for the the genetic genetic control control of of dispersal dispersal is is found found more more easily easily in in organisms organisms that that produce produce offspring offspring with with some some specialized specialized dispersal dispersal morphs. morphs. In In insects, insects, dis dispersal Roff and persal morphs morphs may may be be characterized characterized by by aa winged winged morphology morphology ((Roff and Fairbairn,, 2001 990), or Carriere and Fairbairn 2001),), physiology physiology (Clark, (Clark, 11990), or behavior behavior ((Carri~re and Roitberg, Roitberg, 11995). 995). Roff Roff and and Fairbairn Fairbairn (2001 (2001)) presented presented aa number number of of cases cases where where significant significant heritability has associated with heritability has been been found found for for aa trait trait associated with dispersal dispersal and/or and/or migration. migration. In In plants, plants, seed seed dimorphism dimorphism is is known known in in the the Asteraceae, Asteraceae, and and genetic genetic effects effects have have been demonstrated 989; Imbert, 1 ) . Dispersal been demonstrated (Venable (Venable and and Burquez, Burquez, 11989; Imbert, 200 2001). Dispersal morphs been morphs are are known known in in the the naked naked mole mole rat rat but but genetic genetic effects effects have have not not been described described in in this this case. case. Several Several traits traits must must be be present present simultaneously simultaneously to to facilitate facilitate dispersal; dispersal and dispersal; those those that that trigger trigger dispersal and those those that that subsequently subsequently facilitate facilitate movements movements and and finally finally settlement settlement in in aa new new patch. patch. In In vertebrates, vertebrates, most most studies studies report propensities within report aa strong strong correlation correlation in in dispersal dispersal propensities within families families (Massot (Massot and and Clobert, Clobert, 2000). 2000). Whether Whether aa family family component component of of dispersal dispersal is is due due to to common common genes genes or or environment environment is is undecided undecided in in most most field field studies. studies. Two Two studies, studies, however, however, provide provide good good support support for for aa genetic genetic basis basis of of dispersal. dispersal. The (a monkey), The first first one one reports reports that that in in the the rhesus rhesus macaque macaque (a monkey), the the timing timing of of
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dispersal versus late presence of dispersal (early (early versus late in in life) life) is is conditioned conditioned to to the the presence of aa specific specific allele transporter gene second one one allele at at aa serotonin serotonin transporter gene (Trefilov (Trefilov et et aI., al., 2000). 2000). The The second concerns lizard U. stansburiana, concerns the the side-blotched side-blotched lizard stansburiana, where where offspring offspring sired sired by by males with with different different throat throat color color morph morph (genetically (genetically based) based) have have subsequently subsequently males different different dispersal dispersal patterns patterns (Sinervo (Sinervo et et aI., al., 2003). 2003). There There is is little little doubt doubt that that dis dispersal basis, although persal in in this this case case has has aa genetic genetic basis, although it it is is far far from from being being clear clear what what are example, there are mechanisms mechanisms of of the the genetic genetic control. control. For For example, there are are few few cases cases where where there seems to genetic mechanism there seems to be be aa simple simple genetic mechanism underlying underlying aa state-independent state-independent dispersal strategy. sp. (Roff dispersal strategy. An An exception exception seems seems to to be be the the cricket cricket Gryllus sp. (Roff and and Simons, 997; Roff 997). The Simons, 11997; Roff et et aI., al., 11997). The widespread widespread presence presence of of aa state-dependent, state-dependent, environment-driven nature is, is, however, environment-driven dispersal dispersal in in nature however, incompatible incompatible with with aa sim simple, rigid genetic control. On the ple, rigid genetic control. On the contrary, contrary, �tate-dependent ~ate-dependent dispersal dispersal suggests suggests aa genetic genetic control and heritability control in in terms terms of of dispersal dispersal plasticity plasticity and heritability of of norms norms of of reaction. reaction. Reaction Reaction norms norms may may explain explain the the apparent apparent contrasting contrasting heritabilities heritabilities 1 ) . To for heritabilities in for dispersal dispersal heritabilities in Drosophila Drosophila spp. spp. (Lefranc, (Lefranc, 200 2001). To our our know knowledge, ledge, no no studies studies have have been been done done on on the the genetic genetic control control of of phenotypic phenotypic plasti plasticity city in in dispersal-related dispersal-related traits. traits.
Physiological Physiological and and Behavioral Behavioral Control Control most organisms, organisms, there is ample ample evidence that that many many aspects of the envir envirIn most onment 1 ) can dispersal. It onment (for (for aa review, review, see see Ims Ims and and Hjermann, Hjermann, 200 2001) can condition condition dispersal. It somehow trivial trivial to to say say that that aa state-dependent state-dependent dispersal will be be selected selected if if dispersal will is somehow organisms organisms can can predict predict its its expected expected reproductive reproductive success success in in aa spatial spatial setting. setting. Indeed, there environmental factor Indeed, there is is hardly hardly any any environmental factor that that has has never never been been found found to to affect dispersal in affect dispersal in some some species species or or under under some some circumstances. circumstances. Examples Examples of of individuals individuals using using concurrent concurrent cues cues for for dispersal dispersal decisions decisions are are numerous and almost any numerous and documented documented for for almost any environmental environmental factors factors (Dixon, (Dixon, 11985; 985; Denno aI., 11991; 991; Denno 995; Lidicker Denno et et al., Denno and and Peterson, Peterson, 11995; Lidicker and and Stenseth, Stenseth, 11992; 992; Ims 1 ) . It Ims and and Hjermann, Hjermann, 200 2001). It is, is, however, however, not not clear clear when when and and how how organisms can so. Indeed, organisms can do do so. Indeed, individuals individuals also also use use cues cues at at some some earlier earlier stage stage to to disperse stage (delayed disperse at at aa later later stage (delayed dispersal). dispersal). In In many many species species of of birds, birds, individ individuals assessing habitat uals are are often often assessing habitat quality quality the the year year before before they they actually actually leave leave their their breeding 998; Doligez Doligez et breeding area area to to settle settle in in aa new new one one (Danchin (Danchin et et aI., al., 11998; et aI., al., The maintenance maintenance of information-gathering information-gathering systems, especially on the 11999). 999). The scale several mechanisms scale of of aa lifetime, lifetime, is is most most probably probably costly, costly, especially especially if if several mechanisms are are needed to process the necessary information. needed to acquire, acquire, store, store, and and process the necessary information. To To reduce reduce these evolution might species to these costs, costs, evolution might have have driven driven species to select select integrative integrative cues cues (describing environment) as (describing several several aspects aspects of of the the environment) as well well as as to to use use existing existing physio physiological example, Danchin ai. (200 1 ) proposed logical systems. systems. For For example, Danchin et et al. (2001) proposed that that many many ani animal mal species species evaluate evaluate environmental environmental quality quality through through the the success success of of conspecifics. conspecifics. This obviously integrates dimensions of This parameter parameter obviously integrates several several dimensions of habitat habitat quality. quality. In In other other cases, cases, dispersal-conditioning dispersal-conditioning cues cues entailed entailed modification modification of of the the indi individuals' aI., 1998) viduals' internal internal condition condition (Nunes (Nunes et et al., 1998) or or in in the the development development of of the the phenotype O'Riain et 996), which phenotype ((O'Riain et aI., al., 11996), which then then later later on on will will influence influence dispersal dispersal decisions 1 ) . Effects decisions (for (for aa review, review, see see Ims Ims and and Hjermann, Hjermann, 200 2001). Effects of of the the maternal maternal and environment, for and in in some some cases cases even even the the grandmaternal grandmaternal environment, for instance, instance, in in terms terms of presence of of food food and and presence of predators, predators, on on the the production production of of winged winged offspring offspring are are particularly well exemplified aphids ((Dixon, Dixon, 11985; 985; MacKay MacKay and particularly well exemplified in in aphids and
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Wellington, 977). Modifications Wellington, 11977). Modifications of the phenotypic phenotypic traits traits affecting dispersal propensities propensities are are sometimes sometimes subtle. subtle. For For instance, instance, different different cues cues may may act act on on the the physiology or behavior (de Fraipont Fraipont et al., 2000; 2000; Meylan et al., 2002) 2002) at dif different ferent moments moments of individual individual ontogeny ontogeny (Ims and Hjerman, Hjerman, 2001 2001;; Dufty et al., 2002; 2002; Massot Massot et et al., al., 2002). 2002). Although Although there there are are still still relatively relatively few few empirical empirical examples examples of of both both maternal maternal effects effects and and stage-dependent stage-dependent dispersal dispersal cues, cues, they they are are likely to be quite common common given the rapidly accumulating accumulating evidence for for disper dispersal sal being being largely largely state state dependent. dependent. Combining Combining present present and and past past information information on on several several environmental environmental factors factors might might be be done done through through the the organizational organizational and and activational 1 ), and, activational effect effect of of hormones hormones (Dufty (Dufty and and Belthoff, Belthoff, 200 2001), and, indeed, indeed, hor hormone mone have been demonstrated demonstrated to trigger dispersal decision decision at different different devel developmental opmental stages [Dufty et al. (2002) for for a review on vertebrates, vertebrates, see Zera Zera and and Denno 1 997) for Denno ((1997) for insects] insects].. For For example, example, corticosterone corticosterone during during ontogeny ontogeny has has an an influence influence on on brain brain organization organization and and on on the the distribution, distribution, type, type, and and density density of of hormonal hormonal receptors receptors in in different different part part of of the the body body (organizational (organizational effect) effect).. This This will will iinn turn turn set set the the behavioral behavioral repertoire repertoire and and hormone-mediated hormone-mediated stimuli stimuli reaction in reaction profile profile (activational (activational effect) effect) later later in in life. life. In In other other words, words, variations variations in several several environmental environmental factors factors might might be be translated translated into into the the modification modification of of one one or or aa few few hormones hormones (or (or other other message-carrying message-carrying substances) substances) during during development development and/or mechanism under and/or at adulthood adulthood and and be the proximate proximate and and common common mechanism underpinning pinning a majority of dispersal decisions. The questions questions now now are what what sort sort of state state dependence dependence is prevailing prevailing in a given The situation, what what are the underlying mechanisms mechanisms that that species and ecological situation, shape response, and environment do shape aa given given response, and which which cues cues for for assessing assessing the the environment do the the organisms use? These organisms use? These depend depend on on many many factors, factors, including including which which environmental environmental cues cues are are available, available, what what are are their their predictive predictive powers, powers, how how organisms organisms cope cope with with different degrees of spatial and temporal temporal variability/predictability variability/predictability (autocorrela (autocorreladifferent tion), (and even opposing) cues. tion), and and the the simultaneous simultaneous actions actions of of different different (and even opposing) cues. The The that varying dispersal rates feed back on the social and competitive envir envirfact that onment, instance, adds problem of onment, for for instance, adds to to the the complexity. complexity. The The problem of density-dependent density-dependent dispersal example (see 1 ) . Most dispersal is is aa good good example (see later later and and Ims Ims and and Hjermann, Hjermann, 200 2001). Most of of the species are living in environments environments where where conditions conditions cueing for dispersal various degrees of autocorrelation. autocorrelation. The The individuals' individuals' abilities of gather gatherexhibit various information about about the spatial scaling scaling of critical environmental the spatial of critical environmental factors factors are are ing information determinant optimizing their choice ((Doligez 2002). Species-specific determinant for optimizing Doligez et al., 2002). exploration exploration ranges ranges relative relative to to the the spatial spatial scaling scaling of of the the environment environment are are obvi obviously important important in this context. Very little is known known about about this. present lack of a unified unified theory theory of the evolution evolution of state-dependent state-dependent In the present dispersal, we dispersal, we suggest suggest aa preliminary preliminary framework framework that that centers centers around around the the infor information acquisition process condition-dependent dis mation acquisition process that that has has to to precede precede any any condition-dependent dispersal persal event. event. Indeed, Indeed, the the presence presence of of cues cues that that can can be be sensed sensed and and assessed assessed by by the the organism organism is is aa prerequisite prerequisite for for condition condition dependence dependence to to evolve. evolve. Three Three important 1 ) Information important aspects aspects are are recognized recognized (Fig. (Fig. 13.2): 13.2): ((1) Information accumulated accumulated over an organism lifetime will lead to an increasingly accurate accurate knowledge knowledge of the actual situation, situation, (2) the spatial and temporal temporal autocorrelation autocorrelation in environ environmental mental factors factors will will determine determine the the reliability reliability of of the the information information gathered, gathered, and and ((3) 3 ) the value of the information information gathered gathered is decided decided by its relevance relevance to the organism's stage-dependent reproductive values (Ims (Ims and organism's stage-dependent reproductive values and Hjermann, Hjermann, 2001; 2001; Dufty Dufty et et al., al., 2002). 2002).
JEAN JEAN CLOBERT CLOBERTET ET AL.
322 322 Frustrated disperser Control resident
Control disperser Frustrated resident
Frustrated disperser Frustrated resident
A)
B) B) Mis-directed disperser Control resident
Fig. 3.2 Examples Fig. 113.2 Examples of experiments aiming to to study differences in fitness between dis dispersing and resident individuals. Two patch patch systems systems connected connected by one-way corridors corridors help
identify emigration aI., 11999) 999) by capturing emigration attempts (for an example, example, see see Boudjemadi Boudjemadi et al., individuals at the end (gray square) square) on each each one-way corridor. corridor. This This can be further studied studied by varying population characteristics characteristics in the two patches.
Although Although research research along along these these lines lines has has just just started started to to be be carried carried out, out, it it is is tempting tempting to to think think that that the the apparent apparent complexity complexity of of factors factors acting acting on on dispersal dispersal can can be be reduced reduced to to the the study study of of the the action action of of aa few few proximate proximate mechanisms. mechanisms. If If this dispersal, including evolution and this is is true, true, it it will will then then make make the the study study of of dispersal, including its its evolution and (meta)population (meta)population consequences, consequences, easier easier than than it it might might appear appear based based on on the the pre present sent review. review.
113.7 3.7
DISPERSAL: DISPERSAL: PHENOTYPIC PHENOTYPIC ADAPTATION, ADAPTATION, COST, COST, AND AND BENEFITS BENEFITS
Initial Initial Differences Differences among among Dispersers Dispersers and and Philopatrics Philopatrics There There is is evidence evidence in in many many taxa taxa that that dispersers dispersers often often are are characterized characterized by by aa special apparatus, should enhance enhance their their ability special apparatus, which which should ability to to disperse disperse (see (see Section Section 113.6). 3.6). In (or the In many many cases, cases, the the production production of of such such structures structures (or the proportion proportion of of individuals, individuals, among among the the progeny progeny or or in in the the populations, populations, with with these these specializa specializations) is tions) is condition condition dependent. dependent. For For example, example, when when food food is is lacking, lacking, some some ciliates ciliates are able to are able to elongate elongate their their body body and and their their flagellae, flagellae, reaching reaching 1100 times times the the speed speed of of aa normal normal cell cell (Nelson (Nelson and and DeBault, DeBault, 1978). 1978). In In most most species, species, however, however, such such extreme extreme specializations specializations are are lacking, lacking, and and differences differences between between dispersing dispersing and and nondispersing nondispersing individuals individuals are are often often subtle. subtle. Indeed, Indeed, although although not not yet yet well well docu documented, mented, dispersers dispersers are are found found to to be be aa nonrandom nonrandom sample sample of of the the population population (for (for aa review, land, 11983). 9 8 3 ) . They review, see see Swing Swingland, They might might slightly slightly differ differ in in morphology, morphology, physiology, et aI., 1 ) . In physiology, or or behavior behavior (Murren (Murren et al., 200 2001). In gray-sided gray-sided voles, voles, asocial asocial
1 3. DISPERSAL DISPERSAL 13.
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individuals were were predominant predominant among among dispersers dispersers (Ims, (Ims, 1990). 1 990). Sometimes Sometimes such such individuals differences have have been been taken taken as as evidence evidence for for dispersers dispersers being being inferior inferior individindivid differences uals in in terms terms of of competition competition for for resources resources or or mates mates at at their their natal natal site site (Ims (Ims and and uals Hjermann, 2001; 200 1 ; Stenseth and Lidicker, Lidicker, 1992). 1 992). It It was, was, however, however, demonstrated demonstrated Hjermann, Stenseth and in aa few few cases cases that that dispersers dispersers were were not not inferior inferior individuals. individuals. For For juveniles juveniles of of the the in commonlizard which which dispersed dispersed were were the the biggest biggest individuals individuals (L~na (Una et et al., aI., 1998; 1 998; commonlizard de Fraipont et 2000). de Fraipont et aI., al., 2000). In addition to adaptations related related directly directly to mobility, evolution evolution may may have have In addition to adaptations to mobility, shaped philopatric philopatric and and dispersing dispersing individuals differently according to coadcoad shaped individuals differently according to apted sets of of traits traits (similar (similar or or not not to to syndromes syndromes found found at at the the interspecies interspecies level; level; apted sets see Chapter 1 0 ) maximizing maximizing their respective fitness. fitness. The The presence of disperser disperser see Chapter 10) their respective presence of traits subject subject to to state state dependence dependence offers parents aa possibility possibility of of manipulating manipulating traits offers parents offspring dispersal dispersal propensity. propensity. Although evidence for for the the parental parental control control of of offspring Although evidence offspring dispersal dispersal is is only only starting starting to to appear appear (Massot (Massot and and Clobert, 1 995; offspring Clobert, 1995; Meylan et et al., aI., 2002), 2002), the the existence existence of parental control control on on other other traits traits is is well well Meylan of parental demonstrated 99 8 ) . These might demonstrated (Mousseau (Mousseau and and Fox, Fox, 11998). These considerations, considerations, which which might seem at glance of importance, may, impact seem at first first glance of little little importance, may, however, however, have have aa strong strong impact on essential aspects is very little information information about: about: the on two two essential aspects of of dispersal, dispersal, there there is very little the costs costs associated associated with with dispersal dispersal and and the the settlement/colonization settlement/colonization success. success.
Individual Cost aand Benefits of I n d i v i d u a l Level: Level: Cost n d Benefits o f Dispersal Dispersal The has been been presented recurrently as as the the most most costly costly stage stage The transience transience phase phase has presented recurrently of example, Aars al. (1999) ( 1 999) and Ims and Andreassen (2000) (2000) of dispersal. dispersal. For For example, Aars et et al. and Ims and Andreassen found that dispersal movements in experimental vole vole metapopulations metapopulations found that dispersal movements in experimental increased increased predation predation risk risk quite quite dramatically. dramatically. The The risk risk associated associated with with the the tran transient likely to condition dependent, dependent, and sient stage stage of of dispersal dispersal is is likely to be be species species and and condition and other other studies studies have have not not been been able able to to demonstrate demonstrate aa significant significant cost cost of of movement movement (Belichon 996). However, likely that (B~lichon et et aI., al., 11996). However, it it is is likely that dispersal dispersal in in the the typical typical setting setting of populations with of meta metapopulations with highly highly fragmented fragmented habitats habitats imbedded imbedded in in aa hostile hostile matrix situations. matrix is is more more costly costly than than in in other other situations. Clearly, (as opposed Clearly, specific specific traits traits of of dispersing dispersing individuals individuals (as opposed to to philopatric philopatric individuals) render them less susceptible risks associated with dis individuals) may may render them less susceptible to to the the risks associated with dispersal. persal. Dispersers Dispersers may may be be better better skilled skilled than than other other individuals individuals to to becoming becoming integrated integrated in in aa novel novel population population or or colonizing colonizing an an empty empty patch. patch. For For example, example, Danielson 1 987) showed Danielson and and Gaines Gaines ((1987) showed that that individuals individuals colonizing colonizing an an empty empty habitat dispersing or habitat had had aa higher higher growth growth rate rate and and survival survival than than non nondispersing or frustrated frustrated individuals (dispersers (dispersers forced forced to to be be resident) resident).. It It was was also also demonstrated demonstrated experi experimentally that immigrants immigrants into a population population of Daphnia Daphnia spp. had a higher long-term long-term fitness than local individuals (through heterosis; Ebert et aI., al., 2002). In 1996) found In aa review review based based mostly mostly on on observational observational studies, studies, Belichon B~lichon et et al.( a1.(1996) found that that dispersers, dispersers, when when settled settled in in aa new new population, population, did did not not necessarily necessarily have have lower lower fitness fitness than than residents. residents. However, However, all all these these studies studies have have their their interpreta interpretation tion complicated complicated by by problems problems of of study study designs designs (size (size of of the the study study area, area, type type of of habitats, habitats, etc) etc) and and nature nature of of the the dispersal dispersal events. events. In the very few experimental experimental studies aimed at comparing comparing the fitness of dis dispersers persers and and residents, residents, only only aa restricted restricted number number of of situations situations have have been been explored 3 . 3 ) such explored (see (see Fig. Fig. 113.3) such that that aa same-ground same-ground comparison comparison of of the the two two strat strategies egies has has seldom seldom been been done. done. Gundersen Gundersen et et al. al. (2002) (2002) were were able able to to compare compare
JEAN JEAN CLOBERT CLOBERTET ET AL. AL.
324 3;24
Shape of of the cumulative cumulative accuracy of the collected information
of the current reproductive reproductive values Shape of V
aI u e
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u e
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/stage Age/stage
Time/age Ti Shape of the environmental environmental autocorrelation autocorrelation I n
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Fig. 3.3 Condition-dependent Condition-dependent dispersal. A dispersal decision may be influenced Fig. 113.3 dispersal decision influenced by the pres presenvironment in interaction interaction with information of the past history gathered by various means. ent environment means. The dispersal decision way this information will affect the dispersal decision and phenotype should depend on the shape of the current age- or stage-dependent of stage-dependent reproductive reproductive value, value, the shape shape of the cumulative cumulative accuracy environmental autocorrelation. autocorrelation. The shapes of of the collected information, and the shape of the environmental shapes of results of a model, model, but are just for illustrating these curves are not the results illustrating the way a dispersal dispersal decision could be influenced times in in the development of the the phenotype. influenced by cues cues collected collected at different times phenotype.
that settlement successes on experimental patches patches of of same-age subadult subadult voles that had shown different propensities to disperse from their had shown different propensities to disperse from their natal natal patch. patch. Individuals that already some propensity propensity for had much much Individuals that already had had shown shown some for dispersal dispersal had better success success in in settling, settling, surviving, surviving, and and reproducing reproducing on on patches patches with with aa resident resident better population than than those those who who had had never never attempted attempted to to disperse. disperse. This This experiment experiment population also also showed showed that that the the density-dependent density-dependent nature nature of of the the settlement settlement success success was was sex specific (Fig. 13.4). 1 3.4). Moreover, Moreover, the the cost cost of of forced forced dispersal dispersal in residents residents (important to to consider consider when when patches patches are are suddenly suddenly disappearing) disappearing) may may be be difdif (important ferent than than that that of of forced forced residency residency of of individual individual destined destined to to disperse. disperse. ferent The actual actual cause cause of of dispersal also matters matters when when measuring measuring costs costs and and benebene The dispersal also fits of dispersal. dispersal. Indeed, Indeed, phenotypic phenotypic adaptations adaptations to to disperse, disperse, and and the the associated associated fits of costs and and benefits, benefits, may may be be dependent dependent on on the the cause cause of of dispersal dispersal itself. itself. The The availavail costs able evidence evidence of of aa phenotypic differentiation dependent dependent on on the the dispersal dispersal cause cause able phenotypic differentiation is contradictory. contradictory. For For example, example, in in a study study on on aphids aphids (Dixon (Dixon 1985), 1985), the the disperser disperser phenotype was was not not specific specific to to the the cause cause of of dispersal dispersal (most (most factors factors studied studied did did phenotype induce induce the the production production of of winged winged offspring). offspring). On On the the contrary, contrary, in in the the common common
325 325
113. 3. DISPERSAL DISPERSAL 1
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_Residents
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Population Population density (Individuals/patch) (Individuals/patch) Fig. Fig. 1 33.4 . 4 Settlement Settlement success. Settlement Settlement success measured measured by the weekly weekly survival survival rate of root voles root voles introduced introduced experimentally experimentally to to patches patches with with varying varying densities densities of of resident resident animals animals (modified (modified from from Gundersen Gundersen et ai., al., 2002). 2002). Two Two types of animals were were introduced. introduced. Dispersers Dispersers were were animals that shown an ability to disperse, whereas whereas residents were that earlier had shown were same-age same-age animals (belonging (belonging to to the same cohort) cohort) that that never left their their natal patch. patch. Survival probabilities probabilities were were modeled modeled by by logistic logistic regression. regression.
lizard, lizard, phenotypic phenotypic differences differences between between residents residents and and dispersers dispersers were were only only found (Lena et 998), found in in the the context context of of aa mother-offspring mother-offspring competition competition (L~na et aI., al., 11998), although although (or (or maybe maybe because) because) it it is is predicted predicted that, that, under under kin kin competition, competition, the the evolution of of dispersal could could be achieved achieved even in the presence of of a high cost cost to evolution disperse (Hamilton 977; Murren et aI., 1 ) . Therefore, (Hamilton and May, 11977; al., 200 2001). Therefore, we we hypothesize hypothesize that individuals individuals dispersing dispersing for different causes causes display display different different skill at patches or integrating already skill at colonizing colonizing empty empty patches or integrating already occupied occupied patches. patches. The The aforementioned aforementioned hypothesis hypothesis might might even even be be more more important important to to investi investigate considered. Indeed, Indeed, it been recurrently gate when when dispersal dispersal distances distances are are considered. it has has been recurrently proposed that dispersal distances distances are increasing from socially based based dispersal proposed that dispersal are increasing from socially dispersal to (Krebs, 11992; 992; Ronce 1 ). However, to habitat-based habitat-based dispersal dispersal (Krebs, Ronce et et aI., al., 200 2001). However, the the relationship relationship between between different different dispersal dispersal causes causes imposing imposing different different phenotypic phenotypic adaptations dispersal distances distances may complex than adaptations and and dispersal may in in fact fact be be more more complex than previ previously ously thought. thought. For instance, instance, consider the possibility possibility that, for a given given disper dispersal cause, cause, a mother imposes a specific mother (through (through maternal effects) effects) imposes specific phenotype phenotype to its offspring such as in the case of kin competition in the common lizard; offspring ((such of kin competition in common lizard; Lena 99 8 ) so as to enhance their overall at dispersing. dispersing. L~na et aI., al., 11998) enhance their overall efficiency efficiency at In case, it be predicted In such such aa case, it might might be predicted that that those those individuals individuals going going to to disperse disperse over show particular particular ability empty habitat habitat are are over long long distances distances or or show ability to to colonize colonize empty
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individuals individuals with with the the most most adapted adapted phenotype, phenotype, i.e., i.e., for for the the dispersal dispersal cause cause that that produces produces the the best-adapted best-adapted phenotype phenotype to to disperse. disperse. Although Although the the individual individual consequences consequences of of philopatry philopatry and and dispersal dispersal are are only only starting starting to to be be considered, considered, the the first first reported reported studies studies clearly clearly demonstrate demonstrate the the importance importance of of such such considerations considerations and and their their potential potential impact impact on on dispersal dispersal cost cost and and settlement settlement success. success. Overall, Overall, they they also also pointed pointed out out aa lack lack of of theoretical theoretical stud studies on ies on females females strategies strategies producing producing aa dispersal-dependent dispersal-dependent offspring offspring phenotype. phenotype.
113.8 3.8
DISPERSAL, DISPERSAL, POPULATION POPULATION GENETICS, GENETICS, AND AND GENETIC GENETIC FEEDBACK FEEDBACK MECHANISMS MECHANISMS
Local Local Adaptation Adaptation Low Low dispersal dispersal sets sets the the conditions conditions for for local local adaptation, adaptation, as as predicted predicted repeat repeatedly Nagylaki, 1975). edly by by models models of of evolution evolution on on aa spatial spatial gradient gradient (e.g., (e.g., Nagylaki, 1975). It It fol follows, recolonizations by lows, therefore, therefore, that that recurrent recurrent recolonizations by immigrants immigrants will will reduce reduce the the potential local adaptation adaptation to abiotic factors populations. This potential for for local to abiotic factors in in meta metapopulations. This in in turn local adaptation dis turn feeds feeds back back on on the the evolution evolution of of dispersal, dispersal, as as local adaptation makes makes dispersal persal more more costly. costly. In In aa two-habitat two-habitat model model with with environmental environmental fluctuations, fluctuations, the the most most common common outcome outcome of of the the coevolution coevolution of of dispersal dispersal and and habitat habitat spe specialization cialization was was the the coexistence coexistence of of two two habitat habitat specialists specialists with with low low dispersal dispersal (Kisdi, 2002), (Kisdi, 2002), although although the the relaxation relaxation of of some some model model assumptions assumptions on on habitat habitat patchiness patchiness and and species species dispersal dispersal range range might might change change these these predictions. predictions. A A different different outcome outcome may may result result from from host host parasite parasite coevolution. coevolution. Parasites Parasites may may be be locally locally adapted adapted to to their their hosts hosts in in that that they they bear bear virulence virulence alleles alleles that that best best match match resistance resistance allele allele of of their their local local hosts. hosts. This This implies implies that that hosts hosts will will be be locally locally maladapted maladapted to to their their parasites. parasites. Conversely, Conversely, hosts hosts may may be be locally locally adapted adapted to to their their parasites, parasites, whereas whereas parasites parasites may may be be maladapted. maladapted. With With limited limited disper dispersal between populations, combinations of virulence/resistance populations, different different combinations virulence/resistance alleles may may evolve evolve in in different different populations, populations, with with some some average average tendency tendency for for either either hosts parasites to locally adapted. locally adapted hosts or or parasites to be be locally adapted. The The partner partner that that is is locally adapted is is the the one one that that evolves evolves faster faster in in response response to to changes changes in in the the other other partner's partner's geno genotypes. types. Parasites Parasites tend tend to to be be advantaged advantaged as as they they often often have have shorter shorter generation generation times, but depends on local input times, but the the speed speed of of evolution evolution also also depends on the the local input of of genetic genetic variation variation through through immigration immigration and and mutation. mutation. Thus Thus the partner partner with with higher dispersal and mutation locally adapted dispersal mutation rates tends to be locally adapted (e.g., Gandon, Gandon, 2002). 2002). Some 996; Davies Some studies studies confirm confirm this this trend trend (Dybdhal (Dybdhal and and Lively, Lively, 11996; Davies et et ai., al., 11999; 999; Kaltz 999; Delmotte 999). Kaltz et et aI., al., 11999; Delmotte et et aI., al., 11999).
Genetic Genetic Diversity Diversity Recurrent local and Recurrent recolonizations recolonizations by by aa few few founders founders reduce reduce both both local and global global genetic genetic diversity. diversity. The The impact impact of of demographic demographic process process is is commonly commonly described described terms of genetic diversity and of spatial genetic structure. The total total genetic in terms diversity diversity of of aa species species is, is, in in principle, principle, determined determined by by its its effective effective size, size, which which defined as measuring measuring the rate at which gene lineages in different indi indimay be defined viduals merge in a common common ancestral lineage (asymptotic (asymptotic inbreeding inbreeding effective size, also also known 982, Whitlock known as eigenvalue effective size; Ewens, 11982, Whitlock and and
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Barton, 1997). 1 997). As guessed, guessed, frequent frequent extinctions extinctions and and recolonizations recolonizations reduce reduce Barton, this effective effective size, size, particularly particularly when when recolonization recolonization is is done done by by aa few few number number of of this individuals. This This prediction prediction is supported supported by a series of of simple simple models models [see individuals. Chapter 7 and and Rousset Rousset (2003, (2003, 2004) 2004) for reanalyses and and interpretations interpretations in a Chapter for reanalyses coalescent framework]. framework] . coalescent and spatial genetic structure structure may may bbee viewed aass reflecting Genetic diversity and two different different forms forms of of "inbreeding" "inbreeding" effects. effects. First, First, offspring offspring sired sired by by parents parents two from a same population population may may be less fit than those those of of parents parents coming coming coming from fit than from different different populations populations (heterosis). (heterosis). Second, Second, "inbreeding" "inbreeding" may may reduce reduce from genetic variability as well as promote promote fixation of deleterious deleterious mutation mutation in the the genetic variability fixation of total population population (which does not not result result in heterosis). heterosis). The The importance importance of of such such total effects has has been been investigated investigated by by simulation simulation (Whitlock, (Whitlock, 2000; 2000; Theodorou Theodorou and and effects Couvet, 2002) 2002) and and analytical analytical models models (Gl~min ( Glemin et et al., aI., 2003), 2003), including including some some Couvet, metapopulation models models [Whitlock, (2002) and and Chapter Chapter 7; 7; although see Roze Roze metapopulation [Whitlock, (2002) although see and Rousset, Rousset, (2003) (2003 ) for for alternative analyses of of such such models]. models]. With sufficiently and alternative analyses With sufficiently weak deleterious effects and strong strong density regulation, regulation, inbreeding may may have litlit weak effects and tle or or no no effect on population population demography, demography, but but otherwise population numbers numbers tle effect on otherwise population will be be reduced (see, e.g., e.g., Saccheri et al., aI., 1998). 1998). This This reduced may feedback feedback will reduced (see, Saccheri et reduced size size may on effective size and and genetic diversity so that that the metapopulation meta population becomes less and less fit, fit, eventually to its extinction ("mutational ( "mutational meltdown"). and less eventually leading leading to its extinction meltdown"). Models have have shown shown that that meltdown could occur occur in principle (Lande, (Lande, 1994; 1 994; Models meltdown could in principle Lynch et al., aI., 1995a) 1 995a) and and that that it could could occur occur much much faster if dispersal is restricted restricted to adjacent populations rather rather than than following following an an island island model model (Higgins to adjacent populations (Higgins and and Lynch, 200 1 ) . However, However, the the value value of of the the key key parameters parameters of this process, process, the the Lynch, 2001). of this genomic rate of of deleterious deleterious mutations and the the distribution of deleterious effects genomic rate mutations and distribution of deleterious effects of of individual individual mutations, mutations, is is still still debated debated (e.g., (e.g., Keightley Keightley and and Bataillon, Bataillon, 2000; 2000; Chapter 14). In addition, most of these models neglect the genomic rate of bene beneficial ficial mutations, mutations, although although its its importance importance has has been been demonstrated demonstrated in in other other cases cases on and Otto (2000); data, Shaw et al. (2002)], also a matter [models, Po Poon matter of debate aI., 2003 debate (Keightley (Keightley and and Lynch, Lynch, 2003; 2003; Shaw Shaw et et al., 2003).). Observations Observations of of local local adaptation, adaptation, despite low population sizes and low diversity of molecular molecular mark markers, also raise 1). ers, also raise doubt doubt about about such such processes processes (McKay (McKay et et aI., al., 200 2001). Information oonn population population structure structure may bbee useful iinn aann appreciation appreciation ooff the Information importance between relatives populations. Wright's importance of of competition competition between relatives in in meta metapopulations. Wright's PST FST is here the kin selection is here the relatedness relatedness parameter parameter weighting weighting kin selection effects effects relative relative to to the the direct fitness direct fitness effects effects of of an an individual individual on on its its own own number number of of offspring offspring (see (see Chapter Chapter 10 10 for for an an example example of of kin kin selection selection effects effects in in metapopulations). metapopulations). Whether Whether the the metapopulation metapopulation turnover turnover reduces reduces or or increases increases spatial spatial structure, structure, as as measured by PST, FsT, depends depends on on whether whether recolonizers recolonizers tend tend to to come come from from the the same same or 9 8 8 ) . Data or from from different different populations populations (Wade (Wade and and McCauley, McCauley, 11988). Data have have been been lit little (see, however, 5 ) , although tle analyzed analyzed in in light light of of these these models models (see, however, Chapter Chapter 115), although esti estiparameters have been obtained for two two fungus fungus beetles mates of their parameters (Whitlock, 11992a; 992a; Ingvarsson et aI., 997). Cases where immigrants tend to al., 11997). come come from from the the same same origin origin are are expected expected not not to to be be rare rare because because there there is is aa strong strong family family effect effect of of dispersal dispersal propensity propensity (Massot (Massot and and Clobert, Clobert, 2000) 2000).. Gene genealogies iinn some metapopulation models may bbee understood understood as "structured "structured coalescents" coalescents" in in which which the the coalescence coalescence of of gene gene lineages lineages from from dif different 19 82a) coalescent ferent demes demes is is described described by by Kingman's Kingman's ((1982a) coalescent process process (Wakeley (Wakeley and Aliacar, 2001 ) . This 2001;; Chapter 88). This suggests that simulation algorithms such
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as 1994a) for as those those of of Nielsen Nielsen and and Wakeley Wakeley (2001 (2001)) or or Griffiths Griffiths and and Tavare Tavar6 ((1994a) for maximum likelihood likelihood estimation parameters could maximum estimation of of genetic genetic parameters could be be adapted adapted to to the the metapopulation context. meta population context.
Methodological Methodological Issues Issues A weakness weakness of most most of the aforementioned aforementioned analytical analytical approaches approaches is that that they based on island model model of they are are based on the the island of dispersal, dispersal, which which is is often often poorly poorly suited suited for 997a). At for data data analysis analysis (e.g., (e.g., Hanski, Hanski, 11997a). At the the other other extreme, extreme, simulation simulation can can be be used used to to analyze analyze complex complex demographic demographic scenarios, scenarios, including including localized localized dis dispersal Some genetic persal and and density density dependence dependence (e.g., (e.g., Barton Barton et et aI., al., 2002) 2002).. Some genetic pat patterns terns may may be be interpretable interpretable in in terms terms of of "effective "effective density" density" and and "effective "effective dispersal" 999b). In this case, number of dispersal" parameters parameters (Rousset, (Rousset, 11999b). In this case, the the large large number of parameters parameters of of aa complex complex model model is is reduced reduced to to aa small small number, number, such such as as meta population size, metapopulation size, effective effective density, density, and and effective effective dispersal. dispersal. In In principle, principle, complex causes of complex life life cycle cycle and and various various causes of dispersal dispersal could could be be taken taken in in account account by by such such effective effective parameters, parameters, but but several several difficulties difficulties still still impede impede progress. progress. First, First, there validated and there is is no no validated and practical practical method method for for estimating estimating the the effective effective size size of of aa metapopulation. than are available. metapopulation. Available formulas call for more data data than Second, population Second, the the most most important important demographic demographic parameters parameters for for meta metapopulation processes processes are are not not necessarily necessarily extracted extracted easily easily from from the the reduced reduced set set of of genetic genetic parameters. parameters parameters. Indeed, Indeed, both both effective effective density density and and effective effective dispersal dispersal parameters are complex functions functions of are expected expected to to be be complex of age age structure, structure, age-dependent age-dependent disper dispersal and fecundities, 999b; and fecundities, and so on (Rousset, 11999b; and case study in Sumner et aI., 200 1 ). Furthermore, al., 2001). Furthermore, the the effective effective parameters parameters do do not not describe describe well well patterns patterns of genetic differentiation 999c), yet the latter may differentiation at short short distances (Rousset, 11999c), be neighbors. Thus, be important important for for quantifying quantifying kin kin competition competition between between neighbors. Thus, it it is is still still unclear unclear what what can can offer offer genetic genetic diversity diversity analyses analyses in in the the absence absence of of detailed detailed demographic observations. observations. demographic
113.9 3.9
FEEDBACK FEEDBACK BETWEEN BETWEEN METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS AND AND DISPERSAL DISPERSAL The The role role of of dispersal dispersal as as aa determinant determinant of of colonization-extinction colonization-extinction dynamics dynamics is main topic topic in numer is aa main in metapopulation metapopulation biology, biology, which which has has been been explored explored by by numerous studies (see Chapters 4, and 119-22). 9-22). Naturally, patches ous studies (see Chapters 4, 14, 14, and Naturally, colonization colonization of of patches following extinction can only take place as result of dispersal, and and dispersal affects affects extinction extinction probability probability by by increasing increasing it it through through emigration emigration and and decreas decreasing (Martin et 997; Hanski, Hanski, 1999b, ing it it by by immigration immigration (i.e., (i.e., rescue rescue effect) effect) (Martin et aI., al., 11997; 1999b, 200 1 ) . Indeed, studies directly indirectly aim 2001). Indeed, most most metapopulation metapopulation studies directly or or indirectly aim to to address dispersal as a driver or a cause of metapopulation metapopulation dynamics dynamics through through its effects on colonization colonization and extinction rates. However, the fact that that there is potential potential for for aa dynamic dynamic duality duality in in the the dispersal-metapopulation dispersal-metapopulation dynamics dynamics rela relation, tion, in in the the sense sense that that dispersal dispersal appears appears as as aa consequence consequence of of aa particular particular (meta)population addition to being aa cause, cause, has been less (meta)population dynamics dynamics in in addition to being has been less appre appreciated. In ciated. In particular, particular, the the extent extent to to which which there there is is aa feedback feedback between between spatio spatiotemporal temporal population population dynamics dynamics and and dispersal parameters parameters (e.g., rate rate and distance) has been explored distance) has been explored to to aa limited limited extent. extent. The The potential potential importance importance of of
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the cause-consequence cause-consequence duality duality of of dispersal dispersal in in aa metapopulation metapopulation setting setting is, is, for for the instance, evident evident concerning concerning the the relationship relationship between between spatial spatial population population synsyn instance, chrony and and dispersal. dispersal. The The degree degree of of population population synchrony, synchrony, itself itself an an important important chrony determinant of of metapopulation metapopulation extinction extinction probability, probability, is is expected expected to to increase increase determinant with increasing increasing dispersal dispersal rate rate (Lande (Lande et et al., aI., 1999), 1 999), whereas whereas both both the the rate rate and and with the distance distance of of dispersal dispersal are are likely likely to to be be dependent dependent on on the the degree degree of of synchrony synchrony the (Ims and and Hjermann, Hjermann, 2001). 200 1 ) . (Ims Understanding which which dispersal dispersal cause cause has has the the highest highest impact impact on o n metapopumetapopu Understanding lation dynamics dynamics is still aa largely largely open open question. question. It It is most most likely likely that that the the answer answer lation will depend depend on on species species biology. biology. For For example, example, one one might might imagine imagine that that species species will disturbed or or evolving evolving habitats habitats (high (high level level of of extinction) extinction) will will be living in highly disturbed dominated habitat-driven dispersal dispersal and and what what will be most most important important to to dominated by habitat-driven consider in a metapopulation metapopulation framework framework will be habitat habitat selection, i.e., i.e., the the consider dynamics will be driven driven by the the settlement settlement phase phase of of dispersal. dispersal. In species living living dynamics in a less disturbed disturbed habitat, habitat, social social characteristics of the the local local patch patch are are likely to to in characteristics of important, and and two two types of of dispersal, kin and and density-dependent disdis become important, persal, are are to to be be considered. considered. persal,
Multiple Action Action of of Density Density Multiple The most obvious way by which population population dynamics may may feed back on dispersal is through the common common dispersal is through density-dependent density-dependent dispersal. dispersal. However, However, the assumption positive, linear linear relationship relationship between between population population density density (den assumption of of aa positive, (density as an ultimate ultimate and and proximate proximate cause) and and dispersal (for a review, see Hanski, 11999b) questioned by several empirical empirical studies studies and reviews Hanski, 999b) has been questioned (Hanski, 2001 1 ; Lambin et aI., 1 ; Chapter 2 1; 2001;; Ims and Hjermann, Hjermann, 200 2001; al., 200 2001; 21; Section 3 . 6 ) . Indeed, Section 113.6). Indeed, negative negative density-dependence density-dependence has has been been found found frequently. frequently. Understanding Understanding why such negative density-dependence occurs and what what are the consequences population dynamics consequences for for meta metapopulation dynamics are are critically critically important. important. With With respect to potential proximate proximate causes of negative density-dependent dispersal, one one possibility possibility is is that that density density has has effects effects on on other other traits traits that that are are related related indi indirectly rectly to to dispersal. dispersal. For For example, example, in in species species where where the the ontogeny ontogeny of of dispersal dispersal is is linked 1 ) , aa density linked to to puberty puberty and and sexual sexual maturation maturation (Dufty (Dufty and and Belthoff, Belthoff, 200 2001), densityinduced delayed sexual reproduction at high densities will then also result in delayed dispersal, possibly to time periods with lower densities in temporally fluctuating populations. For individual organisms that require a certain amount of stored energy reserves for emigration to be triggered triggered (e.g., Nunes et 997), aa high et aI., al., 11997), high local local density density resulting resulting in in intense intense resource resource competition competition can can also also affect affect the the rate rate of of emigration emigration negatively, negatively, so so precise precise species species biology biology has has to to be be known known in in order order to to model model the the effect effect of of dispersal dispersal in in aa metapopulation. metapopulation. Unless a high emigration rate rate from low-density patches patches is compensated compensated for by by aa high immigration rate, emigration will be aa likely cause of extinction (see (see Kuusaari et aI., 996; Andreassen and 1 ). During the transient phase, al., 11996; and Ims, Ims, 200 2001). aa high high density density of of territorial territorial individuals individuals may may impede impede movements movements and and thus thus reduce reduce dispersal distance. This This second mechanism of negative density dependence (see (see Section 113.4 3.4 for more more details) details) is probably less likely in a typical metapopula metapopulation tion setting setting in in which which most most of of the the dispersal dispersal trajectory trajectory takes takes place place in in an an empty empty matrix matrix between between suitable patches. Still Still the effect effect of aa social socialfence fence (see (see Hestbeck, Hestbeck, 11982) 982) may may be be relevant relevant for for individuals individuals situated situated near near the the center center of of large large patches patches
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(Stamps, 11987) 987) and for which within-patch, population-level dispersal precedes between-patch metapopulation level dispersal. Thus the social fence effect size of a local population, population, in addition to its den denopens the possibility that the size sity, can determine the per capita emigration rate. However, in a metapopula metapopulasity, tion the impact of social fences is most relevant in the immigration stage of dendispersal in which settlement success is likely to be related negatively to the den sity of conspecifics in putative immigration patches. Experimental studies have resdemonstrated that such a density-dependent immigration rate both tends to res popucue small (extinction prone) populations and evens out spatial variance in popu Ims, 11999; Gundersen et al., lation density among patches (Aars and lms, 999; Gundersen aI., 2001, al., 200 2001; al., 2003; 2003; Chapter 21). 2002; Lambin et ai., 1; Lecomte et aI., 21).
Which Densities? Densities? Which The kind of dispersal stage-specific response response to population population density study of metapopulation metapopulation dynamics dynamicsdescribed previously complicates the study immigradispersal interface. For instance, densities in the emigration and all immigra tion patches within the exploration range of an organism must be mapped and both emigration and immigration probabilities must must the density dependence of both predict the organism's overall transfer transfer probability. probability. Having be estimated to predict information available from from a relatively simple, small-scale and transpar transparsuch information Andreassen and Ims (200 (2001) showed that that root ent experimental model system, Andreassen 1 ) showed root tended to disperse most most frequently from relatively low-density patches patches to voles tended that patches with even lower population density. In this case it appeared that emigration probability from a given patch patch could be modeled as a function of emigration probability from emigration patch coefficient of variation in density density in the emigration patch and the coefficient among patches in the metapopulation (Fig. 13.5). among metapopulation (Fig. Late season
Early season 0.6 0.5
'E Q) 9-
0.4 0.3 0.2 0.1 0
Fig. 13.5 1 3.5 Emigration Emigration probability. probability. The The emigration emigration probability probability from from patches in experimental experimental Fig. metapopulations root voles metapopulations of of root voles (Andreassen (Andreassen et et al., aI., 2002) 2002) depending depending on on local local density, regional regional density
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However, limited limited knowledge knowledge about about the the scale scale and and mode mode of of patch patch and and However, matrix exploration exploration of of dispersing dispersing individuals individuals (Ims, (Ims, 1995; 1 995; Wiens, Wiens, 2001; 2001 ; matrix Ricketts, 2001; 2001 ; Conradt Conradt et et al., aI., 2003) 2003 ) represent represent obstacles obstacles for for partialling partialling out out Ricketts, the effects effects of of density-dependent density-dependent dispersal dispersal in in most most situations. situations. The The exploration exploration the stage of of dispersal dispersal is is important important to to consider consider because because itit imposes imposes aa feedback feedback stage between immigration immigration and and emigration emigration stages stages in in the the dispersal dispersal process. process. In In parpar between ticular, ticular, if if immigration immigration is negatively negatively density-dependent density-dependent and and there there are are only only high-density high-density patches patches within within the the exploratory exploratory range, range, any any departed departed animal animal will will tend to to return return to to its its patch patch of of origin. origin. This This will will markedly markedly decrease decrease colonizacoloniza tend tion of of even even more more distant distant patches. patches. However, However, many many organisms organisms are are not not capable capable tion of exploring exploring their their environment environment before before deciding deciding to to disperse, disperse, either because of either because they entirely entirely lack lack this this ability ability or or because because the the nearest nearest patch patch is out out of of explorexplor they ation range. range. In In this this case, case, density-dependent density-dependent can can only only act act locally locally and and through through ation its influence influence on on emigration emigration probability; probability; i.e., i.e., density density in in potential potential immigration immigration its patches will will only only affect affect whether whether emigration emigration is is succeeded succeeded by by successful successful immiimmi patches gration (i.e., whether whether dispersal dispersal is efficient) and and eventually eventually dispersal dispersal distance distance gration (how far an animal animal must must move move before before finding suitable patch; patch; Ronce Ronce et al., (how far an finding a suitable aI., 200 1 ) . Whether Whether the the immigration immigration rate rate can can be expected expected to to be lower lower or or higher higher 2001). depending on on whether whether an an organism organism is able to to explore explore patches patches is unclear. unclear. depending Nevertheless, this this point point is is important important because because it it concerns concerns the the impact impact of of disdis Nevertheless, persal e.g., the effect of immigration other in situ persal on on local dynamics ((e.g., immigration versus other demographic the degree of coupling ((synchrony) synchrony) of dynamics demographic processes), processes), the of coupling dynamics between populations, and and the possibility for feedback between between metapopulametapopula between populations, for feedback tion level processes processes and and dispersal. dispersal. Many Many more more empirical studies on on the the tion level empirical studies metapopulation-dispersal urgently needed. Central questions metapopulation-dispersal interface interface are urgently Central questions such as what what is the sign, strength, functional functional form, form, and spatial spatial scale of density densitydependent dependent emigration and immigration immigration processes are largely unexplored unexplored even for for well-studied well-studied metapopulations. metapopulations.
Qualitative Qualitative Effects Effects In addition to the form of the density-dependent dispersal rate, metapopu metapopulation dynamics can also be affected by the nature nature of the dispersing or philopatric individuals. As discussed in Section 113.7, 3.7, there is accumulating evi evidence that that dispersers are not a random subset of their population population of origin (BeIichon 996; Murren et ai., (B~lichon et ai., al., 11996; al., 2001 2001).) . Consequently, the success of a propagule and its effectiveness at reenforcing existing populations or coloniz colonizing quality" of ing empty empty patches patches might might be be strongly strongly dependent dependent on on the the ""quality" of dispersers. dispersers. In In other other words, words, there there may may be be an an interplay interplay between between the the cause cause of of dispersal dispersal and and the the effect effect of of colonists colonists and and immigrants immigrants have have on on the the growth growth rate rate in in previously previously empty empty and already already occupied patches. For this reason, the specific cause of dis dispersal may be important important to know when considering considering the consequences of dis dispersal population setting. persal in in aa meta metapopulation setting. Just Just as as high-quality high-quality propagules propagules may may enhance enhance population population growth growth in in recipient recipient patches and populations, the extinction extinction of donor populations may also be enhanced enhanced by by the the poor poor quality quality of of the the remaining remaining individuals, individuals, which which may may be be decreased decreased even even further further by by the the potential potential deleterious deleterious effect effect of of inbreeding. inbreeding. If If organisms organisms use use conspecifics conspecifics as as cues cues when when selecting selecting aa new new patch, patch, aa patch patch con containing individuals individuals in in aa poor poor shape shape is is less less likely likely to to be be rescued rescued by by high-quality high-quality taining
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further increasing extinction immigrants because of habitat selection. This may further departprobabilities (extinction vortex). Thus generally, the interplay between depart understand ure and settlement decisions may be of prime importance to better understand colonization-extinction processes. colonization-extinction 3 . 11 00 11 3.
DISPERSAL AND A N D SPECIES SPECIES INTERACTIONS INTERACTIONS DISPERSAL
As discussed earlier, species are likely to influence each other dispersal predators, predators predators strategies. Indeed, seeds are dispersed by dispersing seed predators, (parasites) have to search for their prey (host), and prey (hosts) try to avoid predation (parasitism) by many ways, including emigration from patches with proa high predation (parasitism) pressure. For example, aphids increase the pro portion of winged offspring when predation predation increases (Weisser et al., portion aI., 11999). 999). Moreover, Moreover, dispersal dispersal was was enhanced enhanced under under high high predation predation rates rates in in experimental experimental metapopulations of voles (Ims and and Andreassen, 2000). Dispersal in host hostmetapopulations parasite, multi trophic systems systems has has therefore therefore received parasite, predator-prey, predator-prey, or or multitrophic received increas increasattention. ing attention. Indeed, the inclusion inclusion of a spatial spatial dimension dimension into systems of interacting interacting species has revealed an extension of the domain of coexistence in many many cases, shown an increased complexity complexity of the overall population population but it has also shown 2001). pardynamics (Bernstein et al., aI., 11999; 999; van Baalen and Hochberg, 200 1 ) . Of par ticular population context ticular interest in a meta metapopulation context is the opposite opposite action action on local population persistence population persistence of the degree of information information a species has about about its environment environment and and the spatial heterogeneity heterogeneity of this environment. environment. In the case of two interacting species, the the degree of of information two interacting species, information leads to to an ideal ideal free distridistri bution for both but to to a low probability probability of coexistence, whereas whereas bution (IFD) for both species, but spatial heterogeneity distributions far from spatial heterogeneity leads to distributions from IFD, but but to to an an increased increased probability of of coexistence coexistence (van Baalen Baalen and and Sabelis, 1993). 1 993). There There is a certain certain probability here with with the the effect of spatial autocorrelation autocorrelation of of environmental environmental varivari analogy here ation earlier discussion). ation (see earlier discussion). Another Another analogy analogy with with the the one-species one-species case is that that habitat selection selection by by one one individual individual depends depends on on the the other individuals' (of (of all all habitat other individuals' species) decisions. individuals to to show dynamical decisions. Thus, Thus, one one expects expects individuals show some some dynamical which may may lead to condition-dependent condition-dependent dispersal in both responsiveness, which both species. However, accessibility and However, constraints constraints on on information information accessibility and the capacity to to disperse typically disperse typically vary vary among among species. For For example, example, the the degree degree of of local adapadap tation strongly depends tation in the the predator predator (parasite) (parasite) and and prey prey (host) (host) strongly depends on on their their respective dispersal dispersal capacities, capacities, as discussed discussed earlier earlier (Hochberg (Hochberg et et al., aI., 1992; 1 992; respective van 2001). van Baalen Baalen and and Hochberg, Hochberg, 200 1). The inclusion inclusion ooff dispersal dispersal into into models models ooff metapopulation meta population o interacting The off interacting species has has therefore therefore deep deep consequences consequences on on the the overall overall dynamic dynamic of of the the syssys tem, but but more more theoretical theoretical and and experimental experimental research research has has to to be be conducted conducted tem, to to measure measure the the actual actual magnitude magnitude of of such such an an effect. effect. In In turn, turn, these these ecological ecological consequences will will feed back on on the the evolution evolution of of dispersal dispersal in both both inter consequences feed back interacting species. species. The The extent to which which such evolutionary feedbacks feedbacks will will be be acting extent to such evolutionary important depends on the the other forces molding molding dispersal well important depends on other forces dispersal evolution evolution as well as on on the the species species capacity capacity to to invest invest in in other other mechanisms, mechanisms, such such as as defensive defensive structures, chemical chemical weapons, weapons, or or immunity, immunity, which which prevent prevent them them to to escape escape structures, without without moving. moving.
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CONCLUSION CONCLUSION Underlying Underlying the the omnibus omnibus term term dispersal, dispersal, there there is is large large set set of of mechanisms mechanisms and and adaptations, each of which is likely to generate different qualitative and quanti quantitative shortshort- and long-term effects on meta metapopulation protative population dynamic. We have pro vided a rather general review, in particular focusing on the many factors promoting promoting the departure of individuals from patches and directing them to arrive at new ones. The various factors may cause different rates of dispersal, different phenotypic phenotypic profiles in dispersing and resident individuals, and different links between leaving and settlement decisions. Despite the fairly long history of dispersal studies and the renewed interest due to the recent focus on landscape ecology, spatial population population biology, all of which population dynamics, and meta metapopulation knoware research disciplines where dispersal is a key parameter, our present know ledge about dispersal is rather scattered. There is no such thing as an unified the theory of dispersal, although dominant causal factors have been proposed. For extinctions has been proposed proposed to be dominant dominant over other other example, the rate of extinctions causes for dispersal evolution based on theoretical considerations (Gandon and Michalakis, 200 1 ), whereas empirically, density dependence 2001), dependence has recurrently been influencing the been found found to to be be one one of of the the most most important important proximate proximate factor factor influencing the al., 200 2001). dispersal rate (Lambin et aI., 1 ). Moreover, there is accumulating evidence, that kin interactions interactions are important important to con conboth theoretically and empirically, that sider sider in in the the study study of of dispersal dispersal (Perrin (Perrin and and Goudet, Goudet, 2001 2001;; Leturque Leturque and and Rousset, Rousset, 2002; Le Galliard and 2003b). Thus, dispersal can quite certainly be found found to respond respond to almost any potential potential cause. Indeed, dispersal, in the majority majority of cases, cases, has has been been found found to to be be state state dependent. dependent. Potentially Potentially this this has has profound profound impli implications population dynamics cations on the role of dispersal in meta metapopulation dynamics and and evolution. evolution. Even though recent models factors (i.e., local though models of metapopulations metapopulations take some local factors population dynamic) into account through density-dependent population account (i.e., through density-dependent dispersal), very few, if any, have an empirical empirical basis for the local factors included factors that that are included and example, the the strengths and shapes and how how they they are are modeled, modeled, for for example, strengths and shapes of of the the func functional relationships and and the local factors. Moreover, tional relationships and between dispersal rate rate and Moreover, it is not made made clear what considered as local factors (within (within the influwhat is to to be considered influ exploration scale of an individual) and and what what are more ence or exploration more regional factors and, eventually, how factors interact. interact. Population Population density is how local and regional factors naturally factor for most individual individual dispersal decisions as it can be naturally a prominent prominent factor seen both both as an an indication indication of intraspecific intraspecific competition competition (density as an an ultimate ultimate and as a sign of of habitat habitat quality quality (density as proximate proximate cue). The The nature nature of of cause) and individuals individuals themselves themselves (age, size, and and relatedness) relatedness) will influence influence the the way way density density is perceived. For effect of For this and and other other reasons, the the effect of population population density density in donor donor and recipient recipient patches patches is likely to to relate relate differently differently to to the the emigration emigration and and immiimmi and gration respectively. That gration probabilities, probabilities, respectively. That population population density density both both indicates indicates and and determines different aspects and processes processes complicates complicates how how density-dependence density-dependence determines different aspects and dispersal We expect dispersal should should be estimated, estimated, interpreted, interpreted, and and modeled. We expect that that the the question affects the question of of how how dispersal affects the density density and and correlated correlated descriptors descriptors of of popupopu lation genetic makeup) lation characteristics characteristics (age structure, structure, sex ratio, ratio, genetic makeup) will be an an importimport ant next decade. decade. ant and and rich field field for research research in the next Throughout our our review review on on the the causes causes and and consequences consequences of of dispersal, dispersal, we we Throughout have have referred referred to to its its three three stages: emigration, emigration, transience, transience, and and settlement settlement (i.e., ( i.e., immigration immigration and and colonization). colonization). While While the the three three stages stages all are are obviously obviously
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important important for for the the likelihood likelihood of of aa successful successful transfer transfer of of the the individual individual in in metapopulations metapopulations (Ims (Ims and and Yoccoz, Yoccoz, 1997), 1 997), itit seems seems that that the the settlement settlement stage stage is is the least least appreciated appreciated by by metapopulation metapopulation biologists. biologists. Dispersers Dispersers are are not not at at ranran the dom within within the the metapopulation. metapopulation. If If we we except except idiosyncratic idiosyncratic models models dom (MacDonald and and Johnson, Johnson, 2001), 200 1 ), settlement settlement decisions decisions are are almost almost never never taken taken (MacDonald into account account in in attempts attempts to to model model the the evolution evolution of of dispersal dispersal and and to to understand understand into its genetic genetic and and population population consequences. consequences. This This is is certainly certainly one one of of the the most most its important weaknesses weaknesses of of the the current current metapopulation metapopulation paradigm. paradigm. There There is is important indeed ample ample evidence evidence that that most most species species are are able able to to assess assess the the quality quality of of their their indeed abiotic and and biotic biotic environment environment and and that that this this assessment assessment serves serves as as aa basis basis for for abiotic settlement strategies. strategies. The The knowledge knowledge of of these these strategies strategies and and an an understanding understanding settlement of environmental environmental and and internal internal cues cues individuals individuals are are using using as as aa basis basis for for aa given given of strategy will certainly certainly prove prove to to be be necessary necessary if if we we want want to to have have aa more more realisrealis strategy will tic view view of of dispersal metapopulations. tic dispersal within within metapopulations. A pervasive pervasive theme theme in in empirical empirical studies studies of of dispersal dispersal is is that that dispersers are not not A dispersers are aa random of the the individuals in the may differ random subset subset of individuals in the source source population: population: they they may differ in morphology, or behavior. behavior. Whether Whether dispersers dispersers have have features features that that in morphology, physiology, physiology, or make is important, especially when the cost cost of of trantran make them them more more successful successful is important, especially when the sience are considered. considered. Indeed, Indeed, although although still still sience and and the the success success at at settlement settlement are open debate, the the success success of of an an immigrant immigrant when compared to the resident resident open to to debate, when compared to the in aa resident is often to be be different. This ""quality" quality" effect effect does does in resident patch patch is often found found to different. This not seem to be be associated associated to to all dispersal causes, causes, as as dispersers are not not found not seem to all dispersal dispersers are found to be less less competitive individuals in in many many cases. cases. For For example, example, the the study on to be competitive individuals study on the dispersers in relation relation to to the the cause cause of of dispersal suggests that, that, at the dispersers phenotype phenotype in dispersal suggests at least in some there are types of on the set least in some species, species, there are two two types of dispersers dispersers depending depending on the settlement tlement conditions conditions that that might might be be important important to to distinguish: distinguish: those those individuals individuals movements movements that that are are completed completed by by settlement settlement in in an an already already occupied occupied patch patch (reenforcement) (reenforcement) and and those those that that end end up up in in an an unoccupied unoccupied patch patch (colonization). (colonization). Consider Consider that that these these two two types types of of dispersal dispersal are are performed performed by by qualitatively qualitatively different individuals that different types types of of individuals that are are preconditioned preconditioned by by the the environmental environmental conditions in conditions in the the patches patches of of departure. departure. Then, Then, in in case case of of aa large-scale large-scale envir environmental global warming, metapopulation survival onmental change, change, for for instance, instance, global warming, metapopulation survival will will be be enhanced enhanced by by individuals individuals tending tending to to leave leave degraded degraded patches patches and and colonize colonize newly suitable suitable patches. patches. In In that that case, case, meta metapopulation survival will will be be very very newly population survival much much dependent dependent on on how how the the conditioning conditioning for for immigration immigration influences influences the the sen sensitivity sitivity for for different different settlement settlement cues. cues. It It is is possible possible that that individuals individuals destined destined for for settlement settlement in in empty empty patches patches are are sensitive sensitive to to cues cues (e.g., (e.g., density density and and quality quality of of the the conspecific conspecific individuals) individuals) other other than than those those more more likely likely to to settle settle in in already already occupied occupied patches patches (e.g., (e.g., resource resource levels levels and and abiotic abiotic conditions). conditions). To To which which extent extent these these "quality" "quality" differences differences are are important important to to the the dynamic dynamic of of extinction extinction and and recolonization recolonization has has just just started started to to be be considered considered and and definitively definitively deserves deserves more more studies. studies. Genetic Genetic models models bring bring us us aa better better understanding understanding of of who who should should disperse disperse and and when, when, and and these these questions questions are are being being investigated investigated with with increased increased realism realism in in meta populations. However, metapopulations. However, other other contributions contributions of of genetic genetic studies studies of of metapopu metapopulations have have been been limited limited (beyond (beyond topics topics that that are are not not specific specific to to metapopula metapopulalations tions tions per per se). se). Dispersal Dispersal surely surely feeds feeds back back on on many many aspects aspects of of the the genetics genetics of of metapopulations, but but claims claims about about the the importance importance of of genetic genetic effects effects on on metapopulations, meta population persistence metapopulation persistence remain remain highly highly speculative. speculative. Attempts Attempts to to estimate estimate
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demographic demographic parameters parameters from from genetic genetic structure structure will will face face conflicting conflicting issues. issues. On On the the one one hand, hand, theory theory aims aims to to analyze analyze the the genetic genetic structure structure in in terms terms of of aa few few syn synthetic On the thetic effective effective parameters parameters (population (population size, size, density, density, and and dispersal dispersal rates). rates). On the other hopes to complex demography demography out other hand, hand, there there are are hopes to uncover uncover complex out of of these these few few parameters. evidence, however, genetically effective parameters. There There is is no no evidence, however, that that the the genetically effective param parameters eters are are related related in in aa simple simple way way to to parameters parameters that that would would appear appear most most import important evolution in populations. In ant for for population population dynamics dynamics or or life life history history evolution in meta metapopulations. In such such aa context, context, we we cannot cannot advise advise putting putting more more effort effort in in studies studies of of genetic genetic struc structure expense of exception of ture at at the the expense of demographic demographic studies, studies, with with the the important important exception of studies studies of of relatedness relatedness between between competing competing individuals, individuals, as as they they may may help help under understand stand many many behavioral behavioral decisions. decisions. There metapopulation dynamic There is is no no doubt doubt that that dispersal dispersal is is important important to to metapopulation dynamic and and evolution. evolution. The The extent extent to to which which aa detailed detailed knowledge knowledge of of dispersal dispersal is is nec necessary essary to to understand understand and and predict predict metapopulation metapopulation dynamics dynamics and and evolution evolution is is still aa largely largely open open question. question. We We nevertheless nevertheless suggest suggest that that model model adjustments adjustments still to population data to actual actual meta metapopulation data should should not not be be used used as as aa demonstration demonstration that that aa more mechanistic knowledge knowledge is Indeed, simple simple models more detailed, detailed, mechanistic is unnecessary. unnecessary. Indeed, models used used for for conservation conservation purposes purposes can can indeed indeed yield yield aa good good fit fit to to the the actual actual dynamics (Schoener et dynamics of of colonization-extinction colonization-extinction observations observations (Schoener et ai., al., 2003), 2003), but but often often lead lead to to false false conclusions conclusions with with respect respect to to the the underlying underlying processes processes gener generating models. ating these these patterns patterns when when compared compared to to more more realistic realistic models.
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MECH ANISMS MECHANISMS OF PULATION OF PO POPULATION EXTINCTION EXTINCTION Oscar E. E. Gaggiotti Gaggiotti and and Ilkka Hanski Hanski
114.1 4. 1
INTRODUCTION INTRODUCTION Population ecologists have traditionally been concerned with questions about about population regulation and the mechanisms that that increase population Elton, 11949; 949; Nicholson, 954, 11957; 95 7; Milne, 11957, 957, 11962; 962; stability ((Elton, Nicholson, 11954, Andrewartha, 11957; 95 7; den Boer, 968; Andrewartha and 984; Sinclair, Boer, 11968; and Birch, 11984; 11989; 989; Hanski, 11990b; 990b; Price and Cappuccino, 11995; 995; Turchin, 11995, 995, 200 3). 2003). Population ecologists tended ttoo study large populations, often ooff recognized "pest" species, which appeared to exhibit great persistence. In fact, until the early 1960s the predominant predominant view in population ecology considered popula population extinctions unlikely in the presence of effective population population regulation, wide dispersal, and generally large population sizes. sizes. This view predominated because little attention was paid to the actual spatial structure structure of popula populations (Allee aI., 11949). 949). Notable exceptions were three (Allee et al., three Australian ecolo ecologists who recognized the possibility of small populations populations with high rate of extinction, although they reached this conclusion for entirely different rea reasons. Nicholson ((1957), 1 957), the principal architect architect of the population regulation paradigm, envisioned spatially structured structured populations and extinctions of small local populations, but principally in the case of host-parasitoid host-parasitoid dynamics with strong density dependence leading to oscillations with increasing amplitude and, therefore, to local extinction (Nicholson, 11933). 93 3 ) . IInn contrast, Andrewartha and Birch ((1954), 1 954) , who were not impressed bbyy the
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effectiveness of population population regulation in preventing extinctions, extinctions, developed proto-meta population ideas of large-scale persistence proto-metapopulation persistence of species with with populations (see (see discussion 999b). One of the ephemeral local populations discussion in Hanski, 11999b). that gradually changed ecologists' views about about more influential studies that the spatial spatial structure structure and dynamics of populations populations was Ehrlich's study on the the a in California, checkerspot checkerspot butterfly Euphydryas Euphydryas edith editha California, showing showing apparently apparently independent populations over short distances in the independent dynamics of similar populations 9 6 1 , 11965; 965; Singer, 11972). 972) . absence of obvious density dependence dependence (Ehrlich, 11961, checkerspot noteworthy for having addressed addressed both both ecological The checkers pot studies are noteworthy populations and for for having contributed contributed many and genetic processes in local populations about the processes of population extinction for the last four four insights about population extinction decades (for a comprehensive comprehensive review, see Ehrlich and Hanski, Hanski, 2004). The perspective in population population biology changed greatly in the 1970s in the wake of the emergence of modern conservation conservation biology and its emphasis on questions about Simberloff, 11988; 988; about reserve design and population population viability ((Simberloff, Questions about about reserve design stemmed from Hanski and Simberloff, 11997). 997). Questions the dynamic theory of island biogeography (MacArthur 963, (MacArthur and Wilson, 11963, 11967), 967), which, of course, was was explicitly concerned concerned with population population extinctions. Early analyses of population population viability in conservation conservation biology emphasized genetic factors, 977; Chesser et ai., 980; Soule factors, inbreeding and drift (Foose, 11977; al., 11980; Soul~ and Wilcox, 11980; 980; Frankel and Soul~, Soule, 11981; 9 8 1 ; O'Brien et ai., 11983; 983; etal., Schonewald-Cox et ai., 986). In the late 11980s, 980s, increasing al., 1983; Soule Soul~ 11986). recognition of habitat loss and fragmentation fragmentation as the main threats to biodiver biodiversity (Wilson, 1988, 11989; 989; Reid and Miller, 11989; 9 8 9; Groombridge, 11992; 992; Ehrlich and Daily, 11993) 993) contributed see contributed to the growth growth of metapopulation metapopulation biology ((see Fig. 11.2 .2 in Chapter 11),), with emphasis on the the spatial structure of populations populations and on the often high rate of extinction extinction of small local populations populations (Gilpin and and Hanski, 11999b). Hanski, 11991; 9 9 1 ; Hanski and Gilpin, 11997; 997; Hanski, 999b). The relative importance importance of ecological versus genetic factors in population population extinction extinction has been the subject of controversy controversy ever since the birth birth of modern modern conservation conservation biology. As already mentioned, mentioned, conservation conservation biology emerged as two foundations, foundations, the island theory and the vision of population population a discipline on two extinction due to genetic deterioration. 1 9 8 8 ) influential paper deterioration. Lande's ((1988) reviewed the issue 1155 years ago. He concluded concluded that focusing focusing primarily on genetic mechanisms of extinction was misguided and would would not provide an adequate adequate basis for understanding understanding the the processes underpinning underpinning the the survival of endangered endangered species. He also stressed the need for a realistic realistic integration of demography and population population genetics that that would be applicable to species in their natural natural environments. Following the publication publication of this this paper, a consen consensus started to form supporting supporting the primary role of ecological factors in extinc extinction. This consensus was later challenged by a series of theoretical see theoretical studies ((see later) of the decrease in fitness due due to the accumulation accumulation of deleterious muta mutations ("genetic meltdown" meltdown").) . These analyses suggested that that even relatively large populations might go extinct extinct due to genetic deterioration. deterioration. Undoubtedly, it has populations been difficult to reach a robust robust understanding understanding about about the mechanisms mechanisms of popu population lation extinction because of the multitude multitude of factors involved involved and the likely interactions interactions among among them, including ecological and genetic factors. Despite these difficulties, there has been substantial progress in this area during during the last decade.
1 4. 14.
MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION MECHANISMS
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Although itit is is appropriate appropriate to to emphasize emphasize interactions interactions among among different different kinds kinds Although of mechanisms mechanisms influencing influencing population population extinction, extinction, itit is is practical practical to to start start with with aa of review of of particular particular ecological ecological and and genetics genetics factors, factors, which which is is done done in in Sections Sections review 1 4.2 and and 14.3. 14.3. One One way way of of integrating integrating the the different different factors factors is is to to relate relate them them to to 14.2 the most most important important correlate correlate of of extinction extinction risk, risk, small small population population size. size. A A comcom the mon surrogate surrogate of of local local population population size size in in metapopulation metapopulation studies studies is is the the size size of of mon the habitat habitat fragment in which which the the population population occurs. occurs. Effects Effects of of population population size size the fragment in and habitat habitat patch patch size size on on extinction extinction risk risk are are reviewed reviewed in in Section Section 14.4. 14.4. The The and range of of significant significant extinction extinction mechanisms mechanisms is is expanded expanded further further when when we we concon range sider local local extinction extinction in in the the metapopulation metapopulation context context (Section (Section 14.5) 14.5) and and extincextinc sider tion of of entire entire metapopulations metapopulations (Section ( Section 14.6). 14.6). Some Some challenges challenges for for further further tion research are are discussed discussed in in Section Section 14.7. 14.7. research It is is customary customary in in reviews reviews like like the the present present one one to to make make the the point point that that the the It reasons why why populations populations and and species species are are currently currently going going extinct extinct at at aa distressdistress reasons ingly high high rate rate have have primarily primarily to to do do with with loss of of habitats habitats and and interactions interactions with the globe. globe. This This is is what with species species that that humans humans have have displaced displaced around around the what Caughley 1 994) , in in an influential paper, the declining-population Caughley ((1994), an influential paper, called called the declining-population paradigm. In In contrast, most of paradigm. contrast, most of the the factors factors reviewed reviewed in in this this chapter chapter belong belong to to Caughley's 1 994) small-population Caughley's ((1994) small-population paradigm paradigm and and relate relate to to the the ecological ecological and and genetic that render the persistence populations precari genetic mechanisms mechanisms that render the persistence of of small small populations precarious even added threats by humans. humans. A major excep ous even without without any any added threats introduced introduced by A major exception population theory, be employed to elucidate the risk tion is is meta metapopulation theory, which which can can be employed to elucidate the risk of of metapopulation habitat loss fragmentation ((examined examined metapopulation extinction extinction due due to to habitat loss and and fragmentation in Chapter also Chapter on landscape landscape ecology). ecology). It It is is important to in Chapter 4; 4; see see also Chapter 22 on important to realize be made, made, but but it it is is equally important to to real realize that that such such aa distinction distinction can can be equally important realize 1 994) dichotomy ize that, that, to to some some extent, extent, Caughley's Caughley's ((1994) dichotomy is is false false (Hedrick (Hedrick et et aI., al., 11996; 996; Holsinger, 2000). The Holsinger, 2000). The dichotomy dichotomy between between small-population small-population and and declining-population declining-population paradigms paradigms is is partly partly false false because because mechanisms mechanisms in in the the two two realms metapopulation realms interact. interact. This This is is especially especially apparent apparent in in the the context context of of metapopulation biology, biology, where where our our interest interest is is focused focused on on species species with with spatially spatially structured structured populations, consisting of small local populations, often often consisting of many many small local populations populations even even if if the the metapopulation large. To properly understand understand the metapopulation as as aa whole whole is is large. To properly the dynamics dynamics and population biology and population biology of of such such species, species, we we need need to to understand understand the the mechan mechanisms populations that main isms of of extinction extinction of of the the local local populations that are are often often small. small. The The main objectives objectives of of this this chapter chapter are are to to provide provide an an update update on on the the status status of of our our under understanding standing of of these these issues issues and and to to outline outline avenues avenues of of future future research research that that could could help help improve improve it. it.
114.2 4.2
POPULATION POPULATION EXTINCTION: EXTINCTION: ECOLOGICAL ECOLOGICAL FACTORS FACTORS
Demographic Demographic and and Environmental Environmental Stochasticlties Stochasticities The The classic classic models models of of population population dynamics dynamics are are deterministic deterministic and and of of little little use use in in the the study study of of population population extinction, extinction, except except in in making making the the trivial trivial but but hugely hugely important important point point that that if if the the population population growth growth rate rate rr is is negative, negative, the the population population will will surely, surely, and and rather rather quickly, quickly, go go extinct. extinct. This This is is important important because because the the human human onslaught onslaught on on the the environment environment introduces introduces changes, changes, such such as as habitat habitat loss loss and and alter alteration, ation, and and spreading spreading of of invasive invasive species, species, which which will will make make rr negative negative in in many many
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populations. populations. In In deterministic deterministic models models without without age age structure, structure, the the time time to to extinc extincfrom initial initial population population size size No No (which (which is is assumed assumed to to be be much much below below the the car cartion from rying capacity) capacity) is is given given by by-ln No/r (Richter-Dyn (Richter-Dyn and and Goel, Goel, 11972). In contrast, contrast, rying -In Nair 972). In populations populations with with r > > 00 will will not not go go extinct extinct in in simple simple deterministic deterministic models. models. Deterministic Deterministic models models are are inadequate inadequate for for real real populations populations because because their their dynamics dynamics are are influenced influenced by by stochastic stochastic effects. effects. It It is is useful useful to to distinguish distinguish between two two forms forms of of stochasticity. stochasticity. Demographic Demographic stochasticity is is due due to to ran ranbetween independent variation variation in the births and deaths of individuals. dom independent Environmental Environmental stochasticity, in in contrast, contrast, is is generated generated by by random random effects effects affect affectpopulation similarly. The label "environmental" "environmental" sig siging all individuals in the population nifies that the effects are caused by the shared environment of the individuals in the the same same population, population, such such as as adverse adverse weather weather effects effects increasing increasing mortality. mortality. in These are are the the exogenous exogenous factors factors of of population population ecologists ecologists (Turchin, (Turchin, 2003 2003).). These popuIn line with the two forms of stochasticity maintaining fluctuations in popu lation lation size, size, the the variance variance in in the the change change in in population population size size ll.N ~tN conditioned conditioned on on population size partitioned into components, which population size N N may may be be partitioned into two two components, which are are demo demographic and and environmental environmental variances (Engen et et aI., al., 11998). Assuming that that these these graphic variances (Engen 998). Assuming components are constant and denoting them by (T cr~ O'ee22 ,, respectively, respectively, i2 and (T components var(ANN) = = (T cr,12N + (T O'e2N iN + /N22.. Engen Engen et al. ai. ((1998) 1 998) presented presented general general definitions var(ll.N[N) of the the demographic demographic and and environmental environmental variances variances in in terms terms of of the the lifetime of lifetime reprorepro ductive ductive contributions contributions of of individuals individuals to to the the next next generation, generation, Ri• Ri. They They showed showed that the demographic variance (T ~i 2 is half of the variance in the difference of that (conditioned on population size). the the Ri values values for for pairs pairs of of individuals individuals (conditioned on current current population size). Thus, Thus, if if all all individuals individuals would would make make exactly exactly the the same same contribution contribution to to the the next next generation, the the demographic demographic variance would be be zero, zero, that that is, is, there there would would be be generation, variance would no "demographic stochasticity. stochasticity."" In of course, course, this no "demographic In reality, reality, of this will will not not happen happen because of the the intrinsic intrinsic uncertainty individual births births and and deaths. deaths. because of uncertainty involved involved in in individual The environmental variance variance is al., The is aa covariance covariance of of the the R Rii values values (Engen et aI., 11998). 998). ""Environmental Environmental stochasticity" is hence hence great great when when Ri values vary vary in stochasticity" is R i values in parallel, as will happen happen if the performance performance of of all individuals is is influenced influenced by by parallel, as will if the all individuals the same environmental) factors. factors. Note Note that that positive positive covariance covariance of of the same common common ((environmental) values means means that that the the population population growth rate exhibits tem the the individual individual Ri values growth rate exhibits temporal variation. It is noteworthy that poral variation. is also also noteworthy that environmental covariance covariance may may be be negative, that is, an environmental environmental effect of the the negative, that is, an effect may may reduce reduce the the variance variance of change population size. ai. (1998) ( 1 998) gave the (hypothetical)example (hypothetical) example change in in population size. Engen Engen et et al. gave the of of space space limitation limitation and and territoriality territoriality leading leading to to aa completely completely constant constant populapopula Ri values values would would be be necessarily necessarily negatively negatively tion size. size. In In this this case, case, the the individual individual Ri tion correlated. correlated. The approach approach developed developed by by Engen Engen et et al. al. (1998) ( 1 998) to to characterize population The characterize population fluctuations can be be applied applied to to real real populations populations to to estimate estimate the the demographic demographic fluctuations can and and environmental environmental variances variances and and to to predict predict changes changes in in population population size, size, includinclud ing ing the the risk risk of of population population extinction. extinction. The The drawback drawback of of this this approach, approach, howhow ever, is is that that one one requires requires data data on on individual individual lifetime lifetime reproductive reproductive ever, contributions, which which data data are Saether et contributions, are not not often often available. available. Saether et al. ai. (1998a) ( 1 998a) anaana lyzed long-term lyzed long-term data data on on the the great great tit tit population population at at Wytham Wytham Wood Wood near near Oxford. Oxford. The The environmental environmental variance variance turned turned out out to to be be large large in in this this case, case, but but the growth rate the population population was was not not expected expected to to go go extinct extinct because because the the growth rate was was also environmental variance also large. large. In In contrast, contrast, in in aa brown brown bear bear population population the the environmental variance was was very very small small and and smaller smaller than than the the demographic demographic variance variance (Saether (Saether et et al., aI.,
MECHANISMS OF POPULATION POPULATION EXTINCTION 114. 4. M ECHANISMS OF
341
11998b). 998b). This This is is consistent consistent with with the the general general expectation expectation that that large-bodied large-bodied ver vertebrates tebrates (like (like the the brown brown bear) bear) are are less less influenced influenced by by environmental environmental stochas stochasticity than than small-bodied small-bodied vertebrates vertebrates (like (like the the great great tit) tit) and and invertebrates. invertebrates. ticity We have have more more to to say say about about this this in in the the next next section. section. We As 01) As aa more more detailed detailed example, example, we we outline outline the the analysis analysis by by Engen Engen et et ai. al. (20 (2001) population dynamics ooff the barn swallow population studied ooff the stochastic population by A.P. A.P. Moller Moller at at Kraghede, Kraghede, Denmark, Denmark, since since 11970. At this this site, site, the the barn barn swal swalby 970. At low population had declined declined from from 1184 pairs in in 11984 to 5588 pairs pairs in in 11999. low population had 84 pairs 9 84 to 999. Reasons for for the the decline decline appear appear to to be be changes changes in in agricultural agricultural practices practices reducing reducing Reasons the reproductive success of the birds. The The model model fitted fitted by by Engen Engen et et ai. al. (2001 (2001)) to to data data on on barn barn swallows swallows assumes assumes that the the stochasticity stochasticity in in the the population population size size is is described described by by aa Markov Markov process process that and that that the the year-to-year year-to-year change change in in the the logarithm logarithm of of population population size size X( X(= In N) N) = In and is normally distributed with the expectation E(AXX = x ) = r -
1/20-e 2 -
1/2o',t2/N
((14.1) 14.1 )
and variance var(kX[X = x) = Ore2+ O'd2/N
((14.2) 14.2)
The The quantity quantity Yo r0 = yr - 1/2U/ 1/2Ore2 is is defined defined as as the the stochastic stochastic growth growth rate rate and and indi indicates the the extent extent to to which which stochastic stochastic fluctuations fluctuations in in population size reduce reduce the the cates population size long-term "long-run" ) growth 982; Lande long-term (("long-run") growth rate rate (Tuljapurkar, (Tuljapurkar, 11982; Lande and and Orzack, Orzack, 11988; 988; Lande, 993). Demographic Lande, 11993). Demographic stochasticity stochasticity also also reduces reduces the the long-term long-term growth rate, and and the the combined combined effects effects of demographic and and environmental sto growth rate, of demographic environmental stochasticity lead lead to expectation in chasticity to the the expectation in Eq. Eq. ((14.1). 14 . 1 ) . ui2 Engen 1 ) obtained estimate ooff the the demographic variance crd Engen eett ai. al. (200 (2001) obtained aann estimate demographic variance from data data on on the the individual individual contributions contributions of from of breeding breeding females females to to the the next next gengen eration, Ri R i (number (number of of female female offspring following gen generation, offspring recorded recorded in in the the next next or or following erations plus erations plus 1 if if the the female itself survived), calculated calculated as as 2 ( R(Ri i- - ~ /( k -- 11 )) L R))22 , 11/(k
((14.3) 14.3)
where the mean mean contribution the individuals the number number of of where R R is is the contribution of of the individuals and and kk is is the recorded data are for several ua is is estiesti recorded contributions contributions in in 11 yr. yr. If If data are available available for several years, years, cr~ mated as as the the weighted average of of the the yearly yearly estimates estimates (Saether (Saether and and Engen, Engen, mated weighted average 2002). In In the the case case of of the the barn barn swallow, swallow, there there were were extensive extensive data data on on individual individual 2002). reproduction and and survival, survival, and and hence hence ~d ui2 was was assumed assumed to to be be accurately accurately known known reproduction as estimated estimated from from data data for several years, years, Crd ui2 == 0.180. 0.1 80. Next Next the the values values of of r0 Yo and and as for several u/ were estimated estimated from from time time series series data data on on yearly yearly population population sizes sizes by by maximaxi O' e2 were mizing function numerically mizing aa likelihood likelihood function numerically (Engen (Engen et et al., aI., 2001). 200 1 ). The The maximum maximum Yo ** == -0.076 - 0.076 and and O" uee2*2 * == 0.024. 0.024. This This likelihood parameter parameter estimates estimates were were ro likelihood barn swallow population population has has thus thus shown shown aa mean mean decline decline of of 7.6% 7.6% per per year. year. barn swallow Figure 14.1 14. 1 shows shows the the lower lower bound bound of of the the prediction prediction interval, interval, which which includes includes Figure the the predicted predicted population population size size with with probability probability I1 -- c~. a. Comparison Comparison between between Figs Figs 14.1A and and 14.1B 14.1 B demonstrates demonstrates that that ignoring ignoring uncertainty uncertainty in in parameter parameter estiesti 14.1A mates (and (and using using their their maximum maximum likelihood likelihood estimates) estimates) increases increases the the predicted predicted mates time to to extinction. extinction. In In other other words, words, acknowledging acknowledging the the uncertainty uncertainty in in the the time
OSCAR OSCAR E. E, GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKA HANSKI HANSKI
3342 42
A A
�
5
5
'w 44 ~ 9
0__.__
4
c o
8
iii 3 'S
3
a. o
e::.. 2
2
c.s:
o /-"----,.-----.--"--"""-T"'-->-r--+---+ o0 -1 0 40 30 10 20 50 o -10 0 10 20 30 40 50
O
Years Years
Ql
B
5
,"g� 44
4
8 '� 33 ~ 'S c o
3
a. o
e::. g- . 22 e-.s:
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o 0 -"----,.-----.---r.....:..-�::...--"'r--+ 0 0 40 10 30 o0 50 20 -10 40 110 0
Years Years
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c C
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,"~� 44
5 4
c o
iii 3
'S
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3 2
c.s:
-,.-----.---"""-r--l---"--r---'---r--"--+ 00 o0 -"--, , " -1 40 30 20 50 o0 -100 110 0 20 30 40 50
Years Years Fig. Fig. 14.1 14.1 Annual Annual variation variation in in the the number number of of breeding breeding pairs pairs of of the the barn barn swallow swallow at at aa study study site 984 until 999 (the period until site in in Denmark Denmark from from 11984 until 11999 (the time time period until zero zero on on the the x x axis), axis), followed followed by by the ifferent the lower lower bound bound of of different different prediction prediction intervals intervals for for the the future future population population size size for for ddifferent values values of of a, e. Results Results when when (A) (A) all all available available information information is is included, included, (B) (B) uncertainty uncertainty in in param parameter eter estimates estimates is is ignored, ignored, and and (C) (C) demographic demographic variance variance is is set set to to zero zero (from (from Engen Engen et et aI., al., 2001 ) 2001). ,
114. 4. MECHANISMS MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION
343 343
parameter parameter values values leads leads to to more more cautious cautious predictions: predictions: the the population population may may go go extinct extinct sooner sooner than than the the maximum maximum likelihood likelihood estimates estimates would would suggest. suggest. In In Fig. C, the Fig. 14.1 14.1C, the demographic demographic variance variance is is assumed assumed to to equal equal zero. zero. Ignoring Ignoring this this component increases the predicted time to extinction. Additionally, ignoring environmental variance reduces the range of variation of the prediction inter interval (Engen et aI., al., 2001 2001).) . In other words, the fate of the population would be much easier to predict without environmental stochasticity. stochasticity.
Scaling Scaling of Extinction Extinction Risk Risk with with Carrying Carrying Capacity Capacity A many ecological A useful useful framework framework for for examining examining many ecological factors factors in in population population extinction is provided by the simple "ceiling" model of population dynamics (Lande, 11993; 993; Foley, 994, 11997; 997; Middleton et aI., 995). Although this Foley, 11994, al., 11995). model model does does not not incorporate incorporate any any details details of of demography demography and and life life history history of of species, species, it it is is helpful helpful in in encapsulating encapsulating in in general general terms terms the the effects effects on on extinction extinction probability of those factors that should always be considered. This theory is also helpful in providing a submodel of local extinction that that can be used in metapopulation models (Hanski, 11998a, 9 98a, 11999b; 999b; Chapter 4). The ceiling model is described in Box 14. 1. 14.1.
BOX 14.1
The Ceiling Model of Popal.tlon Extinction
Population dynamics are assumed to obey the following equations: nr+l
nr + l
nr+ l
= =
r
nr + r
k
if 0
:s nr + 1 :s
if nr+l > k
=0
if
nr+l
k
< 0,
where nt is the natural logarithm of population size (N) at time t, k is the logarithm of the population ceiling (K), and rr is a normally distributed random variable with mean , and variance The model assumes that the population size performs a random walk between the absorbing lower boundary of population extinction and the reflecting upper boundary of population ceiling. Population fluctuations are driven by environ mental stochasticity. Using the diffusion approach to analyze this model (Foley, 1 994; lande, 1 993; Middleton et aI., 1 995), the expected time to extinction of a population with r > 0 and starting at the ceiling K is given by
CT/.
s = 2rICT/.
T(K)
= !(SIs
,
[1 - (1 + sk)/exp(sk)],
(81 )
where For reasonably large values of sk the term in square brackets is close to 1 and hence the result simplifies to
T(K)
...
f<Slsr.
(82)
These results were obtained for a model that ignores demographic stochasticity. Hanski (1 998a) compared the scaling of time to extinction with population ceiling predicted by (81 ) and by the comparable model (from Foley, 1 997) with both demographic and environmental stochasticities. For values of CT ICTe2 less than 1 , which is likely to be valid for most natural populations, the scaling result (81 ) is little affected by the added demo graphic stochasticity (see Hanski, 1 998a).
i
OSCAR E. E. GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKA HANSKI HANSKI OSCAR
3414 344
The most most lucid lucid and and useful useful result result is is obtained obtained by by assuming assuming that that population population flucfluc The tuations are are driven driven solely solely by by environmental environmental stochasticity. stochasticity. In In other other words, words, we we tuations assume, for for simplicity, simplicity, that that the the demographic demographic variance variance equals equals zero. zero. The The key key assume, parameters are are then then the the population population ceiling ceiling (absolute (absolute carrying carrying capacity) capacity) K, which which parameters the population population size size cannot cannot exceed exceed (Box (Box 14.1), 14. 1 ), and and the the stochastic stochastic population population the ro discussed discussed in in the the previous previous section section and and given given by by r0 ro == rr-- 1/2ere 1/2Ue2• growth rate rate r0 growth 2. Note that that if if rr < < 1/2~re 1/2Ue2, the population population will will go go extinct extinct with with probability probability I1 even even in in Note 2, the the absence of any any density density dependence. dependence. For For convenience, convenience, we we denote the ratio ratio the absence of denote the 2r1u/2 by by s. Assuming Assuming that that rr > > 00 and and that that sk sk isis reasonably reasonably large large (where (where k is is the the 2rkre the time time to to extinction extinction scales scales asymptotically asymptotically as as logarithm of of K), the logarithm
T
T ~ = KS/sr. KS/sr.
(14.4) ( 14.4)
Thus, if if population population fluctuations fluctuations are caused solely solely by by environmental environmental stochasstochas Thus, are caused ticity, the the time time to to extinction extinction scales scales as as aa power power function function of of the the population population ceilceil ticity, ing. is no no environmental stochasticity and and ing. In In the the other other extreme, extreme, when when there there is environmental stochasticity population fluctuations caused by by demographic demographic stochasticity stochasticity alone, alone, the the population fluctuations are caused nearly exponential exponential (MacArthur and Wilson, 1967; 1 967; Lande, 1993; 1 993; scaling is nearly (MacArthur and Foley, 1994). 1 994). Exponential Exponential scaling that for for reasonably large r, only only very very Foley, scaling means means that reasonably large extinction. The extreme small populations populations have an appreciable appreciable risk of of extinction. extreme case of only demographic stochasticity is is of academic interest interest only, as all all real real popula only demographic stochasticity of academic only, as populations are more influenced by both tions more or or less influenced both environmental environmental and and demographic demographic to the model leading leading to to stochasticities. stochasticities. Adding Adding demographic demographic stochasticity stochasticity to the model Eq. ((14.4) 14.4) will will shorten time to to extinction extinction (see (see Fig. Fig. 14.1), 14. 1 ), but but the the scaling scaling is is Eq. shorten the the time are very 1 9 97; Hanski, Hanski, little affected little affected unless unless both both the the ceiling ceiling and and ss are very small small (Foley, (Foley, 1997; 11998a; 998a; Box 1 ). Hence 14.4). Box 14. 14.1). Hence we we focus focus on on the the simple simple result result given given by by Eq. Eq. ((14.4). Taking now now the interpretation interpretation of the power-function power-function scaling further, let us inverse measure measure of value of observe that that the the value of s = = 2r1ue2 2r/(re2 is is an an inverse of the the strength strength of of environmental environmental stochasticity, stochasticity, scaled scaled by by r. The The greater greater the the impact impact of of environ environmental stochasticity (the smaller the value of s), stochasticity on the population population growth rate rate (the the the shorter shorter the the expected expected lifetime lifetime of of the the population population and and the the smaller smaller the the increase increase in population ceiling [Eq. ((14.4)]. 1 4.4)]. A in lifetime lifetime with with increasing increasing population ceiling [Eq. A high high growth growth rate rate (r) has increasing population opposite effect (r) has the the net net effect effect of of increasing population lifetime lifetime and and the the opposite effect with population to that of to that of u/ (re2 on on the the scaling scaling with population ceiling. ceiling. A 14.4) is A useful useful feature feature of of Eq. Eq. ((14.4) is that that the the value value of of the the scaling scaling constant constant ss can can be be estimated estimated with with empirical empirical data. data. Recording Recording actual actual extinction extinction rates rates (liT) (l/T) for for particular populations populations is is impractical, impractical, but but in in the the context context of of metapopulations metapopulations particular with with many many local local populations populations in in aa patch patch network, network, one one may may use use the the spatially spatially real realistic istic metapopulation metapopulation theory theory (Chapter (Chapter 4) 4) to to estimate estimate s from from data data on on the the inci incidence 1 998a) applied dence of of patch patch occupancy. occupancy. Hanski Hanski ((1998a) applied aa mainland-island mainland-island metapopulation 993) to data metapopulation model (Hanski 11993) data on the occurrence of four species of of Sorex Sorex shrews shrews on on small small islands. islands. The The key key assumptions assumptions were were that that island island area area multiplied multiplied by by an an estimate estimate of of population population density density is is an an adequate adequate surrogate surrogate of of the the population ceiling ceiling and and that that the the occurrence occurrence of of the the species species on on islands islands represents represents population aa balance balance between between stochastic stochastic extinctions extinctions and and recolonizations recolonizations [as [as supported supported by by the 1 986) and 1991)]. Figure the results results of of Hanski Hanski ((1986) and Peltonen Peltonen and and Hanski Hanski ((1991)]. Figure 14.2 14.2 shows shows the the relationship relationship between between the the expected expected lifetime lifetime of of populations populations and and their their four species based on the parameter parameter values estimated carrying capacity for the four with 993). This with the the metapopulation metapopulation model model (Hanski, (Hanski, 11993). This result result shows shows wide wide
1 4. 14.
MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION MECHANISMS
345 345
100 100
�
�
i
:8 CI1 50 60 � 0
araneUScaecutiens J
c
C~ 0.
¥
al §t
w
cinereus minutus minutus
60 50 Expected population population size size Expected
100 100
Fig. 114.2 Relationship between the the expected expected population population lifetime lifetime and the the carrying carrying capacity Fig. 4 . 2 Relationship Sorex shrews on islands. The result was calcal (island area times average density) in four four species of of Sorex (island culated with with the parameters the incidence function population model fitted fitted to to data on parameters of the function meta metapopulation island 1 993). island occupancy occupancy (from (from Hanski, Hanski, 1993).
variation in the value of of s, which can be interpreted interpreted as as variation variation in in the impact variation in the value s, which can be the impact of the species, species, as their rr values values are are comparcompar of environmental environmental stochasticity stochasticity among among the as their able. positive correlation the able. Furthermore, Furthermore, aa positive correlation exists exists between between the the body body size size of of the s, suggesting that environmental stochasticity plays plays aa species and species and the the value value of of s, suggesting that environmental stochasticity greater role in in the small than than large large species species of shrew [Cook [Cook and and greater role the dynamics dynamics of of small of shrew Hanski ((1995) 1995) reported oceanic islands]. Hanski reported the the same same relationship relationship for for birds birds on on oceanic islands]. This This result result makes makes biological biological sense sense because because the the smallest smallest species species of of shrew, shrew, which which weigh less than hours, are particularly vulnerable weigh less than 33 gg and and starve starve in in aa few few hours, are particularly vulnerable to to temporal 1998a) further temporal variation variation in in food food availability. availability. Hanski Hanski ((1998a) further estimated estimated the the values of common shrew araneus) from values of rr and and U'e O'e22 for for the the common shrew (Sorex (Sorex araneus) from the the parameter parameter values population model, values values of of the the meta metapopulation model, as as r = = 0.75 0.75 and and U'e %22 = = 0.42. 0.42. These These values are shrews, which are consistent consistent with with the the biology biology of of shrews, which live live for for 11 yr yr only only and and produce produce one 989). The one to to three three litters litters of of seven seven young young on on average average (Sheftel, (Sheftel, 11989). The coefficient coefficient of of variation variation calculated calculated from from these these values values of of rr and and U'/ %2 is is 0.86, 0.86, which which is is consistent consistent with calculated from with the the observed observed CV CV calculated from trapping trapping data, data, 0.67 0.67 (average (average of of four four independent 989). These independent estimates; estimates; Hanski Hanski and and Pankakoski, Pankakoski, 11989). These results results are are encouraging encouraging in in highlighting highlighting aa clear clear connection connection between between parameters parameters of of the the extinction parameters of population extinction model model for for single single populations populations and and parameters of the the meta metapopulation model. model. One One general general difficulty, difficulty, however, however, is is that that the the estimates estimates of of rr and and U'e %22 thus thus obtained 998a). obtained are are sensitive sensitive to to the the estimate estimate of of population population density density (Hanski, (Hanski, 11998a). Luckily, Luckily, the the scaling scaling constant constant ss is is not not similarly similarly affected. affected.
Complex Population Dynamics Dynamics and Extinction Population Population dynamics dynamics may may be be called called simple simple if if the the growth growth rate rate is is aa monoton monotonically ically decreasing decreasing function function of of population population density density and and if if the the density-dependent density-dependent feedback feedback itself itself does does not not suffice suffice to to generate generate population population oscillations. oscillations. In In this this case, case, exemplified exemplified among among others others by by the the continuous-time continuous-time logistic logistic model model and and the the ceiling ceiling model model in in Box Box 14.1, 14.1, population population density density would would settle settle to to aa stable stable state state with with
346 346
OSCAR AND ILKKA OSCAR E. E. GAGGIOTTI GAGGIOTTI AND ILKKA HANSKI HANSKI
constant constant population population size size in in the the absence absence of of environmental environmental perturbations perturbations and and demographic extinction is typically caused caused by demographic stochasticity. stochasticity. Population Population extinction is typically by aa low low growth poor habitat growth rate rate (e.g., (e.g., due due to to poor habitat quality), quality), aa high high variance variance in in growth growth rate rate (environmental (environmental stochasticity), or a small population population size size due to low carrying capacity increases extinction capacity or or other other factors, factors, which which increases extinction risk risk for for many many reasons reasons ((Section Section 14.4). Not Not all populations populations exhibit such simple dynamics, however, and and their extinc extinction risk may be affected by the extra complexities of population population dynamics. Most population growth Most commonly, commonly, the the population growth rate rate may may be be expected expected to to be be reduced reduced at at very densities due locating aa mate very low low densities due to to difficulty difficulty of of locating mate or or performing performing other other coop cooperative behaviors; this is called the Allee effect (Allee, (Allee, 1938; Allee Allee et al., aI., 1949). If If the the reduction reduction in in growth growth rate rate is is severe severe enough, enough, aa small small population population will will go go deter deterministically also substantially ministically extinct. extinct. Demographic Demographic stochasticity stochasticity also substantially increases increases the the risk of small populations, risk of extinction extinction of of very very small populations, especially especially if if their their growth growth rate rate is is low, low, and can be threshold population size below below which most likely likely popu and there there can be aa threshold population size which the the most population trajectory is a decreasing population population size. In this sense, demographic demographic sto stochasticity 998; Dennis, chasticity creates creates aa sort sort of of stochastic stochastic Allee Allee effect effect (Lande, (Lande, 11998; Dennis, 2002). 2002). conventional Allee effect and demographic demographic stochasticity, In models with both conventional there is an an inflection there is inflection point point in in the the probability probability of of reaching reaching aa small small population population size size before point, which before reaching reaching aa large large size. size. This This inflection inflection point, which corresponds corresponds to to the the unstable equilibrium equilibrium in unstable in the the underlying underlying deterministic deterministic model, model, represents represents aa thresh threshold prospects for old in in the the probabilistic probabilistic prospects for the the population population (Dennis, (Dennis, 2002). 2002). The The inci incidence dence and and importance importance of of the the Allee Allee effect effect have have been been reviewed reviewed most most recently recently by by Saether et 1996), Kuussaari 1 996), Wells al. ((1998), 1 998), Courchamp Saether et al. al. ((1996), Kuussaari et et al. al. ((1996), Wells et et al. Courchamp et al. ((1999), 1 999), and Sutherland ((1999). 1 999). It et al. and Stephens Stephens and and Sutherland It should should be be recognized recognized that that small small populations populations have have aa high high risk risk of of extinction extinction for for many many reasons, reasons, including including both both ecological ecological and and genetic genetic factors factors (Section (Section 14.4), 14.4), and and factors factors that that reduce reduce the the expected increase the expected growth growth rate rate as as well well as as factors factors that that increase the variance variance in in growth growth rate rate (Stephens 999; Dennis, (Stephens et et aI., al., 11999; Dennis, 2002). 2002). Therefore, Therefore, it it is is generally generally difficult difficult to to con conclusively isolate isolate the operation of any particular particular mechanism, mechanism, including including the Allee clusively effect. effect. Many Many mechanisms mechanisms are are often often likely likely to to operate operate in in concert. concert. A Allee effect unstable equilibrium A strong strong Allee effect creates creates an an unstable equilibrium point point below below which which the the population goes extinct case there population goes extinct in in aa deterministic deterministic model. model. In In this this case there are are two two alternative stable alternative stable equilibria, equilibria, one one corresponding corresponding to to large large population population size size (set (set by by density dependence dependence at high density) and the other other one corresponding corresponding to popu population lation extinction. extinction. If If the the dynamics dynamics exhibit exhibit such such alternative alternative stable stable equilibria, equilibria, aa small population below the unlikely to small population below the unstable unstable equilibrium equilibrium is is unlikely to become become large, large, although thanks to although it it may may do do so so and and cross cross the the unstable unstable equilibrium equilibrium thanks to aa favorable favorable environmental perturbation. perturbation. Likewise, a large population population above the unstable unstable environmental equilibrium equilibrium is expected expected to remain remain large, but but a perturbation perturbation may take it below the the treshold treshold population population size size and and send send it it toward toward extinction. extinction. This This is is aa worrying worrying possibility large populations may have possibility because because it it implies implies that that currently currently large populations may have aa much much greater one might expect and predict with models that greater risk risk of of extinction extinction than than one might expect and predict with models that fail include the mechanism creating alternative equilibria. fail to to include the mechanism creating alternative equilibria. Unfortunately, Unfortunately, it it is is difficult likely this difficult to to assess assess how how likely this scenario scenario is is for for real real populations. populations. Complex Complex population population dynamics dynamics in in the the sense sense of of cyclic cyclic or or chaotic chaotic fluctuations fluctuations maintained maintained by population population dynamic processes (as opposed to environmental environmental effects) effects) have have received received much much attention attention during during the the past past decades decades (May, (May, 1974; 1974; Schaffer, 985; Turchin, Population variability Schaffer, 11985; Turchin, 2003). 2003). Population variability generated generated by by intraspecific intraspecific
347 341
114. 4 . MECHANISMS MECHANISMSOF OF POPULATION POPULATIONEXTINCTION EXTINCTION
and and interspecific interspecific interactions interactions is is expected expected to to increase increase the the risk risk of of extinction extinction just just like like variability variability generated generated by by environmental environmental stochasticity. stochasticity. It It has has even even been been argued argued that that extinctions extinctions caused caused by by chaotic chaotic dynamics dynamics would would exert exert aa (group) (group) selection selection pressure likely and local extinctions pressure that that would would make make chaotic chaotic dynamics dynamics less less likely and that that local extinctions due population persistence because the due to to chaotic chaotic dynamics dynamics would would enhance enhance meta metapopulation persistence because the extinctions 993; Gonzalez-Andujar extinctions would would be be asynchronous asynchronous (Allen (Allen et et aI., al., 11993; Gonzalez-Andujar and and Perry, 993; Bascompte Sole, 11994; 994; Ruxton, 996). Although Perry, 11993; Bascompte and and Sol~, Ruxton, 11996). Although these these issues issues involve involve many many challenges challenges for for further further research, research, it it seems seems unlikely unlikely that that complex complex dynamics major factor dynamics in in this this sense sense would would be be aa major factor in in population population extinctions. extinctions.
114.3 4.3
POPULATION POPULATION EXTINCTION: EXTINCTION: GENETIC GENETIC FACTORS FACTORS Natural also subject genetic factors Natural populations populations are are also subject to to extinction extinction due due to to genetic factors even in human impact impact and posed by even in the the absence absence of of any any human and the the threat threat posed by ecological ecological processes. processes. Genetic Genetic threats threats are are aa function function of of the the effective effective population population size, size, N Ne. e• Strictly speaking, Strictly speaking, N Nee is is defined defined as as the the number number of of individuals individuals in in an an ideal ideal popu population lation that that would would give give the the same same rate rate of of random random genetic genetic drift drift as as observed observed in in the the actual 9 3 1 , 11938). 93 8 ) . The actual population population (Wright, (Wright, 11931, The ideal ideal population population consists consists of of N N individuals ping generations individuals with with nonoverlap nonoverlapping generations that that reproduce reproduce by by aa random random union union of gametes. More More intuitively, N N~e can can be be defined defined as as the the number number of of indi individuals viduals in in aa population population that that contribute contribute genes genes to to the the following following generation. generation. This This number number can can be be much much lower lower than than the the observed observed population population size size because because of of unequal unequal sex ratios, variance variance in family family size, temporal temporal fluctuations fluctuations in population population size, 995). Thus, size, and and so so forth forth (for (for aa review, review, see see Frankham, Frankham, 11995). Thus, apparently apparently large large populations populations may still be quite small in a genetic sense and and hence face genetic genetic problems. problems. Small N N~e can have multiple multiple effects that that include include loss of genetic genetic vari variability, ability, inbreeding inbreeding depression, depression, and and accumulation accumulation of of deleterious deleterious mutations. mutations. The The time time scales scales at at which which these these factors factors operate operate differ differ and, and, to to aa large large extent, extent, deter determine the entail (Table 4. 1 ). mine the risk risk of of population population extinction extinction that that they they entail (Table 114.1).
Loss Loss of of Genetic Genetic Variability Variability Genetic Genetic variation variation comprises comprises the the essential essential material material that that allows allows natural natural popu populations lations to to adapt adapt to to changes changes in in the the environment, environment, to to expand expand their their ranges, ranges, and and even 992). even to to reestablish reestablish following following local local extinctions extinctions (e.g., (e.g., Hedrick Hedrick and and Miller, Miller, 11992). The The types types of of genetic genetic variation variation considered considered most most often often are are the the heterozygosity heterozygosity of of
TABLE TABLE 11 44.. 11 Time Scales Scales at Which Which Genetic Factors Operate and Their IImportance m portance for for Population ExtinctionG Extinction a Factor Factor
Time scale Time
Extinction Extinction risk risk involved involved
Extinction Extinction vortex vortex
Inbreeding depression Loss of genetic diversity Mutational meltdown
Short Long Long Medium/long
High High Low Low Unknown Unknown
F F A A A A
a a
The last column indicates the extinction vortex (as defined by Gilpin and Soule, 9 8 6 ) under Soul~, 11986) which which each each genetic genetic factor factor operates. operates.
348 348
OSCAR E. HAN SKI OSCAR E. GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKA HANSKI
neutral neutral markers, markers, H, H, and and the the additive additive genetic genetic variance, variance, Va, V~, which which underlies underlies polygenic polygenic characters characters such such as as life life history history traits, traits, morphology, morphology, and and physiology. physiology. In drift leads In small small populations, populations, random random genetic genetic drift leads to to stochastic stochastic changes changes in in gene due to to Mendelian and variation gene frequencies frequencies due Mendelian segregation segregation and variation in in family family size. size. In In the the absence absence of of factors factors that that would would replenish replenish genetic genetic variance, variance, such such as as mutation, mutation, migration, selection favoring migration, and and selection favoring heterozygotes, heterozygotes, populations populations lose lose genetic genetic vari variance ance according according to to
Va(t + 1 ) = Va(t)(1- 2@e)'
((14.5) 14.5)
Va(t)
where where Va(t) is is the the additive additive genetic genetic variance variance in in the the tth tth generation. generation. A A similar similar equation equation is is obtained obtained for for heterozygosity heterozygosity by by replacing replacing Va Va with with H. H. When When aa popu population and maintained maintained at at that that size size for for lation is is reduced reduced to to aa small small effective effective size size Ne and more than than 2Ne generations, greatly (Wright, generations, its its genetic genetic variability variability is is reduced reduced greatly (Wright, more 11969). 969). Genetic restored to muta Genetic variability variability can can be be restored to its its original original level level through through mutation original size. tion if if the the population population grows grows back back to to its its original size. The The number number of of genera generations tions required required to to attain attain the the original original level level is is of of the the order order of of the the reciprocal reciprocal of of the the mutation mutation rate, rate, /-1. ix. Thus, Thus, for for aa nuclear nuclear marker marker with with aa mutation mutation rate rate of of 1100-6, -6 genetic genetic variation variation is is restored restored after after 106 106 generations, generations, but but genetic genetic variation variation of of quantitative 000 generations quantitative characters characters can can be be restored restored after after only only 11000 generations because because the the relevant relevant mutation mutation rate rate is is two two orders orders of of magnitude magnitude higher. higher. The genetic variation during aa bottleneck The maximum maximum fraction fraction of of genetic variation lost lost during bottleneck is is aa function 975). Populations function of of the the population population growth growth rate rate (Nei (Nei et et al., al., 11975). Populations that that recover variation even recover quickly quickly after after the the bottleneck bottleneck lose lose little little genetic genetic variation even if if the the popu population example, aa growth lation was was reduced reduced to to aa few few individuals individuals only. only. For For example, growth rate rate of of r 0.5 (A. r = = 0.5 (~ = = e' er = = 1.65) 1.65) allows allows aa population population that that is is reduced reduced to to only only two two indi individuals 0 % of viduals to to retain retain 550% of its its genetic genetic variability variability (Fig. (Fig. 14.3). 14.3). If If the the population population is is (A. = . 10) would reduced 10 individuals, reduced to to 10 individuals, then then aa growth growth rate rate of of r = = 0.1 0.1 (~ = 11.10) would allow allow it it to to retain retain 60% 60% of of its its variability. variability. Additionally, Additionally, generation generation overlap overlap can can buffer environmental fluctuations sizes. In buffer the the effect effect of of environmental fluctuations on on population population sizes. In general, general,
Ne
2Ne
0.8 tl � c .2
�
ll.
0.6 0.4 0.2 o
Growth rate, r Fig. 1 44.3 the genetic Fig. . 3 Fraction of the genetic variation variation lost during during a population population bottleneck bottleneck of N N = 2 or 1100 individuals. Calculated using al. (1 975). individuals. Calculated using Eq. Eq. (8) (8) in in Nei Nei et et al. (1975). =
MECHANISMSOF OF POPULATION POPULATION EXTINCTION EXTINCTION 114. 4. MECHANISMS
349 349
population size are brought about by environmental environmental changes reductions in population fluctuations in vital rate parameters parameters (environmental stochasticity; that cause fluctuations Section Section 14.2). 14.2). The The effect effect of of these these fluctuations fluctuations on on Ne Ne depends depends on on the the life life his hisNe to census size is directly directly proportional proportional to the tory of the species. The ratio of Ne population, but the sensitivity of of this this ratio ratio to to total reproductive value of a population, environmental fluctuations fluctuations is proportional proportional to the generation overlap. The environmental generation overlap, the smaller the effect of environmental environmental fluctua fluctualarger the generation tions on the level of genetic variability maintained maintained by natural populations populations ((Gaggiotti Gaggiotti and Vetter, 11999). 999). Thus, genetic variability is maintained maintained through through the "storage" long-lived stages. Adult "storage" of genotypes in long-lived Adult individuals individuals representing these stages reproduce reproduce many times throughout throughout their lives and, therefore, the genetic variability present in a given cohort cohort is more likely to be transferred transferred to future generations than in the case of organisms with discrete generations. These buffering mechanisms may explain why there are very few clear examples examples of of populations populations that that have have lost lost aa very very large large fraction fraction of of their their genetic genetic vari varithat of the Mauritius Mauritius kestrel, ability due to a bottleneck. One of the few cases is that which 950s. A which was was reduced reduced to to aa single single pair pair in in the the 11950s. A comparison comparison of of microsatellite microsatellite museum specimens collected before before the bottleneck bottleneck and in diversity present in museum extant individuals individuals reveals that that at least 50% 50% of the heterozygosity was lost due extant to the bottleneck (Groombridge (Groombridge et aI., al., 2000). Another Another example example is the northern northern elephant seal, which was exploited exploited heavily during the 119th and reduced elephant 9th century and to a bottleneck 0-30 individuals (Hoelzel et aI., bottleneck population size estimated estimated to be 110-30 al., 2002). A comparison bottleneck and comparison of genetic diversity in pre prebottleneck and postbottleneck postbottleneck samples shows samples shows aa 50% 50% reduction reduction in in mtDNA-haplotype mtDNA-haplotype diversity. diversity. The The reduction reduction heterozygosity at microsatellite microsatellite loci was less pronounced, in heterozygosity pronounced, however. An important important caveat caveat concerning concerning the the effect effect of reductions in in population population size size on on An of reductions genetic diversity diversity is that although although such such reductions may not not have have aa very very large large genetic is that reductions may H, they will have a large impact effect on on H, impact on allelic diversity because random random eliminate low-frequency low-frequency alleles very rapidly rapidly (Nei et al., genetic drift drift will eliminate aI., 1975). 1 975). This is of the long-term response response of population to to of particular particular concern concern because the of a population that remains remains after bottleneck selection is determined determined by the allelic diversity that after the bottleneck or mutations (James, 1971). 1 971 ) . A second that in the the or that that is gained through through mutations second caveat is that case of quantitative quantitative genetic characters, genetic variability variability may may not not always be model with overlapping generations assuming weak stabeneficial. Using a model with overlapping generations and and assuming weak sta bilizing selection, Lande Lande and and Shannon Shannon ((1996) that the effects 1 996) showed showed that effects of additive additive genetic variance on the average deviation on the deviation of the the mean mean phenotype phenotype from from the the optiopti mum, mum, and and the the corresponding corresponding "evolutionary" "evolutionary" load, depend depend on on the the pattern pattern of of environmental change. change. In In an an unpredictable unpredictable (random) (random) environment, environment, additive additive environmental genetic variance contributes contributes to to the the evolutionary evolutionary load load because any any response response to to selection increases increases the the expected between the the mean mean phenotype phenotype and and the the selection expected deviation deviation between optimum. However, However, when when environmental environmental changes changes are are unidirectional, unidirectional, cyclic, cyclic, or or optimum. positively correlated (predictable), additive additive genetic variance allows allows the the mean mean positively correlated (predictable), genetic variance phenotype to track the optimum more closely, closely, reducing the evolutionary evolutionary load. load. phenotype to track the optimum more reducing the Most studies on the effects population bottlenecks Most empirical empirical studies on the effects of of population bottlenecks on on genetic focus on on the the heterozygosity heterozygosity of of neutral neutral markers. Although Although neutral neutral diversity focus variation may may become become adaptive adaptive if the the environment environment changes, the the ability ability genetic variation of of a population population to to respond respond to to novel novel selection selection pressures pressures is proportional proportional to to the the additive genetic genetic variation variation underlying underlying the the traits traits that that are are the the target target of of selection selection additive (Falconer and and Mackay, 1 996). Unfortunately, Unfortunately, direct direct quantification quantification of of the the genetic genetic (Falconer Mackay, 1996).
3350 50
OSCAR OSCAR E. E. GAGGIOTTI GAGGIOTII AND AND ILKKA ILKKA HANSKI HANSKI
variation underlying underlying polygenic polygenic traits traits is is difficult difficult to to measure, measure, and and hence hence heterozyheterozy variation gosity gosity of of nuclear nuclear markers markers is is used used as as an an indicator indicator of of additive additive genetic genetic variation variation [see Pfrender pfrender et et al. ai. (2001) (2001 ) and and references references therein]. therein] . This This practice practice is is unwarranted, unwarranted, [see however, because because of of the the different different rates rates at at which which genetic genetic variation variation is is replenished replenished however, in in neutral neutral and and quantitative quantitative markers markers (Lande (Lande 1988; 1988; see see earlier earlier discussion). discussion). Indeed, Indeed, pfrender et et al. ai. (2001) (200 1 ) detected detected no no significant relationship between between heritability heritability Pfrender significant relationship for reproductive reproductive traits traits and and heterozygosity heterozygosity in in natural natural populations populations of of Daphnia for and D. pulicaria. Thus, Thus, the the absence absence of of genetic genetic diversity diversity in in nuclear nuclear markmark pulex and ers does does not not necessarily necessarily indicate indicate an an immediate immediate genetic genetic threat. threat. ers In general, general, the the loss loss of of genetic genetic variation variation is is detrimental detrimental for for the the long-term long-term sursur In vival of of populations. populations. However, However, as as pointed pointed out out by by Allendorf Allendorf and and Ryman Ryman vival (2002), there is is one one case case where where aa reduction reduction in in genetic genetic variability variability can can represent represent (2002), there an imminent imminent extinction extinction threat. threat. This This is is the the case case for for loci loci associated associated with with disease disease an resistance, such such as as the the major major histocompatibility complex (MHC), (MHC), which is one one resistance, histocompatibility complex which is of the the most most important genetic systems infectious disease disease resistance resistance in in verver of important genetic systems for for infectious tebrates tebrates (Hill, (Hill, 1998; 1 998; Hedrick Hedrick and and Kim, Kim, 2000). 2000). Allelic Allelic diversity diversity at at these these loci loci is is extremely Parham and Otha ((1996) 1 996) documented documented 179 1 79 alle extremely high; high; for for example, example, Parham and Otha allethe MHC MHC class class II locus in humans. humans. However, However, species been les at les at the locus in species that that have have been through known bottlenecks have very very low amounts of of MHC MHC variation. A through known bottlenecks have low amounts variation. A study of the the Arabian Arabian oryx found only only three three alleles alleles present present at the MHC MHC class class II II study of oryx found at the DRB locus locus in in aa sample of 57 57 individuals (Hedrick et et al., ai., 2000). 2000). Hunting DRB sample of individuals (Hedrick Hunting pres pressure led led to to the the extinction of this this species in the the wild wild in in 1972. 1 972. Captive popula sure extinction of species in Captive populations have been susceptible to tuberculosis and foot-and-mouth foot-and-mouth disease, disease, which which tions have been susceptible to tuberculosis and is low genetic variability at at MHC MHC loci. loci. Low Low genetic genetic diversity diversity is consistent consistent with with low genetic variability at the MHC MHC complex complex was was also also observed observed in in the the bison, bison, which which went went through at the through aa bottleneck at end of 9th century 997). In bottleneck at the the end of the the 119th century (Mikko (Mikko et et aI., al., 11997). In the the Przewalski's horse, in entire species species is founders, Przewalski's horse, in which which the the entire is descended descended from from 1133 founders, Hedrick et ai. ((1999) 1 999) observed one locus Hedrick et al. observed four four alleles alleles at at one locus and and two two alleles alleles at at aa second locus. The another example example of MHC second locus. The northern northern elephant elephant seal seal is is another of low low MHC diversity, 1 999) found class II diversity, as as Hoelzel Hoelzel et et al. al. ((1999) found only only two two alleles alleles at at the the MHC MHC class II DQB DQB gene gene in in aa sample sample of of 69 69 individuals. individuals. To To summarize, summarize, we we may may conclude conclude that that loss loss of of genetic genetic variation variation as as measured measured by by heterozygosity heterozygosity and and additive additive genetic genetic variance variance represents represents aa long-term long-term extinc extinction tion threat. threat. In In the the short short term, term, the the loss loss of of allelic allelic diversity diversity can can have have important important consequences consequences if if it it occurs occurs at at loci loci associated associated with with disease disease resistance. resistance.
Inbreeding Inbreeding Depression Depression The The decrease decrease in in fitness fitness due due to to mating mating between between related related individuals individuals is is known known as partially reces as inbreeding inbreeding depression depression and and results results from from the the segregation segregation of of partially recessive sive deleterious deleterious mutations mutations maintained maintained by by the the balance balance between between selection selection and and Deleterious mutations mutations occur occur continuously continuously in in all all populations populations and and mutation. Deleterious most most mutations mutations are are at at least least partially partially recessive. recessive. In In large large populations, populations, selection selection keeps keeps these these detrimental detrimental mutations mutations at at low low equilibrium equilibrium frequencies. frequencies. Thus, Thus, under under random random mating, mating, most most copies copies of of detrimental detrimental alleles alleles are are present present in in aa het heterozygous erozygous state state and and hence hence their their detrimental detrimental effects effects are are partially partially masked. masked. Mating Mating between between relatives, relatives, however, however, increases increases homozygosity homozygosity and, and, therefore, therefore, the the deleterious deleterious effects effects become become fully fully expressed, expressed, decreasing decreasing the the fitness fitness of of inbred inbred individuals. individuals.
351
MECHANISMSOF OF POPULATION POPULATION EXTINCTION EXTINCTION 114. 4. MECHANISMS
Although Although it it is is generally generally agreed agreed that that increased increased expression expression of of deleterious deleterious par parrecessive alleles is the main cause of inbreeding depression, there is an add addtially recessive itional mechanism that can contribute to inbreeding depression. If the fitness fitness of a heterozygote is superior to that of both homozygotes (heterozygous advantage heterozygotes will reduce the or overdominance), the reduced frequency of heterozygotes importopportunities to express heterozygous advantage. This mechanism may be import Drosophila melanogaster) melanogaster) and ant for certain traits (e.g., sperm precedence in Drosophila may contribute contribute to the very high inbreeding inbreeding depression for net fitness observed in Drosophila 999). Drosophila and outcrossing plants (Charlesworth and Charlesworth, 11999). The degree of inbreeding in a population population is measured by the inbreeding coefficient alleles of coefficient F, F, which which can can be be defined defined as as the the probability probability that that the the two two alleles of aa inbreeding in a gene in an individual are identical by descent. The effect of inbreeding population population with with inbreeding inbreeding coefficient coefficient F F can can be be measured measured in in terms terms of of the the log logratio of the mean fitness values for the outbred, outbred, W Wo, 0, and the arithm of the ratio inbred, W], Charlesworth and Charlesworth, 999), WI, populations populations ((Charlesworth Charlesworth, 11999),
( :�) =
In In - ~ o
=
- BF. BF.
((14.6) 14.6)
coefficient B can be interpreted interpreted as the reduction reduction in log log fitness associated associated The coefficient with complete complete inbreeding (F ( F - 11). ). with populations, the opportunities opportunities for mating mating are restricted, restricted, even under under In small populations, random random mating. mating. Thus, mating among among relatives is common common and the proportion proportion individuals that that are homozygous homozygous at many many loci increases, which which results in of individuals inbreeding inbreeding depression. depression. The The amount amount of of inbreeding inbreeding depression depression manifested manifested by by aa population but also on the selection to to population depends not not only on F, but the opportunity opportunity for selection purge recessive lethal and mutations. Gradual inbreeding by increincre and semilethal semilethal mutations. Gradual inbreeding mental reductions in population population size over many many generations mental reductions generations allows allows selection to to eliminate and sublethal mutations when become homozygous eliminate the lethal lethal and sublethal mutations when they become homozygous ((Falconer, Falconer, 11989). 989). However, However, the the component component of of inbreeding inbreeding depression depression due to due to more mutations of of small effect to purge purge by inbreedinbreed more nearly additive additive mutations effect is difficult difficult to 1 995). As to to empirical results, recent indicate that that purg ing (Lande, 1995). empirical results, recent reviews indicate purgreducing inbreeding inbred populations ing is inefficient inefficient in reducing inbreeding depression depression in small inbred populations Allendorf and and Ryman Ryman (2002) and references [see Allendorf (2002) and references therein]. therein]. Most of of the the evidence for for inbreeding inbreeding depression from domesticated Most depression comes from domesticated or captive This, together captive populations. populations. This, together with with the the theoretical theoretical expectation expectation that that a fraction of of inbreeding inbreeding depression depression can be purged purged in small small populations populations and and large fraction the the numerous numerous mechanisms mechanisms of of inbreeding inbreeding avoidance avoidance observed in many many species, has importance of has led many many researchers researchers to to question question the the importance of inbreeding inbreeding depression depression the persistence persistence of of natural natural populations populations (Keller and and Waller, 2002). 2002). However, However, for for the in the the last last decade decade there there has has been been aa rapid rapid accumulation accumulation of of evidence evidence showing showing in that that many many populations populations do do exhibit exhibit inbreeding inbreeding depression. depression. For For example, example, the the Soay sheep suffer of sheep on on the the island island of of Hirta Hirta (Saint ( Saint Kilda Kilda archipelago, archipelago, UK) suffer of sigsig nificant nificant inbreeding inbreeding depression depression in survival (Coltman ( Coltman et et al., a!., 1999). 1 999). More More homozygous sheep sheep suffered higher rates rates of of parasitism parasitism and, and, in turn, turn, lower lower overover homozygous suffered higher winter winter survival survival than than heterozygous heterozygous sheep. Another Another example example comes comes from from song sparrows sparrows living living on on Mandarte Mandarte Island Island (western (western Canada). Canada). In In this this case, inbred inbred birds birds died died at at a much much higher higher rate rate during during a severe severe storm storm than than outbred outbred birds birds (Keller et et al., a!., 1994). 1 994). A more more recent recent study study (Keller, 1998) 1998) was was able able to to quantify quantify
352 352
OSCAR SKI OSCAR E. E. GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKA HAN HANSKI
inbreeding depression in this population and estimated that that inbreeding depres depres49%.. The sion in progeny from a mating between first-degree relatives was 49% effect of inbreeding has also been documented in the red-cockaded negative effect woodpecker living in the southeastern United States. Inbreeding reduced egg hatching rates, fledgling survival, and recruitment to the breeding population (Daniels and Walters, Wahers, 2000). Extensive long-term data sets can help uncover inbreeding depression in large populations with a low rate of inbreeding. An population of the collared flycatcher revealed that 118-yr 8-yr study of a large population inbreeding was rare, but when it did occur it caused a significant reduction in suregg hatching rate, in fledgling skeletal size, and in postfledging juvenile sur vival (Kruuk et aI., al., 2002). This study also found that the probability of mat mat(F = 0.25 0.25)) inc-eased inczeased throughout the breeding ing between close relatives (F season, possibly reflecting increased costs of inbreeding avoidance. Inbreeding depression 1 999) documented depression is is also also evident evident in in plants. plants. Byers Byers and and Waller Waller ((1999) documented many examples of inbreeding depression in natural populations and indicated that purging does not not appear to act consistently as a major force in natural natural that populations. plant populations. Evidence shows shows that that stressful environmental environmental conditions conditions can amplify Crnokrak and Roff ((1999) inbreeding depression. Crnokrak 1 999) gathered and analyzed a that included seven bird species, nine mammal species, four species of data set that poikilotherms, and 15 plant species. They were able to show that that conditions experienced in the wild increase the cost of inbreeding. A more recent study by that the magnitude of inbreeding depression in Keller et al. ai. (2002) showed that juvenile and adult survival of cactus finches living in Isla Daphne Daphne Major Major (Galapagos (Galfipagos Archipelago) was strongly modified by two two environmental environmental condi conditions; food food availability and number of competitors. and number competitors. In juveniles, inbreeding present only in years with depression was present with low food food availability, whereas in adults, inbreeding inbreeding depression was with low was five times times more more severe in years years with food availability and and large population population size. Demonstrating the Demonstrating the importance importance of inbreeding inbreeding depression depression in the wild wild does not necessarily imply that natural populations to decline (Caro (Caro not that it will cause natural and that this and Laurenson, 11994). 994). However, recent papers have demonstrated demonstrated that al. (1998) may happen. happen. Saccheri et ai. ( 1 998) studied the the effect of inbreeding on local extinction in a large metapopulation metapopulation of the Glanville fritillary butterfly (Melitaea cinxia) cinxia) and found found that with that extinction extinction risk increased significantly with decreasing heterozygosity due to to inbreeding, inbreeding, even after after accounting accounting for the the effects of of ecological factors. Larval survival, survival, adult adult longevity, and and egg hatching hatching rate affected adversely by inbreeding rate were were all affected inbreeding and and seem to to be the the fitness fitness comcom ponent responsible responsible for the relationship relationship between between inbreeding inbreeding and and extinction. extinction. An ponent for the experiment experiment by Nieminen Nieminen et et al. ai. (2001) (200 1 ) provided further further support support to to the results of of Saccheri et al.'s aI.'s (1998) ( 1 99 8 ) field study. Nieminen Nieminen et al. (2001) (2001 ) established established inbred inbred and and outbred outbred local populations populations of of the Glanville fritillary fritillary at at previously unoccuunoccu pied sites using the the same numbers numbers of of individuals. individuals. The The extinction rate rate was sigsig higher in populations populations established established with with inbred inbred individuals. individuals. Similar nificantly higher for plants plants is provided provided by Newman Newman and and Pilson (1997). ( 1 997). They estab evidence for They established experimental Clarkia pulcbella pulchella that that experimental populations populations of the annual annual plant plant Clarkia differed in the differed the relatedness relatedness of of the the founders. founders. All populations populations were were founded founded by the same same number number of of individuals individuals but but persistence persistence time time was was much much lower lower in those those the populations whose whose founders founders were were related. related. Additional Additional evidence for for inbreeding inbreeding populations
1 4. 14.
MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION MECHANISMS
353 353
influencing population population dynamics dynamics comes comes from from the the study study of of an an isolated isolated populapopula influencing tion of of adders adders in in Sweden Sweden (Madsen (Madsen et et al., aI., 1999), 1 999), which which declined declined dramatically dramatically tion in in the the late late 1960s 1 960s and and was was on on the the brink brink of of extinction extinction due due to to severe severe inbreeding inbreeding depression. The The introduction introduction of of 20 20 adult adult male male adders adders from from aa large large and and geneticgenetic depression. ally variable variable population population led led to to aa rapid rapid population population recovery recovery due due to to aa dramatic dramatic ally increase in in recruitment. recruitment. increase The evidence evidence discussed discussed here here indicates indicates that that inbreeding inbreeding depression depression is is common common The in natural natural populations populations and and can can represent represent aa short-term extinction threat threat to to in short-term extinction small populations, populations, especially especially if if populations populations are are subject subject to to stressful stressful conditions conditions small or to to sharp sharp population population declines. declines. or
Accumulation of of Slightly Slightly Deleterious Deleterious Mutations Mutations Accumulation Under or less less constant constant environmental environmental conditions, conditions, mutations mutations with with phephe Under more more or notypic effects are are usually usually deleterious deleterious because because populations populations tend tend to to be be well well notypic effects adapted to to the the biotic biotic and and abiotic abiotic environmental conditions which they experiexperi adapted environmental conditions which they ence. A A random random mutation mutation is is likely likely to to disrupt disrupt such adaptation. In In populations populations ence. such adaptation. with or large effective sizes, is very in eliminating eliminating with moderate moderate or large effective sizes, selection selection is very efficient efficient in detrimental mutations with with large large effects effects on on fitness. fitness. However, However, mildly detrimental mutations mildly deleteri deleterious mutations are difficult to remove remove mutations with coefficient 5s < ous with selection selection coefficient < 1I2Ne 1/2Ne are difficult to because they behave almost almost as neutral mutations 1 9 3 1 ) . Thus, small because they behave as neutral mutations (Wright, (Wright, 1931). Thus, small population size the role role of of genetic genetic drift in population size hampers hampers selection selection and and increases increases the drift in determining allele frequencies and fates. fates. This increases the the chance fixation of determining allele frequencies and This increases chance fixation of some alleles supplied by mutation mutation and and results in some of of the the deleterious deleterious alleles supplied constantly constantly by results in the population mean leads to to population population the reduction reduction of of population mean fitness, fitness, which which eventually eventually leads extinction 964). Initially, extinction (Muller, (Muller, 11964). Initially, this this process process was was assumed assumed to to represent represent aa threat to only because threat to asexual asexual populations populations only because in in the the absence absence of of recombination, recombination, their their offspring offspring carry carry all all the the mutations mutations present present in in their their parent parent as as well well as as any any newly 964). Mathematical models of newly arisen arisen mutation mutation (Muller, (Muller, 11964). Mathematical models of this this process process ((Lynch Lynch and 990; Lynch 993, 11995a) 995a) show and Gabriel, Gabriel, 11990; Lynch et et aI., al., 11993, show that that the the process process of of mutation mutation accumulation accumulation can can be be divided divided into into three three phases. phases. During During the the first first two two phases, but population phases, deleterious deleterious mutations mutations accumulate accumulate and and fitness fitness declines, declines, but population size size remains remains close close to to carrying carrying capacity. capacity. During During the the third third phase, phase, fitness fitness drops drops below 11 and population size population decline decline increases increases the below and population size declines. declines. This This population the effect effect of of random random genetic genetic drift, drift, which which enhances enhances the the chance chance fixation fixation of of future future deleterious decline and deleterious mutations, mutations, leading leading to to further further fitness fitness decline and reduction reduction in in popu popusize. Due Due to to this this positive positive feedback, feedback, the the final final phase phase of of population population decline decline lation size. (when (when growth growth rate rate is is negative) negative) occurs occurs at at an an accelerating accelerating rate, rate, aa process process known known as as "mutational "mutational meltdown. meltdown."" Although recombination recombination can can slow slow down down the the mutational mutational meltdown meltdown ttoo some some Although extent, extent, sexual sexual populations populations are are also also at at risk risk of of extinction extinction due due to to mutation mutation accu accumulation 994; Lynch 995a). Lande 1 994) modeled mulation (Lande, (Lande, 11994; Lynch et et aI., al., 11995a). Lande ((1994) modeled aa ran randomly population with domly mating mating population with no no demographic demographic or or environmental environmental stochasticity stochasticity and and considered considered only only unconditionally unconditionally deleterious deleterious mutations mutations of of additive additive effects. effects. He He derived derived analytical analytical approximations approximations for for the the mean mean time time to to extinction extinction for for two two cases: cases: (a) (a) when when all all mutations mutations had had the the same same selection selection coefficient coefficient 5s and and (b) (b) when when there 1 995a) provided there was was variance variance in in 5. s. Lynch Lynch et et ai. al. ((1995a) provided aa more more detailed detailed analysis analysis of of scenario scenario (a) (a) and and checked checked the the analytical analytical results results using using computer computer simulations. simulations. With With constant constant 5, s, the the mean mean time time to to extinction, extinction, te, te, is is an an approximately approximately
354 354
OSCAR LKKA HAN SKI OSCAR E. E. GAGGIOTII GAGGIO1-FI AND AND IILKKA HANSKI
exponential exponential function function of of the the effective effective population population size. size. Because Because the the mean mean time time to to extinction extinction increases increases very very rapidly rapidly with with increasing increasing Ne, Ne, the the fixation fixation of of new new muta mutations 00 tions poses poses little little risk risk of of extinction extinction for for populations populations with with Ne Ne of of about about 1100 (Lande, 994) . However, variance in mean time (Lande, 11994). However, with with variance in s, the the mean time to to extinction extinction increases as increases as aa power power of of Ne. N~. For For instance, instance, if if s is is distributed distributed exponentially, exponentially, te is is asymptotically proportional to N N �2.. As As an an increase increase in in te with with population population size size is is asymptotically proportional to now more gradual than for risk of much elevated. now more gradual than for constant constant s, the the risk of extinction extinction is is much elevated. variation around For reasonable variance For reasonable variance in in s (coefficient (coefficient of of variation around 11),), the the mutational mutational meltdown pose aa considerable considerable risk populations meltdown is is predicted predicted to to pose risk of of extinction extinction for for populations with 994). If, with Ne N~ as as large large as as aa few few thousand thousand individuals individuals (Lande, (Lande, 11994). If, as as is is gener generally population size ally agreed, agreed, the the ratio ratio of of Ne N~ to to census census population size is is around around 0.1 0.1 to to 0.5, 0.5, mod moderately populations of erately sized sized populations of several several thousand thousand individuals individuals may may face face extinction extinction due due to to genetic genetic stochasticity. stochasticity. Unfortunately, Unfortunately, there there is is aa paucity paucity of of empirical empirical evidence evidence for for or or against against the the mutational experimental evidence mutational meltdown. meltdown. What What we we have have is is experimental evidence for for the the accumu accumulation deleterious mutations mutations due these studies lation of of deleterious due to to genetic genetic drift, drift, but but these studies do do not not directly 1 ) . As directly address address the the risk risk of of extinction extinction (Zeyl (Zeyl et et aI., al., 200 2001). As of of today, today, only only Zeyl Zeyl et aI. (2001 explicitly explored meltdown. et al. (2001)) explicitly explored the the plausibility plausibility of of the the mutational mutational meltdown. They yeast Saccharomyces cerevisiae They established established 12 12 replicate replicate populations populations of of the the yeast from rates differed from two two isogenic isogenic strains strains whose whose genome-wide genome-wide mutation mutation rates differed by by approximately protocol that approximately two two orders orders of of magnitude. magnitude. They They used used aa transfer transfer protocol that resulted around 250. more than resulted in in an an effective effective population population size size of of around 250. After After more than 100 100 daily daily bottlenecks, bottlenecks, yeast yeast populations populations with with elevated elevated mutation mutation rates rates showed showed aa tendency tendency to size, whereas wild-type mutation mutation rates to decline decline in in size, whereas populations populations with with wild-type rates remained remained constant. popu constant. Moreover, Moreover, there there were were two two actual actual extinctions extinctions among among the the mutant mutant populations. These lations. These results results provide provide support support for for the the mutational mutational meltdown meltdown models. models. Despite Despite this this preliminary preliminary empirical empirical support, support, there there are are aa number number of of issues issues that that remain one relates controversy about remain unresolved. unresolved. The The first first one relates to to aa controversy about the the estimates estimates of of per-genome mutation, s, used per-genome mutation mutation rates, rates, U, and and the the average average fitness fitness cost cost per per mutation, used values that assumed were the meltdown meltdown models. models. The The values that have have been been assumed were based based on on in the mutation experiments using mutation accumulation accumulation experiments using Drosophila meianogaster, melanogaster, suggest suggesting values of 994; ing values of U = = 11 and and aa reduction reduction in in fitness fitness of of about about 1-2% 1-2% (Lande, (Lande, 11994; Lynch aI., 11995a). 995a). Studies aI. ((1999) 1 999) on Lynch et etal., Studies reviewed reviewed by by Garcia-Dorado Garcia-Dorado et etal. on D. melanogaster, as as well well as as on on Caenorhabditis Caenorhabditis elegans and and S. cerevisiae, yielded yielded values values of of U orders orders of of magnitude magnitude less less than than 11.. However, However, some some mutation mutation accumu accumulation aI., 2002; 997) lation experiments experiments (Caballero (Caballero et et al., 2002; Keightley Keightley and and Caballero, Caballero, 11997) reported reported average average fitness fitness effects effects one one order order of of magnitude magnitude higher higher than than those those reported previously. previously. The The assumption assumption of of additive additive effects effects is is also also questioned questioned by by reported Garcia-Dorado 1 999), who estimates of Garcia-Dorado et et al. al. ((1999), who reported reported estimates of 0.1 0.1 for for the the average average coeffi coeffimuch lower cient of dominance. The new new estimates of U and cient of dominance. The estimates of and s would would lead lead to to much lower rates decline, making rates of of fitness fitness decline, making the the mutational mutational meltdown meltdown less less likely. likely. Caballero Caballero et et al. al. (2002) (2002) used used aa combination combination of of mutation mutation accumulation accumulation experiments experiments and and computer computer simulations simulations and and concluded concluded that that aa model model based based on on few few mutations mutations of of large large effect effect was was generally generally consistent consistent with with their their empirical empirical observations. observations. Finally, Finally, an an additional additional criticism criticism of of the the existing existing mutational mutational meltdown meltdown models models relates models ignore relates to to the the fact fact that that the the models ignore the the effect effect of of beneficial beneficial and and back back mutations. Models including mutations. Models including these these types types of of mutations mutations suggest suggest that that only only very very small would face genetic stochasticity small populations populations would face the the risk risk of of extinction extinction due due to to genetic stochasticity (Poon Otto, 2000; Estimates of (Poon and and Otto, 2000; Whitlock, Whitlock, 2000). 2000). Estimates of mutational mutational effects effects using using
114. 4. MECHANISMS MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION
355 355
mutation mutation accumulation accumulation experiments experiments with with Arabidopsis thaliana thaliana indicate indicate that that roughly roughly half half of of the the mutations mutations reduce reduce reproductive reproductive fitness fitness (Shaw (Shaw et et aI., al., 2002). 2002). The 1-0.2. These The genome-wide genome-wide mutation mutation rate rate was was around around 0. 0.1-0.2. These new new results results suggest suggest that that the the risk risk of of extinction extinction for for small small populations populations may may be be lower lower than than initially initially thought. thought. This This issue issue is is reviewed reviewed in in greater greater detail detail in in Chapter Chapter 7. 7. At possible to At the the moment moment it it is is not not possible to draw draw definite definite conclusions conclusions about about the the importance process. This importance of of the the mutational mutational meltdown meltdown process. This will will only only be be possible possible once once the the existing existing controversy controversy over over the the rate rate and and nature nature of of spontaneous spontaneous muta mutations resolution of tions is is resolved resolved (Poon (Poon and and Otto, Otto, 2000). 2000). The The resolution of this this question question in in turn turn requires requires knowledge knowledge of of the the distribution distribution of of mutational mutational effects effects and and the the extent extent to to which which these these effects effects are are modified modified by by environmental environmental and and genetic genetic background. background. Additionally, contribution of Additionally, it it is is necessary necessary to to better better understand understand the the contribution of basic basic bio biological logical features features such such as as generation generation length length and and genome genome size size to to interspecific interspecific dif differences mutation rate 999). ferences in in the the mutation rate (Lynch (Lynch et et aI., al., 11999).
114.4 4.4
POPULATION POPULATION SIZE, SIZE, HABITAT HABITAT PATCH PATCH SIZE, SIZE, AND AND EXTINCTION EXTINCTION RISK RISK The The most most robust robust generalization generalization that that we we can can make make about about population population extinction extinction is populations face is that that small small populations face aa particularly particularly high high risk risk of of extinction. extinction. Holsinger Holsinger (2000) 1 859), (2000) digged digged up up statements statements to to this this effect effect from from the the writings writings of of Darwin Darwin ((1859), E.B. 1 945), and 1954). More E.B. Ford Ford ((1945), and (not (not surprisingly) surprisingly) Andrewatha Andrewatha and and Birch Birch ((1954). More recent empirical support recent empirical support for for the the extinction-proneness extinction-proneness of of small small populations populations has has been 1984), been found found practically practically whenever whenever this this issue issue has has been been examined; examined; Diamond Diamond ((1984), Newmark 1 991, 1995), 1 993), Burkey 1 995), and Newmark ((1991, 1995), Ouborg Ouborg ((1993), Burkey ((1995), and Fischer Fischer and and Stocklon 1 997) represent St6cklon ((1997) represent aa small small sample sample of of the the literature literature covering covering different different kinds kinds of of taxa taxa and and spatial spatial scales. scales. The The high high extinction extinction risk risk of of small small populations populations is is not not sur surprising prising because because this this is is the the expectation expectation based based on on several several mechanisms mechanisms of of extinction: extinction: demographic demographic and and environmental environmental stochasticity, stochasticity, Allee AUee effect, effect, inbreeding inbreeding depression, depression, mutational mutational meltdown, meltdown, and and so so forth. forth. Furthermore, Furthermore, as as the the different different mechanisms mechanisms tend tend to to make make populations populations ever ever smaller, smaller, they they reinforce reinforce the the effect effect of of each each other other and and lead 1986) termed lead to to what what Gilpin Gilpin and and Soule Soul~ ((1986) termed extinction extinction vortices. vortices. Gilpin Gilpin and and Soule 1 986) identified Soul~ ((1986) identified four four extinction extinction vortices. vortices. Two Two of of them, them, the the R R and and D D vor vortices, tices, involve involve only only demographic demographic and and ecological ecological factors factors (demographic (demographic stochasticity stochasticity and ones, F and population population fragmentation). fragmentation). The The two two other other ones, F and and A A vortices, vortices, consider consider the the feedback feedback among among demographic, demographic, ecological, ecological, and and genetic genetic factors. factors. One One way way of of gauging gauging how how much much our our understanding understanding of of the the interactions interactions among among demographic, demographic, ecological, ecological, and and genetic genetic factors factors has has improved improved in in the the last last decade decade or or so so is is to to evalu evaluate ate to to what what extent extent the the current current knowledge knowledge calls calls for for aa reformulation reformulation or or refinement refinement of of the the F F and and A A vortices. vortices. As formulated, the As originally originally formulated, the F F vortex vortex is is the the consequence consequence of of reduced reduced fitness fitness due heterozygosity in in initially due to to inbreeding inbreeding depression depression and and loss loss of of heterozygosity initially large large popu populations lations that that have have been been reduced reduced to to aa small small size. size. The The decrease decrease in in fitness fitness further further reduces population reduces population size, size, which which in in turn turn further further increases increases inbreeding inbreeding depression depression and and loss loss of of heterozygosity, heterozygosity, increasing increasing the the probability probability of of extinction extinction via via this this and and and empirical advances made in the last few years all other vortices. Theoretical and and and reviewed reviewed earlier earlier indicate indicate that that the the enhanced enhanced vigor vigor that that is is often often associated associated with with increased increased heterozygosity heterozygosity is is most most likely likely due due to to aa reduced reduced homozygosity homozygosity of of
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OSCAR SKI OSCAR E. E. GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKAHAN HANSKI
deleterious ). deleterious alleles alleles rather rather than than to to heterozygosity heterozygosity per per se se (see (see Section Section 14.3 14.3). Furthermore, it it is is becoming becoming increasingly increasingly clear clear that that purging purging the the genetic genetic load load Furthermore, leading to to inbreeding inbreeding depression depression is is generally generally not not that that efficient efficient in in natural natural popu populeading lations lations (Section (Section 14.3). 14.3). Therefore, Therefore, the the FF vortex vortex in in the the form form of of inbreeding inbreeding depres depression remains aa likely likely mechanism of of population population extinction. extinction. sion The The A A vortex vortex was was also also attributed attributed to to genetic genetic drift drift and and loss loss of of genetic genetic vari variance, 1986) proposed ance, but but in in this this case, case, Gilpin Gilpin and and Soule Soul~ ((1986) proposed that that aa reduction reduction in in population population size size and and the the increased increased genetic genetic drift drift that that ensues ensues could could reduce reduce the the efficiency efficiency of of stabilizing stabilizing and and directional directional selections, selections, in in turn turn causing causing an an increas increasing and and accelerating accelerating "lack "lack of of fit" fit" between between the the population population phenotype phenotype and and the the ing environment environment it it faces. faces. This This was was hypothesized hypothesized to to reduce reduce population population size size and and growth growth rate rate even even further further until until the the population population goes goes extinct. extinct. This This mechanism mechanism was was not not formulated formulated very very precisely, precisely, but but it it is is related related to to the the mutational mutational meltdown meltdown discussed discussed in in Section Section 14.3. 14.3. The The reduction reduction in in the the efficiency efficiency of of stabilizing stabilizing and and directional directional selections selections leads leads to to an an accumulation accumulation of of slightly slightly deleterious deleterious muta mutations, tions, which which will will progressively progressively reduce reduce population population growth growth rate rate until until it it becomes becomes negative. Once Once this this happens, happens, the the population population size size will will decrease decrease and and the the rate rate at at negative. which which deleterious deleterious mutations mutations accumulate accumulate will will increase increase further. further. This This feedback feedback mechanism mechanism will will eventually eventually lead lead to to population population extinction. extinction. Another Another mechanism mechanism that that was was proposed proposed for for this this vortex vortex is is loss loss of of genetic genetic variance, variance, which which will will impair impair populations to track track environmental environmental changes. the capacity of populations An An additional additional short-term short-term mechanism mechanism could could be be added added to to the the A A vortex. vortex. The The loss of of habitat habitat reduces reduces population population sizes sizes and and may may lead lead to to aa loss loss of of variation variation at at loss MHC MHC loci, loci, making making individuals individuals less less able able to to resist resist infectious infectious diseases. diseases. At At the the same cases, lead initial increase same time, time, habitat habitat destruction destruction might, might, in in some some cases, lead to to an an initial increase in local density, density, as as individuals crowd in habitat. High High in local individuals crowd in the the remaining remaining suitable suitable habitat. density increase the the disease disease transmistransmis density following following fragmentation fragmentation might might in in turn turn increase sion rate (McCallum (McCallum and and Dobson, Dobson, 2002). Additionally, land degradation sion rate 2002). Additionally, land degradation increases increases the the opportunity opportunity for for contact contact among among humans, humans, domesticated domesticated animals, animals, et al., aI., wildlife, also also possibly possibly increasing increasing the the transmission transmission of of diseases diseases (Deem (Deem et and wildlife, 200 1 ) . An An increased increased transmission transmission rate rate and lowered disease disease resistance resistance will will 2001). and aa lowered further decrease population decrease in genetic varifurther decrease population size and and lead to to a further further decrease vari ability at at MHC MHC loci. loci. This This feedback feedback loop loop will will increase increase progressively progressively the extinc ability the extinction tion probability probability via this and and all other other vortices. vortices.
Population Responses Responses to to Environmental Environmental Deterioration Deterioration Delayed Population Although itit is is abundantly abundantly clear clear that that small small populations populations exhibit exhibit aa high high rate rate of of Although extinction, extinction, we we cannot cannot rest rest assured assured that that large large populations populations have have aa low low risk risk of of extinction. Consider Consider the the familiar familiar deterministic deterministic continuous-time continuous-time logistic logistic model, model, extinction. and carrying carrying capacity capacity K. K. The The equilibrium equilibrium population population size, size, with growth growth rate rate rr and with without without any any consideration consideration for for stochasticity, stochasticity, is is given given by by K. Now, Now, many many forms forms of of deterioration in in habitat habitat quality quality affecting affecting the the birth birth and and death death rates rates may may be be deterioration reflected reflected in in aa reduction reduction in in the the value value of of rr while while K K remains remains unchanged unchanged (or (or is is only only little affected). affected). In In this this case, case, the the deteriorating deteriorating environmental environmental conditions conditions are are not not little drops below below zero zero and and the the popupopu expected to to be be reflected reflected in in population population size size until until rr drops expected lation collapses collapses rather rather abruptly abruptly to to extinction extinction or, or, in in aa metapopulation metapopulation context, context, lation turns from from aa source source population population to to aa sink sink population. population. Incidentally, Incidentally, the the genetic genetic turns meltdown models models discussed discussed in in the the previous previous section section envision envision aa similar similar gradual gradual meltdown
114. 4.
MECHANISMS MECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION
357 357
decline decline in r, r, although although now now because of an accumulation accumulation of deleterious mutations. mutations. Although Although the the deterministic deterministic logistic logistic model model can can hardly hardly be be considered considered aa realistic realistic description of the dynamics of real populations, the phenomenon phenomenon we have just outlined occurs in all population population models. Things can be even worse, worse, from the perspective perspective of of aa manager manager who who is is trying trying to to read read the the early early signs signs of of approaching approaching trouble, in multi species models, in which multispecies which interspecific interspecific interactions interactions can compensate compensate for environmental line is that environmental deterioration deterioration (Abrams, 2002). The bottom bottom line that a large population size necessarily aa reliable population size is is not not necessarily reliable indicator indicator of of aa small small risk risk of of extinction. extinction. Even if the equilibrium equilibrium population population size would would fairly reflect the environmen environmental conditions, population would conditions, such that that a large population would indicate favorable favorable conditions conditions and and aa low low risk risk of of extinction, extinction, there there are are still still two two other other concerns concerns that that should should not not be 1 ) the be ignored: ignored: ((1) the possibility possibility of of alternative alternative stable stable states, states, which which was was discussed discussed for the population respond to in Section 14.2, and (2) the time it takes for population to respond changing environmental environmental conditions. conditions. In other words, words, in a changing environment environment the population to the current current size size of of the the population to some some extent extent reflects reflects the the past past rather rather than than the the present present environmental environmental conditions. conditions. If If the the environment environment has has deteriorated deteriorated rapidly, rapidly, the the population population size size is is therefore therefore larger larger than than the the long-term long-term expected expected (equilibrium) population size, and evaluation of extinction extinction risk based on popu population size only would would lead to an overly optimistic optimistic assessment. Ovaskainen Ovaskainen and and Hanski's (2002; Hanski and Ovaskainen, Ovaskainen, 2002) analysis of transient transient dynamics in metapopulation metapopulation models demonstrates demonstrates that that the time lag is especially long when close to when the the environment environment is is close to the the extinction extinction threshold threshold of of the the species species fol folenvironmental change (see (see Section 4.4 in Chapter Chapter 4). Thus, whenever whenever lowing environmental the environmental conditions conditions lead lead to relatively quick in the the changing changing environmental to aa relatively quick change change in the parameters that extinction threshold, threshold, we parameters that set set the the extinction we may may expect expect long long transient transient times in exactly those species that that we are most concerned concerned about. about.
Effect Effect of of Habitat H a b i t a t Patch Patch Size Size on on Extinction Extinction Assuming Assuming constant constant population population density, density, which which implies implies uniform uniform habitat habitat quality, quality, larger larger habitat habitat patches patches have have larger larger expected expected population population sizes sizes than than smaller patches. patches. Therefore, being equal, we could could expect large Therefore, other other things being habitat patches populations with popu habitat patches to to have have populations with aa lower lower risk risk of of extinction extinction than than populations lations in small patches. Although Although other things are usually by no means equal, and population population density density varies because of variation variation in habitat habitat quality quality and for other reasons, patch size and extinction other reasons, aa relationship relationship between between habitat habitat patch size and extinction risk risk has has typically typically been been documented documented whenever whenever this this relationship relationship has has been been examined examined (Hanski, 11994a,b, 994a,b, 11999b). 999b). This (Hanski, This finding finding has has been been employed employed in in the the dynamic dynamic the theory island biogeography Wilson, 11967) 967) and, ory of of island biogeography (MacArthur (MacArthur and and Wilson, and, more more recently, recently, in the spatially realistic metapopulation Chapter 4). More metapopulation theory ((Chapter More generally, the relationship between patch patch size size and provides aa key key rule rule of relationship between and extinction extinction risk risk provides of thumb being equal, conserve aa large large thumb for for conservation: conservation: other other things things being equal, it it is is better better to to conserve than small patch patch of preserve as patch as than aa small of habitat habitat or or to to preserve as much much of of aa particular particular patch as possible. One important important caveat caveat relates relates to possible. One to the the position position of of aa habitat habitat patch patch in in aa patch Section 4.4). information exists vari patch network network ((Section 4.4). Naturally, Naturally, if if empirical empirical information exists on on variation information should should be relative ation in in patch patch quality, quality, such such information be used used in in assessing assessing the the relative values of values of different different patches patches (most (most simply simply by by multiplying multiplying true true patch patch area area by by the the population estimated on basis of habitat quality; quality; for population density, density, estimated on the the basis of habitat for an an example, example, see see Chapter Chapter 20). 20).
3 58 358
OSCAR AND ILKKA OSCAR E. E. GAGGIOTII GAGGIO1-FI AND ILKKA HANSKI HANSKI
If If habitat habitat patches patches of of very very different different sizes sizes are are compared, compared, there there are are likely likely to to be be many many complementary complementary reasons reasons why why large large patches patches have have populations populations with with aa low low 1 999b) discussed of extinction. extinction. Hanski Hanski ((1999b) discussed three three different different scenarios. scenarios. In In the the risk of small-population small-population scenario, scenario, the the reason reason for for aa low low rate rate of of population population extinction extinction patches is discussed in in in large large patches is large large population population size size itself, itself, as as discussed in Section Section 14.2 14.2 [Eq. 1 4.4)] . In changing environment patches support support [Eq. ((14.4)]. In the the changing environment scenario, scenario, large large patches populations small extinction greater environmental populations with with aa small extinction risk risk because because the the greater environmental heterogeneity small patches heterogeneity in in large large than than small patches reduces reduces the the risk risk of of population population extinc extinction. Examples discussed by 1 996) for species of tion. Examples are are discussed by Kindvall Kindvall ((1996) for aa species of bush bush cricket cricket and several chapters Hanski (2004) checkerspot butter and by by several chapters in in Ehrlich Ehrlich and and Hanski (2004) for for checkerspot butterpatches in consist of Finally, in in the the metapopulation metapopulation scenario, scenario, large large patches in fact fact consist of flies. Finally, patch networks for dynamics increase patch networks for the the focal focal species, species, and and metapopulation metapopulation dynamics increase the population in the lifetime lifetime of of the the population in the the patch patch as as aa whole whole (Holt, (Holt, 1993). 1993). Regardless Regardless of habitat support of the the actual actual reason reason why why large large patches patches of of habitat support populations populations with with aa low remain the low risk risk of of extinction, extinction, the the conservation conservation implications implications remain the same. same.
114.5 4.5
LOCAL ETAPOPULATION CONTEXT LOCAL EXTINCTION EXTINCTION IN IN THE THE M METAPOPULATION CONTEXT The The previous previous sections sections discussed discussed the the ecological ecological and and genetic genetic processes processes that that oper operate populations. Although ate in in the the extinction extinction of of isolated isolated populations. Although habitat habitat fragmentation fragmentation increases the increases the isolation isolation of of populations, populations, few few populations populations are are completely completely isolated. isolated. In innumerable local populations interact In contrast, contrast, innumerable local populations interact regularly regularly via via migration migration with local populations populations. It with other other local populations in in meta metapopulations. It is is appropriate appropriate to to ask ask what what new new processes processes influencing influencing the the extinction extinction risk risk of of local local populations populations might might operate operate in in metapopulations. metapopulations. Not Not surprisingly, surprisingly, these these new new processes processes relate relate to to migration migration and gene flow. Migration and gene flow can both increase and decrease local extinction extinction risk. risk.
Migration Decreasing Extinction Migration and and Gene Gene Flow Flow Decreasing Extinction Risk Risk The The beneficial beneficial effect effect of of migration migration arises arises because because immigrants immigrants from from surround surroundpopulations may prevent populations, a ing populations prevent the extinction extinction of small local local populations, known as the rescue effect. In the literature on metapopulations, the process known rescue effect cover recolonization following extinc rescue effect is is occasionally occasionally extended extended to to cover recolonization following extinction, but tion, but more more properly properly the the rescue rescue effect effect refers refers to to processes processes that that reduce reduce the the demographic rescue risk in in the the first first place. place. A A demographic rescue occurs occurs because because immi immiextinction risk population size, thereby making extinction less likely gration increases the population (Brown 977). An (Brown and and Kodric-Brown, Kodric-Brown, 11977). An extreme extreme case case is is presented presented by by source-sink source-sink systems, where systems, where aa (true) (true) sink sink population population has has aa negative negative growth growth rate rate (e.g., (e.g., due due to to poor habitat only survive poor habitat quality) quality) and and may may only survive with with sufficient sufficient immigration immigration from from one or more source one or more source populations populations (Chapter (Chapter 16). 16). Immigration Immigration reducing reducing extinction extinction risk also common common in case of inhabiting small risk is is also in the the case of small small populations populations inhabiting small habitat habitat patches patches located located close close to to large large populations, populations, aa common common situation situation in in many many metapopulations. example on metapopulations. Table Table 14.2 14.2 gives gives an an example on the the Glanville Glanville fritillary fritillary but butland Islands, Islands, Southwest Southwest Finland, terfly terfly (M. cinxia) cinxia) in in the the A Aland Finland, where where the the butterfly butterfly has population consisting local populations has aa meta metapopulation consisting of of several several hundred hundred local populations (Hanski, (Hanski, 11999b). 999b). Larvae population sizes Larvae live live gregariously, gregariously, and and population sizes are are often often very very small small in in terms of even though though populations tens of terms of the the number number of of larval larval groups, groups, even populations have have tens of
359 359
114. 4. MECHANISMS MECHANISMSOF OF POPULATION POPULATION EXTINCTION EXTINCTION TABLE The Rescue T A B L E 11 44.2 .2 Rescue Effect Reduces the Risk Risk of Extinction in Small Local Populations of the Glanville Fritillary Butterfly (Melitaea (Melitaea cinxia)G cinxia) a Number Number of of larval groups groups larval 1 2 2 3-5 3-5 > > 5 5
aa
The The rescue rescue effect effect Extinct Extinct
n
n
Average Average S S
Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No
150 150 76 76 46 46 58 58 46 46 202 202 114 4 204 204
2.55 2.55 2.84 2.84 2.78 2.78 3 .12 3.12 2.88 2.88 2.75 2.75 3.31 3.31 2.83 2.83
tt
P P
-2.97 -2.97
0.003 0.003
-2.24 -2.24
0.025 0.025
- 00.63 .63
0.527 0.527
11.42 .42
0.155 0.155
Sizes 993, Sizes of of local local populations populations are are given given in in terms terms of of the the number number of of larval larval groups groups in in autumn autumn 11993, the the numbers numbers of of these these populations populations that that went went extinct extinct and and survived, survived, aa measure measure of of connectivity connectivity (5) (S) to to nearby nearby populations, populations, and and aa t test test of of the the rescue rescue effect, effect, which which was was measured measured by by the the effect effect of of 5 S on which also included the on extinction extinction (from (from aa logistic logistic regression, regression, which also included the effects effects of of patch patch area area and and regional population sizes 9 99b). regional trend trend in in population sizes on on extinction; extinction; Hanski, Hanski, 11999b).
butterflies. butterflies. Comparing Comparing the the numbers numbers of of populations populations of of given given size size that that did did or or did did not apparent that well connected not go go extinct extinct in in 11 yr, yr, it it is is apparent that populations populations that that were were well connected to to other other populations populations had had aa lower lower risk risk of of extinction extinction than than more more isolated isolated popula populations tions (Table (Table 14.2). 14.2). It It also also makes makes sense sense that that this this effect effect was was statistically statistically signifi significant smallest populations because the cant in in the the case case of of the the smallest populations only only because the influence influence of of aa given given amount in increasing population size amount of of immigration immigration in increasing population size is is greatest greatest in in the the case case of of the the smallest smallest populations. populations. Note Note that that large large populations populations have have aa much much smaller smaller risk risk of of extinction extinction than than small small populations populations in in Table Table 14.2. 14.2. Local Local populations populations may may be be rescued rescued demographically, demographically, as as we we have have just just dis discussed, cussed, but but they they may may also also be be rescued rescued genetically. genetically. Gene Gene flow flow may may increase increase the the mean population mean population fitness fitness due due to to heterosis heterosis and and the the arrival arrival of of immigrants immigrants with with (outbred vigor). Heterosis refers to increased fitness among among off offhigh fitness (outbred spring spring from from crosses crosses among among local local populations; populations; different different populations populations tend tend to to fix fix different each other different random random subsets subsets of of deleterious deleterious alleles, alleles, which which mask mask each other when when populations Crow, 11948; 948; Whitlock, initially populations are are crossed crossed ((Crow, Whitlock, 2000). 2000). Therefore, Therefore, initially rare rare immigrant immigrant genomes genomes are are at at aa fitness fitness advantage advantage compared compared to to resident resident genomes genomes because because their their descendants descendants are are more more likely likely to to be be heterozygous heterozygous for for dele deleterious terious recessive recessive mutations mutations that that cause cause inbreeding inbreeding depression depression in in the the homo homozygous zygous state state (Ingvarsson (Ingvarsson and and Whitlock, Whitlock, 2000; 2000; Whitlock Whitlock et et aI., al., 2000). 2000). Several Several studies studies have have provided provided fairly fairly conclusive conclusive evidence evidence supporting supporting this this expectation. Brakefield (2002) expectation. Saccheri Saccheri and and Brakefield (2002) carried carried out out an an experimental experimental study study with with the the butterfly butterfly Bicyclus anynana. They They focused focused on on the the consequences consequences of of aa sin single equally inbred local populations. gle immigration immigration event event between between pairs pairs of of equally inbred local populations. The The experiment experiment involved involved transferring transferring aa single single virgin virgin female female from from an an inbred inbred (donor) (donor) population population to to another another inbred inbred (recipient) (recipient) population. population. The The spread spread of of the the immi immigrant's grant's and and all all the the residents' residents' genomes genomes was was monitored monitored during during four four consecutive consecutive generations keeping track pedigree of generations by by keeping track of of the the pedigree of all all individuals individuals in in the the treatment treatment populations. replicated this experimental design populations. They They replicated this experimental design and and observed observed aa rapid rapid
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OSCAR E. OSCAR E. GAGGIOTTI GAGGIOTTI AND AND ILKKA ILKKA HANSKI HANSKI
increase in the share of the initially rare immigrant immigrant genomes in local popula populations. Ball et a!. al. (2000) reported reported similar evidence for D. melanogaster, measur measurmarker alleles in the first and second ing the relative frequency of immigrant marker generations following a transfer to inbred populations. populations. When immigrants were outbred, the mean frequency of the immigrant allele in the first and second gen generation after migration was significantly higher than its initial frequency. They attributed attributed this result to the initial outbred vigor of immigrant males, but the possibility of heterosis having played a role was not excluded completely. Ebert et a!. (2002) out experiments water Ebert et al. (2002) carried carried out experiments using using aa natural natural Daphnia water flea metapopulation metapopulation in which local extinctions and recolonizations, genetic bottlenecks, and and local inbreeding are common events. Their results indicate that that because of heterosis, gene flow was several times greater than than would be predicted predicted from from the the observed observed migration migration rate. Somewhat Somewhat less less conclusive conclusive evi evidence comes from Richards's (2000) experiments with the dioecious plant plant Silene alba, in which isolated populations suffer suffer substantial inbreeding depres depression. Richards (2000) measured gene flow among among experimental populations populations separated by 20 m and used paternity analysis to assign all seeds to either local males or to immigrants immigrants from from other other nearby experimental experimental populations. populations. When When the recipient populations populations were inbred, unrelated males from from the experimental population population 20 m away sired more offspring than than expected under under random random mat mating. This may be due to some form form of pollen discrimination that that may be influ influenced by early acting inbreeding depression (Richards, 2000) or to heterosis per se. Incidentally, the rescue effect in Table 14.2 for the Glanville fritillary butterfly butterfly could also involve a genetic component, as it is known known that that inbreed inbreeding depression increases the risk of extinction of small populations populations of this but butterfly (Saccheri et a!., 998; Nieminen et a!., 1). al., 11998; al., 200 2001). Migration can have a long-term beneficial effect on population population persistence. The arrival of migrants from from large populations can increase genetic variability in the recipient populations and, thereby, enhance the evolutionary evolutionary potential of the species as a whole. The extent to which migration can replenish genetic population dynamics and the pattern pattern of migration variability depends on population among among populations. populations. Populations Populations with with positive growth growth rates can recover lost genetic variability rapidly, but sink populations populations will only be able to maintain maintain genetic variability when the variance in the migration process is low (Gaggiotti, 11996; 996; Gaggiotti 996). Gaggiotti and Smouse, 11996).
Migration Migration and and Gene Gene Flow Flow Increasing Increasing Extinction Extinction Risk Risk Migration may increase the extinction risk of local populations for several main reasons. In the landscape ecological literature, the role of corridors corridors in main(meta)populations in fragmented landscapes has been discussed taining viable (meta)populations for a long time. Corridors Corridors enhance recolonization and the rescue effect (Bennett, 11990; 990; Merriam, 995; Andreassen et a!., 1996b; Merriam, 1991; Haas, 11995; etal., Haddad, 999a), but it has been pointed out Haddad, 11999a), out that corridors may also facilitate the spread of disease agents and predators that might actually increase the extinction risk of the focal populations (Simberloff and Cox, 11987; 987; Hess, 11994). 994). More generally, it is well established both theoretically theoretically (Hassell et a!., al., 1991; Comins et a!., 992; Nee et a!., 997) and 958; al., 11992; al., 11997) and empirically (Huffaker, 11958; Nachman, 99 1 ; Eber and Brandl, 11994; 994; Lei and Hanski, 11998; 998; Schops Nachman, 11991; Sch6ps et a!., al.,
1 4. 14.
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M ECHANISMS OF OF POPULATION POPULATION EXTINCTION EXTINCTION MECHANISMS
1 998) that that specific specific natural natural enemies enemies in in prey-predator prey-predator metapopulations metapopulations may may subsub 1998) stantially increase increase the the extinction extinction risk risk of of local local prey prey populations. populations. stantially Just like like immigration immigration into into small small populations populations may may reduce reduce their their risk risk of of extincextinc Just tion, emigration emigration from from small small populations populations may may increase increase extinction extinction risk risk (Thomas (Thomas tion, and Hanski, Hanski, 1997; 1 997; Hanski, Hanski, 1998b). 1 998b). Theoretical Theoretical studies studies have have elucidated elucidated the the critcrit and minimum size of of habitat habitat patches patches that that would would allow the the persistence of of viable ical minimum populations (Okubo, (Okubo, 1980); 1 980); populations populations in patches smaller than than this this critical populations patches smaller size go go extinct extinct because because they they lose lose individuals individuals too too fast fast in in comparison comparison with with the the size rate of of reproduction. reproduction. However, However, just just like like with with the the rescue rescue effect in saving saving small small rate effect in populations, itit is is hard hard to to prove prove conclusively conclusively that that small small populations populations go go extinct extinct populations, because of of emigration, emigration, as as small small populations populations are are likely likely to to go go extinct extinct for for many many because other reasons reasons as as well. well. Nonetheless, Nonetheless, emigration emigration compromising compromising the the viability viability of of other local populations populations is is aa potentially potentially important important consideration consideration in in the the conservation conservation local of some some species. species. For For instance, instance, itit has has been been suggested that small small reserves reserves for for of suggested that butterflies should should not not be be surrounded surrounded by by completely completely open open landscape landscape because because this this butterflies will increase increase the the rate rate of emigration greatly greatly (Kuussaari (Kuussaari et et al., aI., 1996). 1 996). will of emigration Migration can also have negative genetic effects on on population population persistence. In Migration principle, gene flow may may reintroduce to prevent principle, reintroduce genetic load fast fast enough to prevent the purging inbreeding depression, depression, although we are are not not aware aware of of any any clear clear evievi purging of of inbreeding although we dence for this. More More importantly, the long-term long-term beneficial beneficial effects effects of of migration migration dence for this. importantly, the may be be offset by the introduction of of maladapted maladapted genes, genes, which which may may lead lead to to aa loss loss may offset by the introduction of local local adaptation in some some populations, populations, the the appearance appearance of of source-sink source-sink dynamics, of adaptation in dynamics, and the the evolution evolution of of narrow narrow niches niches (Kirkpatrick and Barton, Barton, 1997; 1 997; Ronce Ronce and and and (Kirkpatrick and Kirkpatrick, 2001). 2001 ). This This process, process, called called migrational migrational meltdown meltdown (Ronce (Ronce and and Kirkpatrick, Kirkpatrick, 1 ) because because small small populations populations experience spiral of of Kirkpatrick, 200 2001) experience aa downward downward spiral maladaptation in the maladaptation and and shrinking shrinking size, size, is is discussed discussed in the next next section. section. The immigrant genomes genomes from highly divergent The introduction introduction of of immigrant from aa highly divergent popula population tion can can reduce reduce mean mean population population fitness, fitness, aa phenomenon phenomenon known known as as outbreed outbreeding (Fig. 14.4). be expressed ing depression depression (Fig. 14.4). Outbreeding Outbreeding depression depression will will be expressed in in the the Fl F1
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Generation Fig. 114.4 4 . 4 Potential effects effects of migration on population fitness: (a) heterosis increases increases fitness (solid line and diamonds), (b) heterosis followed by outbreeding depression leads leads to aa short· shortlived lived fitness fitness increase increase followed followed by by aa decline, decline, and and (c) (c) outbreeding outbreeding depression depression leads leads to to aa steady steady decline decline in in fitness. fitness.
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OSCAR SKI OSCAR E. E. GAGGIOTTI GAGGIO-I-FI AND AND ILKKA ILKKA HAN HANSKI
generation generation if if the the favorable favorable between-population between-population dominance dominance effects effects (masking (masking effect of deleterious recessive genes present in the homozygote state in the effect parental parental lines lines but but in in the the heterozygote heterozygote state state in in the the F1 F1)) are are outweighed outweighed by by the the loss in favorable additive xx additive interactions within populations (Lynch occur, outbreeding depres depresand Walsh, 11998). 9 9 8 ) . However, even if this does not occur, sion may still be expressed in the FF22 generation or later. The reason for this is segthat Fls F1s carry a haploid set of chromosomes from each parental line, and seg sinregation and recombination begin to break apart coadapted genes from a sin Dobzhansky, 11950, 950, 11970). 970). Thus, outbreeding F2 generation ((Dobzhansky, gle line in the F2 depression is demonstrated when the performance of F2s is less than the aver average of immigrants and residents (Lynch and Walsh, 11998). 998). Unfortunately, only few studies of natural populations have tracked the contribution of immigrants beyond the Fl F1 generation (Marr et aI., al., 2002). The few studies that F1 indicate that outbreeding depression may be common in the go beyond Fl wild. Marr et aI. al. (2002) showed that the same population of song sparrows Mandarte Islands that manifested heterosis among immigrant offspring in the Mandarte also displayed signs of outbreeding depression in the F2 F2 generation. Studies of the tidepool copepod Tigriopus californicus show that crosses between popu popuF1 hybrid vigor and F2 hybrid breakdown for a lations typically result in Fl number of measures related to fitness (Burton 11987, 987, 11990a,b; 990a,b; Edmands and 998; Burton et aI., 999). Edmands ((1999) 1 999) showed that the detri Burton, 11998; al., 11999). detrimental effects of breaking up co coadaptation adaptation are magnified by increasing genetic distance between populations. This same effect was shown for the shrub Lotus scoparius, but in this case out outbreeding breeding depression was already 2001). present in the F1 Fl generation (Montalvo and Ellstrand, 200 1 ) . Other plant demonstrated include species for which outbreeding depression has been demonstrated lpomopsis aggregata Silene diclinis (Waldmann, Ipomopsis aggregata (Waser et aI., al., 2000) and Silene 1999). 1 999).
114.6 4.6
METAPOPULATION EXTINCTION EXTINCTION METAPOPULATION Not only populations but also metapopulations consisting of many local Not populations possess a smaller or greater risk of eextinction x t i n c t i o n- the metapopulametapopula tion is extinct when the last remaining local population is extinct. Chapter 4 metapopulation theory, albeit albeit largely from presents a thorough account of the metapopulation from particular class of models, stochastic patch occupancy the perspective of one particular models. A primary focus of this theory is to to dissect the conditions conditions of long-term metapopulation persistence (in deterministic models) and the factors deterdeter metapopulation mining the expected lifetime lifetime of metapopulations metapopulations (in stochastic models). models). Chapter the perspective of landscape ecolChapter 2 complements complements this analysis from the ecol metapopulation theory in Chapter Chapter 44 is concerned ogy. The spatially realistic metapopulation primarily with just just one factor factor in increasing the risk of of metapopulation metapopulation extincextinc tion, namely habitat loss and and fragmentation, fragmentation, but but as we all know, this is curcur rently the main cause of of population, population, metapopulation, metapopulation, and species extinctions. Rather Rather than than repeating what what has already been written written in Chapters Chapters 2 and 4 and and discussed in the the context context of of particular particular metapopulations metapopulations in Chapters Chapters 20 and and 21, 21, meta population we highlight here one ecological factor that is often critical in metapopulation we factor that i s often in extinction. extinction. We also discuss two two genetic processes that that have have been proposed proposed to to
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increase the the risk risk of of metapopulation metapopulation extinction, extinction, mutational mutational and and migrational migrational increase meltdowns, both both of of which which stem stem from from an an interaction interaction between between demographic demographic and and meltdowns, genetic processes processes in in metapopulation metapopulation dynamics. dynamics. genetic
Regional Stochasticity The counterpart counterpart of of environmental environmental stochasticity stochasticity in local populations populations is The regional stochasticity stochasticity in metapopulations meta populations m - spatially spatially correlated correlated environmenenvironmen regional tal stochasticity affecting local local populations populations in in metapopulations metapopulations (Hanski, (Hanski, tal stochasticity affecting 1 9 9 1 ) . Just Just as as environmental environmental stochasticity stochasticity amplifies amplifies population population fluctuations fluctuations in in 1991). local populations populations and and is is the the major major cause cause of of population population extinction, regional local extinction, regional stochasticity amplifies fluctuations in in the the size size of of metapopulations metapopulations (Fig. ( Fig. 4.11 4. 1 1 in in stochasticity amplifies fluctuations Chapter 4 gives a theoretical theoretical example example and and Chapter Chapter 21 reviews regional Chapter reviews regional stochasticity in small mammal mammal metapopulations). metapopulations). There There is a large literature literature on on stochasticity population dynamics dynamics (Ranta (Ranta et al., aI., 1998; 1 998; Bjornstad Bj0rnstad et al., aI., spatial synchrony in population 11999; 999; Paradis Paradis et 1 999; Engen Engen et et aI., 2002a) with with the same general message. et aI., al., 1999; al., 2002a) the same general message. The two two mechanisms mechanisms of spatial synchrony synchrony that that have been most most discussed are migration weather migration and regional stochasticity stochasticity (typically spatially correlated correlated weather ( 1 999), conditions influencing birth birth and death death rates). As shown shown by Lande et al. (1999), conditions even low rates rates of may affect even low of short-distance short-distance migration migration may affect population population synchrony synchrony greatly if population population regulation regulation is weak. Engen et al. (2002b) examined the the probability of quasiextinction for continuously in probability for a population distributed distributed continuously space and affected by regional stochasticity (quasiextinction was was defined defined as the the population below 10% 1 0 % of carrying capacity) expected population size size dropping dropping below of the the carrying capacity).. The The expected time decreases with increasing strength time to to quasiextinction quasiextinction decreases with increasing strength of of environmental environmental stochasticity, with decreasing rate of migration, migration, and with with increasing area within which changes in population population size are recorded. The expected popula population extinction increases, tion density density decreases, decreases, and and hence hence the the probability probability of of quasi quasiextinction increases, with with increasing increasing spatial spatial scale scale of of regional regional stochasticity. stochasticity.
Metapopulation Metapopulation Meltdown Meltdown accumulation of slightly deleterious deleterious mutations mutations can have detrimental detrimental The accumulation effects level. Higgins 1 ) extended effects at at the the metapopulation metapopulation level. Higgins and and Lynch Lynch (200 (2001) extended the the mutational meltdown meltdown theory described in Section 14.3 to metapopulations metapopulations using using an an individual-based individual-based model model that that includes includes demographic demographic and and genetic genetic mech mechanisms population structure anisms and and environmental environmental stochasticity. stochasticity. The The meta metapopulation structure was was modeled as a linear array of patches connected by nearest-neighbor (stepping (steppingstone), global (island), or intermediate dispersal. The mutational mutational effect was modeled in such a way that mutations of large effect are almost recessive, whereas whereas those those of of small small effect effect are are almost almost additive. additive. Results Results show show that that for for metapopulations with more than a few patches, an accumulation of deleteri deleterious ous mutations mutations accelerates accelerates extinction extinction time time by by many many orders orders of of magnitude magnitude compared population without compared to to aa globally globally dispersing dispersing meta metapopulation without mutation mutation accu accumulation. Moreover, extinction due to mutation mutation accumulation can be quite rapid, rapid, on on the the order order of of tens tens of of generations. generations. In In general, general, results results indicate indicate that that the the mutational meltdown may be a significant threat to large metapopulations and would exacerbate exacerbate the effects of habitat loss or fragmentation on metapopula metapopulation viability. These conclusions were reached under the assumptions of an
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expected expected genome-wide genome-wide mutation mutation rate rate of of 11 per per generation generation and and unconditionally unconditionally deleterious deleterious mutational mutational effects. effects. As As mentioned mentioned before, before, these these two two assumptions assumptions have have been been placed placed under under close close scrutiny, scrutiny, and and preliminary preliminary evidence evidence indicates indicates that that they may not be generally valid.
Migrational Meltdown Meltdown Migrational Another Another genetic genetic mechanism mechanism for for metapopulation metapopulation extinction extinction stems stems from from the the idea that that peripheral peripheral populations populations receive receive gene gene flow flow from from the the center center of of the the idea species' typically be species' range. range. These These immigrant immigrant genes genes will will typically be adapted adapted to to the the condi conditions at at the the range range center center and and could could inhibit inhibit adaptation adaptation in in the the periphery periphery (Mayr, (Mayr, tions 11963). 963). Kirpatrick 1 997) used Kirpatrick and and Barton Barton ((1997) used aa quantitative quantitative genetic genetic model model to to study study the the evolution evolution of of aa species species range range in in aa linear linear habitat habitat with with local local migration. migration. The The model model tracks tracks evolutionary evolutionary and and demographic demographic changes changes across across space space and and time time and and assumes assumes that that variation variation in in the the environment environment generates generates patterns patterns of of selec selection tion that that change change in in space space but but are are constant constant in in time. time. Among Among other other things, things, results results show species' range show that that aa species' range may may contract contract as as the the dispersal dispersal rate rate increases increases and and extinction extinction may may follow follow if if conditions conditions change change too too rapidly rapidly as as one one moves moves across across space, space, even even if if the the species species remains remains perfectly perfectly adapted adapted to to the the habitat habitat at at the the range range center. Ronce Ronce and and Kirpatrick Kirpatrick (2001 (2001)) also studied the the maladaptive maladaptive effect effect of of center. also studied migration migration but but they they considered considered aa model model with with two two discrete discrete habitat habitat types types con connected nected by by migration. migration. In In this this case, case, an an increasing increasing migration migration rate rate above above aa thresh threshold value results in in the the collapse collapse of of the the total total population population size size and and the the complete old value results complete loss of loss of one one of of the the populations. populations. However, However, in in contrast contrast to to Kirpatrick Kirpatrick and and Barton's Barton's metapopulation extinction. Kirpatrick ((1997) 1 997) analysis, there is no metapopulation extinction. Ronce and Kirpatrick (200 1 ) attributed this disagreement disagreement between the two two models models to to the the assumption assumption (2001) attributed this between the of infinite infinite space space made made by by Kirpatrick Kirpatrick and and Barton: Barton: the the distance distance traveled traveled by by migrants and thus the the maladaptation of such such migrants migrants to to local local conditions conditions migrants and thus maladaptation of increase indefinitely the migration migration rate. rate. This assumption is is unlikely unlikely to to be be increase indefinitely with with the This assumption valid for real situations metapopulation extinction extinction valid for real situations and, and, therefore, therefore, complete complete metapopulation due to migrational migrational meltdown meltdown is is unlikely unlikely to to occur.
114.7 4.7
CONCLUDING REMARKS REMARKS CONCLUDING The major causes causes of of population and species extinctions worldwide worldwide are are habi The major population and species extinctions habitat loss loss and and interactions interactions among among species. species. The The models models discussed in this chapter tat discussed in this chapter address the the adverse adverse effects of habitat habitat loss loss in in terms terms of of the the reduced reduced sizes sizes of of address effects of populations and and metapopulations metapopulations that that are are the the inevitable inevitable and and direct of populations direct result result of habitat loss. loss. With With metapopulation metapopulation models, models, we we may may additionally examine habitat additionally examine the consequences consequences of of habitat habitat loss loss that that occur occur in in the the surroundings surroundings of of the the focal focal the population, and and which which consequences consequences influence influence the the focal focal population population via via population, metapopulation metapopulation dynamics dynamics (Chapter ( Chapter 4). 4). Considering interactions interactions with with other other species, species, it first appear appear surprissurpris Considering it may may at at first ing x t i n c t i o n- if ing that that this this would would be be an an important important cause cause of of population population eextinction if this this were the the case, case, would would such such extinctions extinctions not not have have already already happened happened aa long long time time were ago? This This argument argument does does not not hold hold in in two two situations: situations: in in metapopulations metapopulations with with ago? recurrent extinctions extinctions and and colonizations colonizations (Section (Section 14.5) 14.5) and and when when species species are are recurrent spreading into into areas areas where where they they did did not not use use to to occur occur and and become become hence hence spreading
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engaged engaged in in novel novel interactions. interactions. We We all all know know that that such such invasions, invasions, with with often often adverse native species, have become adverse consequences consequences for for native species, have become rampant rampant in in the the modern modern world, innumerable species world, where where humans humans have have helped, helped, in in one one way way or or another, another, innumerable species to to spread spread beyond beyond their their past past geographical geographical ranges. ranges. The The actual actual mechanisms mechanisms of of extinction include hybridization extinction of of native native species species include hybridization with with the the invasive invasive species species (Simberloff, 994; Wolf aI., 2001 (Simberloff, 11994; Wolf et et al., 2001;; Levin, Levin, 2002; 2002; Perry Perry et et aI., al., 2002 2002).) . The The spreading spreading of of Homo Homo sapiens itself, itself, in in the the far far past, past, was was the the likely likely cause cause of of extinc extinction tion of of aa large large fraction fraction of of the the megafauna megafauna in in North North America, America, Australia, Australia, and and many 9 84; Caughley 996) at many large large islands islands (Martin (Martin and and Klein, Klein, 11984; Caughley and and Gunn, Gunn, 11996) at aa time humans could among other other animals their lack time when when humans could be be placed placed among animals in in their lack of of con concern cern for for the the survival survival of of other other species. species. No No wonder, wonder, then, then, that that modern modern humans humans are are able able to to hunt hunt and and drive drive many many species species to to extinction extinction or or near near extinction. extinction. Harvesting major threat Harvesting of of populations populations has has been been and and continues continues to to be be aa major threat to to both both terrestrial ecological knowledge terrestrial and and marine marine populations. populations. Models Models and and ecological knowledge could could and economically valuable valuable populations and should should be be used used to to guide guide harvesting harvesting of of economically populations (Getz 989; Lande 995), but (Getz and and Haight, Haight, 11989; Lande et et aI., al., 11995), but generally generally this this is is not not what what happens in in reality. reality. happens Interactions with invasive and harvesting, Interactions with invasive species, species, persecution, persecution, and harvesting, along along with with habitat loss, major ultimate populations and species, and habitat loss, are are the major ultimate threats threats to to populations and species, and the the threats threats with with which which most most practical practical conservation conservation efforts efforts have have to to be be concerned. concerned. mechanFrom this perspective, many of the population population ecological and genetic mechan isms isms discussed discussed in in this this chapter chapter may may appear appear insignificant. insignificant. Nonetheless, Nonetheless, the the mat matter ter of of fact fact is is that that increasing increasing numbers numbers of of species species are are being being reduced reduced to to aa state state in in which Caughley, 11994) 994) covered which the the small-population small-population issues issues ((Caughley, covered here here are are relevant relevant and 996). Clearly, and interact interact with with the the primary primary causes causes of of threat threat (Hedrick (Hedrick et et aI., al., 11996). Clearly, population current extinction population biologists biologists alone alone cannot cannot solve solve the the current extinction crisis, crisis, but but we we can provide improved knowledge knowledge of many specific specific biological can provide improved of many biological issues. issues. Finally, Finally, of of course, course, just just like like the the study study of of population population regulation regulation has has been been of of great great intrinsic intrinsic interest population ecologists inevitable interest to to population ecologists for for more more than than aa century, century, so so are are the the inevitable "failures" "failures" of of regulation regulation in in finite finite populations. populations. One One of of the the largely largely open open scientific scientific issues issues in in the the study study of of population population extinc extinction relates to genome-wide mutation tion relates to the the current current controversy controversy surrounding surrounding genome-wide mutation rates Section 14.3). rates and and the the average average effect effect of of deleterious deleterious mutations mutations ((Section 14.3). Before Before these these questions questions have have been been resolved, resolved, it it is is premature premature to to draw draw definite definite conclu conclusions sions about about the the importance importance of of mutational mutational meltdown meltdown in in population population and and metapopulation metapopulation extinctions. extinctions. More More research research on on the the mutation mutation process process underly underlying empirical research ing the the mutational mutational meltdown meltdown and and more more extensive extensive empirical research on on the the feasibility models such that feasibility of of this this phenomenon phenomenon are are needed. needed. Additionally, Additionally, models such as as that of should be include beneficial well of Higgins Higgins and and Lynch Lynch (2001 (2001)) should be extended extended to to include beneficial as as well as carried out as deleterious deleterious mutations. mutations. Likewise, Likewise, additional additional work work has has to to be be carried out to to evaluate heterosis and, evaluate the the importance importance of of the the genetic genetic rescue rescue effect effect due due to to heterosis and, in in particular, breeding influences particular, to to understand understand how how out outbreeding influences the the mean mean fitness fitness of of nat natpopulations. It is likely that that the extent extent of outbreeding outbreeding depression depression depends depends ural populations. on populations that are. Highly on how how inbred inbred the the local local populations that receive receive the the migrants migrants are. Highly inbred populations whose positively to inbred populations whose fitness fitness is is very very low low may may react react positively to the the influx influx of outbreeding depression depression at all. However, of migrants migrants and and show show no no signs signs of of outbreeding at all. However, less less inbred inbred populations populations whose whose fitness fitness has has not not been been impaired impaired dramatically dramatically may may show show heterosis heterosis in in the the Fi F1 generation generation but but outbreeding outbreeding depression depression in in the the F F22 and and subsequent subsequent generations generations or or outright outright outbreeding outbreeding depression. depression. Unraveling Unraveling the the
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effects effects of of immigration immigration on on fitness fitness will will require require carrying carrying out out experiments experiments that that fol follow immigrants beyond beyond the generation and low the the fate fate of of the the descendants descendants of of immigrants the F F22 generation control for the inbreeding level of the target populations. populations. control We have commented commented in the introduction introduction and in later sections sections of this chapter on the changing views about the relative roles of ecological and and genetic factors factors in extinction. The theoretical and in population population and and metapopulation metapopulation extinction. The theoretical and empirical empirical work done in the past decade decade makes makes it clear clear that that genetic factors can contribute contribute significantly significantly to to population population extinction. extinction. In In particular, particular, there there is is aa rapidly rapidly expand expanding body of literature literature demonstrating demonstrating that that inbreeding inbreeding depression in natural popu populations is often sufficiently severe to have significant significant consequences lations consequences for population population dynamics dynamics and and thereby thereby for for extinction. extinction. The The most most clear-cut clear-cut demon demonstrations inbreeding increasing increasing the population extinction, extinction, such strations of of inbreeding the risk risk of of population such as as in in the aI., 11998; 998; Nieminen aI., 200 1 ), the Glanville Glanville fritillary fritillary butterfly butterfly (Saccheri (Saccheri et et al., Nieminen et et al., 2001), relate small populations. dismiss the relate to to very very small populations. For For this this reason, reason, some some might might dismiss the new new evidence little general general importance. evidence as as of of little importance. However, However, this this is is not not so so in in the the metapopu metapopulation context, context, where small small populations lation populations are often frequent and matter matter for the dynamics also the dynamics of of the the metapopulation metapopulation as as aa whole. whole. This This is is also the context context that that shows shows very clearly clearly how 1994) declining-population how Caughley's ((1994) declining-population paradigm paradigm and small smallpopulation population paradigm paradigm interact. Very often, habitat habitat loss and fragmentation fragmentation are the the root root causes causes of of metapopulation metapopulation decline decline (declining-population (declining-population paradigm), paradigm), but but the actual meta population response response to environmental metapopulation environmental changes is largely deter determined by what happens in the often small local local populations what happens populations (small-population (small-population paradigm). paradigm). The relative roles of genetic and ecological factors factors in extinction extinction are also likely to also likely to vary vary among among taxa taxa with with different different biologies. biologies. For For instance, instance, environ environmental generally the mental stochasticity stochasticity is is generally the overriding overriding cause cause of of extinction extinction in in insects insects and and other other invertebrates, invertebrates, whereas whereas inbreeding inbreeding might might be be expected expected to to play play aa rela relatively tively greater greater role role in in vertebrate vertebrate populations populations that that are are less less influenced influenced by by random random variation variation in environmental environmental conditions.
15
MULTILOCUS MULTILOCUS GENOTYPE ODS G ENOTYPE METH M ETHODS FOR THE STUDY STUDY OF OF META PO PULATION M ETAPO PU LATIO N PROCESSES Oscar Oscar E. Gaggiotti
115.1 5. 1
INTRODUCTION INTRODUCTION Three Three fundamental fundamental processes processes are are at at the the heart heart of of metapopulation metapopulation biology: biology: local (Hanski, 11999a). 999a). local population population extinction, extinction, (re)colonization, (re)colonization, and and migration migration (Hanski, The The problems problems being being addressed addressed in in the the metapopulation metapopulation context context are are diverse diverse and and range from population dynamics from the effect of migration migration on local and and global population dynamics (Chapters 3 , 116, 6, and colonization, and (Chapters 4, 4, 113, and 20) 20) to to the the effect effect of of extinction, extinction, colonization, and migration Chapters 7, migration on the evolutionary evolutionary potential potential of metapopulations metapopulations ((Chapters 110-12, 0-12, and and 16). 16). In In all all these these studies, studies, aa common common interest interest is is the the estimation estimation of of the rates rates at which these three three events events take take place. More More detailed detailed information information is also required required when studying colonization colonization and and migration migration processes. In these cases, cases, it it is is also also necessary necessary to to estimate estimate additional additional parameters, parameters, such such as as the the size size of founding/migrant founding/migrant groups groups and and their composition, composition, and and to to identify identify the factors factors that Chapter 113). 3). that force force individuals individuals to to move move away away from from their their place place of of birth birth ((Chapter these different different problems problems can bbee studied studied using purely purely ecological ecological All these approaches, approaches, such such as as mark-release-recapture mark-release-recapture methods methods (MRR), (MRR), but but only only on on aa
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Copyright 2004, Elsevier, Elsevier, Inc. 0-12-323448-4 0-12-323448-4
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limited methods are and limited number number of of species. species. Moreover, Moreover, these these methods are time-consuming time-consuming and cannot large and/or cannot be be applied applied to to study study large and/or spatially spatially extended extended metapopulations. metapopulations. Population Population genetic genetic approaches, approaches, however, however, are are easier easier to to implement implement in in these these situ situations, require a carefully planned ations, as, in general, they only require planned sampling sampling program program aimed at collecting tissue samples for aimed at collecting tissue samples for DNA DNA extraction extraction and and analysis. analysis. The population con The application application of of population population genetic genetic methods methods in in the the meta metapopulation context text is not not problem problem free. The population population turnover turnover that that characterizes characterizes many many metapopulations metapopulations can can decrease decrease genetic genetic variability variability greatly greatly and, and, therefore, therefore, mole molecular markers markers such as allozymes allozymes and mtDNA mtDNA may not not be polymorphic polymorphic enough. microsatellites markers Goldstein and enough. The The more more recently recently developed developed microsatellites markers ((Goldstein and Schlotterer, 999), however, variable and Schl6tterer, 11999), however, are are much much more more variable and useful useful in in this this context. context. Another Another type type of of problem problem that that can can be be found found concerns concerns the the power power of of the the statisti statistical Classical population population genetic cal methods methods that that are are available. available. Classical genetic methods methods for for the the inference of inference of demographic demographic or or ecological ecological parameters parameters have have relied relied on on measures measures that that summarize summarize the information information contained contained in genetic data. data. Among Among these we have have FST, FST, sample sample heterozygosity, heterozygosity, the the distribution distribution of of pairwise pairwise differences differences between sequences, and between DNA DNA sequences, and the the number number of of segregating segregating sites. sites. A A serious serious draw drawback back of of these these approaches approaches is is that that they they assume assume constancy constancy in in demographic demographic parameters and genetic equilibrium conditions. These assumptions assumptions are vio parameters equilibrium conditions. violated by all natural natural metapopulations. metapopulations. Additionally, Additionally, most most of these methods methods lack statistical because they statistical power power because they use use only only information information provided provided by by frequency frequency dis distributions sophisticated methods tributions of of alleles alleles or or haplotypes. haplotypes. More More sophisticated methods that that make make bet better various types information contained ter use use of of the the various types of of information contained in in genetic genetic data data have have been been developed. developed. These These methods methods can can be be grouped grouped into into two two types types of of approaches: approaches: ((1) 1 ) coalescent coalescent or or genealogical genealogical approaches approaches that that use use the the genealogical genealogical informa information contained contained in multilocus genotype tion in DNA DNA sequences sequences and and (2) (2) multilocus genotype approaches approaches that that use information (see (see later). important to use gametic gametic disequilibrium disequilibrium information later). It It is is important to realize realize that that these these two two types types of of methods methods differ differ not not only only in in the the type type of of information information they they use, use, but but also also on on the the nature nature of of the the parameters parameters they they estimate. estimate. Coalescent Coalescent methods methods (and those based on summary summary statistics) estimate estimate long-term evolu evolutionary parameters, whereas tionary parameters, whereas multilocus multilocus genotype methods methods estimate estimate short-term short-term ecological Chapter 88 discusses ecological parameters. parameters. Chapter discusses the the application application of of the the coalescent coalescent in in metapopulations metapopulations and and explains explains how how this this approach approach can can be be used used to to make make infer inferences about demographic demographic processes. discusses the second type ences about processes. This This chapter chapter discusses the second type of of approaches, those based based on multilocus genotype data. It provides some approaches, those on multilocus genotype data. It first first provides some examples examples of of applications applications of of classical classical population population genetic genetic approaches approaches and and explains limitations. Then locus genotype explains their their limitations. Then the the chapter chapter introduces introduces multi multilocus genotype approaches brief account approaches with with aa brief account of of their their short short history history and and some some details details about about their their implementation implementation followed followed by by some some examples. examples. Finally, Finally, it it discusses discusses the the need need to to integrate integrate the the information information provided provided by by genetic genetic data data with with that that coming coming from from demographic demographic and environmental environmental data and provides provides examples of how how to achieve this goal.
115.2 5.2
CLASSICAL CLASSICAL POPULATION POPULATION GENETIC GENETIC APPROACHES APPROACHES Until recently, the the most widely used used genetic approach approach in population population biology statistic (Wright, was was the the estimation estimation of of migration migration rates rates from from Wrights' Wrights' FST FST statistic (Wright, 11931). 93 1 ). This method is based on on the This method is based the island island model model of of population population structure structure
115. 5.
369 369
MULTILOCUS MULTILOCUS GENOTYPE METHODS METHODS
(see which leads and Nm, Nm, the the (see Chapter Chapter 7), 7), which leads to to aa simple simple relationship relationship between between FST FsT and effective effective number number of of migrants: migrants:
), FST FST = ~ 1/(4Nm 1/(4Nm + + 11),
(15.1) (15.1)
where where N N is is the the local local population population size size and and m m iiss the the migration migration rate. rate. The The power power of limited, as of this this method method is is very very limited, as aa small small amount amount of of migration migration is is enough enough to to wipe (Fig. 15. 1 ). Additionally, wipe out out the the genetic genetic signal signal (Fig. 15.1). Additionally, the the use use of of this this method method has has 999) been been criticized criticized repeatedly repeatedly (for (for aa review, review, see see Whitlock Whitlock and and McCauley, McCauley, 11999) because because it it is is based based on on aa large large number number of of unrealistic unrealistic assumptions, assumptions, such such as as con constant stant and and equal equal local local population population sizes, sizes, symmetric symmetric migration, migration, and and probability probability of of migration migration between between populations populations independent independent of of geographic geographic distance. distance. Most Most of of these these assumptions assumptions are are violated violated in in the the case case of of metapopulations metapopulations and and therefore therefore its use is its use is unwarranted. unwarranted. Some about the Some few few studies studies have have tried tried to to make make inferences inferences about the composition composition of of 998). This colonizing propagules e.g., Giles colonizing propagules ((e.g., Giles and and Goudet, Goudet, 1997; 1997; Ingvarsson, Ingvarsson, 11998). This is problem because because the propagules has is an an important important problem the composition composition of of colonizing colonizing propagules has aa substantial populations (Whitlock substantial effect effect on on the the genetic genetic structure structure of of meta metapopulations (Whitlock and and McCauley, 1 977) introduced McCauley, 1990). 1990). Slatkin Slatkin ((1977) introduced two two extreme extreme models models of of colon colonizing izing group group formation: formation: the the propagule pool model, model, in in which which colonizers colonizers are are drawn population, and model, in drawn from from aa single single source source population, and the the migrant pool model, in which which colonizers metapopulation. Intermediate colonizers are are drawn drawn at at random random from from the the entire entire metapopulation. Intermediate cases cases can can be be considered considered if if the the formulation formulation includes includes the the probability, probability,
1
0.8 0.8
0.6 0.6 F FST ST
0.4
0.2
00
T
0
4
~
T
I
112 2
8
!
T
1
116 6
20 20
Nm Nm Fig. 5.1 Degree Fig. 115.1 Degree of of population population subdivision, subdivision, FSf, FST, as as aa function function of of the the effective effective number number of of migrants under the migrants under the island island model. model.
370 370
OSCAR E. OSCAR E. GAGGIOTII GAGGIOTTI
((1977) 1 977) models population models show show that that the the decrease decrease in in genetic genetic diversity diversity due due to to population turnover pronounced under pool model (Pannell and turnover is is more more pronounced under the the propagule propagule pool model (Pannell and Charlesworth, 999). Charlesworth, 11999). Giles and Goudet 1 997) investigated the genetic structure of a metapopu Goudet ((1997) metapopuinhabiting the lation lation of of Silene dioica inhabiting the Skeppsvik Skeppsvik Archipelago Archipelago in in Sweden, Sweden, where where new new islands islands are are created created due due to to rapid rapid land land uplift. uplift. They They used used information information on on the the age of the local populations populations inhabiting the different islands to make inferences inferences about common origin about the the probability probability of of common origin among among migrants. migrants. They They argued argued that that isolation isolation by by distance distance among among young young populations populations or or among among both both young young and and intermediate would indicate indicate that islands were intermediate populations populations would that colonizers colonizers of of new new islands were drawn from a limited number number of source populations, populations, in which case the propag propagule pool model would best describe the colonization colonization process in the archipe archipelago. isolation by lago. However, However, isolation by distance distance among among intermediate intermediate but but not not among among young young populations colonizers represent represent aa sample sample of populations would would indicate indicate that that colonizers of the the whole whole metapopulation, metapopulation, in in which which case case the the migrant migrant pool pool model model would would be be more more approp appropriate. Because Because they only detected isolation by riate. they only detected isolation by distance distance among among populations populations of of intermediate most appropriate intermediate age, age, they they concluded concluded that that the the most appropriate colonization colonization population was pool model. model model for for the the S. dioica meta metapopulation was the the migrant migrant pool model. Ingvarsson 1 998) investigated Ingvarsson ((1998) investigated how how mating mating patterns patterns of of female female Phalacrus substriatus beetles both the size of beetles influenced influenced both the effective effective size of newly newly colonized colonized popu popuprobability of common common origin of individuals in the founding lations and the probability groups. He Eq. (7) 1 990) to groups. He used used Eq. (7) in in Whitlock Whitlock and and McCauley McCauley ((1990) to estimate estimate the the probability of common origin of two alleles in newly founded populations populations (<1» (+).. This equation equation requires estimates of the probability probability of common common origin of diploid founders, which estimated from mark-recapture experiments, diploid founders, which he he estimated from mark-recapture experiments, and and of of the the effective effective size size of of founding founding groups, groups, which which he he estimated estimated using using F F statistics. statistics. The appropriate col The estimate estimate of of + = 0.8 0.8 thus thus obtained obtained indicated indicated that that the the most most appropriate colmodel. The onization onization model model for for P. substriatus was was the the propagule propagule pool pool model. The differ different approaches Giles and 1 997) and and Ingvarsson ent approaches used used in in the the studies studies of of Giles and Goudet Goudet ((1997) Ingvarsson ((1998) 1 998) lead lead to to rough rough estimates estimates of of colonization colonization parameters parameters and and rely rely on on equi equilibrium librium models models that that make make unrealistic unrealistic assumptions assumptions similar similar to to those those used used by by F FsT ST approaches. approaches. methods based on summary measures have been Additionally, genetic methods developed developed for for the the estimation estimation of of the the effective effective size size of of ancestral ancestral and and descendant descendant populations and populations and divergence divergence times times between between descendant descendant populations populations (e.g., (e.g., Gaggiotti 997) These Gaggiotti and and Excoffier, Excoffier, 2000; 2000; Wakeley Wakeley and and Hey, Hey, 11997) These methods methods have have also population declines also been been used used to to detect detect population declines or or bottlenecks bottlenecks (for (for references, references, see see al., 1998) and for the estimation estimation of the effective size size of panmictic Luikart et aI., population methods have population (for (for aa review, review, see see Schwartz Schwartz et et aI., al., 1998). 1998). These These methods have low low statistical provide point statistical power power and, and, in in general, general, they they only only provide point estimates estimates of of the the parameters of interest, although although approximate confidence intervals intervals can parameters of interest, approximate confidence can be be obtained using randomization randomization techniques. =
11 55.3 .3
METHODS M E T H O D S BASED B A S E D ON O N MULTILOCUS M U L T I L O C U S GENOTYPES GENOTYPES As approaches that As opposed opposed to to classic classic population population genetics genetics and and coalescent coalescent approaches that are are aimed aimed at at studying studying processes processes that that take take place place on on an an evolutionary evolutionary time time scale, multilocus multilocus genotype methods methods can be used to study processes that that
115. 5. MULTILOCUS MULTILOCUS GENOTYPE GENOTYPE METHODS METHODS
3lll 371
occur occur on on an an ecological ecological time time scale. scale. The The realization realization that that the the use use of of multilocus multilocus genotype genotype data data was was aa powerful powerful tool tool for for the the genetic genetic study study of of populations populations dates 970s and 980s (Smouse, 978). These dates back back to to the the late late 11970s and early early 11980s (Smouse, 11978). These methods methods were applied applied in in many different contexts, which the the most relevant were many different contexts, among among which most relevant to to us us is is that that of of subdivided subdivided populations populations where where they they were were used used to to allocate allocate individuals to groups groups and and to to estimate estimate the the genetic genetic contribution of different different individuals to contribution of source admixed populations Smouse et al. ((1982) 1 982) and source populations populations to to admixed populations [e.g., [e.g., Smouse et al. and references therein] underlying these these latter that references therein].. The The rationale rationale underlying latter applications applications is is that aa multilocus multilocus or or gametic gametic disequilibrium disequilibrium approach approach uses uses the the information information provided provided by by the the correlations correlations between between alleles alleles at at different different loci loci (Waples (Waples and and Smouse, Smouse, 1990). 1990). In In aa population population in in gametic gametic equilibrium, equilibrium, the the expected expected frequency containing allele allele i at allele jj at frequency of of gametes gametes containing at locus locus 11 and and allele at locus locus 2 2 is is uncorrelated uncorrelated in in the the sense sense that that an an individual's individual's genotype genotype at at locus locus 1I provides provides no pools no information information about about its its genotype genotype at at locus locus 2. 2. In In aa mixture mixture of of gene gene pools with different different frequencies frequencies of of i and and j, j, observed observed gametic gametic frequencies frequencies will will with depart depart from from the the independence independence expectations, expectations, resulting resulting in in gametic gametic disequilib disequilibrium. rium. Thus, Thus, the the presence presence of of allele allele i at at locus locus 11 may may indicate indicate that that allele allele jj is is more 2. Note more or or less less likely likely to to be be present present at at locus locus 2. Note that that the the term term linkage linkage disequilibrium disequilibrium is is also also used used to to describe describe this this phenomenon phenomenon but but is is rather rather misleading because the association of alleles across misleading because the nonrandom nonrandom association of alleles across loci loci is is not not caused by by physical physical linkage. linkage. caused One One of of the the most most successful successful multilocus multilocus genotype genotype methods methods for for making making infer inferences ences about about demographic demographic parameters parameters was was introduced introduced two two decades decades ago ago by by fishery geneticists interested in fishery geneticists interested in estimating estimating the the contribution contribution of of different different popu populations lations to to the the salmon salmon mixed mixed fishery fishery operating operating off off the the northwest northwest coast coast of of North GSI) method North America. America. This This genetic genetic stock stock identification identification ((GSI) method was was described described in aa series series of of unpublished unpublished manuscripts manuscripts dating dating back back to to 11979 and cited cited by by in 979 and Milner 1 985). Smouse 1 990) developed Milner et et al. al. ((1985). Smouse et et al. al. ((1990) developed aa statistically statistically rigorous rigorous formulation formulation of of this this method method that that allowed allowed for for the the estimation estimation of of allele allele fre frequencies quencies in in the the source source populations. populations. The The underlying underlying model model assumes assumes that that aa sample mixed population, sample is is taken taken from from aa mixed population, composed composed of of unknown unknown propor proporXl, X tions tions Xa, x2, xss from from aa known known or or partially partially known known set set of of panmictic panmictic source source 2, 9 , X Xl, X , Xss are populations, 11,, 2, populations, 2 , ... . , , s, s, respectively. respectively. The The proportions proportions Xa, x 22,, . . . , X are treated (see Fig. 5 .2 ) . The treated as as parameters parameters that that need need to to be be estimated estimated (see Fig. 115.2). The source source populations populations are are assumed assumed to to be be at at linkage linkage and and Hardy-Weinberg Hardy-Weinberg equilibrium. equilibrium. The The gametic gametic disequilibrium disequilibrium generated generated by by the the mixing mixing of of individuals individuals coming coming from from different different source source populations populations is is used used to to estimate estimate the the proportionate proportionate con contribution tribution of of each each source source population population to to the the genetic genetic mixture. mixture. As As it it was was shown shown by al. (2002), ideally suited by Gaggiotti Gaggiotti et et al. (2002), this this method method is is ideally suited to to determine determine the the com composition position of of colonizing colonizing groups groups and, and, therefore, therefore, for for the the study study of of the the coloniza colonization process process in in aa metapopulation. metapopulation. This This application application is is described described in greater tion in greater detail detail later. later. Multilocus have been Multilocus genotype genotype methods methods have been applied applied recently recently to to identify identify immi immigrants grants and and assign assign them them to to aa particular particular source source population. population. The The first first assign assignment 1 995) and ment test test was was developed developed by by Paetkau Paetkau et et al. al. ((1995) and simply simply calculated calculated the the likelihood of drawing drawing aa single single multilocus multilocus genotype genotype from from several several potential potential likelihood of sources sources based based on on the the observed observed allele allele frequencies frequencies at at each each locus locus in in each each source. source. A Mountain A more more rigorous rigorous assignment assignment test test developed developed by by Rannala Rannala and and Mountain ((1997) 1 997) can can identify identify individuals individuals that that are are immigrants immigrants or or have have recent recent immigrant immigrant •
•
•
.
.
•
.
.
OSCAR OSCAR E. E. GAGGIOTTI GAGGIOTTI
372 372 Source 1
Source 2
Source
s
Genetic mixture Fig. 115.2 5.2 Schematic description of the genetic stock identification method. Basic Basic inputs GSI method are allele frequency distributions, p, p, in the source source population and needed by the GSI genetic mixture, Gm. Gin. The genotype frequencies iinn source source populations, Gj, Gi, are genotypes in the genetic obtained under the assumption of Hardy-Weinberg equilibrium (HWE) and llinkage inkage equilibrium (LE) across across loci and are used used as as the probability of observing each each mixture mixture genotype in that population.
ancestry using using likelihood likelihood ratio ratio tests. tests. A general approach approach by ancestry A more more general by Pritchard Pritchard et al. (2000) be used to identify migrants but but assumes that the the migration migration et al. (2000) can can be used to identify migrants assumes that rate main objective this latter latter method method was was to to rate is is known known and and small. small. The The main objective of of this infer structure and and assign infer population population structure assign individuals individuals to to populations. populations. This This is is an an important as in in many cases the demarcation of important problem, problem, as many cases the demarcation of local local population population based on on the the geographic geographic location location of of sampled sampled individuals individuals is is not not possible. possible. based Other for the panmictic populations Other methods methods for the identification identification of of panmictic populations and and assignassign ment ment of of individuals individuals are are provided provided by by Corander Corander et et al. al. (2003) (2003) and and Dawson Dawson and and Belkhir Belkhir (2001). (200 1 ) . Although Although these these approaches approaches may may be be able able to to identify identify immigrants, immigrants, they they are are not not appropriate appropriate to to estimate estimate migration migration rates. rates. In In principle, principle, one one could could repeat repeat the the test test for for all all the the individuals individuals in in aa sample sample and and then then simply simply count count those those individuals individuals identified identified as as migrants. migrants. However, However, this this would would be be erroneous erroneous because because making making many many pairwise pairwise comparisons comparisons between between populations populations for for each each of of aa large large number number of individuals of individuals means means that that some some individuals individuals will will appear appear to to be be immigrants immigrants purely by by chance, chance, which which would would lead lead to to overestimation overestimation of of the the number number of of immiimmi purely grants. grants. Wilson Wilson and and Rannala Rannala (2003) (2003) developed developed aa multilocus multi locus genotype genotype method method that estimates estimates rates rates of of recent recent immigration immigration among among local local populations. populations. This This that method method detects detects asymmetric asymmetric migration migration between between pairs pairs of of populations populations and and estiesti mates mates the the total total number number of of nonimmigrants, non immigrants, first-generation first-generation immigrants, immigrants, and and second-generation second-generation immigrants. immigrants. This This detailed detailed information information can can be be very very useful useful for studies studies of of metapopulation metapopulation dynamics dynamics and and for for the the design design of of management management for strategies strategies for for conservation. conservation. All All the the methods methods just just described described use use likelihood-based likelihood-based approaches approaches that that rely rely on on either either maximum maximum likelihood likelihood or or Bayesian Bayesian estimation estimation procedures. procedures. These These approaches approaches are are described described in in Box Box 15.1. 15.1 .
313 373
115. 5. MULTILOCUS MULTILOCUSGENOTYPE GENOTYPE METHODS METHODS
BOX 1 S.l
Likelihood-Based Approaches
Likelihood-based methods proceed by assuming that observed data arose from some probabilistic model with unknown parameters. Their objective is to use data to estimate the parameters of the model and to assess the degree of uncertainty associated with these estimates. The core of the method consists in the calculation of the probability P(Gl 6) of observing genetic data G if the parameters of the model take the va lue 6. This probability is the so-called likelihood function, L«(jIG), which, by definition, is a function of 6. I n the context of a GSI method, data G are the individual genotypes observed in the sample from the genetic mixture and the al lele frequency distributions observed in the source populations. The aim of GSI is to estimate the vector x lXi}, where Xi is the con tribution of sou rce population i. One possible formulation for the GSI l i kelihood func tion is =
PilOn"
L ( xI G ) =
IT 2:: xi(rrOPil, allk Pil, a2Ik)'
k=l
1= 1
I
(Bl )
where, is the frequency of the nth (n = 1 , 2) allele at the Ith locus of individual k in population i, m is the n umber of individuals in the sample from the genetic mixture, L is the number of loci scored, and if if
Ol /k
=
021k
otherwise '
Maximum Likelihood Inference
Maximum l i kelihood (ML) inference consists of finding the value of 6 that maximizes the likelihood fu nction L(6 I G). One problem with this approach is that the u ncertainty associated with the estimate e is expressed by a 95% confidence region that has a rather obscure interpretation. The precise interpretation is that the probability that the confi dence region contains the true value of 6 is 0.95. Note that this is not equivalent to say ing that the probability that 6 lies in the confidence region is 0.95. An i mportant advantage of ML inference is that for a large sample size, the maximum likelihood esti mate, e, will have an approximate normal distribution centered on the true parameter value 6. Thus, an approximate 95% confidence i nterval can be calcu lated as the range of 6 values that are within two log l i keli hood units of the maximum log l i kelihood . Additionally, we can test whether the maximum likelihood estimate is significantly d if ferent from another fixed value, 60, using the likelihood ratio test. This test uses the fact that the log-likelihood ratio statistic, A
=
L( 60 ) - 210g-. L(6 ) ,
(B2)
has asymptotically a x2 distribution, if 60 is the "true" value of 6. A can then be assessed for statistical significance using standard X2 significance levels. ML estimates can be obtained analytically for simple models, but the appl ication of this method in population genetics leads to complex likelihood functions that need to be explored using computer approaches such as the expectation maximization algor ithm. Equation (Bl ) is a good example of this situation.
OSCAR E. OSCAR E. GAGGIOTTI GAGGIO1-FI
374 374
Bayesian Inference
In order to make probability statements about the parameter e given data G, we m ust begi n with a model providing a joint probability distribution for e and G. The joint probability mass, p(e, G), can be written as a product of two probability distributions: the prior d istribution pee) and the sampling distribution, g iven by the likelihood function L( qe ): p( e, G ) p( e ) L( G Ie ) . Using Bayes' theorem, we obtain postdata or posterior distribution: =
p( eIG )
=
L( eIG ) p( e ) P( G )
.
(B3)
The posterior distribution represents our knowledge about the parameters, taking into account both our prior information (represented by the prior d istri bution) and observed data. The primary task of any specific application is to develop the model p(e,G) and perform the necessary computations to summarize p(eIG) in appropriate ways. Visual inspection of the posterior distribution provides information that is unavailable when using Ml estimation. Additionally, this distribution can be described by point esti mates such as the mode or the mean. The uncertainty around the estimate is expressed by the 95% credible region for e. The intuitive interpretation of this region is that the probability that e lies in it is 0.95. Another advantage of Bayesian over Ml estimation approaches is that the former does not rely on asymptotic arguments and, therefore, is valid i n situations where the standard likelihood theory fails. Simple problems in estimation lead to closed form solutions for the posterior distri bution, but typical applications i n population genetics require the use of numerical integration methods, such as Markov chain Monte Carlo (e.g., Brooks, 1 998).
115.4 5.4
INTEGRATION OF GENETIC, INTEGRATION OF GENETIC, DEMOGRAPHIC, DEMOGRAPHIC, AND AND ENVIRONMENTAL ENVIRONMENTAL DATA DATA There is an an increasing in finding of efficiently There is increasing interest interest in finding ways ways of efficiently combining combining genetic, genetic, demographic, demographic, and and other other sources sources of of information information in in order order to to make make infer inferences processes. Some ences about about demographic, demographic, evolutionary, evolutionary, and and ecological ecological processes. Some exam examples 1 ) , Burland al. (200 1 ), and ples are are the the works works of of Estoup Estoup et et al. al. (200 (2001), Burland et etal. (2001), and Charbonnel et al. (200 1 ) use Bayesian approach Charbonnel et al. al. (2002). (2002). Estoup Estoup et et al. (2001) use aa Bayesian approach that that combines information about about demo combines microsatellite microsatellite and and enzyme enzyme data data with with information demographic phases of introduction history graphic parameters parameters describing describing the the major major phases of the the introduction history of of the in various various Caribbean Caribbean and and Pacific Pacific islands. islands. The The the cane cane toad toad Bufo marinus in parameters parameters for for which which limited limited prior prior information information was was available available are are the the size size of of founding founding groups, groups, effective effective size size during during population population expansion, expansion, and and population population size 1 ) combined size at at equilibrium. equilibrium. Burland Burland et et al. al. (200 (2001) combined genetic genetic data data with with infor information mation obtained obtained from from mark-release-recapture mark-release-recapture studies studies in in order order to to identify identify the the evolutionary social organization evolutionary determinants determinants of of social organization in in brown brown long-eared long-eared bat bat Pletocus auritus. Charbonnel Charbonnel et et al. al. (2002) (2002) combined combined demographic demographic and and genetic genetic data population dynamics data in in order order to to study study evolutionary evolutionary aspects aspects of of the the meta metapopulation dynamics of of
Biomphalaria pfeifferi.
115. 5. MULTILOCUS MULTILOCUSGENOTYPE GENOTYPE METHODS
375 375
It It is is clear clear that that understanding understanding complex complex population population histories histories requires requires the the combination combination of of all all available available information. information. Additionally, Additionally, such such aa strategy strategy can can lack of sufficient information help overcome the problems generated by the lack genetic methods in genetic data, a frequent problem when applying population genetic for estimating population parameters. This is typically the case, for example, when studying migration patterns patterns in species with with high dispersal capabilities. when In this case, genetic differentiation is low and, therefore, the estimates of large variance. In order to solve this problem dispersal rates can have a very large demowe need to complement genetic data with other types of data, such as demo graphic, graphic, geographic geographic distance, distance, or or environmental environmental data. data. The The incorporation incorporation of of these these data data should should decrease decrease the the variance variance of of the the estimates estimates without without biasing biasing the the framework needed for results of the analysis. Bayesian methods provide the framework achieving these goals. important limitation of the statistical genetic methods developed is that An important they are are simply simply aimed aimed at at estimating estimating the the parameters parameters of of population genetics they population genetics obtained, the models (Nm, models (Nm, Ne, Ne, etc.) etc.) Once Once the the estimates estimates of of the the parameters parameters are are obtained, the researcher proposes alternative researcher proposes alternative hypotheses hypotheses that that focus focus on on specific specific processes processes and and that are consistent consistent with with these estimates. Under Under these circumstances it is diffi diffithat cult importance of the different ecological cult to obtain a clear clear idea idea of the relative importance genetic processes processes responsible responsible for for the the results. results. Hierarchical Hierarchical Bayesian Bayesian methods methods or genetic ((e.g., e.g., Gelman 995) are ideal for problem because Gelman et et a!., al., 11995) are ideal for addressing addressing this this problem because they they relationship between the likelihood funccan be used to explicitly model the relationship likelihood func tion parameters and relevant ecological or genetic processes. This approach approach alternative models that that consider consider different different processeslfactors processes/factors and whose whose leads to alternative significance can can be be evaluated evaluated using using computationally computationally intensive intensive methods methods such such as as significance deviance information information criterion DIC) (Spiegelhalter (Spiegelhalter et et al., a!., 2002) 2002) or deviance criterion ((DIC) or reversible reversible jump MCMC ((Green, j ump MCMC Green, 1995). 1 995). This section section exemplifies the implementation implementation of these ideas ideas using using aa study study This exemplifies the of these of al., of colonization colonization patterns patterns in a metapopulation metapopulation of of gray gray seals (Gaggiotti (Gaggiotti et a!., 2002). 2002 ) .
The Gray Seal Metapopulation The Gray Seal Metapopulation iinn the the Orkney Orkney Isles Isles The grypus) is is aa colonially colonially breeding breeding marine mammal gray seal The gray seal (Halicoerus (Halicoerus grypus) marine mammal each year. year. During During autumn, gray seals that produces produces only only aa single single offspring offspring each autumn, gray seals that gather at for the the females gather at breeding breeding colonies colonies for females to to give give birth birth and and suckle suckle the the pups. pups. Toward the end of of lactation, lactation, the the females females are are mated mated by by one one or or more more males males and and Toward the end then return to then return to sea, sea, leaving leaving their their weaned weaned pup pup on on land land (Anderson (Anderson et et al., a!., 1975). 1 975) . Pups Pups remain remain oonn land land for for 11 or or 22 more more weeks weeks and and then then go go to to sea sea where where they they spend 4-6 yr spend the the next next 4-6 yr without without returning returning to to the the breeding breeding grounds. grounds. The The average average age age of of recruitment recruitment to to the the breeding breeding population population is is 55 yr yr and and new new recruits recruits usually usually arrive at at the the breeding after the the first first pups pups have have been been born. born. One-third One-third arrive breeding grounds grounds after of of the the world world population population of of gray gray seals breeds breeds at at 48 48 colonies colonies around around the the British British Isles, with with the the majority majority on on offshore offshore islands islands to to the the north north and and west west of of Scotland. Scotland. Isles, Although the the whole whole population population is is growing exponentially, individual individual colonies colonies Although growing exponentially, exhibit diverse dynamics. Some of constant exhibit diverse of them them fluctuate fluctuate around around a long-term long-term constant value, whereas whereas others others are are increasing increasing exponentially exponentially or or logistically logistically in in size. size. value, Furthermore, some some colonies colonies have have become become extinct extinct and and others others have have been been coloncolon Furthermore, ized ized recently.
376 376
OSCAR F. E. GAGGIO'I-FI GAGGIOTII
The The focus focus of of the the case case study study is is the the Orkney Orkney Isles, Isles, aa group group of of 50 50 islands islands lying lying off off northeast northeast Scotland Scotland (see (see map map in in Fig. Fig. 15.3). 15.3). In In about about 1992, 1 992, three three vacant vacant islands (Stronsay, (Stronsay, Copinsay, Copinsay, and and Calf Calf of of Eday) Eday) were were colonized. colonized. At At more more or or islands less the the same same time, time, two two large large colonies colonies reached reached their their carrying carrying capacity capacity (Holm (Holm less of Huip Huip and and Faray). Faray) . Therefore, Therefore, the the question question arises arises as as to to whether whether densitydensity of dependent effects in these these populations populations may may have have played played aa role role in in the the colocolo dependent effects in nization previously unoccupied nization of of the the three three previously unoccupied islands, islands, in in which which case case most most of of the the colonizers would would come come from from the the colonies colonies that that are are at at or or close close to to their their carrying carrying colonizers capacity. Additionally, Additionally, itit is is important important to to determine determine the the composition composition of of foundfound capacity. ing effect on structure of ing groups groups because because of of its its substantial substantial effect on the the genetic genetic structure of meta populations (see earlier discussion). Finally, there there is is an interesting behavbehav metapopulations (see earlier discussion). Finally, an interesting ioral question question that that could could be be investigated: investigated: the the possibility possibility that individuals in in the the ioral that individuals newly founded colonies mate depending on on their their origin. origin. In In other newly founded colonies mate assortatively assortatively depending other words, it it is is possible possible that individuals that come from from the words, that individuals that come the same same source source colony colony are likely to to mate mate among among themselves than with with individuals individuals from other are more more likely themselves than from other source source populations. populations. The identification method (see earlier earlier discussion) discussion) is is the the loglog The genetic genetic stock stock identification method (see ical approach approach to answer these these questions because the the vector vector of mix ical to answer questions because of genetic genetic mix= {xi}, {Xi} ' can can be be used used to to describe the composition composition of ture ture coefficients, coefficients, xx = describe the of colonizing groups. groups. Thus, Thus, genetic genetic samples from the three newly newly colonizing samples were were obtained obtained from the three founded populations that founded colonies colonies and and from from seven seven potential potential source source populations that were were deemed the most likely sources colonies deemed to to be be the most likely sources of of founders. founders. Some Some few few smaller smaller colonies
0 35
·3° 00'
.HH
0.3 .HS
0.25 E(x,)
0.2 0.15 01 0.05
FA
•
MG
.RU
o +---�--�-� o 0.1 0.2 03 .SW .ST
Productivity
0.35
HH
0.3 0.25 E(x,)
.HS
0.2 0.15 0.1 0.05
MG.
.FA . RU SW 5T
o +---_�-�-� o 20 40 60 80 ••
Orkney Islands
Distance
Fig. 5.3 Results Fig. 115.3 Resultsof the the hierarchical Bayesian Bayesian analysis analysis for the colonization event observed in Stronsay. Triangles identify new new colonies, and circles circles identify potential source populations. The black triangle identifies Stronsay, Stronsay, and gray circles identify the two two main contributors to its col colonizing group. The source colonies are Faray Faray (FA), (FA), Holm of Huip (HH), Holm of Spurness Spurness (HS), (HS), Muckle Greenholm (MG), Ruskholm (RU), (RU), Swona (SW), (SW), and Stroma (ST). (ST). The three newly founded colonies are Stronsay Stronsay (SR), (SR), Calf of Eday Eday (CE), (CE), and Copinsay (CO). The plot of E(x;) E(xi) ver versus sus productivity shows aa weak association, whereas that of E(x;) E(xi) versus versus distance shows aa strong association. (see ). (see Color Plate Plate 11).
377 377
115. 5. MULTILOCUS MULTILOCUSGENOTYPE GENOTYPE METHODS METHODS
were were not not sampled sampled because because they they were were very very difficult difficult to to reach. reach. Due Due to to the the diffi difficulties adult gray gray seals, and in order to culties associated associated with with the the sampling sampling of of adult seals, and in order to obtain samples (at least 1150 5 0 individuals individuals per per colony), obtain large large samples (at least colony), samples samples were were obtained from obtained from pups pups and and were were scored scored for for nine nine highly highly variable variable microsatellite microsatellite loci. All colonies were loci. All samples samples from from the the recently recently founded founded colonies were collected collected suffi sufficiently soon after sure that that they were from FI generation. ciently soon after founding founding to to be be sure they were from the the F1 generation. The The standard standard GSI GSI method method assumes assumes that that individuals individuals in in the the genetic genetic mixture mixture are are adults Our samples, adults that that came came directly directly from from the the source source populations. populations. Our samples, how however, consisted of could include include ever, consisted of descendants descendants of of the the original original colonizers colonizers and and could hybrid individuals. Thus, the [see Box 5.1, hybrid individuals. Thus, the GSI GSI likelihood likelihood function function [see Box 115.1, Eq. B l ) l needs Eq. ((B1)] needs ttoo bbee modified. modified. The The Likelihood Likelihood Function Function
There There are are $s source source populations populations contributing contributing to to each each newly newly founded founded pop population. ulation. Baseline Baseline data data come come from from samples samples of of ni n i pups pups from from each each of of the the con contributing tributing populations populations and and consist consist of of the the multilocus multilocus genotypes genotypes for for each each pup, pup, denoted denoted by by Yij, Yij, the the genotype genotype of of the the jth jth juvenile juvenile in in the the ith ith source source population population where 1, 2, newly founded where ii - 11,, 2, 2 , ... .. ,. , $s and and jj -= 1, 2 , ... .. ,. , ni' n i. Data Data for for the the newly founded populations mixtures) also pups and populations (i.e., (i.e., the the genetic genetic mixtures) also come come from from pups and are are denoted denoted n(m), where by by Y(m)j Y(m)j,' with with jj = 11,, 2, 2 , ... .. ,. , n(m), where m m denotes denotes mixture. mixture. The The complete complete data data set set is is denoted denoted by by Yy = = {Yij: {Yi? ii = = 1, 1, 2, 2 , . ... ,. . , $; s; jj = = 1, 1, 2, 2 , . ... ,. . , ni}U ni}U {Y(m)j: {Y(m)j: jj = = 11,, 22 ,, ... .. , , n(m)}' n(m)}. The The allele allele frequencies frequencies at at each each locus locus iinn each each of of the the $s source source pop populations ulations are are parameters parameters that that need need to to be be estimated. estimated. We We denote denote these these param parameters i , where is the frequency of allele h at locus I in population i eters Pi Pi = {{Phli}, where Phli is the frequency of allele h at locus I in population Phl } P hl ii and and let let P p - {{PI> P l , .. .. . ,, pJ. Ps}. In In order order to to include include the the possibility possibility that that some some pups pups in in the the newly newly founded founded colonies parents that colonies have have parents that came came from from two two different different source source populations, populations, we we assume tendency w for same colony mate assume that that there there is is aa tendency for individuals individuals of of the the same colony to to mate interpreted as mating coefficient. together. Thus, w can together. Thus, can be be interpreted as an an assortative assortative mating coefficient. The The conditional conditional probability probability of of genotype genotype kk in in the the mixture mixture given given that that both both its its =
.
=
.
L L
If
parents source ii is parents came came from from source is P P ((KKlii ii)) = = H~)Pil,allkPil,a21k, where Pil,anlk Pil,anlk is is the the OPil,allkPi/,a2Ik' where lth locus locus of frequency frequency of of the the nth nth (n (n = - 11,, 2) 2) allele allele at at the the/th of individual individual k k in in popu population lation i, i, and and
{�
1 2
o =
=
if 2lk if a allk = a a21k ilk = if a a if allk 4~ a21k 2lk ' llk *'
Conversely, genotype kk in Conversely, the the conditional conditional probability probability of of genotype in the the mixture mixture given given that that its came from its parents parents came from two two different different populations populations ii and and j, j, is is L L
P jl,a2Ik jl,allk' P (( k k lij q)) = II I-[ Pil,allkP Pil,allkPjl, a21k + + 'YPil,a2IkP ~lPil, a2tkPjl,allk, =
where where
7 =
Il
0 1
if if a allk = a a2tk 21k llk = ' a if a altk 4= a21k if k ll *' 21k
378
OSCAR OSCAR E. E. GAGGIOTTI GAGGIO-I-FI
Given Given the the aforementioned aforementioned assumptions assumptions and and notations, notations, the the probability probability of of find finding genotype of a given individual kk in the mixture is
l -w ) w~(xiP(kii)) � (xiP( klii ) ) ++ ((l-w) PP(kw,x,p) ( k lw,x,p ) == w
(x2p(kii)) ~ �. ~. (XiXjP(klij)) (xixjP(kij))]. k l ii)) + � [[ �~ (x'tP( ]. 1l
1
+
1
1 *1
((15.2) 15.2)
The 1 5.2) represents The first first term term on on the the right-hand right-hand side side of of Eg. Eq. ((15.2) represents the the probability probability of genotype genotype kk given given that that its its parents parents mated mated assortatively, assortatively, whereas whereas the the second second of term term represents represents the the same same probability probability given given that that the the parents parents mated mated at at random. random. Using Using the the aforementioned aforementioned expression, expression, we we define define the the likelihood likelihood of of w w,, x x,, and and p p given data, yy,, n(m) n(ml
lw,x,p ) == IJ/ P(yw,x,p) ~=lP(kw,x,p), ( k 1 w,x,p ), LL(w,x,py) ( w,x,ply ) == P(Y
((15.3) 15.3)
where where n(m) n(m) IisS the the number number of of individuals individuals sampled sampled from from the the newly newly founded founded colony.
The Hierarchical Hierarchical Bayesian Approach We We want want to to develop develop aa method method for for testing testing the the hypothesis hypothesis that that density-dependent density-dependent effects effects in in the the source source populations populations are are responsible responsible for for the the new new colonizations. colonizations. If If this this is the the case, we expect expect that that the of colonizing groups, which is case, we the composition composition of colonizing groups, which is is described by by the vector x {Xi} , will will be be dominated dominated by by individuals individuals from from colonies described the vector x = colonies = {xi}, may also be aa func at their carrying We note, note, however, however, that at or or near near their carrying capacity. capacity. We that xx may also be function of the geographic the newly newly founded founded colonies colonies and and the the tion of the geographic distance distance between between the potential nearby colonies colonies may may send potential source source populations; populations; nearby send more more colonizers colonizers than than far far away sources. Additionally, we want to reduce the the variance variance of of the estimate of of xx away sources. Additionally, we want to reduce the estimate by combining combining genetic genetic data data with with other other sources sources of of information. of these by information. Both Both of these goals, goals, testing of of hypothesis of the estimates, can testing hypothesis and and improving improving the the precision precision of the estimates, can be be achieved incorporating demographic demographic and and geographic geographic distance distance data data in in the con achieved by by incorporating the context of aa hierarchical hierarchical Bayesian data available text of Bayesian approach. approach. Demographic Demographic data available consist consist of of time series series of of pup pup production production estimates estimates for for the the different different Orkney Orkney colonies. colonies. From From time these these time time series series itit is is possible possible to to calculate calculate aa colony-specific colony-specific productivity productivity index index ~ri, 'ITi, which strong the within aa source which describes describes both both how how strong the density-dependent density-dependent effects effects within source are are and and how how large large the the source source population population is. is. Details Details of of its its calculation calculation are are presented presented in Gaggiotti et in Gaggiotti et al. ai. (2002). (2002). The The geographic geographic distance, distance, 8i, 0i, is is obtained obtained by by measuring measuring the the distance distance along along the the path path that that aa seal seal would would use use to to move move between between colonies. colonies. The integration of geographic data The integration of demographic demographic and and geographic data is is achieved achieved in in aa natunatu values are are viewed viewed as as samsam ral Xi values ral way way if if we we use use aa prior prior distribution distribution in in which which the the xi ples from aa distribution functions of ples from distribution whose whose parameters parameters are are functions of productivity productivity and and geographic distance. distance. Because Because xx is is aa vector vector of of proportions, proportions, we we can can assume assume that that geographic it with parameters it follows follows aa Dirichlet Dirichlet distribution distribution (e.g., (e.g., Gelman Gelman et et al., aI., 1995) 1 995) with parameters (Xi(i == 1,1, 22,, ... .. ,.s, )s). There are are two two alternative alternative ways ways of of implementing implementing this this oti(i . There approach. approach. The The first first one, one, employed employed by by Gaggiotti Gaggiotti et et al. ai. (2002), (2002), uses uses aa model model that that and productivity productivity makes specific specific assumptions assumptions about about the the dependence dependence between between xi makes Xi and
379 379
115. 5. MULTILOCUS MULTILOCUSGENOTYPE GENOTYPE METHODS
and and distance. distance. The The second second one, one, more more recently recently developed developed ((Gaggiotti Gaggiotti et et al., aI., in in press), press), uses uses aa more more general general linear linear model model that that links links the the Cii cxi with with any any pair pair of of fac factors that could be and productivity. tors that could be but but are are not not restricted restricted to to distance distance and productivity. Figure 5.4 shows shows aa graph the model used by Figure 115.4 graph describing describing the model used by the the Gaggiotti Gaggiotti et et ai. al. (2002) (2002) approach. approach. Commonly Commonly used used models models of of dispersal, dispersal, such such as as the the normal normal and and Laplace 995), assume Laplace dispersal dispersal kernels kernels (e.g., (e.g., Neubert Neubert et et aI., al., 11995), assume that that the the propor propori-j units units away, away, tion of of individuals individuals that that move move from from patch patch ii to to patch patch j, j, located located i-j tion decays these models, decays exponentially exponentially with with distance. distance. Following Following these models, we we assume assume that that the proportion of colony that the proportion of individuals individuals in in aa newly newly founded founded colony that came came from from aa source population distance units units away decays exponentially source population 0i 8i distance away decays exponentially with with distance. distance. However, assume that However, we we assume that the the effect effect of of productivity productivity is is linear linear so so that that more more pro productive ductive source source colonies colonies contribute contribute more more individuals individuals to to the the founding founding groups. groups. Thus Thus the the expected expected contribution contribution of of aa given given source source population population to to the the founding founding groups, E(Xi), groups, E(xi), is is
S'rri]e-RSi [ (( 11 -- 8S)) + S1T i]e-Rbi
,
E(xi) = 4 s [( [ ( 11 -S - S )) + + S'rri]e-RSi Srri]e- Rbi ' Iz
( 15.4) (15.4)
where R R is is the the rate rate of of decay decay with with distance distance and and S is is the the contribution contribution of of where productivity. productivity. Because Because x x is is assumed assumed to to follow follow aa Dirichlet Dirichlet distribution, distribution, the the expect expectation E(xi)i ) = = Ci c~Jcx0, where CiO c~0 = = 2"ai' E,~xi.Thus, Thus, Cii c~i = = ation of of its its elements elements can can be be written written as as E(X i/CiO, where CioE(Xi) nuisance parameter parameter that otoE(xi) and and CiO cx0 is is aa nuisance that needs needs to to be be estimated. estimated. Equation 15.4) can Equation ((15.4) can be be used used to to formulate formulate four four alternative alternative models models (Table (Table 15.1). 15.1). The The first first one one can can be be considered considered as as the the null null model model because because it it assumes assumes that that the the composition composition of of founding founding groups groups is is independent independent of of the the two two factors. factors. Thus, Thus, the = 1 the prior prior distribution distribution is is the the Dirichlet Dirichlet with with Cii c~i-1 for for all all i.i. The The following following two two models one factor and the last one one is model that that includes models include include only only one factor and the last is the the full full model includes both both factors. factors.
Fig. Fig. 1 55.4 . 4 Schematic description of Bayesian Bayesian approach 11.. Genetic data are combined with demographic demographic and genotypic information information by focusing on the expected contribution contribution of each source population.
380 380
OSCAR OSCAR E. E. GAGGIOTII GAGGIOTTI
TABLE 115.1 5. 1
Alternative Models Obtained from Eq. (4)
Model Model
E(x;) E(xi)
Founding group composition is independent of both density and distance
E(Xi) E(xi)
= = s$
Founding group composition depends only on density
E E ( (Xi) xi)
= = --"-------"-� E [ ( 1[(- S1) +-S S " I)T + i l S7Ti]
11
[([(1-S)-I1 -S) + S7T;] S'lT i ]
l
t
E E ( (Xi) xi)
Founding group composition depends only on distance
e - RR ~ i&i ee - RR g i&i �.~ e-
= -- --- l
t
Founding group composition depends on both density and distance
E E ( (Xi) xi)
R &i [([(l-s) 1 -S ) + S7Ti]eS~i]e-R~, = = ------R&i � l -S ) + E [( I(1-S) + S7Ti]eSITi]e -R~i tl
The formulation for model is The Bayesian Bayesian formulation for the the full full model is P( w,S,R,ao,xIY ) a ( w )P ( ao )P( R )P( S )P( xIS,R,ao )P( ylw,x ) . P(w,S,R,oto,x]y) ot P P(w)P(oto)P(R)P(S)P(x]S,R,oLo)P(yw,x).
((15.5) 15.5)
15 . 3 ) . The The is given given by by Eq. Eq. ((15.3). The prior prior for for P(ao) P(cl0) is is uniform uniform The likelihood likelihood P(ylw,x) is (noninformative) from s to 1 00, whereas those for P(w) and P(S) are and P(S) are uniform uniform (noninformative) from s to 100, whereas those for from zero to one. The ( R ) is that, P(R) is uniform uniform from from zero zero to to five. five. Note Note that, from zero to one. The prior prior for for P for the sake of simplicity and given the large sample sizes used, this formu for the sake of simplicity and given the large sample sizes used, this formulation assumes [thus, the lation assumes that that allele allele frequencies frequencies are are known known [thus, the parameter parameter p p is is not not included in Eq. ( 15 . 5 ) ] . Thus, the estimation is carried out using their included in Eq. (15.5)]. Thus, the estimation is carried out using their maximum-likelihood more general formulation allowing allowing the maximum-likelihood estimates. estimates. A A more general formulation the use use of sizes would place aa Dirichlet Dirichlet prior prior on allele frequency of smaller smaller sample sample sizes would place on the the allele frequency distributions approach of Mountain ((1997) 1 997) and distributions following following the the approach of Rannala Rannala and and Mountain and would estimate estimate them would them at at the the same same time time as as all all other other parameters. parameters. The The Bayesian Bayesian formulations formulations for for each each of of the the three three remaining remaining models models are are obtained by 15.4) and 1 5.5) the obtained by eliminating eliminating from from Eqs. Eqs. ((15.4) and ((15.5) the factors factors that that are are not not included in in the the respective respective model. model. The The posterior posterior distribution distribution under under each each model model included is MCMC approach 998). is estimated estimated separately separately using using an an MCMC approach (e.g., (e.g., Brooks, Brooks, 11998). The The simplest simplest approach approach for for comparing comparing the the fit fit of of the the different different models models to to genetic genetic data described in 5.2. data is is the the use use of of the the DIC DIC (Spiegelhalter (Spiegelhalter et et a!., al., 2002), 2002), described in Box Box 115.2.
BOX 1 5.2
Bayesian Model Choice
We wish to compare a lternative model formulations with the aim of identifying a model that appears to describe the information in data adequately. More precisely, we want to know if the incorporation of the effect of productivity and/or geographic dis tance leads to a better fit to data. Model choice is a relative measure: we choose the best-fitting model from those that are available A model may be the best choice but it may still be inadequate by a bsolute standards. The likelihood ratio test used in maxi mum likel ihood inference (see Box 1 5 . 1 ) is a model choice test: it measures relative merits of competing models but revea ls little about their overall adequacy. Two alterna tive model choice approaches used in Bayesian statistics are described briefly. .
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115. 5. MULTILOCUS MULTILOCUS GENOTYPE GENOTYPE METHODS METHODS
Deviance Information Criterion
Spiegel halter et al. (2002) developed Bayesian measures of model complexity (Po, the "effective number of parameters") and fit (5, the posterior mean deviance) and used them to obtain a deviance information criterion that can be used for model comparison. The measure of model complexity is estimated as the difference between the sam ple mean of the simulated values of the deviance (mean deviance) minus an estimate of the deviance using the simulated values of the parameters 6 (deviance of the means): PD
=
0( 6 )
-
0(6).
(B1 )
The function 0(6) is the Bayesian deviance given by
0( 0 )
=
(B2)
- 210gP( )10 ) + 210g f( y ),
where P(Yl6) is the likelihood function and f(y) is a standardizing function of data, y, alone. In the gray seal example presented in the text, we used the null standardization obtained by assuming f(y) is the perfect predictor that gave probability 1 to each observation. The deviance information criterion is defined as the estimate of fit plus twice the effective number of parameters: ole
= 0(6) + 2 PD.
(B3)
More complex models may be preferred if they give a sufficient i mprovement of fit or, equivalently, the preferred model will have the lower value of the DIe. This approach requires running a separate MCMC for each model from which we calculate the quantities required to obtain the model-specific DIe. Reversible Jump Markov Chain Monte Carlo (RJMCMC)
The Bayesian paradigm provides a very natural framework for considering several models simultaneously, assigning probabilities to each model. This involves moving between parameter spaces with different dimensions, as the alternative models may include different num bers of parameters. Green (1 995) extended the basic Metropolis-Hastings a lgorithm to deal with jumps between states of different dimen sions. R)MCMC al lows for the estimation of the joint probability distribution of , Okm} is (M,em), where M = (1 ,2, . . ., K) is a "model i ndicator" and em = {61 , 021 a real stochastic vector whose dimension, km, depends on each model m. We assign a prior probability to each model, commonly assuming that all models are equally likely unless there is prior information that may suggest some models are more likely than others. Priors for the model and their corresponding parameters can be com bined with the likelihood to obtain a full joint posterior d istribution over both the model and the parameter space. R) MCMC al lows us to sample from this joint poster ior distribution, thereby providing estimates of model probabilities within the MCMC simulation itself by simply observing the number of times that the chain visits each distinct model. .
•
.
382 382:
OSCAR OSCAR E. E. GAGGIOTII GAGGIOTTI
Results of Gaggiotti et Results of this this analysis analysis are are presented presented by by Gaggiotti et al. al. (2002), (2002), who who show show that importance of distance and productivity in that the the relative relative importance of distance and productivity in each each of of the the col colonization onization events events appears appears to to have have been been determined determined by by the the location location of of poten potential Copinsay is less tial sources sources around around the the vacant vacant site. site. The The Isle Isle of of Copinsay is more more or or less equidistant from from all all potential sources, reducing reducing the the evidence evidence of of any any associa associaequidistant potential sources, tion tion with with distance distance and and leading leading to to aa better better fit fit for for models models that that include include pro productivity. ductivity. Conversely, Conversely, some some potential potential source source colonies colonies are are much much closer closer than than others others to to Stronsay Stronsay and and the the main main contributors contributors are are the the closest closest leading leading to to aa bet better ter fit fit for for the the models models that that include include distance. distance. For For Calf Calf of of Eday, Eday, results results indicate indicate that population density purpose that both both population density and and distance distance act act concurrently. concurrently. For For the the purpose of 5 .3 shows of illustration, illustration, Fig. Fig. 115.3 shows the the results results for for Stronsay. Stronsay. For sake of Gaggiotti et results for For the the sake of brevity, brevity, Gaggiotti et al. al. (2002) (2002) did did not not present present results for the shows the the assortative assortative mating mating coefficient. coefficient. Figure Figure 15.5 15.5 shows the posterior posterior distribu distribution tion for for each each new new colonization colonization event. event. The The posterior posterior distributions distributions obtained obtained are are fairly fairly similar similar to to the the uniform uniform distribution distribution used used prior prior for for the the assortative assortative mating indicates that contained in mating coefficient. coefficient. This This indicates that information information contained in the the genetic genetic samples question of samples is is not not enough enough to to provide provide clear clear answers answers to to the the question of assortative assortative mating. mating. Note, Note, however, however, that that in in the the case case of of Stronsay, Stronsay, the the posterior posterior distribu distribution 1, suggesting tion of of w is is skewed skewed toward toward w = 1, suggesting that that there there is is aa tendency tendency for for
Stronsay Stronsay
f
I
I
0.0 0.0
0.2 0.2
0.4 0.4
I
E(w) E(w) = = 0.596 0.596 sd = = 0.275 0.275 sd I
I
I
0.6 0.6
0.8 0.8
11.0 .0
Copinsay Copinsay - - - -
E(w) 0.522 E( ==O.522 sd = = 00.286 .286 n--i
0.0 0.0
i
i
0.2 0.2
0.4 0.4
0.6 0.6
i
r-
0.8 0.8
11.0 .0
Calf Calf of of Eday Eday
E(w) E(w) = = 0.596 0.596 sd = 0.287 sd=0.287 I
0.0 0.0
I
0.2 0.2
I
0.4 0.4
I
w
0.6 0.6
I
0.8 0.8
I
11.0 .0
5.5 Posterior Posterior distribution Fig. 115.5 distribution for for the the assortative assortative mating mating coefficient. coefficient. Also Also shown shown are are the the posterior means posterior means of of w w and and the the standard standard deviation. deviation.
383 383
115. 5. MULTILOCUS MULTILOCUS GENOTYPE GENOTYPE METHODS METHODS
individuals individuals from from the the same same source source colony colony to to mate mate more more often often among among them themselves number of selves than than with with individuals individuals from from another another source source colony. colony. A A larger larger number of highly highly polymorphic polymorphic microsatellites microsatellites might might provide provide aa more more definitive definitive answer answer to to this this question. question. A Gaggiotti A more more sophisticated sophisticated approach approach to to the the study study of of colonization colonization ((Gaggiotti et assume aa more general relationship et aI., al., in in press) press) is is to to assume more general relationship for for the the vector vector x x and and the determine the the factors factors that that are are hypothesized hypothesized to to determine the composition composition of of colonizing colonizing groups. groups. A A full full description description of of this this approach approach is is published published elsewhere, elsewhere, but but aa of com brief Box 15.3 as brief description description is is presented presented in in Box as another another possible possible way way of combining different different types types of of data data and and testing testing for for alternative alternative hypothesis. hypothesis. bining
BOX 1 5.3
A More Sophisticated Hierarchical Bayesian Approach
In the second hierarchical Bayesian approach, we no longer focus on the expected contribution of each source population to the founding g roups, E(x;). Instead, we focus directly on the parameters, ax;, of the Dirichlet distribution and assume that they are dis tributed lognormally (Fig. 1 5.6). In other words, the log of the ith element of the vec tor ax = {ax;} has a normal distribution with mean fl.; and variance (12. We further assume that means fl.; values are linear functions of the productivity of each source and the dis tance between the newly founded colony and the source populations, fl.;
= a + b8; + C'Tr; + d8rTT;.
(B1 )
Although we are using productivity and distance i n this example, this approach could be used to study the effect of any other factors, such as inbreeding avoidance, frequency of environmental perturbations, and habitat quality. Additionally, this approach allows us to address possible interactions between different factors. Equation (B1 ) can be used to generate nine alternative models (see Table B1 5.1 ). In order to discriminate among them, we derive probabilities associated with each model using reversible jump MCMC techniques (Green, 1 995; for a simple derivation, see Waagepetersen and Sorensen, 2001 ). TABLE B1 5.1
Nine Alternative Models obtained from Eq. (B1 )
Model
Constant effect Distance effect only Density effect only
fl., /Lj = a fJ.; = c8; fJ.; = C'Tr;
Constant and distance effects
fJ.j = a + b&;
Distance and density effects Constant, distance, and density effects
/L; = a + C'IT;
Constant and density effects
Distance, density, and interaction effects
Full model
/L; = b&; + C'Tr; /L; = a + b&; + C'ITj /Lj = b&; + C'IT; + d8,'ITj fJ.j = a + b8; + C'IT; + d8,'IT;
Dimension of e 6 6 6 7 7 7 8 8 9
384 384
115.5 5.5
OSCAR E. E. GAGGIOTTI GAGGIOTTI OSCAR
POTENTIAL POTENTIAL PROBLEMS PROBLEMS This This review review would would be be incomplete incomplete without without discussing discussing the the problems problems that that may may be Bayesian approaches. approaches. As be found found when when applying applying Bayesian As already already mentioned, mentioned, the the result approaches can result of of Bayesian Bayesian approaches can be be highly highly influenced influenced by by the the prior prior distribu distributions 5 . 1 ) . This potential drawback tions used used in in their their formulation formulation (see (see Box Box 115.1). This potential drawback has has elicited statisticians that elicited harsh harsh criticism criticism from from statisticians that use use classical classical statistical statistical inference inference or maximum likelihood numerous tools tools can or maximum likelihood methods. methods. However, However, numerous can be be used used to to investigate the potential prior distributions. Such knowledge investigate the potential biasing biasing effect effect of of prior distributions. Such knowledge allows researcher to bias. More allows the the researcher to change change priors priors so so as as to to eliminate eliminate the the bias. More import importantly, use of priors minimizes antly, the the use of aa modeling modeling approach approach for for the the formulation formulation of of priors minimizes least, it equalizes it the selection Ccr,l', at the subjectivity subjectivity involved involved in in their their selection at the the very very least, it equalizes it with that involved the likelihood with that involved in in formulation formulation of of the likelihood function. function. Indeed, Indeed, researchers researchers commonly commonly make make subjective subjective judgements judgements about about the the parameters parameters that that should should be be included included in in the the likelihood likelihood function. function. Moreover, Moreover, the the use use of of approaches approaches such DIC and such as as DIC and R]MCMC RJMCMC provides provides measures measures of of fit fit for for the the alternative alternative mod models considered els considered by by the the different different prior prior distributions. distributions. The The choice choice of of the the prior prior can can therefore therefore be be based based on on an an objective objective measure, measure, namely namely the the fit fit of of the the model model to to genetic data. data. genetic The The fact fact that that all all these these strategies strategies are are available available invalidates invalidates the the criticism criticism con concerning cerning the the subjectivity subjectivity of of Bayesian Bayesian approaches, approaches, but but it it is is necessary necessary to to acknow acknowledge the less than ledge the fact fact that that their their implementation implementation is is less than straightforward. straightforward. These These complications complications may may limit limit the the use use of of Bayesian Bayesian methods methods to to scientists scientists with with sub substantial stantial training training in in statistics. statistics. The The development development of of sophisticated sophisticated computer computer soft software functions implementing ware that that includes includes easy-to-use easy-to-use functions implementing diagnostic diagnostic tests tests to to detect biases should make Bayesian detect potential potential biases should make Bayesian methods methods more more accessible accessible to to aa wider range of wider range of users. users. Another Another problem problem associated associated with with Bayesian Bayesian methods methods is is that that they they require require sub substantial stantial computing computing power. power. This This may may not not be be the the case case when when they they are are applied applied to to answer population answer simple simple questions, questions, but but the the type type of of problems problems that that arise arise in in meta metapopulation
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O~xs)
Source Allele Frequencies
/ /
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Geographic data, ~i
Mixture Genotypes
Fig. 115.6 5.6 Schematic Schematicdescription description of the Bayesian Bayesian approach approach 2. The genetic data data is combined focusing directly on the parameters parameters of the with demographic and genotypic information by focusing Dirichlet distribution used used as as the prior of the vector of genetic mixture coefficients.
15. 15.
MULTILOCUS GENO-FYPE GENOTYPE METHODS METHODS MULTILOCUS
385 3 85
biology do do lead lead to to complex complex statistical statistical models models and and their their analysis analysis requires requires subsub biology stantial computing computing time. time. Luckily, Luckily, the the power power of of desktop desktop workstations workstations increases increases stantial substantially every every year year and and will will soon soon be be enough enough to to allow allow the the study study of of complex complex substantially problems in in metapopulation metapopulation biology. biology. problems
1 5.6 15.6
CONCLUDING REMARKS REMARKS CONCLUDING Recent developments in in the the field field of of statistical statistical genetics genetics have paved the the way way Recent developments have paved for the development development of of new new statistical statistical approaches approaches for studying the the fundamenfundamen for the for studying tal processes processes that that characterize characterize metapopulations, metapopulations, in in particular particular those those that that tal involve dispersal dispersal of of individuals. individuals. The The possibility possibility of of combining combining genetic genetic data data with with involve other sources sources of of information information in in aa single single statistical statistical framework is particularly particularly other framework is promising in in the the context context of of metapopulation metapopulation studies, studies, as as population population turnover turnover promising and dispersal dispersal tend tend to to decrease decrease the the amount amount of of information information contained contained in in genetic genetic and data. This due to to the predominantly negative negative effect that these these processes processes have have data. This is is due the predominantly effect that on effective metapopulation metapopulation size size and the degree degree of of differentiation differentiation among among on the the effective and the local populations populations ((Chapter Chapter 7). local 7). The potential for the the development development of new methods methods of inference depends depends on the The potential for of new of inference on the particular process process that that we we envision envision to to study. In the the case of colonization particular study. In case of colonization processes, processes, the extension of the the GSI method described in this this chapter chapter should prove very very useuse the extension of GSI method described in should prove ful. The The challenge here lies in the the building up of the databases databases needed needed for ful. challenge here lies in building up of the for its its appli application. Nevertheless, Nevertheless, as the example example of of the the gray seal illustrates, demographic data cation. as the gray seal illustrates, demographic data could be be simple simple estimates of abundance. abundance. Additionally, molecular ecologists rou could estimates of Additionally, molecular ecologists routinely collect distance information. information. Application Application of of the the GSI GSI method method to to tinely collect geographic geographic distance study study the the effect effect of of other other factors factors (e.g., (e.g., kin kin competition competition and and habitat habitat quality) quality) that that can can also also influence influence the the movement movement of of individuals individuals away away from from their their patch patch of of origin origin may may prove prove more more difficult. difficult. Still, Still, as as Chapter Chapter 1133 highlights, highlights, there there is is aa need need for for multi multifactorial dispersal; the discussed earlier earlier is factorial studies studies of of dispersal; the method method discussed is likely likely to to be be very very useful analysis of collected in useful for for the the analysis of data data collected in such such studies. studies. Studying population context context is complicated than Studying migration migration in in aa meta metapopulation is more more complicated than studying colonization processes. available maximum studying colonization processes. There There are are available maximum likelihood likelihood methods methods based based on on the the coalescent coalescent that that can can be be used used to to estimate estimate the the effective effective number of 1 ) . However, number of migrants migrants (Beerli (Beerli and and Felsenstein, Felsenstein, 200 2001). However, this this is is aa param parameter eter that that measures measures the the long-term long-term effect effect of of migration migration under under equilibrium equilibrium condi conditions tions and, and, therefore, therefore, is is of of little little relevance relevance in in aa metapopulation metapopulation context context where where nonequilibrium nonequilibrium dynamics dynamics are are pervasive. pervasive. The The Bayesian Bayesian method method described described by by Wilson because it Wilson and and Rannala Rannala (2003) (2003) is is more more appropriate appropriate because it does does not not assume assume equilibrium equilibrium conditions. conditions. One One of of the the limitations limitations of of this this method method is is that that accurate accurate estimates estimates of of migration migration rates rates are are only only possible possible when when levels levels of of genetic genetic differen differentiation tiation among among local local populations populations are are large. large. This This is is rarely rarely the the case case in in metapopu metapopulations lations with with high high turnover turnover rates, rates, but but it it may may be be possible possible to to extend extend the the method method in in order order to to incorporate incorporate different different sources sources of of information information much much in in the the same same way way as as the the GSI GSI method method has has been been modified modified for for the the same same purpose. purpose. A A related related problem problem that that could could be be addressed addressed using using some some of of the the methods methods dis discussed cussed earlier earlier is is the the study study of of the the effect effect of of different different factors factors such such as as geographic geographic distance, distance, environmental environmental factors, factors, and and cultural cultural affinities affinities on on the the degree degree of of genetic genetic differentiation differentiation between between pairs pairs of of local local populations. populations. The The test test generally generally used used for for these these purposes purposes is is the the partial partial Mantel Mantel test test (Smouse (Smouse et et aI., al., 1986), 1986), the the generality generality
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of which has been recently questioned (Raufaste and Rousset, 2001). When factor is included in the analysis, the Monte Carlo randomiza randomizamore than one factor tion tion procedure procedure used used to to test test for for the the significance significance of of the the correlation correlation is is inadequate. inadequate. The The use use of of hierarchical hierarchical Bayesian Bayesian approaches approaches that that relate relate the the prior prior distribution distribution FsT, or a related parameter, with the different factors would provide a for FST, proper way of testing for their effects. Another Another related related problem problem is is the the detection detection of of sex-biased sex-biased dispersal. dispersal. Goudet Goudet et et al. al. (2002) described described methods methods based based on on biparental biparental inherited inherited genetic genetic markers markers and summary statistics and concluded that they have a limited power, being Methods that that able to detect biased dispersal only when the bias is extreme. Methods make full use of the information contained in genetic data may prove more powerful. Additionally, it may be possible to devise statistical tests to identify powerful. the factors or attributes (e.g., size, colour, social status) status) that may influence the probability of dispersal. In summary, summary, the the development development of of statistical statistical methods methods that that make make full full use use of of all all In available available data data is is an an area area that that will will expand expand in in the the coming coming years years and, and, as as already already shown by some of the existing studies, will be of great help in the study study of shown metapopulation meta population processes.
AL AND AND 16o ECOLOGIC ECOLOGICAL EVOLUTIONA RY EVOLUTIONARY CONSEQUENCES OF SOURCE-SINK PO PULATION POPUtATION DYNAMICS DYNAMICS Tadeusz Tadeusz J. J. Kawecki Kawecki
116.1 6. 1
INTRODUCTION INTRODUCTION The change of population population density at a given area area reflects the balance among local births, local deaths, immigration, and emigration. A local population population may thus remain stable even though though births do not not equal deaths, the difference being compensated compensated by net emigration to, or immigration from, neighboring populations. Trivial as this statement populations. statement may be, its ecological and evolutionary consequences consequences became appreciated appreciated only relatively recently. Although Although several earlier papers considered consequences of differences between emigration and con975; Keddy, 11982; 982; Holt, 11983, 983, 11985), 985), the con immigration (e.g., Lidicker, 11975; cept of source-sink population population structure was brought brought to the general attention attention of ecologists by Pulliam ((1988). 1 988). He saw it as a consequence of differences in
Ecology, Genetics, Genetics, and Evolution of Metapopulations Metapopulations
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Copyright 2004, Elsevier, Inc. 0-12-323448-4 0-12- 323448 -4
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habitat usually high habitat quality. quality. Local Local births births on on average average exceed exceed deaths deaths in in some some ((usually highquality) quality) habitats, habitats, with with surplus surplus individuals individuals dispersing dispersing to to other other (usually (usually low lowquality) quality) habitats; habitats; the the latter latter become become net net importers importers of of individuals. individuals. Hence Hence the the definition definition of of source source and and sink sink habitats habitats based based on on the the difference difference between between emi emigration and and immigration: immigration: in in source source habitats, habitats, emigration emigration exceeds exceeds immigration; immigration; gration the reverse reverse holds holds in in sink sink habitats habitats (Pulliam, (Pulliam, 1988). This This is is the the definition definition used used the in this chapter. However, However, in in the the same same paper paper Pulliam Pulliam implied implied that that aa sink sink habitat habitat cannot cannot sus sustain aa population population in in the the absence absence of of dispersal. dispersal. This This will will generally generally not not be be the the case case tain except in in models models with with no no population population regulation. regulation. A A given given habitat habitat may may well well be be except able able to to sustain sustain aa population population of of aa certain certain density, density, but but immigration immigration from from aa nearby habitat can nearby higher higher quality quality habitat can lead lead to to aa state state of of permanent permanent overcrowding overcrowding (Holt, (Holt, 1985), in in which which births births will will not not compensate compensate for for deaths. deaths. When When the the dis dishabitinction is necessary, I refer to such a habitat as a relative sink, and to a habi tat tat unable unable to to sustain sustain aa population population as as an an absolute absolute sink sink [Watkinson [Watkinson and and 1 995) refer Sutherland Sutherland ((1995) refer to to them them as as "pseudosinks" "pseudosinks" and and "true "true sinks," sinks," respect respectively]. ively]. Differentiating Differentiating between between relative relative and and absolute absolute sinks sinks in in natural natural hetero heterogeneous environments environments will geneous will often often be be impossible impossible without without actually actually preventing preventing immigration specified otherwise, immigration and and emigration. emigration. Except Except where where specified otherwise, the the results results dis discussed later are valid for both relative and absolute absolute sinks. This consequences of This chapter chapter reviews reviews the the consequences of imbalance imbalance between between immigration immigration and emigration for population population dynamics and distribution and for adaptive dynamics. First evolution. Section evolution. Section 16.2 reviews reviews models models of of source-sink source-sink dynamics. First aa sim simple patch patch model model is is introduced introduced and and used used to to discuss discuss the the concept concept and and meaning meaning of of ple habitat-specific value; then extensions of habitat-specific reproductive reproductive value; then extensions of the the basic basic model model are are dis discontains aa discussion discussion of of main main theoretical theoretical predictions predictions con cussed. Section Section 16.3 contains cussed. concerning the effect effect of of source-sink on population cerning the source-sink population population structure structure on population dynamics, size, size, distribution, distribution, persistence, and stability, by aa review review of dynamics, persistence, and stability, followed followed by of 16.5 focuses focuses on the reasons reasons for for which indi relevant empirical data. relevant empirical data. Section Section 16.5 on the which individuals sink habitats. of source-sink source-sink structure viduals may may disperse disperse into into sink habitats. Consequences Consequences of structure for evolution are discussed in Section 16.6. Integration Integration of of the the concept concept for adaptive adaptive evolution are discussed in Section of source-sink population structure structure with extinction-colonization dynamics of source-sink population with extinction-colonization dynamics and metapopulation concept concept is the focus focus of of Section Section 16.7. 16.7. The section and the the metapopulation is the The final final section includes some about future research directions and neglected neglected applied includes some thoughts thoughts about future research directions and applied source-sink population structure. aspects of source-sink population structure.
16.2 1 6.2
MODELS OF OF SOURCE-SINK SOURCE-SINK POPULATION POPULATION DYNAMICS DYNAMICS MODELS Throughout this chapter the word "habitat" is used used to to describe describe aa certain certain set set Throughout this chapter the word "habitat" is of of environmental environmental conditions conditions (including (including abiotic abiotic conditions, conditions, available available resources, resources, and and predator predator pressure), pressure), whereas whereas "patch" "patch" refers refers to to an an actual actual physical physical space. space. Thus, numerous fine-grained numerous patches patches of of two two habitats habitats may may form form aa more more or or less less fine-grained Thus, mosaic in in the the physical physical landscape. landscape. If If variation variation in in the the environmental environmental conditions conditions mosaic is is continuous continuous (e.g., ( e.g., along along aa gradient), gradient), there there would would be be no no discrete discrete patches, patches, but but still still each each point point in in the the landscape landscape can can be be defined defined as as aa certain certain habitat, habitat, charactercharacter ized ized by by given given environmental environmental parameters. parameters. A A spatially spatially explicit explicit approach approach (con(con sidered sidered in in one one of of the the following following subsections) subsections) would would be be more more appropriate appropriate than than aa patch patch model model in in such such aa case. case. This This section section first first uses uses aa simple simple patch patch model model to to
SOURCE-SINKPOPULATION POPULATION DYNAMICS 116. 6. SOURCE-SINK
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define the concept concept of source-sink population dynamics; dynamics; some of introduce and define assumptions are relaxed in the subsequent subsections. its assumptions
A A Patch Patch Model M o d e l of o f Source-Sink S o u r c e - S i n k Population P o p u l a t i o n Structure Structure Consider Consider aa species species with with discrete discrete generations, generations, inhabiting inhabiting an an environment environment composed composed of of patches patches of of two two types types of of habitats. habitats. Assume Assume that that habitat habitat 11 is is of of bet betpopulation density the net ter quality than habitat 2, i.e., at any particular population reproductive reproductive rate rate (the (the expected expected lifetime lifetime reproductive reproductive success) success) is is greater greater in in habitat 11 (Fig. 116.1). habitat 6. 1 ). population dynam dynamIf the local populations are isolated from each other, the population in each each habitat habitat is is fully fully determined determined by by the the respective respective density-dependent density-dependent net net ics in reproductive reproductive rate. rate. Assuming Assuming that that aa stable stable equilibrium equilibrium exists, exists, each each population population is is expected expected to to equilibrate equilibrate at at the the local local carrying carrying capacity, capacity, i.e., i.e., the the density density at at which which births balance deaths, and thus the net reproductive rate equals 11 (Fig. 16.1a). not any more be the case if local populations populations in the two two habitat habitat This will not types types are are connected connected by by dispersal. dispersal. Because Because dispersal dispersal usually usually reduces reduces variation variation better habitats habitats below the local carrying in density, it tends to keep density in better capacity, capacity, whereas whereas poor poor habitats habitats tend tend to to be be overcrowded overcrowded relative relative to to the the density density they absence of 6. 1 b illustrates they would would support support in in the the absence of dispersal. dispersal. Figure Figure 116.1b illustrates the the extreme case of complete mixing, mixing, whereby dispersing individuals individuals (propagules) both habitats habitats form a common pool, which then becomes distributed distributed from both pool, which between between the the two two habitats habitats in in proportion proportion to to their their relative relative area. area. In In this this case case the the population just after population density, density, censused censused just after dispersal, dispersal, will will be be the the same same in in both both habi habipopulation in the better habitat habitat will be tats. Consequently, at equilibrium, the population below local carrying and its will be below the the local carrying capacity capacity and its reproductive reproductive rate rate will be greater greater than than with the excess of births births over deaths deaths compensated compensated by an excess of of emigra emigra11,, with tion over over immigration. The The reverse reverse will be the for the poor The the case for poor habitat. habitat. The tion equilibrium will thus thus have with aa net equilibrium population population will have aa source-sink source-sink structure, structure, with net flow flow of dispersers from from habitat to habitat habitat 2 ((sink). sink). of habitat 1 (source) to An that of limited dispersal, dispersal, where where aa certain fraction of of An intermediate intermediate case case is is that of limited certain fraction than 50%) 50%) exchange exchange their generation. In this their habitats habitats each generation. this individuals (smaller (smaller than each other other and case, population population densities will, in general, general, be be different different from from each and from the carrying capacities. propensity to disperse is habitat inde indefrom the local local carrying capacities. If If the the propensity to disperse is habitat pendent, the the equilibrium density density will be greater pendent, greater in habitat habitat 1 than than in habitat habitat 2, (a)
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Fig. 16.1 1 6.1 A A simple simple graphical graphical model model of of the the source-sink source-sink population population structure; structure; for for explanaexplana Fig. tions, tions, see the the text. text.
TADEUSZ I.j. KAWECKI
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but it it still still will will be be below below the the local local carrying carrying capacity capacity in in habitat habitat 11 and and above above the the but carrying capacity capacity in in habitat habitat 22 (Fig. (Fig. 16.1c). 1 6. 1 c ) . Thus, Thus, the the population population as as aa whole whole carrying will still still have have aa source-sink source-sink structure, structure, although although the the net net flow flow of of dispersers dispersers from from will the source source to to the the sink sink habitat habitat will will be be smaller smaller than than under under complete complete mixing. mixing. the This graphical graphical model model can can be be formalized formalized and and extended extended to to an an arbitrary arbitrary numnum This ber of of habitat habitat patches by application of the the general general matrix approach to to popupopu ber patches by application of matrix approach lation dynamics ( Caswell, 1989): 1 98 9 ) : lation dynamics (Caswell, A[n(t)]n(t), n(t ++ 1) 1 ) == A[n(t)]n(t), n(t
(16.1) (16.1)
where n(t) n(t) iiss the the (column) (column) vector vector o population sizes sizes nnii at at time time (generation) (generation ) tt where off population in the the respective respective habitats, habitats, and and A[n(t)] is is aa density-dependent density-dependent transition transition matrix. matrix. in The element element in in the the ith row and and jth jth column column of of A[n(t)] A[n( t)] is is given given by by The ith row ( 1 6.2) (16.2)
[aii( ni) ] = [l~(ni)mii],
where �(nj) is is the the net reproductive rate rate (expected (expected lifetime lifetime reproductive reproductive success) success) where/~(nj) net reproductive of individual living habitat j, j, and mji is is the the dispersal from habitat habitat jj of an an individual living in in habitat and mji dispersal rate rate from This model thus assumes that population population regulation takes place place to to habitat habitat i. This model thus assumes that regulation takes within followed by by dispersal; census takes place within each each habitat habitat independently, independently, followed dispersal; census takes place after probability that individ after dispersal. dispersal. The The dispersal dispersal rate rate is is defined defined as as the the probability that an an individual present in habitat the dispersal phase will will end up in in habitat habitat ii after ual present in habitat jj before before the dispersal phase end up after dispersal. assume here here that that this probability is is the the same same for for all all patches patches of of aa dispersal. II assume this probability given in patch patch connectivity connectivity is is negligible. negligible. This This model model given habitat habitat type, type, i.e., i.e., variation variation in also assumes that parameters are are constant, discrete, and and each each also assumes that parameters constant, generations generations are are discrete, individual individual spends spends most most of of its its life life in in aa single single habitat habitat and and only only this this habitat habitat affects affects its sense of its survival survival and and reproduction reproduction (coarse-grained (coarse-grained environment environment in in the the sense of Levins, 968a). Note that this this last last assumption would be Levins, 11968a). Note that assumption would be violated violated if if there there were were nonnegligible nonnegligible habitat-related habitat-related maternal maternal effects effects on on fitness, fitness, e.g., e.g., if if the the viability viability of of seedlings seedlings was was affected affected by by the the habitat habitat from from which which the the seeds seeds originated. originated. Assume Assume that that the the system system has has aa stable stable nonzero nonzero equilibrium equilibrium n. The The dominant dominant eigenvalue eigenvalue Ak of of matrix matrix A(n) A(fi) equals equals 11,, and and the the equilibrium equilibrium population population sizes sizes are are aa corresponding corresponding right right eigenvector. eigenvector. The The normalized normalized eigenvector eigenvector u u = i1!(lni) fi/(2~ni) describes describes the the distribution distribution of of individuals individuals among among habitats. habitats. The The corresponding corresponding 1, consists left left eigenvector eigenvector v, v, normalized normalized so so that that the the scalar scalar product product < u . v > - 1, consists of of the the reproductive reproductive values values of of individuals individuals in in the the respective respective habitats habitats (Caswell, (Caswell, 11989; 989; Rousset, 999a). The Rousset, 11999a). The importance importance of of habitat-specific habitat-specific reproductive reproductive values values is is discussed discussed in in the the following following section. section. The The number number of of individuals individuals that that disperse disperse from from habitat habitat h to to other other habitats habitats at at equilibrium equilibrium is is =
=
hfh( fzh) L Ehh = nnhfh(nh) E ~ mhi mhi d i :/:hh =
((16.3) 16.3)
while while the the number number of of immigrants immigrants to to habitat habitat h from from other other habitats habitats is is
Ih - ~
nifi(ni)mih.
((16.4) 1 6.4)
i :/:h
According 1 9 8 8 ) definition, According to to Pulliam's Pulliam's ((1988) definition, habitat habitat h is is aa sink sink if if E Ehh < < h. Ih. Noting 1 6.2) {;(ni)mi Noting that, that, from from Eq. Eq. ((16.2) fi(~i)mihh = aahi(ni), and that, that, from from the the definition definition hi(ni), and
SOURCE-SINKPOPULATION POPULATION DYNAMICS DYNAMICS 116. 6. SOURCE-SINK
= h (fh( fzh) � Eh Eh--- Ih Ih == n;~b(fb(~tb) ~ mmhi b i - --1 ) )
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of of right right eigenvector, eigenvector, lia ] ~ i a hh ii(ni)ni ( ~ i ) ~ i = nh, one one can can show show that that the the difference difference between between the numbers numbers of of emigrants emigrants and and immigrant immigrant at at equilibrium equilibrium equals equals the 1
((16.5) 1 6.5)
l
Thus, Thus, habitat habitat hh is is aa sink sink (following (following Pulliam's Pulliam's definition) definition) if if ffh(~h)Eimhi < 1. 1. h(nh)limhi < (;(ni), lim If If mortality mortality during during dispersal dispersal is is negligible negligible or or is is absorbed absorbed into into/~(ni), ~,imhi hi == 1. In In this this case, case, Pulliam's Pulliam's definition definition of of aa sink sink implies implies that that ffh(~/#) < 11,, i.e., i.e., that that the the h (nh) < local local density density in in aa sink sink habitat habitat is is above above the the local local carrying carrying capacity capacity [defined [defined as as the density density at at which which fft,(nl,) the ]. h (nh ) = 11].
=
Reproductive Value Value and and the the Definition Definition of of Sources Sources and and Sinks Sinks Reproductive Equation 1 6.5) formalizes Equation ((16.5) formalizes the the definition definition of of source source and and sink sink habitats habitats as as net net exporters exporters and and importers importers of of dispersing dispersing individuals. individuals. However, However, as as noted noted by by Kawecki 1 993) and Rousset ((1999a), 1 999a), habitat-specific Kawecki and and Stearns Stearns ((1993) and Rousset habitat-specific reproductive reproductive values values may may be be more more closely closely related related to to the the ecological ecological and and evolutionary evolutionary conse consequences quences of of environmental environmental heterogeneity heterogeneity than than the the difference difference between between emigra emigration and immigration. First, First, the the reproductive reproductive value value measures measures the the expected expected long-term long-term contribution contribution of Caswell, of an an individual individual to to population population growth growth and and the the future future gene gene pool pool ((Caswell, 11989). 9 8 9 ) . The The asymptotic asymptotic contribution contribution of of the the local local population population in in habitat habitat h h to to aa future gene pool is is U u~vt, (note the the analogy analogy between this quantity and the the patch patch future gene pool between this quantity and hVh (note value value as as defined defined in in Chapter Chapter 4; 4; the the two two are, are, however, however, not not identical identical as as the the latter latter focuses focuses on on colonization colonization of of empty empty patches) patches).. Local Local populations populations in in habitats habitats with with > I1 contribute contribute more expected based Vv~h > more to to future future generations generations than than would would be be expected based on on their share habitats with with v~ Vh << 11 contribute less. Similarly, Similarly, the the their share of of individuals; individuals; habitats contribute less. reproductive value the notion notion of of "phylogenetic "phylogenetic envelope" envelope" intro reproductive value quantifies quantifies the introduced Holt and and Gaines Gaines (1992) ( 1 992) to tracing the the ancestry of individ duced by by Holt to describe describe tracing ancestry of individuals alive by looking looking ~lt !1t generations back. As As At !::. t increases increases (the ancestry uals alive at at time time t by generations back. (the ancestry h to to the the is traced traced into into increasingly increasingly distant distant past), past), the the contribution contribution of of habitat habitat h is UhVh (assuming ( assuming constant constant conditions). conditions) . In In ancestry of of an an individual individual converges converges to to u~vh ancestry sexual population, population, the the contribution contribution of of habitat habitat hh to to an an individual's individual's ancestry ancestry aa sexual is interpreted interpreted as as the the proportion proportion of of genes genes descended descended from from genes genes present present in in habihabi is tat h h at at time time tt - At; !1t; in in aa clonal clonal lineage as the the probability probability that that the the ancestor was tat lineage as ancestor was !1t in in habitat habitat h. h. The The effective effective population population size size of of aa subsub present at at time time tt - At present divided population is is also also aa function function of of the the reproductive reproductive values values of of the the local local divided population populations (Whitlock and populations (Whitlock and Barton, Barton, 1997). 1 997). Second, the the definition definition based based on on the the difference difference between between emigration emigration and and Second, immigration is is difficult difficult to to apply apply in in aa nonequilibrium nonequilibrium situation, situation, when when the the popupopu immigration lation sizes sizes and and the the numbers numbers of of migrants migrants fluctuate, fluctuate, whether whether due due to to inherent inherent lation instability of of population population dynamics dynamics or or due due to to fluctuations fluctuations in in the the environment. environment. instability If emigration emigration from from aa given given habitat habitat exceeds exceeds immigration immigration into into that that habitat habitat in in If some generations, generations, but but the the reverse reverse is is true true in in others, others, how how should should they they be be averaver some aged over over generations, generations, especially especially that that the the population population sizes sizes fluctuate fluctuate as as well? well? In In aged contrast, the the concept concept of of reproductive reproductive value value can can be be generalized generalized to to transition transition contrast, matrices that that vary vary in in time time (Kim, (Kim, 1987; 1 9 8 7; Tuljapurkar, Tuljapurkar, 1990). 1 99 0 ) . matrices A logical logical consequence consequence of of the the aforementioned aforementioned arguments arguments would would be b e to t o base base A the definition definition of of source source and and sink sink habitats habitats on on the the reproductive reproductive value value (Rousset, ( Rousset, the
--
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11999a); 999a); habitats habitats with would be those with with Vh vh > > 11 would be sources, sources, whereas whereas those with Vh vh < < 11 would be avoid contributing would be sinks. sinks. To To avoid contributing to to terminological terminological confusion, confusion, this this redef redefinition is not advocated here. here. However, However, it it is is useful useful to to see see when the definition definition inition is not advocated when the based would classify based on on the the reproductive reproductive value value would classify habitats habitats differently differently than than the the one based on First, note one based on net net immigration. immigration. First, note that that from from definition definition of of the the left left eigen eigenh, then vector Hence, if vector Vh vh = kiviaih. ~,iviaih. Hence, if Vh vh = = 11 for for all all h, then kiaih Eiaih = = fhkimhi fhEimhi = = 11 for for all all h. In words, if emigration in habitats ((i.e., i.e., the h. In words, if immigration immigration balances balances emigration in all all habitats the system system does not reproductive values values in does not have have aa source-sink source-sink structure), structure), then then the the reproductive in all all habitats immigration balances habitats are are 11.. However, However, if if immigration balances emigration emigration in in some some habitats habitats but others, the but not not in in others, the reproductive reproductive values values in in those those habitats habitats will will generally generally be Second, if be different different from from 11.. Second, if there there are are only only two two habitats, habitats, then then ((11 and and 2) 2) ff1(m11 (m + mu) > 1 > h(m2I + m ) implies VI > 1 > V , i.e., the reproductive 22 implies Vl > 1 > v2, 2 i.e., the reproductive I ll + m12) > 1 > f2(m21 + m22) value habitat classified source according according to value is is greater greater in in the the habitat classified as as source to Pulliam's Pulliam's defi definition 999a). This nition (Rousset, (Rousset, 11999a). This is is not not any any more more the the case case when when there there are are more more than than two 1 999a) illustrated two habitats. habitats. Rousset Rousset ((1999a) illustrated this this with with an an example example with with one-way one-way dispersal. However, dispersal. However, aa discrepancy discrepancy between between Pulliam's Pulliam's source-sink source-sink definition definition and values may and the the pattern pattern of of habitat-specific habitat-specific reproductive reproductive values may also also occur occur when when i). Such a case is illustrated in dispersal dispersal rates rates are are symmetric symmetric (i.e., (i.e., mi mii; = = m mji). Such a case is illustrated in ; Fig. 6 .2. In example, habitat habitat 11 and Fig. 116.2. In that that example, and habitat habitat 2 2 are are both both sources sources according according to excess of emigration over to Pulliam's Pulliam's definition, definition, with with the the excess of emigration over immigration immigration being being greater habitat 2 (in absolute greater for for habitat 2 than than for for habitat habitat 11 (in absolute terms terms and and relative relative to to equilibrium equilibrium population population sizes) sizes).. Consistent Consistent with with this, this, the the net net reproductive reproductive rate rate habitat 2. 2. Yet, Vb reflecting at at equilibrium, equilibrium, fh(nh), fh(t/h), is is largest largest in in habitat Yet, Vv22 < < 11 < < vl, reflecting the the fact which is fact that that most most emigrants emigrants from from habitat habitat 2 2 end end up up in in habitat habitat 33,, which is aa strong strong sink. sink. To summarize, summarize, the the application application of of the the concept concept of of source-sink source-sink population population To structure habitat types. structure is is most most straightforward straightforward when when there there are are only only two two habitat types. When When there habitats, the habitat emigration there are are more more than than two two habitats, the fact fact that that for for aa given given habitat emigration exceeds exceeds immigration immigration does does not not necessarily necessarily imply imply that that habitat habitat contributes contributes more more to pool than based on to aa future future gene gene pool than would would be be expected expected based on its its share share Uh uh of of the the total total population. reason for population. The The reason for this this discrepancy discrepancy is is that that emigration emigration and and immigra immigration movements of tion refer refer to to the the movements of individuals individuals within within aa single single generation, generation, whereas whereas reproductive values take into account the consequences of chains of migration reproductive values take into account the consequences of chains of migration events generations. The events among among habitat habitat types types happening happening over over many many generations. The following following subsections model described 16.1 ) subsections summarize summarize some some special special cases cases of of the the model described by by Eq. Eq. ((16.1) and ((16.2), as well well as as its its extensions extensions to to include age structure structure and and explicit spatial and 1 6.2), as include age explicit spatial dimensions. dimensions.
Habitat H a b i t a t Area Area versus versus Habitat H a b i t a t Quality Quality At can be At the the first first approximation, approximation, aa spatially spatially heterogeneous heterogeneous environment environment can be characterized terms of of the area and of the characterized in in terms the area and quality quality of the habitats habitats it it consist consist of. of. A A rea reasonable way sonable way to to describe describe the the quality quality of of different different habitats habitats would would be be to to compare compare the the reproductive reproductive success success that that is is expected expected in in each each of of them them at at the the same same popula population tion density. density. Because Because the the aforementioned aforementioned model model is is formulated formulated in in terms terms of of local local population population sizes sizes rather rather than than densities, densities, fI(n) fl(n) > > h(n) f2(n) does does not not imply imply that that habitat habitat 11 is areas. If is of of higher higher quality quality if if the the habitats habitats cover cover different different areas. If spatial spatial variation variation in in density within aa habitat density within habitat is is negligible, negligible, it it is is straightforward straightforward to to reformulate reformulate the the model by where bh is area of model by setting setting fh(nh) ft,(nh)== Fh(nh1bh), Fh(nffbh), where is the the area of habitat habitat h h and and
116. 6. SOURCE-SINK SOURCE-SINK POPULATION POPULATION DYNAMICS DYNAMICS
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(a) (a)
Hab. 1 f1 (n1 ) = 1 0 1 + cn1
--
Hab. 2
Hab. 3
m23 == m32 _..,m23 m32 = = 0.3 0.3 If
•
2 t3 (n3) - -1 + cn3
(b) (b) n1 = 835 f1 (n1) = 1 .07 V1 = 1 .39
~,
9
n2 = 401 f2(n2) = 1 .20 v2 = 0.75
_.. ..
44.4 44 4
144.1 "~
1 44. 1
t73 = 286 f3(n3) = 0.52 v3 = 0.23
Fig. 6.2 A three-patch habi Fig. 116.2 three-patch model model illustrating illustrating the the discrepancy discrepancy between between the the classification classification of of habitats sink and and the value. (a) tats as as source source or or sink the habitat-specific habitat-specific reproductive reproductive value. (a) Parameters Parameters of of the the model; model; cc = = 0.01 0.01.. (b) (b) Properties Properties of of the the equilibrium; equilibrium; numbers numbers next next to to arrows arrows indicate indicate the the number number of of indi individuals viduals dispersing dispersing from from one one habitat habitat to to the the other other each each generation. generation.
h(nh1bh) F ~ ( n f f b h ) is is the the net net reproductive reproductive rate rate in in habitat habitat h h as as aa function function of of the the local local pop population ulation density. density. Other Other things things being being equal, equal, habitats habitats covering covering aa larger larger area area are are likely likely to to receive receive more more immigrants, immigrants, especially especially with with passive passive dispersal. dispersal. One One way way to to implement implement such such aa relationship relationship is is to to assume assume that that aa fraction fraction 11 - JL I~ of of potential potential dispersers habi dispersers remain remain in in the the habitat habitat of of origin origin while while the the rest rest end end up up in in various various habitats tats (including (including the the habitat habitat of of origin) origin) in in proportion proportion to to their their area, area, i.e., i.e., -
m miiii = 11 -- JL i~ + + ~bi/E~bh
= JLb/Ihbh mi m i jj = = JLb;lIhbh I~bi/Ehbh
((16.6a) 1 6.6a) ((16.6b) 1 6.6b)
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This This model model of of dispersal dispersal was was implemented implemented to to study study the the effects effects of of habitat habitat qual quality versus versus habitat habitat area area on on adaptive adaptive evolution evolution (Kawecki (Kawecki and and Stearns, Stearns, 11993; ity 993; Kawecki, 995). More Kawecki, 11995). More realistically, realistically, the the dispersal dispersal rates rates will will be be also also affected affected by by the size and and arrangement the size arrangement of of habitat habitat patches patches and and their their connectivity connectivity (see (see later). later). For For the the sake sake of of the the argument, argument, unless unless specified specified otherwise, otherwise, most most of of the the chapter chapter assumes area. assumes that that all all habitats habitats have have the the same same area.
Asymmetric Asymmetric Dispersal Dispersal Rates, Rates, "Reverse" "Reverse" Source-Sink Source-Sink Structure, Structure, and Black Hole Sinks and Black Hole Sinks In In the the model model described described in in the the preceding preceding paragraph, paragraph, individuals individuals in in all all habi habitats tats show show the the same same propensity propensity to to disperse, disperse, and and the the source-sink source-sink structure structure results from habitat quality. individuals may results from differences differences in in habitat quality. However, However, individuals may change change their response to their propensity propensity to to disperse disperse in in response to their their habitat. habitat. Environmental Environmental factors factors such ocean current, current, wind, wind, or also lead lead to such as as river river or or ocean or gravity gravity may may also to an an asymmetry asymmetry of dispersal dispersal rates, increasing the the probability probability of of dispersing dispersing from from an an "upstream" "upstream" of rates, increasing habitat habitat to to aa "downstream" "downstream" habitat habitat and and reducing reducing the the probability probability of of dispersing dispersing in equilibrium properties in the the opposite opposite direction. direction. The The equilibrium properties of of aa set set of of populations populations connected connected by by dispersal dispersal depend depend on on both both habitat-specific habitat-specific net net reproductive reproductive rates rates {;(ni) and ~(ni) and dispersal dispersal rates rates mij' mij. Asymmetries Asymmetries of of dispersal dispersal rates rates will will thus thus have have consequences consequences for for source-sink source-sink population population dynamics. dynamics. In particular, particular, asymmetric asymmetric dispersal dispersal rates rates can can create create aa source-sink source-sink structure structure in in In the absence of differences differences in in habitat habitat quality. quality. In In aa system system of of two two habitats habitats of of the absence of equal (;(n), habitat equal size, size, characterized characterized by by the the same same/~(n), habitat 1I will will be be aa source source and and habitat habitat m12 > m2b and m21 will 2 2 aa sink sink if if m12 > m21, and vice vice versa. versa. More More generally, generally, m12 m12 > > m21 will reinforce reinforce the the source-sink source-sink structure structure if if fl(n) fl(n) > > !2(n). f2(n). Conversely, Conversely, m m12 > m21 m21 will will make make 12 > the the source-sink source-sink structure structure less less pronounced pronounced if if fl(n) fl(n) < < !2(n) f 2 ( n )~ up up to to aa point. point. If If fl(n) fl(n) < < !2(n), f2(n), but but m12 m12 exceeds exceeds m21 m21 by by aa sufficient sufficient margin, margin, the the source-sink source-sink structure habitat 11 will become aa sink. sink. In structure will will become become rreversed e v e r s e d- habitat will become In other other words, words, an upstream upstream habitat habitat of of lower lower quality quality ((but still good good enough enough to to sustain sustain aa popu popuan but still lation lation despite despite the the drain drain due due to to emigration) emigration) may may become become aa source source if if the the asym asymmetry whereas the metry of of dispersal dispersal rates rates is is sufficient, sufficient, whereas the better better downstream downstream will will act act as as (relative) sink. sink. For For specific specific models models of of such such populations, populations, see see Doebeli Doebeli ((1995) aa (relative) 1 99 5 ) and and Kawecki Kawecki and and Holt Holt (2002) (2002).. Ann extreme extreme case case of of asymmetric asymmetric dispersal dispersal iiss one-way one-way dispersal, dispersal, resulting resulting in in A what 1 997) termed black hole hole sink" what Holt Holt and and Gomulkiewicz Gomulkiewicz ((1997) termed aa ""black s i n k "- - aa habitat habitat that that receives receives immigrants immigrants but but sends sends no no emigrants emigrants back back to to the the source. source. Within Within the framework of 1 6 . 1 ) and and ((16.2), 1 6.2), the the framework of the the model model described described by by Eq. Eq. ((16.1) the existence existence of hole sinks Caswell, 11989). 98 9 ) . This of black black hole sinks implies implies that that matrix matrix A(n) is is reducible reducible ((Caswell, This means corresponding to hole means that that eliminating eliminating the the rows rows and and columns columns corresponding to the the black black hole sinks sinks would would have have no no effect effect on on the the equilibrium equilibrium population population sizes sizes and and repro reproductive ductive values values in in the the remaining remaining habitats. habitats. In In other other words, words, population population dynam dynamics ics in in the the source source habitat habitat is is unaffected unaffected by by what what happens happens in in the the sink; sink; from from the the viewpoint viewpoint of of the the source source habitat, habitat, emigration emigration to to the the sink; sink; is is not not different different from from O. For such aa mortality. value of hole sink mortality. The The reproductive reproductive value of black black hole sink habitats habitats is is 0. For such system to exist, the must be good enough sustain aa popu system to exist, the source source habitat(s) habitat(s) must be good enough to to sustain population, lation, despite despite the the drain drain imposed imposed by by emigration. emigration. Note Note that that aa black black hole hole sink sink may may still still send send some some migrants migrants to to another another black black hole hole sink, sink, as as in in the the example example given 1 999a). given by by Rousset Rousset ((1999a).
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Balanced Dispersal A special special case case worth worth considering considering in in the the context context of of asymmetric asymmetric dispersal dispersal A rates rates is is the the balanced balanced dispersal dispersal scenario, scenario, whereby whereby asymmetries asymmetries in in the the dispersal dispersal rate exactly exactly compensate compensate for for differences differences in in habitat habitat quality quality (Doebeli, (Doebeli, 11995; rate 995; Lebreton Lebreton et et aI., al., 2000). 2000). Under Under the the balanced balanced dispersal dispersal scenario, scenario, Vh v h -= 11 and and fh(nh)kimhi fh(~h)~,imhi == 11 for for all all habitats, habitats, i.e., i.e., there there is is no no source-sink source-sink structure. structure. Dispersal Dispersal rates rates leading leading to to aa balanced balanced dispersal dispersal situation situation are are expected expected to to be be favored Doebeli, 11995; 995; Lebreton favored when when dispersal dispersal is is cost cost free free ((Doebeli, Lebreton et et aI., al., 2000). 2000). This Fretwell and 970). This is is equivalent equivalent to to the the ideal ideal free free distribution distribution ((Fretwell and Lucas, Lucas, 11970). Reasons Reasons why why the the evolution evolution of of balanced balanced dispersal dispersal may may be be prevented, prevented, and and thus thus the source-sink source-sink population population structure structure may may persist persist over over evolutionary evolutionary time, time, are are the discussed in in Section Section 116.5. discussed 6.5.
Stage-Structured Populations Populations Age- or Stage-Structured Generalization 1 6 . 1 ) and 1 6.2) to Generalization of of the the model model described described by by Eq. Eq. ((16.1) and ((16.2) to multiple multiple age classes classes (or stages)) is, is, in in principle, principle, straightforward, straightforward, provided provided that that the the vital vital age (or stages rates ((survival and fecundity) fecundity) are are assumed to be be aa function function of of age age ((stage) and rates survival and assumed to stage) and the current current habitat habitat only only (Lebreton, Nevertheless, the the consequences consequences of of the (Lebreton, 11996). 996). Nevertheless, source-sink source-sink population population structure structure in in age-structured age-structured populations populations remain remain rather rather unexplored. unexplored. The The definition definition of of sources sources versus versus sinks sinks based based on on the the number number of of emigrants emigrants versus versus immigrants immigrants can can still still be be upheld upheld if if dispersal dispersal occurs occurs at at aa well welldefined case, e.g., defined prereproductive prereproductive stage, stage, as as is is the the case, e.g., in in perennial perennial plants plants or or corals. corals. However, However, this this definition definition does does not not seem seem appropriate appropriate if if an an individual individual can can change change its habitat habitat at at different different ages stages, and it repeatedly, as is birds its ages or or stages, and do do it repeatedly, as is the the case case in in birds and mammals. mammals. This This can can be illustrated by equivalent of of the and be illustrated by considering considering an an equivalent the balbal anced dispersal in the the previous 1 99 6 ) anced dispersal scenario scenario discussed discussed in previous paragraph. paragraph. Lebreton Lebreton ((1996) has shown shown that that under under cost-free dispersal, natural natural selection selection should should favor favor a has cost-free dispersal, a combination of age-specific dispersal rates that that would would equalize equalize the the vector vector of of combination of age-specific dispersal rates age-specific reproductive reproductive values values across habitats. However, However, in in contrast to the the age-specific across habitats. contrast to discrete generations this case case does does imply imply balanced dispersal (Lebreton ( Lebreton discrete generations case, case, this balanced dispersal et al., aI., 2000). 2000). It It is is thus thus difficult difficult to to derive derive general general predictions this model model et predictions from from this and will become complicated if, and more more work work is is needed. needed. The The problem problem will become even even more more complicated if, as is is biologically biologically realistic, realistic, survival survival and and fecundity fecundity depend depend not not only only on on the the as current habitat, habitat, but but on the habitats individual has has experienced experienced in in the past. current on the habitats an an individual the past. Nonetheless, incorporating incorporating both both age age structure structure and and habitat heterogeneity will, will, Nonetheless, habitat heterogeneity in many many cases, cases, substantially substantially improve improve the the predictive predictive power power of of managementmanagement in oriented oriented models models of of specific specific natural natural populations populations (e.g., (e.g., Doak, Doak, 1995). 1 995).
Spatially Spatially Explicit Explicit Models Models The above above discussion assumed environmental environmental variation in the the form form of of aa The discussion assumed variation in set of of discrete discrete habitats, habitats, such such that that within within aa given given habitat habitat individuals individuals become become set mixed thoroughly thoroughly and and density density is is the the same same everywhere. everywhere. This may be be aa suffisuffi mixed This may cient approximation approximation for for systems systems such such as as herbivorous herbivorous insects insects that that use use two two cient host plant plant species species occurring occurring in in the the same same area area or or in in other other cases cases where where wellwell host defined discrete habitat patches patches form form aa relatively relatively fine-grained fine-grained mosaic mosaic (e.g., ( e.g., defined discrete habitat Blondel et et al., aI., 1992). 1 992 ) . However, However, the the spatial spatial location location of of individuals individuals must must be be Blondel
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explicitly considered considered if if variation variation in in environmental environmental factors factors is is continuous. continuous. This This explicitly can be be done done with with aa diffusion diffusion approximation approximation (e.g., (e.g. , Kirkpatrick Kirkpatrick and and Barton, Barton, can 1 997) or or with with an an individual-based individual-based model. model. A A spatially spatially explicit explicit approach approach will will 1997) also be be necessary necessary if if there there are are discrete discrete habitat habitat types, types, but but the the patches patches are are large large also relative to to the the dispersal dispersal distance distance (e.g., (e.g., Boughton, Boughton, 2000). 2000 ) . Such Such aa case case is is illusillus relative trated in in Fig. Fig. 16.3, 1 6 . 3 , where where high-quality high-quality habitat habitat 11 borders borders low-quality low-quality habitat habitat trated along aa sharp sharp ecotone ecotone (model (model details details in in the the figure figure legend). legend). As As expected, expected, at at 22 along equilibrium, habitat habitat 11 is is aa source source and and habitat habitat 22 aa sink, sink, but but the the spatial spatial model model equilibrium, reveals that that the the source-sink source-sink nature nature of of the the two two habitats habitats is is most most pronounced pronounced reveals close to to the the ecotone. ecotone. That That is, is, in in habitat habitat 11 the the excess excess of of births births over over deaths deaths close (fh(fih) > > 1), 1 ), and and of of thus thus emigration emigration over over immigration, immigration, is is greatest greatest just just left left of of (fh(~) the ecotone ( light solid solid line line in in Fig. Fig. 16.3). 1 6. 3 ) . The The same same holds holds for for the the excess excess of of the ecotone (light deaths over over births births (fh(~h) (fh(fih) < < 1) 1 ) on on the the other side of of the the ecotone. ecotone. As As one one moves moves deaths other side away from from the the ecotone, ecotone, the the population population density density (heavy (heavy line) line ) converges converges to to the the away local carrying carrying capacity capacity and and fh(~) fh( fih ) converges converges to to 1. 1 . Note, Note, however, however, that that the the local reproductive value value (dotted (dotted line) line) does does not not follow follow the the pattern of fh(~h) fh( fih) within within reproductive pattern of the habitats. habitats. Instead, Instead, in in the the better better habitat habitat it it declines declines somewhat somewhat as as the the ecoeco the tone is is approached, approached, indicating indicating that that the the improved improved lifetime lifetime reproductive reproductive sucsuc tone cess due due to to lower lower density density does does not not quite quite compensate compensate for for the the fact fact that that some some cess of the the offspring offspring will will end end up up in in the the poor poor habitat. habitat. This This is is thus thus another case of another case where the pattern emigration versus where the pattern based based on on births births versus versus deaths deaths and and emigration versus immi immigration does not pattern of values. gration does not agree agree with with the the pattern of reproductive reproductive values. habitats, Even if if the the environment environment consists consists of of discrete discrete patches patches of of different different habitats, Even their connectivity will differ their size, size, shape, shape, spatial spatial arrangement, arrangement, and and connectivity will often often cause cause different patches the same habitat type different dispersal rates. Such Such aa ent patches of of the same habitat type to to have have different dispersal rates. patch modeled within patch network network may may be be modeled within the the framework framework of of the the patch patch model model described earlier [Eq. 1 6 . 1 ) ] . However, described earlier [Eq. ((16.1)]. However, the the dispersal dispersal rates rates would would now now have have to to be simple model pre be defined defined on on aa patch-to-patch patch-to-patch basis. basis. Thus, Thus, in in contrast contrast to to the the simple model presented at patches of habitat type sented at the the beginning beginning of of this this section, section, patches of the the same same habitat type could could not not be be lumped lumped together. together. Instead, Instead, the the vector vector of of population population sizes sizes n(t) n(t) would would have have to to have have an an entry entry for for each each patch, patch, not not only only for for each each habitat habitat type. type. Consequently, Consequently, the the definition definition of of source source versus versus sink sink could could now now be be applied applied to to individual individual patches; patches; depending depending on on their their connectivity, connectivity, size, size, and and shape, shape, some some patches patches of of aa given given 1100 88 'S(
66
. . . ......
.......
1: 4 4
22
00 -20 -20
I
.................... habitat habitat 11 habitat habitat 22 +-- ---. ~ -1 00 110 0 -100 Spatial Spatial distance distance xx
11.6 .6 11.4 .4 11.2 .2 11 0.8 0.6 0.4 0.4 0.2 0.2 00 20 20
�
x X 2 ~. A C x
Fig. Fig. 116.3 6.3 A A spatially spatially explicit explicit source-sink source-sink model model with with two two habitat habitat patches. patches. The The population population density density n(x) n(x) (heavy (heavy line, line, left left axis), axis), net net reproductive reproductive rate rate f(n(x» f(n(x)) (light (light line), line), and and the the reproductive reproductive value value (dotted (dotted line) line) at at equilibrium equilibrium are are plotted plotted as as aa function function of of spatial spatial location location x. x. Discrete Discrete genera generations tions are are assumed, assumed, with with census census after after dispersal. dispersal. The The net net reproductive reproductive rate rate f(n(x» f(n(x)) = = Rhl(l Rh/(1 + + n(x» n(x)),, where O in where Rh Rh = l10 in habitat habitat 11 and and Rh Rh = 4 4 in in habitat habitat 2. 2. Dispersal Dispersal distances distances follow follow aa normal normal distribu distribution tion with with mean mean 00 and and acr = = 5. 5. =
=
1 6. 16.
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habitat type type may may act act as as sources sources whereas whereas others others may may act act as as sinks. sinks. Taking Taking into into habitat account such such aa spatial effect is is of of particular particular importance importance in in applied applied models models account spatial effect developed for for the the management management and and conservation conservation of of particular particular species. species. developed
1 6.3 16.3
ECOLOGICAL CONSEQUENCES CONSEQUENCES OF OF SOURCE-SINK SOURCE-SINK ECOLOGICAL DYNAMICS: THEORY THEORY DYNAMICS: In addition addition to to the the defining defining feature feature of of source-sink source-sink structure structure m - net net flow flow of of In dispersing individuals individuals from from source source to to sink habitats m - aa number number of of other other ecoeco dispersing sink habitats logical consequences consequences of of source-sink source-sink population population structure structure have been predicted predicted logical have been by mathematical mathematical models. models. These These predictions predictions are are summarized summarized in in this this section, section, by whereas Section Section 16.4 1 6.4 reviews reviews relevant relevant empirical empirical examples. examples. whereas
Species Range Range Species Immigration stable local local population in aa habitat, in which which Immigration can can maintain maintain aa stable population in habitat, in deaths exceed low density sinks ) . Unless Unless limited limited by by deaths exceed births births even even at at low density (absolute (absolute sinks). barriers ranges will will therefore therefore as as aa rule rule extend extend beyond beyond the the barriers to to dispersal, dispersal, species species ranges areas where where habitat habitat quality quality is is sufficient to sustain sustain aa population population without without immiimmi areas sufficient to gration where the local conditions conditions satisfy satisfy the the species' species' niche niche requirements; requirements; gration (i.e., (i.e., where the local Pulliam, 11988, 988, 2000). 2000). This both to to the the geographical range of the Pulliam, This applies applies both geographical range of the species and distribution on the scale scale of of local local habitat habitat variation variation (habitat species and to to its its distribution on the (habitat occupancy) In practice, will often be difficult to distinguish distinguish between between an an occupancy).. In practice, it it will often be difficult to absolute habitat that that is not quite and acts acts as as relative absolute sink sink and and aa habitat is not quite optimal optimal and relative sink, sink, but but still still satisfies satisfies the the species' species' niche niche requirements. requirements. Successful Successful reproduction reproduction may may take take place place in in absolute absolute sinks sinks and and population population density density may may be be relatively relatively high high and and stable; stable; it it may may not not be be apparent apparent that that the the population population would would deterministically deterministically go go extinct extinct without without immigration. immigration.
Population Population Size Size and and Distribution Distribution What (global) What is is the the effect effect of of source-sink source-sink population population structure structure on on the the total total (global) population population size size?? An An answer answer will will depend depend on on the the precise precise formulation formulation of of this this question. question. First, First, one one may may compare compare aa set set of of habitat habitat patches patches connected connected by by dispersal dispersal (and (and thus thus potentially potentially having having source-sink source-sink structure) structure) with with the the same same set set of of patches patches each each inhabited inhabited by by an an isolated isolated population. population. This This perspective perspective thus thus focuses focuses on on the the effect effect of of changing changing the the dispersal dispersal rate(s) rate(s) while while keeping keeping the the landscape landscape unchanged. unchanged. In 1 98 5 ) showed In aa two-patch two-patch model model with with symmetric symmetric passive passive dispersal, dispersal, Holt Holt ((1985) showed that that no no simple simple general general prediction prediction about about the the effect effect of of dispersal dispersal on on the the total total population population size size can can be be made. made. Whether Whether the the total total population population size size will will increase increase or or decrease decrease as as aa result result of of dispersal dispersal will will depend depend on on the the shape shape of of the the functions functions relating relating local local density density to to the the local local birth birth and and death death rates. rates. This This applies applies even even if if the the poorer 1 99 5 ) considered poorer habitat habitat is is an an absolute absolute sink. sink. Doebeli Doebeli ((1995) considered two two patches patches of of the the same same habitat habitat quality quality and and showed showed that that asymmetric asymmetric dispersal, dispersal, which which resulted resulted in in aa source-sink source-sink structure, structure, led led to to an an increase increase of of the the total total population population size. size. It It is is not not clear clear how how general general this this result result is is (only (only aa numerical numerical example example is is presented). presented). A A more more general general prediction prediction concerns concerns the the effect effect of of dispersal dispersal on on the the distribution distribution of of
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the the population population among among the the habitats: habitats: with with increasing increasing dispersal dispersal the the fraction fraction of of the total total population population living living in in the the source source habitats habitats tends tends to to decrease decrease (e.g., (e.g., Holt, Holt, the Pulliam, 2000; 2000; Kawecki Kawecki and and Holt, Holt, 2002 2002).) . 11985; 985; Pulliam, Second, Second, one one may may ask ask how how adding adding aa sink sink habitat habitat oorr changing changing its its size size affects affects the total total population population size size and and the the population population in in aa high-quality high-quality source source habitats, habitats, the assuming assuming that that the the dispersal dispersal pattern pattern is is unchanged. unchanged. It It is is not not surprising surprising that that replacing replacing some some good good habitat habitat with with poor poor habitat habitat will will reduce reduce the the overall overall popula population size. size. It It is is more more interesting interesting to to ask ask how how the the population population size size is is affected affected if if tion some some sink sink habitat habitat patches patches are are eliminated eliminated (converted (converted into into hostile hostile "nonhabitat" "nonhabitat" or "matrix" "matrix")) while while keeping keeping the the amount amount of of the the source source habitat habitat constant. constant. Under Under or passive passive dispersal dispersal this this will will lead lead to to aa greater greater fraction fraction of of propagules propagules perishing perishing in in the the "nonhabitat," "nonhabitat," causing causing aa reduction reduction of of the the total total population population size. size. This This is is not not necessarily the the case case if if dispersal dispersal is is active active and and thus thus the the dispersing dispersing individuals individuals necessarily avoid the the ""nonhabitat." In one one model model that that made made this this assumption assumption ((Pulliam and avoid nonhabitat." In Pulliam and Danielson, 99 1 ), the Danielson, 11991), the number number of of individuals individuals in in the the source source habitats habitats increased increased as the the area area of of sink sink habitat habitat was was reduced. The effect effect on on the the total total population population size size as reduced. The in Pulliam Pulliam and and Danielson's Danielson's model depended depended on on the the degree degree of of habitat habitat selection. selection. in With poor habitat selection selection ability, ability, the the total total population increased as as the the With poor habitat population size size increased area decreased. With area of of sink sink habitat habitat decreased. With aa better better habitat habitat selection selection ability, ability, the the total total population size peaked peaked at habitat. In population size at an an intermediate intermediate amount amount of of sink sink habitat. In contrast, contrast, an individual-based individual-based model model (Wiegand (Wiegand et et aI., al., 11999) predicted that that eliminating eliminating an 999) predicted sink habitat will lead lead to to aa reduction of the the total total population This discrep discrepsink habitat will reduction of population size. size. This ancy suggests suggests that that no no simple simple general general predictions predictions can can be be made the effect effect ancy made about about the of eliminating eliminating patches patches of of sink sink habitat habitat on on the the overall overall population population size. size. of
P o p u l a t i o n Stability S t a b i l i t y and a n d Persistence Persistence Population If If too too many many dispersing dispersing individuals individuals end end up up in in aa habitat habitat that that is is an an absolute absolute sink, sink, the entire entire population population will will go go deterministically deterministically extinct (Pulliam, 1988; Donovan 1 988; Donovan and Thompson, Thompson, 200 2001). This is of source-sink source-sink populaand 1 ) . This is the the most most obvious obvious effect effect of popula tion structure on population population persistence. as extinction extinction risk, tion structure on persistence. More More generally, generally, as risk, at at least on the the short least on short term, term, tends tends to to be be correlated correlated negatively negatively with with population population size size ((Chapter Chapter 14), 14), the the effects effects of population structure equilibrium of source-sink source-sink population structure on on equilibrium population size likely to to have have implications implications for for population population persistence. persistence. population size are are likely However, the the existence existence of sink habitats habitats may may affect population persistence persistence However, of sink affect population by affecting affecting the by the population population dynamics dynamics independently independently of of their their effects effects on on the the equilibrium population population size. size. Several Several models models (Holt, (Holt, 1984, 1 984, 1985; 1 985; McLaughlin McLaughlin equilibrium and Roughgarden, Roughgarden, 1991) 1 9 9 1 ) predict predict that that adding adding aa habitat habitat that that is is aa sink sink for for the the and prey prey can can stabilize stabilize an an otherwise otherwise unstable unstable or or neutrally neutrally stable stable predator-prey predator-prey model. The The source-sink source-sink structure structure also also tends tends to to have have aa stabilizing stabilizing effect effect on on the the model. dynamics of of aa host-parasitoid host-parasitoid model model (Holt (Holt and and Hassell, Hassell, 1993). 1 99 3 ) . Finally, Finally, dynamics Doebeli (1995), ( 1 995), generalizing generalizing results results of of Hastings Hastings (1993), ( 1 993), showed showed that that dispersal dispersal Doebeli between between two two patches patches of of the the same same quality quality tends tends to to stabilize stabilize intrinsically intrinsically chaotic chaotic population dynamics dynamics (see ( see also also Gyllenberg et al., aI., 1996). 1 996). The The stabilizing stabilizing effect effect population Gyllenberg et is is stronger stronger if if dispersal dispersal rates rates are are asymmetric asymmetric so so that that at at equilibrium equilibrium there there is is aa source-sink population population structure. structure. One One intuitive intuitive explanation explanation of of those those results results is is source-sink that that sink habitats habitats act act as a buffer, buffer, absorbing absorbing surplus surplus individuals individuals produced produced in in source habitats. habitats. This This prevents prevents the the population population from from greatly overshooting the the source greatly overshooting equilibrium density, density, thus thus reducing reducing or or averting averting aa population population crash crash due due to to equilibrium
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overcompensating overcompensating density density dependence. dependence. In In contrast, contrast, dispersal dispersal to to aa sink sink habitat habitat that that is is available available only only seasonally seasonally can can destabilize destabilize population population dynamics; dynamics; this this mechanism (Lomnicki, 11995). 995). mechanism has has been been proposed proposed to to contribute contribute to to rodent rodent cycles cycles (Lomnicki, Existence of of aa sink sink habitat habitat may may make make the the population population less less sensitive sensitive to to envir envirExistence onmental onmental fluctuations fluctuations affecting affecting birth birth and and death death rates, rates, provided provided that that the the sink sink habitat is (Holt, 11997). 997). An case of habitat is less less affected affected by by the the fluctuations fluctuations (Holt, An extreme extreme case of this this type subject to occasional catastrophes type involves involves aa source source habitat habitat subject to occasional catastrophes that that wipe wipe out out the the local local population. population.
Age Age Structure Structure In structure in In organisms organisms with with overlapping overlapping generations, generations, the the age age structure in source source and and sink sink habitats habitats may may differ differ as as aa consequence consequence of of differences differences in in local local survival. survival. More More interestingly, dispersal dispersal into into sink sink habitats habitats may may be be age age dependent. dependent. In In territorial territorial interestingly, species, young species, young individuals individuals may may be be more more likely likely to to be be excluded excluded from from breeding breeding in in optimal, optimal, source source habitats. habitats. Sink Sink habitats habitats will will then then contain contain aa disproportionately disproportionately large large fraction fraction of of young young adults. adults.
116.4 6.4 ECOLOGICAL ECOLOGICAL CONSEQUENCES CONSEQUENCES OF OF SOURCE-SINK SOURCE-SINK DYNAMICS: DYNAMICS: EMPIRICAL EMPIRICAL EVIDENCE EVIDENCE Basic Basic Source-Sink Source-Sink Structure Structure There There is is increasing increasing evidence evidence of of source-sink source-sink structure structure in in natural natural popula populations, tions, involving involving habitat habitat variation variation at at various various spatial spatial scales. scales. At At aa continental continental scale, wolf scale, it it has has been been reported reported in in reindeer, reindeer, in in which which low low recruitment recruitment due due to to wolf predation predation causes causes boreal boreal forests forests to to act act as as sink sink habitats; habitats; the the tundra tundra is is the the source source ((Bergerud, Bergerud, 11988). 9 8 8 ) . Similarly, Similarly, the the reproductive reproductive success success of of pied pied flycatchers flycatchers (Ficedula does not, on average, average, com (Ficedula hypoleuca) at at the the northern northern range range limit limit does not, on compensate pensate for for mortality mortality (although (although it it may may do do so so in in good good years) years);; these these northern northernmost populations must most populations must thus thus be be maintained maintained by by immigration immigration (Jarvinen (J~rvinen and and Vasainen, 984). In Vfisfiinen, 11984). In black-throated black-throated blue blue warbler warbler (Dendroica (Dendroica caerulescens), caerulescens), population population density density and and estimated estimated habitat habitat quality quality decline decline aass one one moves moves away away in in either Graves, 11997). 997). either direction direction from from the the Appalachian Appalachian mountains mountains ((Graves, The The source-sink source-sink structure structure at at aa more more local local spatial spatial scale scale has has been been well well charac characterized terized in in blue blue tits tits (Parus (Parus caeruleus) caeruleus) in in southern southern France, France, where where patches patches of of good good (deciduous) (deciduous) and and poor poor (sclerophyllous) (sclerophyllous) habitat habitat form form aa mosaic mosaic landscape landscape with with aa patch 992). Even patch size size on on the the order order of of 11 to to 100 100 km2 km 2 (Blondel (Blondel et et aI., al., 11992). Even though though the the breeding less than half that breeding density density in in the the sclerophyllous sclerophyllous habitat habitat is is less than half that in in the the decidu deciduous ous habitat, habitat, birds birds in in the the sclerophyllous sclerophyllous habitat habitat have have aa smaller smaller clutch clutch size size and and aa lower 996). The lower breeding breeding success success (Dias (Dias and and Blondel, Blondel, 11996). The breeding breeding performance performance in see in the the sink sink is is impaired impaired additionally additionally by by aa locally locally maladaptive maladaptive laying laying date date ((see Section 6.6) and (Dias Section 116.6) and possibly possibly by by aa smaller smaller size size of of individuals individuals breeding breeding there there (Dias and 996). Genetic and Blondel, Blondel, 11996). Genetic marker marker data data are are also also consistent consistent with with an an asymmetric asymmetric gene deciduous to gene flow flow from from the the deciduous to the the sclerophyllous sclerophyllous habitat habitat patches patches (Dias (Dias et et aI., al., 11996). 996). A A number number of of North North American American migratory migratory songbirds songbirds suffer suffer extreme extreme rates rates of of nest nest parasitism parasitism and and predation predation in in fragmented fragmented forest forest patches patches of of agricultural agricultural and suburban landscapes. and suburban landscapes. These These highly highly fragmented fragmented habitats habitats constitute constitute sinks sinks
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supported supported by by immigration immigration from from more more extensive extensive forests forests (Robinson (Robinson et et aI., al., 1995). 1995). A A similar similar source-sink source-sink structure structure related related to to habitat habitat fragmentation fragmentation has has been been observed observed in in the the reed reed warbler warbler in in The The Netherlands Netherlands (Foppen (Foppen et et aI., al., 2000) 2000).. A A source-sink structure local scale been found pot source-sink structure at at aa more more local scale has has been found in in the the checkers checkerspot clearings and butterfly butterfly (Euphydryas (Euphydryas editha), where where forest forest clearings and rocky rocky outcrops outcrops con constitute spatially separated stitute two two spatially separated habitats, habitats, each each with with aa different different host host plant plant for for the the caterpillars caterpillars (Boughton, (Boughton, 2000) 2000).. A A source-sink source-sink structure structure at at the the scale scale ooff meters meters occurs occurs iinn the the snow snow buttercup buttercup (Ranunculus adoneus), aa perennial plant confined perennial alpine-zone alpine-zone plant confined to to deep deep snow snow beds beds of of the the Rocky Rocky Mountains. Mountains. The The beginning beginning of of the the vegetative vegetative season season and and flowering flowering time snowmelt (Stanton 997). As time are are determined determined by by the the snowmelt (Stanton and and Galen, Galen, 11997). As the the pattern pattern of patches of of snow snow accumulation accumulation is is fairly fairly constant constant from from year year to to year year and and patches of old old snow snow tend tend to to melt melt from from the the edges, edges, the the relative relative timing timing of of snowmelt snowmelt at at different different localities fairly constant year to only tens localities is is fairly constant from from year to year. year. Localities Localities separated separated by by only tens of of meters may weeks apart. is correlated meters may become become clear clear of of snow snow several several weeks apart. As As seed seed size size is correlated positively with with season length, plants plants at at later-melting later-melting sites sites produce produce smaller smaller seeds seeds positively season length, (Galen 993). These establishment rate, (Galen and and Stanton, Stanton, 11993). These small small seed seed have have aa low low establishment rate, and and most seeds produced produced in most individuals individuals at at all all localities localities come come from from large large seeds in early early melting melting sites Galen, 11997). 997). The sites (Stanton (Stanton and and Galen, The source-sink source-sink population population structure structure in in this this species is least partially mediated by species is thus thus at at least partially mediated by maternal maternal effects. effects. A A source-sink source-sink structure structure dominated dominated by by asymmetry asymmetry in in dispersal dispersal rates rates imposed imposed 981, by by wind wind has has been been described described in in the the sand sand dune dune plant plant Cakile edentula edentula (Keddy, (Keddy, 11981, 11982; 982; Watkinson, 985). In Watkinson, 11985). In that that system, system, the the base base of of aa dune dune oonn the the seaward seaward side side is is the the source source habitat habitat where where most most seeds seeds are are produced. produced. However, However, because because most most seeds closer to seeds are are transported transported by by wind wind to to the the sink sink habitat habitat closer to the the dune dune crests, crests, plant plant density considerably higher density in in the the latter latter habitat habitat is is considerably higher than than in in the the source source habitat. habitat. At At the same time, time, seed seed emigration emigration from from the the source source habitat habitat reduces reduces competition competition and and the boosts 985). In boosts the the reproductive reproductive output output from from that that habitat habitat (Watkinson, (Watkinson, 11985). In this this case case the the source source and and sink sink habitats habitats are are only only separated separated by by several several meters. meters. The above review The above review of of examples examples of of source-sink source-sink structure structure in in natural natural popula populations tions is is not not meant meant to to be be exhaustive, exhaustive, and and as as the the interest interest in in this this aspect aspect of of spatial spatial ecology increases, more ecology increases, more evidence evidence will will accumulate. accumulate. Relatively Relatively unexplored unexplored remain cases of caused by remain cases of potential potential source-sink source-sink dynamics dynamics caused by biotic biotic interactions, interactions, particularly particularly the the source-sink source-sink structure structure of of parasite parasite populations populations caused caused by by varia variation Jokela, 11996). 996). tion in in host host susceptibility susceptibility (e.g., (e.g., Lively Lively and and Jokela,
Other Other Ecological Consequences Despite accumulating Despite accumulating evidence evidence for for the the ubiquity ubiquity of of source-sink source-sink structure structure in in natural populations, populations, data data directly directly addressing addressing specific specific predictions predictions concerning concerning natural its ecological consequences scarce. Addressing its ecological consequences are are scarce. Addressing these these predictions predictions directly directly would would involve involve experimental experimental intervention, intervention, e.g., e.g., changing changing the the amount amount of of source source or or sink sink habitat habitat or or altering altering the the dispersal dispersal pattern. pattern. Applying Applying this this approach approach to to natural populations also question natural populations may may not not only only be be technically technically difficult, difficult, but but also questionable able on on ethical ethical or or legal legal grounds. grounds. For For example, example, it it is is likely likely that that some some of of the the examples earlier involve involve populations examples mentioned mentioned earlier populations persisting persisting in in absolute absolute sinks, sinks, unable 98 8 ; unable to to sustain sustain aa population population without without immigration immigration (e.g., (e.g., Bergerud, Bergerud, 11988; Robinson 99 5 ) . However, Robinson eett aI., al., 11995). However, definitive definitive confirmation confirmation would would require require "clos "closing" ing" the the population, population, i.e., i.e., preventing preventing immigration immigration and and emigration. emigration.
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Given direct experimental Given the the problems problems with with aa direct experimental approach, approach, monitoring monitoring the the consequences natural experiments, consequences of of ""natural experiments,"" i.e., i.e., natural natural or or anthropogenic anthropogenic changes changes in in the the environment, environment, has has aa particular particular value. value. For For example, example, the the importance importance of of aa sink habitat was demonstrated sink habitat for for population population persistence persistence was demonstrated clearly clearly in in aa source source992, an unusual sink population of checkers pot butterfly sink population of the the checkerspot butterfly E. editha. editha. In In 11992, an unusual summer popu summer frost frost killed killed all all larvae larvae in in the the source source habitat habitat (forest (forest clearings) clearings).. The The population persisted only because larvae (rocky outcrops) lation persisted only because larvae in in the the sink sink habitat habitat (rocky outcrops) sur survived (Thomas et 996; Boughton, 999) (the vived (Thomas et aI., al., 11996; Boughton, 11999) (the outcrops outcrops were were presumably presumably the the main main habitat habitat of of this this species species before before humans humans created created forest forest clearings) clearings).. A A popu population subject to lation structure structure with with the the source source populations populations subject to repeated repeated catastrophes catastrophes has (Frouz and 1 ). Another has also also been been reported reported for for aa midge midge (Frouz and Kindlmann, Kindlmann, 200 2001). Another study Luttrell et 99 9 ) suggested study ((Luttrell et aI., al., 11999) suggested that that extinction extinction of of numerous numerous local local popu populations lations of of aa cyprinid cyprinid fish fish was was due due to to disruption disruption of of dispersal dispersal between between source source and natural and sink sink habitats habitats by by artificial artificial reservoirs. reservoirs. The The problem problem with with such such ""natural experiments" experiments" is is often often the the lack lack of of replication replication and and controls. controls. An An alternative alternative approach approach involves involves spatial spatial analysis analysis of of landscape landscape ecology ecology (Chapter 2), populations can (Chapter 2), whereby whereby the the properties properties of of local local populations can be be correlated correlated not not only only with with the the local local habitat habitat conditions, conditions, but but with with the the composition composition of of the the regional regional habitat habitat matrix matrix (e.g., (e.g., the the presence presence and and size size of of nearby nearby source source and and or or sink sink patches) patches).. Foppen Foppen et et ai. al. (2000) (2000) used used this this approach approach to to show show that that the the existence existence of of sink sink habitat habitat patches patches leads leads to to aa greater greater size size and and stability stability of of reed reed warbler warbler popu populations 1 997) has lations in in source source patches. patches. Graves Graves ((1997) has shown shown that that the the proportion proportion of of year yearlings lings among among breeding breeding males males of of black-throated black-throated blue blue warbler warbler (D. caerulescens) caerulescens) is is correlated correlated negatively negatively with with habitat habitat quality, quality, indicating indicating an an effect effect of of source-sink source-sink dynamics population age dynamics on on the the population age structure. structure. The The influence influence of of source-sink source-sink dynam dynamics population size been demonstrated ics on on the the population size structure structure has has been demonstrated in in blue blue tits tits in in south southern France, where males breeding habitat are ern France, where males breeding in in the the source source habitat are larger larger than than those those breeding breeding in in the the sink sink habitat habitat (size (size measured measured as as tarsus tarsus length; length; Dias Dias and and Blondel, Blondel, 11996). 996). However, However, because because male male fledglings fledglings produced produced in in the the two two habitat habitat types types do do not (Dias and 996), the not differ differ in in tarsus tarsus length length (Dias and Blondel, Blondel, 11996), the difference difference with with respect respect to smaller individuals to breeding breeding males males must must reflect reflect the the displacement displacement of of smaller individuals from from the the source. source. As As any any approach approach based based on on correlations, correlations, this this approach approach does does not not directly directly address causation potentially confounded included in address causation and and can can be be potentially confounded by by factors factors not not included in the analysis. This problem can the analysis. This problem can be be illustrated illustrated by by results results from from the the same same study study of of blue blue tits. tits. The The population population density density in in the the sclerophyllous sclerophyllous habitat habitat in in southern southern France, France, where where it it acts acts as as aa sink, sink, is is much much lower lower than than in in the the same same habitat habitat in in Corsica, Corsica, where where it it is is aa dominant dominant habitat habitat not not affected affected by by immigration immigration (Dias (Dias and and Blondel, 996). These Blondel, 11996). These results results seem seem to to contradict contradict the the prediction prediction that that immigra immigration should boost boost the see earlier earlier discussion). tion from from aa source source should the density density in in the the sink sink ((see discussion). The The discrepancy discrepancy is is explained explained by by the the fact fact that that reproductive reproductive success success in in the the sclero sclerophyllous phyllous habitat habitat in in Corsica Corsica is is higher higher than than in in the the same same habitat habitat in in southern southern France Blondel, 11996). 996). France (Dias (Dias and and Blondel, A A powerful powerful but but rarely rarely used used approach approach to to study study consequences consequences of of the the source sourcesink setting up sink population population structure structure involves involves setting up controlled controlled experimental experimental source-sink source-sink systems systems in in the the laboratory laboratory or or in in outdoor outdoor enclosures enclosures or or "mesocosms. "mesocosms."" Davis 1 99 8 ) used Davis and and collaborators collaborators ((1998) used this this approach approach to to study study the the effect effect of of dispersal dispersal on on population population size size and and distribution distribution along along an an environmental environmental gradient. gradient. Their population cages, Their system system involved involved four four Drosophila population cages, arranged arranged along along aa series series of 10, 15, 20, and to of temperatures temperatures ((10, 15, 20, and 25°C), 25~ to simulate simulate four four habitat habitat patches patches along along
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aa thermal thermal gradient. gradient. In In one one treatment, treatment, adjacent adjacent cages cages were were connected connected with with plas plastic tubes, tubes, enabling enabling dispersal dispersal (dispersal (dispersal rate rate about about 66% per day) day) and and thus thus creating creating tic % per conditions under under which which the the source-sink source-sink structure structure was was expected. expected. This This could could be be conditions contrasted contrasted with with aa no-dispersal no-dispersal treatment, treatment, which which simulated simulated isolated isolated populations populations living living at at different different temperatures. temperatures. Three Three Drosophila species species were were tested tested separately. separately. As As predicted predicted by by source-sink source-sink models, models, in in D. melanogaster and and D. simulans per permitting mitting dispersal dispersal led led to to reduced reduced density density in in patches patches that that had had high high density density under under no no dispersal dispersal and and to to increased increased density density at at marginal marginal temperatures. temperatures. The The pattern pattern was was less less clear clear in in D. subobscura, in in which which aa reduction reduction of of density density at at the the optimal optimal temperature temperature was was not not accompanied accompanied by by aa marked marked increase increase of of population population size size at at suboptimal suboptimal temperatures. temperatures. In In all all three three species, species, dispersal dispersal led led to to maintenance maintenance of of local local populations populations in in absolute absolute sinks, sinks, i.e., i.e., at at temperatures temperatures at at which which local local popu popul Ooe for lations lations went went extinct extinct in in the the absence absence of of dispersal dispersal ((10~ for D. melanogaster and and D. simulans, 25 25~e for for D. subobscura). Although Although Davis Davis and and colleagues colleagues did did not not address address this this question question statistically, statistically, in in all all three three species species the the overall overall (global) (global) popula population size size tended tended to to be be larger larger in in the the absence absence of of dispersal. dispersal. This This study study points points to to the the tion potential potential usefulness usefulness of of experimental experimental source-sink source-sink model model systems systems to to study study eco ecological and and evolutionary evolutionary consequences consequences of of the the source-sink source-sink structure. structure. Because Because of of logical scale scale issues, issues, it it can can only only be be used used with with some, some, mostly mostly invertebrate, invertebrate, model model systems. systems. However, However, use use of of such such model model laboratory laboratory systems systems enabled enabled important important advances advances in in other areas of ecology their use other areas of ecology and and evolutionary evolutionary biology, biology, and and their use to to address address source-sink-related questions questions should should be be promoted. promoted. source-sink-related °
116.5 6.5
NATURAL SELECTION SELECTION ON ON DISPERSAL DISPERSAL AND AND EVOLUTIONARY EVOLUTIONARY NATURAL STABILITY OF SOURCE-SINK POPULATION STRUCTURE STABILITY OF SOURCE-SINK POPULATION STRUCTURE Given the expected reproductive success is lower lower in in aa sink sink than than in in a Given that that the expected reproductive success is a source habitat, one one would would expect that dispersal dispersal from from source to sink sink habitats source habitat, expect that source to habitats should be countered by natural selection. As a result, the the dispersal dispersal pattern pattern should be countered by natural selection. As a result, should evolve evolve toward toward retaining retaining more more individuals individuals in in the the source, source, up up to to the the point point should at which which differences differences in in local local density density compensate compensate for for differences differences in in habitat habitat at quality and and the the source-sink source-sink structure disappears (balanced ( balanced dispersal dispersal scenario, scenario, quality structure disappears Section 16.2). intuitive argument 1 6 .2). This This intuitive argument has has been been supported supported by formal formal analysis analysis of of a patch patch model model assuming assuming passive dispersal dispersal (Doebeli, ( Doebeli, 1995; 1 995; Lebreton Lebreton et al., aI., 2000 ); it it also also underlies underlies the the ideal ideal free distribution model model for for actively actively disperdisper 2000); free distribution sing To explain sing organisms organisms (Fretwell ( Fretwell and and Lucas, Lucas, 1970). 1 970 ) . To explain why why the the source-sink source-sink population structure structure should should persist persist over over evolutionary evolutionary time, time, one one must must find find population reasons why why the the above above prediction prediction should should not not hold. hold. These These reasons reasons are are likely likely to to reasons be be different different for for passively passively and and actively dispersing dispersing organisms. organisms.
Passive Passive Dispersal Dispersal definition, passively passively dispersing dispersing individuals individuals cannot cannot choose choose their their destindestin By definition, ation. Dispersal Dispersal to to sink sink habitats habitats in in such such organisms organisms can can be be understood understood easily easily as as ation. consequence of of aa general general propensity propensity to to disperse. disperse. The The balanced balanced dispersal dispersal aa consequence as defined defined in in Section Section 16.2) 1 6.2) from from aa scenario requires requires that that the the dispersal dispersal rate rate (mij as scenario high to to aa low-quality low-quality habitat habitat is is lower lower than than the the dispersal dispersal rate rate in in the the opposite opposite high direction (Doebeli, (Doebeli, 1995). 1 995). Such Such an an asymmetry asymmetry of of dispersal dispersal rates rates is is possible possible if if direction
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propagules propagules produced produced in in poorer poorer habitats habitats have have aa greater greater propensity propensity to to disperse, disperse, reflecting reflecting the the plasticity plasticity of of behavioral behavioral and and morphological morphological traits traits affecting affecting dispersal. dispersal. However, However, the the evolution evolution of of such such plasticity plasticity is is likely to to be be constrained, constrained, in in particular particular because because the the probability probability of of dispersing dispersing from from aa source source to to aa sink sink not not only only depends depends on on the the propensity propensity to to disperse, disperse, but but also also on on the the relative relative area area of of different different habitats habitats types types within within an an individual's individual's dispersal dispersal shadow. shadow. If If plasticity plasticity of of dispersal dispersal rates rates is is constrained, constrained, simple simple source-sink source-sink models models predict predict that that natural natural selection Feldman, 11973; 973; selection should should drive drive dispersal dispersal to to minimum minimum (e.g., (e.g., Balkau Balkau and and Feldman, Holt, 9 8 5 ) . However, Holt, 11985). However, this this tendency tendency will will be be counteracted counteracted by by advantages advantages of of dispersing dispersing within within aa given given habitat habitat type, type, such such as as avoidance avoidance of of inbreeding inbreeding and and sib sib competition competition or or assurance assurance against against temporal temporal unpredictability unpredictability of of the the environ environment see Chapter 0 ) . The ment ((see Chapter 110). The optimal optimal dispersal dispersal propensity propensity will will reflect reflect aa balance balance between between these these two two forces. forces.
Active Active Dispersal Dispersal with with Habitat Habitat Choice Choice Three Three general general reasons reasons have have been been proposed proposed to to explain explain deviations deviations from from an an ideal ideal free free distribution distribution and and dispersal dispersal into into sink sink habitats habitats in in actively actively dispersing dispersing organisms capable of (Holt, 11997). 997). First, organisms capable of habitat habitat choice choice (Holt, First, territoriality territoriality or or other other forms forms of of contest contest competition competition may may prevent prevent some some individuals individuals from from breeding breeding in in the the source source habitat. habitat. It It will will often often pay pay for for such such individuals individuals to to attempt attempt breeding breeding in floaters" in in sink sink habitats habitats rather rather than than be be nonbreeding nonbreeding ""floaters" in source source habitats habitats (Pulliam, 11988; 9 8 8 ; Pulliam 99 1 ). Thus, (Pulliam, Pulliam and and Danielson, Danielson, 11991). Thus, in in this this scenario scenario individ individuals uals breeding breeding in in aa sink sink do do the the best best of of aa bad bad job. job. Second, Second, ideal ideal free free distribu distribution tion requires requires that that individuals individuals can can assess assess not not only only the the quality quality of of different different habitats, also the among habitats. Gaining this this habitats, but but also the distribution distribution of of individuals individuals among habitats. Gaining information likely to cognitive abilities information is is likely to be be constrained constrained by by the the cognitive abilities of of the the species, species, particularly particularly if if the the environment environment is is changing changing in in time time (Remes, (Remes, 2000) 2000).. Even Even if if the the species species is is capable capable of of evaluating evaluating habitats habitats accurately, accurately, inspecting inspecting many many habitat habitat patches patches will will be be costly costly in in terms terms of of energy, energy, time, time, and and mortality. mortality. Thus, Thus, it it may may pay pay to to settle settle in in the the first first more more or or less less suitable suitable habitat habitat patch patch (van (van Baalen Baalen and and Sabelis, 9 9 3 ) . Third, Sabelis, 11993). Third, if if the the environment environment is is temporally temporally variable variable in in such such aa way way that habitat occasionally that fitness fitness in in the the sink sink habitat occasionally exceeds exceeds that that in in the the source habitat habitat and and dispersal dispersal back back from from the the sink sink to to the the source source is is possible, possible, genotypes genotypes that that choose sink choose sink habitat habitat with with aa small small but but nonzero nonzero probability probability will will have have advantage advantage over (Holt, 11997; 997; Wilson, over those those that that avoid avoid sink sink completely completely (Holt, Wilson, 2001 2001).) . In In this this scenario, dispersal Seger and scenario, dispersal into into aa sink sink habitat habitat is is thus thus aa form form of of bet bet hedging hedging ((Seger and Brockmann, 987). Brockmann, 11987).
116.6 6.6 EVOLUTIONARY EVOLUTIONARY CONSEQUENCES CONSEQUENCES OF OF SOURCE-SINK SOURCE-SINK STRUCTURE STRUCTURE From From an an evolutionary evolutionary perspective, perspective, "habitat "habitat quality," quality," which which determines determines whether whether aa habitat habitat is is aa source source or or sink, sink, reflects reflects an an interaction interaction between between the the prop properties erties of of the the habitat habitat and and the the characteristics characteristics of of the the species; species; the the latter latter can can evolve. evolve. It It is is thus thus of of interest interest to to know know how how the the relative relative performance performance of of aa population population in in source source and and sink sink habitats habitats should should change change over over evolutionary evolutionary time. time. Adaptation Adaptation to to initially initially marginal marginal sink sink habitats habitats has has important important implications implications for for the the evolution evolutionary ary dynamics dynamics of of species species distributions. distributions.
TADEUSZ J.l. KAWECKI
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Adaptation lack of Adaptation to to marginal marginal habitats habitats may may be be constrained constrained by by aa lack of genetic genetic variation Lewontin and and Birch, 966; Parsons, 975; Blows Blows and Hoffmann, variation ((Lewontin Birch, 11966; Parsons, 11975; and Hoffmann, 11993), 99 3 ) , which which in in turn turn may may reflect reflect biochemical, biochemical, physiological, physiological, and and develop developmental species' evolutionary evolutionary history mental constraints constraints resulting resulting from from the the species' history (Stearns, (Stearns, 11994). 994). This This factor factor is is not not specific specific to to source-sink source-sink populations populations and and is is not not dis discussed cussed here. here. Instead Instead this this section section focuses focuses on on predictions predictions concerning concerning the the effect effect of source-sink source-sink population population structure on on adaptive adaptive evolution, evolution, assuming assuming that that of genetic variation fitness in both source and sink sink habitats genetic variation for for fitness in both source and habitats exists. exists. Two Two intuitive as to to why why source-sink intuitive arguments arguments have have been been made made as source-sink dynamics dynamics make make it it difficult difficult for for aa population population to to evolve evolve improved improved performance performance in in habitats habitats that that function function as as sinks. sinks. The The first first argument argument notices notices that that sink sink habitats habitats contribute contribute relatively reproduction of relatively little little to to the the reproduction of the the entire entire population. population. Therefore, Therefore, their their contribution to contribution to the the overall overall fitness, fitness, averaged averaged over over habitats, habitats, is is relatively relatively small, and selection on small, and natural natural selection on performance performance in in sink sink habitats habitats is is relatively relatively weak. second argument stresses gene weak. The The second argument stresses gene flow flow swamping swamping locally locally adapted adapted genotypes habitats. These genotypes in in sink sink habitats. These two two arguments arguments and and the the relationship relationship between between them them are are discussed discussed in in the the following following two two subsections. subsections. The The third third sub subsection section discusses discusses the the predictions predictions of of the the theory, theory, while while the the last last subsection subsection reviews reviews the the empirical empirical evidence. evidence.
Reproductive Reproductive Value Value and and Sensitivity Sensitivity of of Fitness Fitness In In the the classic classic model model of of quantitative quantitative traits traits under under natural natural selection selection (Price, (Price, 11970; 970; Lande 98 3 ) , the Lande and and Arnold, Arnold, 11983), the expected expected direct direct response response of of aa trait trait to to selection selection is is proportional proportional to to the the strength strength of of selection, selection, measured measured as as the the deriva derivative of of fitness fitness with with respect to the the trait trait value. For aa source-sink source-sink population population at at aa tive respect to value. For density equilibrium, density equilibrium, the the dominant dominant eigenvalue eigenvalue A.k of of the the transition transition matrix matrix A(Ii A(ri)) is Caswell, 11989; 989; Charlesworth, 994). is an an appropriate appropriate measure measure of of fitness fitness ((Caswell, Charlesworth, 11994). Thus Thus the the strength strength of of selection selection on on trait trait zz can can be be partitioned partitioned according according to to its its h (nh ) in each habitat: effect effect on on the the net net reproductive reproductive rate rate ffh(~/h) in each habitat:
aA. 2: aA. a f a;- h a fh --;;; '
O}t z - ~ OfhO-~Xo bfzOh , _
((16.7) 1 6.7)
all all derivatives derivatives are are evaluated evaluated at at Ii; ri; the the arguments arguments of of ffhh are are left left out out for for trans transparency formula. From parency of of the the formula. From the the general general equation equation for for eigenvalue eigenvalue sensitivity sensitivity ((Caswell, Caswell, 11989, 989, Eq. Eg. 6 . 6 ) , one 6.6), one gets gets
OX
UhVi
O-~h= ~z. < H i ; >
Oaih
Uh
~9-~h= 2~a vimhi" z
((~6.8) 1 6. 8 )
To 1 6.2) mhi = aih1fh, To proceed proceed further, further, note note that that from from Eg. Eq. ((16.2) aih/fh , and and that that kiviaih Eiviaih = Vh (this the definition these relationships (this follows follows from from the definition of of left left eigenvector). eigenvector). Using Using these relationships 1 6. 8 ) into in 1 6. 8 ) , noting in Eg. Eq. ((16.8), noting that that < u . v > - = 11 and and substituting substituting Eg. Eq. ((16.8) into Eg. Eq. ((16.7), 1 6.7), one one arrives arrives at at
0 X _- ~_~ ul~vh~__ofh. Oz
h
fhOZ
((16.9) 1 6.9)
116. 6. SOURCE-SINK SOURCE-SINK POPULATION POPULATION DYNAMICS DYNAMICS
405 405
Thus the relative on the the reproductive i.e., local Thus the relative effect effect of of trait trait zz on reproductive rate rate ((i.e., local fitness) fitness) in in each each habitat habitat is is weighed weighed by by the the pooled pooled reproductive reproductive value value of of individuals individuals present (Rousset, 11999a; 999a; see 993; present in in that that habitat habitat (Rousset, see also also Kawecki Kawecki and and Stearns, Stearns, 11993; Holt, 996b). The Holt, 11996b). The reproductive reproductive value value tends tends to to be be smaller smaller in in sink sink habitats habitats ((Section Section 116.2), 6 .2), and and sink sink habitats habitats tend tend to to harbor harbor fewer fewer individuals individuals than than sources. The The evolution evolution of of trait trait zz will will thus thus be be affected affected more more strongly strongly by by its its impact impact on on performance performance in in source source habitats. habitats. If If increasing increasing zz has has aa positive positive effect effect on sink, but negative effect on performance performance in in the the sink, but aa negative effect on on performance performance in in the the source, toward smaller source, the the trait trait will will evolve evolve toward smaller values values unless unless the the positive positive effect effect in larger than than the sink (Holt in the the source source is is considerably considerably larger the negative negative effect effect in in the the sink (Holt and 992; Kawecki, 995; Holt, 996b). Following and Gaines, Gaines, 11992; Kawecki, 11995; Holt, 11996b). Following this this logic, logic, one one laz = would would predict predict that that the the optimal optimal trait trait value value will will satisfy satisfy a}l.. aX/c3z = 00 (Holt (Holt and and Gaines, 992). Gaines, 11992).
Gene Gene Flow Flow versus versus Local Local Selection Selection The The approach approach just just given given is is simple simple and and elegant elegant and and has has been been used used to to gener generate Gaines, 11992; 992; Houston ate interesting interesting predictions predictions (e.g., (e.g., Holt Holt and and Gaines, Houston and and McNamara, 992; Brown 992; Kawecki 993; McNamara, 11992; Brown and and Pavlovic, Pavlovic, 11992; Kawecki and and Stearns, Stearns, 11993; Kawecki, 995; Holt, 996b). It Kawecki, 11995; Holt, 11996b). It is, is, however, however, problematic problematic because because it it neglects neglects genetic populations, which genetic differentiation differentiation between between populations, which may may be be substantial substantial if if the the dispersal dispersal rate rate is is low low in in relation relation to to selection selection coefficients coefficients operating operating on on individ individual 976; Chapter ual genetic genetic loci loci (Felsenstein, (Felsenstein, 11976; Chapter 77 of of this this volume) volume).. The The importance importance of of accounting accounting for for genetic genetic differentiation differentiation can can bbee illustrated illustrated by adaptation to black hole hole sink habitat ((i.e., i.e., aa habitat habitat that by considering considering adaptation to aa black sink habitat that receives repro receives immigrants immigrants but but sends sends no no dispersers dispersers back back to to the the source) source).. As As the the reproductive see Section 6.2), the approach ductive value value in in aa black black hole hole sink sink is is 00 ((see Section 116.2), the above above approach would allele beneficial deleterious in would predict predict that that an an allele beneficial in in the the sink sink and and deleterious in the the source source should should never never be be maintained maintained in in the the population. population. In In contrast, contrast, explicit explicit genetic genetic 997; Gomulkiewicz 99 9 ) demon models (Holt and and Gomulkiewicz, 11997; Gomulkiewicz et et aI., al., 11999) demonstrate strate that, that, although although eliminated eliminated deterministically deterministically from from the the source source habitat, habitat, such such an an allele allele will will be be maintained maintained in in the the sink sink if if the the local local net net reproductive reproductive rate rate of of its its carriers carriers exceeds exceeds 11.. The The effect effect ooff aa passive passive dispersal dispersal rate rate oonn adaptive adaptive evolution evolution in in aa source-sink source-sink system system is is another another issue issue where where qualitative qualitative discrepancies discrepancies arise arise between between the the predictions explicit genetic predictions of of fitness fitness sensitivity sensitivity approach approach and and explicit genetic models. models. In In aa two-patch model with symmetric dispersal m12 = m two-patch model with aa symmetric dispersal rate rate ((m12 m21), the pooled pooled 2 1 ), the reproductive reproductive value value of of the the subpopulation subpopulation in in the the sink sink (U (u~v~) typically increases increases hVh ) typically monotonically monotonically with with increasing increasing dispersal dispersal rate rate (for (for aa numerical numerical example, example, see see Fig. 6.4) . This Fig. 116.4). This is is largely largely because because aa greater greater dispersal dispersal rate rate shifts shifts the the spatial spatial distribution Section 116.3), 6 . 3 ) , exposing distribution of of the the population population ((Section exposing aa greater greater fraction fraction of of the the total total population population to to natural natural selection selection in in the the sink. sink. The The fitness fitness sensitivity sensitivity approach that high approach would thus thus suggest that high dispersal dispersal rates are most favorable favorable and and low low dispersal dispersal rates rates least least favorable favorable for for adaptation adaptation to to aa sink sink habitat habitat (Holt (Holt and and Gaines, 992; Kawecki, 995; Holt, 996a). Gaines, 11992; Kawecki, 11995; Holt, 11996a). However, However, the the dispersal dispersal rate rate also also affects affects the the amount amount of of gene gene flow flow between between habitats, habitats, and and thus thus the the degree degree of of genetic genetic differentiation differentiation between between source source and and sink sink habitats. The approach is habitats. The fitness fitness sensitivity sensitivity approach is likely likely to to provide provide aa reasonable reasonable approximation approximation if if gene gene flow flow is is already already strong strong enough enough to to prevent prevent any any substantial substantial
TADEUSZ I.j. KAWECKI
406 406
4.0 4.0 0.5 0.5 .., 3.5 3.5 3.0 L_--------==== 3.0 2.5 2.5 2.0 2.0 11.5.5 -------
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0.4 o'4
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=
Fig. in aa sink Fig. 11 6.4 6 . 4 The The pooled pooled reproductive reproductive value value of of individuals individuals in sink habitat, habitat, U2VZ, u21)2, as as aa function function of and habitat and (1 of dispersal dispersal rate rate and habitat quality. quality. The The model model follows follows Eq. Eq. (1 (1 6.1 6.1)) and (1 6.2) 6.2) with with two two patches patches all results and symmetric passive and symmetric passive dispersal dispersal m1 m122 = = m21; m21; fh(nh) fh(nh)== Rh/(l Rh/(1 + + nh)' nh). For For all results Rl R1 = = 4 4 is is assumed; ifferent values 2 < assumed; the the different different lines lines are are for for d different values of of R R2 indicated on on the the right. right. If If R R2 < 4, 4, habi habi2 indicated tat sink. tat 2 2 is is aa sink.
genetic genetic differentiation. differentiation. In In contrast, contrast, if if dispersal dispersal is is low, low, and and thus thus gene gene flow flow restricted, some degree degree of in restricted, some of local local adaptation adaptation may may be be possible: possible: alleles alleles beneficial beneficial in the sink but while remaining the sink but deleterious deleterious in in the the source source may may increase increase in in the the sink sink while remaining rare case, increasing result rare in in the the source. source. In In this this case, increasing the the dispersal dispersal rate rate will will first first of of all all result in swamping of pool in in greater greater swamping of the the local local gene gene pool in the the sink sink by by gene gene flow flow from from the the source. This increased dispersal dispersal on is source. This negative negative effect effect of of increased on adaptation adaptation to to the the sink sink is likely exposing aa greater likely to to outweigh outweigh any any positive positive effect effect due due to to exposing greater fraction fraction of of the the population sink habitat. This argument argument predicts population to to the the sink habitat. This predicts that, that, at at least least under under some some circumstances, dispersal rate circumstances, the the relationship relationship between between dispersal rate and and the the expected expected degree degree of of adaptation adaptation to to aa sink sink habitat habitat will will be be U-shaped U-shaped rather rather than than monotonic, monotonic, with with an an intermediate intermediate dispersal dispersal rate rate being being least least favorable. favorable. Furthermore, Furthermore, for for aa given given amount can maintain allele frequency amount of of gene gene flow, flow, selection selection can maintain greater greater allele frequency differen differentiation tiation between between the the habitats habitats at at loci loci with with larger larger effects effects (Felsenstein, (Felsenstein, 1976). 1976). For For that that reason reason the the range range of of dispersal dispersal rates rates over over which which the the conditions conditions for for adapta adaptation become more dispersal should tion to to aa sink sink become more favorable favorable with with increasing increasing dispersal should be be greater greater when when the the adaptation adaptation involves involves loci loci with with small small effects effects (Kawecki, (Kawecki, 2000). 2000). These These predictions predictions are are confirmed confirmed by by the the results results of of aa polygenic polygenic model model of of evo evolution described in 6 .5. This lution in in aa two-patch two-patch source-sink source-sink system system described in Fig. Fig. 116.5. This model model assumes habitats, mediated assumes aa fitness fitness trade-off trade-off between between the the habitats, mediated by by aa quantitative quantitative trait each with total trait determined determined by by up up to to eight eight additive additive loci loci each with two two alleles. alleles. The The total variability constant by variability range range of of the the trait trait is is kept kept constant by adjusting adjusting the the effects effects of of single single loci. results of model (symbols) compared to predic loci. The The results of the the genetic genetic model (symbols) are are compared to the the predictions of model based based on lines) . tions of an an optimality optimality model on the the fitness fitness sensitivity sensitivity approach approach ((lines). The The latter latter approach approach predicts predicts that that the the mean mean fitness fitness iinn the the sink sink habitat habitat should should increase monotonically monotonically with with the the dispersal dispersal rate rate ((lower line in in each each panel) panel).. increase lower line When optimality approach When the the trade-off trade-off is is mediated mediated by by eight eight loci, loci, the the optimality approach accur accurately model except less ately predicts predicts the the outcome outcome of of the the genetic genetic model except for for dispersal dispersal rates rates less than Fig. 116.5a). 6.5a). Only dispersal rates local populapopula than 0.05 0.05 ((Fig. Only at at such such low low dispersal rates can can the the local tions local population population in tions differentiate, differentiate, which which allows allows the the local in the the sink sink to to adapt adapt locally. local populations populations causes causes the locally. Genetic Genetic differentiation differentiation between between the the local the mean mean
116. 6.
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11 locus / locus O +---.---r---.--. o o0 0.1 0.2 0:2 0:3 014 0.5 0;5 0.3 0.4 Dispersal Dispersal rate rate m m12 m21 21 12 = m =
Fig. in aa source-sink Fig. 11 6.5 6 . 5 Results Results of of aa genetic genetic model model of of adaptive adaptive evolution evolution in source-sink system system contrasted contrasted with model is Eq. (1 6.1 ) and and (1 6.2) and with predictions predictions of of an an optimality optimality approach. approach. The The model is based based on on Eq. (16.1) (16.2) and assumes assumes aa trade-off trade-off in in relative relative fitness fitness across across habitats, habitats, mediated mediated by by aa quantitative quantitative trait trait z, which which under antagonistic antagonistic directional directional selection in the the two two habitats. This is implemented implemented by setting setting is under fh fh = = Rhwh(z)/(l RhWh(Z)/( 1 + + nh), nh), where where Rl R1 = 4 4 and and RR22 = = 2 2 and and the the relative relative fitness fitness is is Wl wl = = 11 - zz33 in in the the sink. Symmetric m1 2 = assumed. source source and and W w22 = = 11 - (1 (1 - Z)3 z) 3 in in the the sink. Symmetric dispersal dispersal rates rates ((m12 = m2l mzl)) are are assumed. In In the the genetic genetic model, model, trait trait z z is is determined determined by by one one to to eight eight freely freely recombining recombining loci loci with with equal equal and and additive additive effects, effects, and and codominance. codominance. The The phenotypic phenotypic effect effect of of each each locus locus is is inversely inversely pro proranges from portional portional to to the the number number of of loci loci so so that that z z always always ranges from 0 0 (maximum (maximum possible possible adaptation adaptation in the in the the in the source, source, zero zero fitness fitness in in the the sink) sink) to to 11 (maximum (maximum adaptation adaptation to to the the sink, sink, zero zero fitness fitness in source). obtained using using deterministic computer iterations in Kawecki source). Results Results were were obtained deterministic computer iterations (details (details in Kawecki and and Holt, Holt, 2002) 2002) until until an an evolutionary evolutionary equilibrium equilibrium was was reached; reached; initial initial allele allele frequencies frequencies at at all all loci loci have 0.5 (slightly among loci). have been been set set to to about about 0.5 (slightly different different among loci). Plots Plots show show mean mean relative relative fitnesses fitnesses (Wh) in (X) and (I-3) habitats as functions the dispersal rate and the number number of of loci (Wh) in source (X) and sink sink (0) functions of the and the loci coding in the (Z*) and and sink sink W coding for for trait trait z. z. Solid Solid lines lines show show relative relative fitnesses fitnesses in the source source Wl wl(z*) w2(z*) pre2(Z*) pre ddicted icted with approach (the same for all panels). value z* satisfies with an an optimality optimality approach (the same for all panels). The The optimal optimal trait trait value satisfies 2A/az2z < Ak = and ac32;k/az evaluated at -- 11,, aA/az 0X/c3z = = 0, 0, and < 0, 0, where where the the derivatives derivatives are are evaluated at the the equilibrium. equilibrium. =
fitnesses fitnesses in in the the two two habitats habitats to to become become less less bound bound by by the the trade-off. trade-off. As As the the number loci that decreases, and number of of loci that mediate mediate the the trade-off trade-off decreases, and thus thus the the effect effect of of each each single single locus locus increases, increases, the the range range of of dispersal dispersal rates rates permitting permitting local local adaptation adaptation in in the the sink sink increases. increases. This This causes causes the the results results of of the the genetic genetic model model to to deviate deviate increasingly increasingly from from the the predictions predictions of of the the optimality optimality approach; approach; the the minimum minimum of of the the mean mean fitness fitness in in the the sink sink habitat habitat is is shifted shifted toward toward higher higher dispersal dispersal rates rates 6.5b-1 6.5d). With locus, the (Fig. 116.5d). 6.5d). With only only aa single single locus, the fit fit is is very very poor poor (Fig. ((Figs. Figs. 116.5b-16.5d). An An additional additional factor factor that that reduces reduces the the mean mean fitness fitness in in the the one-locus one-locus model model is is the the segregational segregational load load -~ as as the the trade-off trade-off is is convex, convex, variance variance reduces reduces mean mean fitness. fitness. It It is is also also interesting interesting to to note note that that the the two-locus two-locus version version of of the the model model predicts the the same same mean mean relative relative fitness fitness in in both both habitats habitats at at high high dispersal dispersal predicts
408 408
TADEUSZ j.J. KAWECKI KAWECKI
rates r a t e s- at at equilibrium equilibrium the the two two loci loci are are fixed fixed for for the the alleles alleles with with opposite opposite effects effects and and no no genetic genetic variation variation remains. remains. Analyzing Analyzing the the properties properties of of equilib equilibria in in polygenic polygenic models models goes goes beyond beyond the the scope scope of of this this chapter, chapter, but but it it should should be be ria kept kept in in mind mind that that details details of of the the genetic genetic system system will will affect affect the the outcome outcome of of adap adaptive populations. tive evolution evolution in in source-sink source-sink populations. This This example example illustrates illustrates the the importance importance of of using using explicit explicit genetic genetic models models to to study study evolution evolution in in source-sink source-sink systems. systems. The The overall overall effect effect of of dispersal dispersal on on adap adaptive tive evolution evolution in in aa sink sink habitat habitat will will depend depend on on the the relative relative importance importance of of the the demographic effect demographic effect of of dispersal dispersal and and the the homogenizing homogenizing effect effect of of gene gene flow. flow.
Source-Sink Population Population Dynamics and Evolutionary Dynamics of of Ecological Niches In In the the model model described described above above the the mean mean relative relative fitness fitness in in the the sink sink is is typi typically habitat quality. cally lower lower than than in in the the source, source, thus thus magnifying magnifying differences differences in in habitat quality. Similar by many Similar predictions predictions have have been been reached reached by many published published models. models. Alleles Alleles with with aa small source habitat small positive positive effect effect on on fitness fitness in in the the source habitat will will tend tend to to be be favored favored even sink (Holt (Holt and 992; even if if they they have have large large negative negative effects effects in in the the sink and Gaines, Gaines, 11992; Holt, 996a; Kawecki, Holt, 11996a; Kawecki, 2000) 2000).. An An allele allele beneficial beneficial in in aa black black hole hole sink sink (no (no dis dispersal persal back back to to the the source source)) may may be be eliminated eliminated deterministically deterministically even even if if neutral neutral in source (e.g., 948; Nagylaki, 975; Slatkin, Slatkin, 11995; 995; Holt in the the source (e.g., Haldane, Haldane, 11948; Nagylaki, 11975; Holt and and Gomulkiewicz, 997). Source-sink Gomulkiewicz, 11997). Source-sink populations populations are are prone prone to to accumulate accumulate mutations source (Kawecki mutations deleterious deleterious in in the the sink sink but but neutral neutral in in the the source (Kawecki et et aI., al., 11997). 997). A A quantitative quantitative trait trait affecting affecting fitness fitness may may remain remain far far from from its its local local opti optimum habitat is mum in in aa sink sink habitat habitat if if the the optimum optimum in in the the source source habitat is different different (Garcia-Ramos 997; Kirkpatrick 997). To (Garcia-Ramos and and Kirkpatrick, Kirkpatrick, 11997; Kirkpatrick and and Barton, Barton, 11997). To summarize, natural natural selection selection is is expected expected to to maintain maintain or or improve improve adaptation adaptation in in summarize, habitats, where where the the population population is is already already well well adapted, adapted, and and be be ineffective ineffective in in habitats, improving marginal habitats. improving adaptation adaptation to to marginal habitats. This This implies implies that that ecological ecological niches niches should (Holt and 992; Kawecki, should usually usually be be evolutionarily evolutionarily conserved conserved (Holt and Gaines, Gaines, 11992; Kawecki, 11995; 995; Holt, 996b). Holt, 11996b). This This conclusion conclusion has has also also been been reached reached in in models models in in which which habitat-specific habitat-specific parameters are priori differences parameters are symmetric symmetric so so there there are are no no aa priori differences in in habitat habitat quality. quality. A A symmetric symmetric model model will will usually usually have have aa symmetric symmetric evolutionary evolutionary equi equilibrium, librium, at at which which the the mean mean fitness fitness in in all all habitats habitats would would be be the the same. same. However, However, such such an an equilibrium equilibrium may may be be unstable, unstable, and and even even when when it it is is stable, stable, alternative alternative asymmetric asymmetric equilibria equilibria may may exist; exist; which which equilibrium equilibrium is is reached reached will will depend population. Such depend on on the the initial initial genetic genetic composition composition of of the the population. Such alternative alternative asymmetric model asymmetric and and symmetric symmetric equilibria equilibria exist exist in in aa symmetric symmetric two-patch two-patch model by Kirkpatrick (200 1 ) . If initially well by Ronce Ronce and and Kirkpatrick (2001). If the the population population is is initially well adapted adapted to habitat 11 and poorly adapted adapted to habitat 2, remain so to habitat and poorly to habitat 2, it it will will tend tend to to remain so or or may may even even evolve evolve toward toward even even greater greater adaptation adaptation in in habitat habitat 11 and and reduced reduced fitness fitness in 2. The happens if population is in habitat habitat 2. The reverse reverse happens if the the population is initially initially adapted adapted to to habi habitat tat 2. 2. A A symmetric symmetric equilibrium equilibrium is is only only reached reached if if the the allele allele frequencies frequencies are are initially initially intermediate intermediate so so that that the the population population is is initially initially moderately moderately well well adapted species range adapted to to both both habitats. habitats. Similarly, Similarly, in in aa model model of of aa species range evolving evolving on on an an environmental environmental gradient, gradient, source-sink source-sink population population dynamics dynamics lead lead to to evolu evolution along the tion of of aa limited limited range, range, centered centered at at the the point point along the gradient gradient to to which which the the population was initially best adapted 997). This population was initially best adapted (Kirkpatrick (Kirkpatrick and and Barton, Barton, 11997). This
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effect is augmented effect is augmented by by character character displacement displacement caused caused by by interspecific interspecific compe competition Case and tition ((Case and Taper, Taper, 2000 2000).) . At At the the population population genetic genetic level level it it implies implies that that the the source-sink source-sink population population structure structure generates generates epistasis epistasis among among fitness fitness effects effects of of different different loci, loci, such such that that aa positive positive effect effect of of aa particular particular allele allele on on perform performance background adapted ance in in aa given given habitat habitat is is augmented augmented by by aa genetic genetic background adapted to to that that habitat. Conversely, Conversely, selection selection against against an an allele with with aa habitat-specific habitat-specific delete deleterious effect becomes weaker alleles with rious effect becomes weaker as as alleles with similar similar effects effects increase increase in in fre frequency, sink habitat quency, which which may may lead lead to to aa mutational mutational erosion erosion of of fitness fitness in in aa sink habitat (Kawecki aI., 11997). 997). (Kawecki et et al., Nonetheless, Nonetheless, the the prediction prediction that that ecological ecological niches niches should should be be conserved conserved evo evolutionarily lutionarily is is not not absolute. absolute. A A number number of of evolutionary evolutionary changes changes of of ecological ecological niches niches have have been been directly directly observed, observed, including including host host shifts shifts in in herbivorous herbivorous insects insects or or repeated repeated evolution evolution of of tolerance tolerance of of plants plants to to high high concentrations concentrations of of heavy heavy metals. This raises the question about about environmental environmental factors properties of metals. This raises the question factors and and properties of the population will the organism, organism, which which make make it it more more likely likely that that aa population will adapt adapt to to aa novel habitat, novel habitat, which which is is initially initially aa sink. sink. Dispersal Dispersal rate rate and and pattern pattern are are obviously obviously of of crucial crucial importance. importance. Given Given the the tension tension between between local local adaptation adaptation and and gene gene flow, flow, one-time one-time colonization colonization of of the the novel novel habitat habitat followed followed by by complete complete isolation isolation would would seem seem most most favor favorable. persistent population locally malmal able. However, However, foundation foundation of of aa persistent population by by aa few few locally adapted exceptions like Darwin's adapted colonizers colonizers must must be be rare, spectacular exceptions Darwin's finches finches notwithstanding. notwithstanding. If If the the population population initially initially performs performs poorly, poorly, it it will will likely become extinct Gomulkiewicz and likely become extinct before before it it has has time time to to adapt adapt ((Gomulkiewicz and Holt, Holt, 11995), 99 5 ) , especially especially that that aa single single colonization colonization event event will will typically typically be be associated associated with with aa bottleneck bottleneck causing causing loss loss of of heritable heritable variation. variation. If If so, so, gene gene flow flow fol following lowing the the initial initial colonization colonization may may facilitate facilitate adaptation adaptation to to the the novel novel habi habitat Caprio and 992; tat by by replenishing replenishing genetic genetic variation variation ((Caprio and Tabashnik, Tabashnik, 11992; Gaggiotti, 996; Gaggiotti 996; Chapter 5 ) . Finally, Gaggiotti, 11996; Gaggiotti and and Smouse, Smouse, 11996; Chapter 115). Finally, com complete plete elimination elimination of of gene gene flow flow may may be be impossible. impossible. The The above above model model suggests suggests that dispersal rates favorable for adaptation to that high high dispersal rates will will often often be be more more favorable for adaptation to aa marginal than intermediate marginal habitat habitat than intermediate dispersal dispersal rates, rates, particularly particularly if if genes genes with with small see also Kawecki small effects effects are are involved involved ((see Kawecki and and Holt, Holt, 2002 2002).) . This This conclu conclusion sion is, is, however, however, contradicted contradicted by by spatially spatially explicit explicit models models of of populations populations adapting Kirkpatrick and 99 7; adapting to to an an environmental environmental gradient gradient ((Kirkpatrick and Barton, Barton, 11997; Salathe and Kawecki, high dispersal Salathe and Kawecki, unpublished unpublished results) results),, where where high dispersal rates rates are are most Garcia most unfavorable unfavorable for for adaptation adaptation to to sink sink habitats. habitats. Another Another model model ((GarciaRamos Ramos and and Rodriguez, Rodriguez, 2002 2002)) predicts predicts aa nonlinear nonlinear relationship relationship between between dis dispersal persal and and evolutionary evolutionary invasions invasions of of novel novel habitats. habitats. It It is is not not clear clear which which of of the the differences differences in in assumptions assumptions of of these these models models were were responsible responsible for for these these different different predictions. predictions. Gene Gene flow flow can can occur occur through through both both sexes, sexes, but but in in species species without without paternal paternal care, care, only only female female dispersal dispersal contributes contributes to to the the maintenance maintenance of of local local populations populations in in sink sink habitats. habitats. One One would would therefore therefore expect expect that that female-biased female-biased dispersal dispersal would would be be more more favorable favorable for for adaptation adaptation to to aa sink sink habitat habitat than than sex-independent sex-independent or or male-biased dispersal. dispersal. A A genetic genetic model model assuming assuming independent independent male male and and female female dispersal dispersal rates rates confirms confirms this this intuition, intuition, although although depending depending on on the the parameters, parameters, the the conditions conditions for for adaptation adaptation to to the the sink sink may may be be least least favor favorable able under under moderately moderately rather rather than than extremely extremely male-biased male-biased dispersal dispersal (Kawecki, (Kawecki, 2003 2003).) .
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Finally, Finally, Kawecki Kawecki and and Holt Holt (2002) (2002) considered considered the the evolutionary evolutionary effect effect of of the the reverse source-sink structure, whereby reverse source-sink structure, whereby an an environment-imposed environment-imposed asymmetry asymmetry of of dispersal rates causes habitat to dispersal rates causes an an "upstream" "upstream" poorer poorer habitat to act act as as an an effective effective source downstream" high-quality sink (Section 6.2). In source and and aa ""downstream" high-quality habitat habitat as as aa sink (Section 116.2). In their model, selection selection tended tended to source habitat their model, to be be more more effective effective in in the the source habitat even even if if it it was was of of lower lower quality quality than than the the sink sink habitat. habitat. They They concluded concluded that, that, assuming assuming sufficient sufficient genetic genetic variance, variance, over over evolutionary evolutionary time time the the population population should should adapt adapt to to the the upstream upstream habitat habitat at at the the expense expense of of reduced reduced fitness fitness in in the the downstream downstream habitat. population dynamics habitat. In In this this case, case, source-sink source-sink population dynamics would would thus thus promote promote an an evolutionary evolutionary shift shift of of the the ecological ecological niche. niche. The dispersal rates The effect effect of of factors factors other other than than dispersal rates on on adaptation adaptation to to aa sink sink habitat habitat has has not not been been investigated investigated systematically. systematically. Fitness Fitness sensitivity sensitivity analysis analysis of 1 6 . 6 ) suggested area of aa model model described described by by Eq. Eq. ((16.6) suggested that that increasing increasing the the relative relative area of makes the of the the sink sink habitat habitat makes the conditions conditions for for adaptation adaptation to to the the sink sink more more favorable, favorable, but but only only when when the the differences differences in in habitat habitat quality quality are are not not large large ((Kawecki, Kawecki, 11995). 995 ) . This This conclusion conclusion still still needs needs to to be be supported supported by by aa genetic genetic model. habitat should model. Several Several models models suggest suggest that that adaptation adaptation to to aa sink sink habitat should be be more involved compared compared to more likely likely if if few few majo majorr loci loci are are involved to many many loci loci with with small small effects e.g., Holt Gomulkiewicz, 11997; 997; Kawecki, Kawecki, 2000) effects ((e.g., Holt and and Gomulkiewicz, 2000).. Density Density dependence dependence in in the the sink sink makes makes the the conditions conditions for for adaptation adaptation to to the the sink sink habi habitat Holt, 11996a; 996a; Gomulkiewicz aI., 11999). 99 9 ) . It tat less less favorable favorable ((Holt, Gomulkiewicz et et al., It is, is, however, however, not how general Most of them were derived from not clear clear how general these these predictions predictions are. are. Most of them were derived from two-patch, turn, spatially two-patch, spatially spatially implicit implicit models. models. In In turn, spatially explicit explicit models models com combining bining source-sink source-sink population population dynamics dynamics and and evolution evolution have have been been based based on on the the diffusion diffusion equation equation and and infinitesimal infinitesimal quantitative quantitative genetic genetic approximation approximation (e.g., 997; Case (e.g., Kirkpatrick Kirkpatrick and and Barton, Barton, 11997; Case and and Taper, Taper, 2000 2000).) . Future Future model modeling should combine ing of of evolution evolution in in source-sink source-sink systems systems should combine spatially spatially explicit explicit and and genetically genetically explicit explicit approaches. approaches.
Evidence Evidence for for Maladaptation M a l a d a p t a t i o n in in Sink Sink Habitats Habitats The The average average reproductive reproductive success success in in aa sink sink habitat habitat is is poor. poor. The The difficult difficult part part is poor at is to to show show that that it it is is poor at least least partially partially because because of of gene gene flow flow from from source source habitats. habitats. This This has has been been demonstrated demonstrated convincingly convincingly in in only only aa few few cases. cases. The The best best evidence evidence for for gene gene flow flow hampering hampering adaptation adaptation in in aa sink sink habitat habitat comes 6.4. Populations comes from from the the blue blue tit tit system system described described in in Section Section 116.4. Populations in in main mainland land southern southern France France have have aa high high breeding breeding success success in in the the deciduous deciduous habitat, habitat, whereas whereas in in the the sclerophyllous sclerophyllous habitat habitat the the breeding breeding success success and and population population density density are are low. low. However, However, on on the the island island of of Corsica, Corsica, where where the the sclerophyllous sclerophyllous forest is is the the dominant dominant habitat habitat the the breeding breeding success success in in that that habitat habitat type type is is forest higher mainland, despite higher than than on on the the mainland, despite much much higher higher local local density density (Blondel (Blondel et et aI., al., 11992; 992; Dias 996). Furthermore, Dias and and Blondel, Blondel, 11996). Furthermore, the the breeding breeding success success of of the the Corsican pockets of deciduous habitat Corsican population population in in small small pockets of deciduous habitat on on the the island island is is poorer poorer than than in in the the sclerophyllous sclerophyllous habitat; habitat; i.e., i.e., the the deciduous deciduous habitat habitat tends tends to to act (Dias and 996). act as as aa sink sink (Dias and Blondel, Blondel, 11996). It It could could still still be be argued argued that that the the difference difference in in breeding breeding success success in in the the sclero sclerophyllous phyllous habitat habitat between between Corsica Corsica and and the the mainland mainland reflects reflects different different product productivity ivity of of the the sclerophyllous sclerophyllous habitat habitat on on the the island island than than on on the the mainland, mainland, rather rather than differential differential adaptation. adaptation. However, However, the the argument argument of of maladaptation maladaptation is is also also than
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supported supported by by data data on on breeding breeding phenology. phenology. The The breeding breeding phenology phenology is expected to the main main to be be synchronized synchronized with with the the availability availability of of caterpillars, caterpillars, which which are the food for food food for for the the young, young, so so that that the the peak peak demand demand of of the the brood brood for food coincides with occurs with the the peak peak of of caterpillar caterpillar availability. availability. This This peak peak of of food food availability availability occurs about about aa month month earlier earlier in in the the deciduous deciduous than than in in the the sclerophyllous sclerophyllous habitat. habitat. Rather Rather than than showing showing aa pattern pattern of of local local adaptation, adaptation, the the breeding breeding phenology phenology of birds birds on on the the mainland mainland does does not not differ differ between between habitats habitats and and is synchronized synchronized with the with caterpillar caterpillar availability availability in in the the source source (deciduous) (deciduous) habitat. habitat. The The birds birds in the sink sink (sclerophyllous) ( sclerophyllous) habitat habitat lay lay their their eggs eggs almost almost aa month month too too early and, and, as a consequence, consequence, suffer suffer additional additional reduction reduction of of breeding breeding success. success. The The reverse holds holds in in Corsica, Corsica, where where the the breeding breeding phenology phenology is is synchronized synchronized with with caterpilcaterpil lar lar availability availability in in the the sclerophyllous sclerophyllous habitat; habitat; birds birds breeding breeding in in small small pockets pockets of of deciduous deciduous habitat habitat lay lay their their eggs eggs much much too too late late (Dias (Dias and and Blondel, Blondel, 1996). 1 996). Thus Thus inin both both regions regions the the breeding breeding phenology phenology is is well well adapted adapted to to the the source source habitat habitat and and maladapted maladapted to to the the sink sink habitat. habitat. The The difference difference in in the the laying laying date date is genetic genetic (Blondel (Blondel et et al., ai., 1990), 1 990), and and itit isis unlikely unlikely that that the the lack lack of of adaptation adaptation to to the the sink sink habitat habitat isis due due to to aa lack lack of of heritable heritable variation variation for for the the laying laying date. date. The The conclusion conclusion about about maladaptation maladaptation of of the the blue blue tits tits in in the the sink sink habitats habitats is thus thus supported supported both both by by an an optimality optimality analysis analysis and and by by comparison comparison of of the the perper formance formanceof ofthe thelocal local populations populations in in patches patches of of the the same same habitat habitat located located in in difdif ferent ferentlandscapes. landscapes. AAsimilar similar oDtimalitv optimality armroach approach has has been been used used to to demonstrate demonstrate mmaladaptation a l ~' - - - d ~ a u u' u -, ~ a t "tit~ uof l , -l~clutch . ~ u t. ~ in ~size i ~ 'ifi i S ~ t u [ u l ~t i . , i1~9 g b8~8) ) aand u d rreproduc ~prudocin.~-great tits ((Pettifor et aai.,
tive tiveeffort effortand andoffspring offspringsize size of ofmosquitofish mosquitofish in in aa marginal marginal population population (Stearns (Stearns and and Sage, Sage, 1980). 1 9 8 0 ) . AA spectacular spectacular counterexample counterexample is is the the repeated repeated evolution evolution of of heavy heavy metal metal tolerance tolerance by by numerous numerous plant plant species species that that colonized colonized abandoned abandoned heavy heavy metal metal mining mining sites sites and and zinc-polluted zinc-polluted areas areas around around the the bases bases of of electrielectri city city pylons pylons (e.g., (e.g., Jain Jain and and Bradshaw, Bradshaw, 1966; 1 966; Coulaud Coulaud and and McNeilly, McNeilly, 1992; 1 992; Alhiyaly Alhiyalyetetal., ai., 1993; 1 993; Nordal Nordal et et al., ai., 1999). 1 999). Initially, Initially, these these sites sites must must have have concon stituted small small pockets pockets of of aa sink sink habitat habitat surrounded surrounded by by aa large large source source habitat. habitat. stituted However, However, the the colonizers colonizers were were in in aa short short time time able able to to adapt adapt to to the the toxic toxic envirenvir onment, onment, despite despite continuous continuous gene gene flow. flow. Genetic Genetic studies studies reveal reveal that that in in most most cases, cases,heavy heavymetal metaltolerance tolerance in in plants plants involves involves several several major major loci, loci, although although the the contribution contribution of of minor minor loci loci isis not not excluded excluded (e.g., (e.g., MacNair, MacNair, 1993; 1 993; Schat Schat et et al., ai., 1996). 1 996). This This finding finding isis consistent consistent with with the the prediction prediction that that adaptation adaptation to to aa sink sink habitat habitatwould would be be more more likely likely ifif itit involved involved few few major major genes genes rather rather than than many many genes geneswith withsmall small effects effects (see (see earlier earlier discussion). discussion). Using Using reciprocal reciprocal transplants transplants of of seeds, seeds, seedlings, seedlings, and and adults, adults, Stanton Stanton and and Galen Galen (1997; ( 1 997; see see Section Section 16.4) 1 6 .4) have have shown shown that that snow snow buttercup buttercup populations populations living livingatatearly earlyand and late late melting melting sites sites do do not not show show aa pattern pattern of of local local adaptation adaptation tototheir theirrespective respective sites. sites. They They do do not not seem seem to to be be differentiated differentiated genetically genetically with with respectto to any any fitness-related fitness-related character. character. Instead, Instead, irrespective irrespective of of the the destination destination respect habitat, seeds seedsoriginating originating from fromlate late melting melting sites sites are are 2.5 25% less likely likely to to germingermin habitat, % less ate despite despite being being only only 8% 8 % smaller. smaller. One One can can speculate speculate that that in in the the absence absence of of ate gene flow, flow, local local populations populations at at late-melting late-melting sites sites would would evolve evolve toward toward propro gene ducing ducingfewer fewer larger larger seeds, seeds, and and that that this this change change is is prevented prevented by by the the gene gene flow. flow. Research on on the the checkerspot checkerspot butterfly butterfly (see (see Section Section 16.4) 1 6.4) provides provides some some evievi Research dence dence for for alternative alternative equilibria, equilibria, similar similar to to those those predicted predicted by by Ronce Ronce and and Kirkpatrick (2001). (200 1 ) . After After the the local local populations populations in in the the original original source source habitat habitat Kirkpatrick (forestclearings) clearings) had had been been wiped wiped out out by by aa frost, frost, in in several several localities localities the the original original (forest
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source-sink source-sink structure structure was was not not recreated. recreated. Instead, Instead, the the population population density density became became much rocky outcrops), much higher higher in in the the former former sink sink habitat habitat ((rocky outcrops), whereas whereas individuals individuals attempting recolonize the poor reproductive attempting to to recolonize the former former source source habitat habitat had had poor reproductive suc success. cess. Thus, Thus, the the source-sink source-sink structure structure became became reversed. reversed. This This reversal reversal was was not not due due to consequence of phenological differences to an an evolutionary evolutionary change change but but was was aa consequence of phenological differences between between the the habitats: habitats: migrants migrants from from the the outcrops outcrops arrived arrived too too late late to to reproduce reproduce successfully clearings. Once resident population established in successfully in in the the clearings. Once aa resident population was was established in aa clearing, it 999). Nonetheless, clearing, it expanded expanded quickly quickly (Boughton, (Boughton, 11999). Nonetheless, this this example example illus illustrates trates aa potential potential for for alternative alternative source-sink source-sink equilibria. equilibria. A A promising promising approach approach to to study study evolutionary evolutionary consequences consequences of of aa source-sink source-sink population structure population structure would would be be to to set set up up laboratory laboratory source-sink source-sink systems systems and and let populations evolve let experimental experimental populations evolve in in them them for for generations. generations. This This "experi "experimental mental evolution evolution"" approach approach has has been been applied applied successfully successfully to to other other evolution evolutionary questions, concerning, 990), ary questions, concerning, e.g., e.g., reproductive reproductive isolation isolation (Rice (Rice and and Salt, Salt, 11990), life life history history (Stearns (Stearns et et aI., al., 2000), 2000), or or learning learning ability ability (Mery (Mery and and Kawecki, Kawecki, 2002 2002).) . Although Although many many studies studies involved involved experimental experimental evolution evolution in in novel novel habi habitats, tats, few few included included experimental experimental populations populations evolving evolving in in heterogeneous heterogeneous envir environments, onments, with with different different habitats habitats connected connected by by dispersal. dispersal. Several Several of of those those studies studies focused focused on on the the role role of of environmental environmental heterogeneity heterogeneity in in the the maintenance maintenance of (McDonald and of genetic genetic variation variation at at allozyme allozyme loci loci (McDonald and Ayala, Ayala, 1974; 1974; Powell Powell and and Wistrand, 978; Haley 9 8 3 ) and Wistrand, 11978; Haley and and Birley, Birley, 11983) and quantitative quantitative traits traits (MacKay, (MacKay, 11981; 9 8 1 ; Garcia-Dorado 9 9 1 ; Hawthorne, 997). In Garcia-Dorado et et aI., al., 11991; Hawthorne, 11997). In those those studies studies the the habitats contributed equally equally to soft selection). habitats contributed to the the total total reproduction reproduction ((soft selection). This This design relationship between design eliminated eliminated the the relationship between mean mean performance performance in in aa habitat habitat and and this habitat's habitat's contribution contribution to to the the total total reproduction, reproduction, which which is is an an important important this characteristics studies were characteristics of of source-sink source-sink populations. populations. Other Other studies were focused focused on on the the evolution 986; Rice 990). evolution of of habitat habitat choice choice (Bird (Bird and and Semeonoff, Semeonoff, 11986; Rice and and Salt, Salt, 11990). Only Only aa few few compared compared adaptation adaptation to to aa novel novel habitat habitat between between lines lines exposed exposed only only to to the the novel novel habitat habitat and and lines lines exposed exposed to to both both habitats habitats (Wasserman (Wasserman and and Futuyma, 9 8 1 ; Mark, 982; Verdonck, 987; Taper, 990). Because Futuyma, 11981; Mark, 11982; Verdonck, 11987; Taper, 11990). Because these these studies studies were were also also concerned concerned with with habitat habitat choice, choice, the the adults adults could could choose choose the the habitat habitat for for oviposition, oviposition, and and the the amount amount of of gene gene flow flow was was not not controlled. controlled. Verdonck 1 987) let Verdonck ((1987) let D. melanogaster melanogaster populations populations evolve evolve in in cages cages containing containing two media: aa standard two media: standard medium medium and and aa medium medium supplemented supplemented with with NaCI. NaC1. The The latter latter medium medium created created aa sink sink habitat, habitat, with with low low larval larval survival survival (although (although not not an an absolute absolute sink) sink).. Despite Despite the the asymmetric asymmetric gene gene flow, flow, the the experimental experimental popula populations tions did did evolve evolve improved improved tolerance tolerance to to NaCl, NaCI, but but to to aa lesser lesser degree degree than than con control trol populations populations bred bred exclusively exclusively to to the the NaCI-supplemented NaCl-supplemented medium. medium. Thus, Thus, in in this this case, case, asymmetric asymmetric gene gene flow flow slowed slowed down, down, but but did did not not completely completely prevent prevent adaptation to 1 990) maintained populations of adaptation to aa sink sink habitat. habitat. Taper Taper ((1990) maintained populations of the the cow cowpea weevil weevil (Callosobruchus (Callosobruchusmaculatus) maculatus) on on aa mixture mixture of of two two host host seed seed species, species, pea either either on on its its own, own, or or together together with with aa competing competing species species specializing specializing on on one one of of hosts. In the hosts. In this this latter latter treatment treatment the the competition competition pressure pressure from from the the other other the species species caused caused that that host host to to become become effectively effectively aa sink sink habitat. habitat. As As predicted, predicted, caused the competition competition with with the the specialist specialist competitor competitor caused the generalist generalist species species to to become less well host species used by become less well adapted adapted to to the the host species used by the the competitor competitor and and better better adapted host species adapted to to the the other other host species (character (character displacement) displacement).. These These studies studies suggest suggest that that the the "experimental "experimental evolution" evolution" approach approach has has aa great great potential potential to to provide insights into provide insights into evolution evolution in in heterogeneous heterogeneous environments. environments.
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SOURCE-SINK SOURCE-SINK METAPOPULATIONS METAPOPULATIONS The The concept concept of of source-sink source-sink population population structure structure emphasizes emphasizes the the effect effect of of dis dispersal population concept persal on on the the local local population population dynamics, dynamics, whereas whereas the the meta metapopulation concept has has originally originally been been motivated motivated by by local local extinctions extinctions and and colonizations colonizations (Levins, (Levins, 1968a). 1968a). Source-sink Source-sink population population structure structure results results from from differences differences in in habitat habitat qual quality, population structure ity, whereas whereas meta metapopulation structure reflects reflects patchiness patchiness of of the the environment. environment. Both Both concepts concepts are are concerned concerned with with the the role role of of dispersal, but but the the source-sink source-sink structure structure requires much much greater greater dispersal dispersal rates rates (i.e., (i.e., greater greater connectivity connectivity of of habi habitat tat patches), patches), which which would would prevent prevent habitat habitat patches patches from from remaining remaining unoccupied. unoccupied. However, many real spatially structured structured populations populations are likely to be affected by both both processes. processes. First, First, in in aa "classic" "classic" metapopulation, metapopulation, immigration may may signifi significantly cantly reduce the the local local extinction extinction rate rate (rescue (rescue effect; effect; Brown Brown and and Kodric-Brown, Kodric-Brown, 11977; 977; Chapters Chapters 44 and and 14). 14). It It may may also also boost the the local local population size size and and the the number of of propagules propagules it it produces, thus thus potentially potentially increasing the the colonization colonization rate. rate. Second, Second, some some local local populations populations (those (those in in large large habitat habitat patches, patches, or or in in the the vicinity vicinity thereof) thereof) may may show show typical typical source-sink source-sink dynamics dynamics with with negligible extinc extinction tion probability, probability, whereas whereas the the fate fate of of others others (those in in small small and and more more isolated isolated patches) patches) will will be be dominated dominated by by extiction-recolonization extiction-recolonization dynamics. dynamics. This This idea idea is is explicit explicit in in metapopulation metapopulation models models of of limits limits of of species species ranges ranges (e.g., Lennon Lennon et et aI., al., 11997; 997; Holt Holt and and Keitt, Keitt, 2000). The The concept concept of of source-sink source-sink dynamics dynamics can can also also be be extended extended to to extinction-recolonization extinction-recolonization dynamics dynamics by by allowing allowing the the extinction extinction rate rate or or the the contribution Chapter 4). contribution to to the the pool pool of of colonizers colonizers to to vary vary among among patches ((Chapter 4). Most empty empty patches patches would would then then be be colonized colonized by by individuals originating from from patches with with more more persistent and and larger larger populations populations (sources). (sources). Colonizers Colonizers from from such such source source patches patches may may maintain maintain aa significant significant level level of of patch patch occupancy occupancy in in neigh neighboring boring sink sink patch patch networks, networks, in in which which otherwise otherwise extinction extinction rate rate would exceed exceed colonization. metapopulation model, in colonization. The The mainland-island mainland-island metapopulation in which which all all colon colonizing "mainland" population, izing individuals individuals originate originate in in aa permanent permanent "mainland" population, is is an an extreme 6.2). Such extreme case, case, analogous analogous to to the the black black hole hole sink sink (Section (Section 116.2). Such source-sink source-sink extinction-colonization extinction-colonization dynamics dynamics is is implicit implicit in in most most structured structured or or spatially spatially explicit 993; Hanski, explicit metapopulation metapopulation models models (e.g., (e.g., Hanski Hanski and and Gyllenberg, Gyllenberg, 11993; 11994; 994; Chapters Chapters 44 and and 55).) . The The distinction distinction between between source-sink source-sink dynamics dynamics at at the the level level of of extinction-colonization extinction-colonization dynamics versus at at the the level level of of local population population dynamics aI., 11999). 999). dynamics disappears disappears in in individual-based individual-based models models (e.g., (e.g., Wiegand Wiegand et et al.,
116.8 6.8
CONCLUSIONS CONCLUSIONS AND A N D PROSPECTS PROSPECTS This necessarily incomplete review of ecological and and evolutionary evolutionary aspects of the the source-sink source-sink population population structure structure elucidates elucidates its its importance importance for for population population dynamics, dynamics, size, size, distribution, distribution, and and persistence, persistence, as as well well as as for for the the understanding understanding of of evolutionary evolutionary dynamics dynamics of of ecological niches niches and and species ranges. The The import importance manage ance of of source-sink source-sink dynamics dynamics for for biodiversity biodiversity conservation conservation and and pest pest management ment has has been been widely widely recognized. recognized. As As in in many many other other areas areas of of population population biology, biology, the the development development of of theory theory has has outpaced outpaced the the accumulation accumulation of of empirical empirical data. data. In In particular, particular, direct direct experi experimental data data addressing addressing ecological ecological and evolutionary evolutionary consequences consequences of
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source-sink population dynamics reason is source-sink population dynamics are are scarce. scarce. One One reason is the the fact fact that that most most research motivated concept has concentrated on birds, research motivated by by the the source-sink source-sink concept has concentrated on birds, mammals, and long-lived flowering plants. Experimental mammals, and long-lived flowering plants. Experimental manipulations manipulations of of spatial population structure spatial population structure (e.g., (e.g., preventing preventing dispersal, dispersal, changing changing the the amount amount of habitat) should should be of source source or or sink sink habitat) be more more feasible feasible in in insects insects or or mites. mites. Their Their shorter allow one those manipula shorter generation generation time time would would allow one to to see see the the effects effects of of those manipulations ideal model tions sooner. sooner. Some Some insects insects or or mites mites are are also also ideal model organisms organisms for for labora laboratory could be tory source-sink source-sink systems. systems. Such Such systems systems could be combined combined with with the the ""experimental experimental evolution" evolution" approach approach to to study study the the evolutionary evolutionary consequences consequences of of the source-sink source-sink population population structure. structure. This This approach approach should should be be promoted. promoted. the Some concept have also remained Some applied applied aspects aspects of of the the source-sink source-sink concept have also remained neg neglected. lected. In In particular, particular, the the concept concept has has important important implications implications for for epidemiology epidemiology and human population sink habitat and public public health; health; the the human population is is aa sink habitat for for numerous numerous para parasites sites and and pathogens pathogens (e.g., (e.g., the the rabies rabies virus) virus).. The The concept concept also also applies applies to to the the dynamics dynamics of of pathogens pathogens within within the the host's host's body, body, whereby whereby some some organs organs may may be be sources others sinks sinks for has medical sources and and others for the the pathogen. pathogen. This This has medical implications, implications, as as anti pathogen drugs only target antipathogen drugs will will be be ineffective ineffective if if they they only target pathogens pathogens in in sink sink organs. human diseases organs. Some Some dangerous dangerous human diseases are are caused caused by by pathogens pathogens invading invading organs organs are hole organs from from which which they they cannot cannot transmit; transmit; such such organs are thus thus black black hole sinks. sinks. Finally, Finally, our our own own population population has has aa source-sink source-sink structure, structure, with with important important economic and social consequences. economic and social consequences. To To summarize, summarize, although although much much progress progress has has been been made made since since Pulliam's Pulliam's ((1988) 1 9 8 8 ) seminal seminal paper, paper, much much work work remains remains to to be be done done before before we we can can fully fully understand evolutionary consequences understand the the ecological ecological and and evolutionary consequences of of the the source-sink source-sink population population structure. structure.
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META PO PULATION M ETAPO PU LATIO N DYNAMICS DYNAMICS OF OF INFECTIOUS ASES INFECTIOUS DISE DISEASES Matt Matt J. Keeling, Ottar Ottar N. N. Bjornstad, Bjornstad, and and Bryan Bryan T. Grenfell
117.1 7. 1
INTRODUCTION INTRODUCTION John John Donne's Donne's famous famous line line "No "No man man is is an an island, island, entire entire of of itself" itself" has has deep deep resonances resonances for for the the dynamics dynamics of of parasites. parasites. This This is is particularly particularly true true for for microparasitic microparasitic infections, infections, such such as as viruses viruses and and bacteria, bacteria, for for which which each each suscep susceptible tible host host is is aa potential potential patch patch of of favourable favourable habitat. habitat. Propagules Propagules from from infected infected ""patches" patches" can others, followed parasitic multiplication multiplication and can colonize colonize others, followed by by parasitic and ""local" local" growth parasite population. scale of host popu growth of of the the parasite population. Thus, Thus, at at the the scale of the the host population, infectious infectious dynamics dynamics bears bears strong strong analogies analogies to to metapopulation metapopulation dynam dynamlation, ics. Furthermore, ics. Furthermore, host host individuals individuals are, are, more more often often than than not, not, structured structured into into populations, within local populations, within which which contact contact among among hosts hosts may may be be very very frequent frequent and and between between which which contacts contacts may may be be less frequent. frequent. In In this this way, way, the the spatiotem spatiotemporal dynamics dynamics and and persistence persistence of of parasites parasites are are determined determined at at two two scales: scales: the the infrapopulation (a local local population population scale; parasites within infrapopulation scale scale (a scale; parasites within hosts) hosts) and and the the metapopulation spatial and/or metapopulation scale scale ((spatial and/or social social aggregation aggregation of of hosts hosts).) . The The spa spatiotemporal tiotemporal dynamics dynamics of of infection infection in in human human and and domestic domestic systems systems are are of of par particular combined with ticular academic academic interest interest because because of of the the wealth wealth of of data data combined with well-described histories. well-described natural natural histories. As As aa result result of of the the dual dual spatial spatial scales scales of of regulation, regulation, an an extended extended metapopu metapopulation disease dynamics lation paradigm paradigm is is central central to to infectious infectious disease dynamics in in two two important important
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ways. First, First, the population approach help us ways. the meta metapopulation approach can can help us understand understand disease disease dynamics dynamics at at the the different different spatial spatial scales. scales. This This topic topic is is the the main main concern concern here, here, we we use realistic dynamic models to discuss the use extensive extensive data data sets sets and and realistic dynamic models to discuss the metapopu metapopulation dynamics Second, there lation dynamics of of infectious infectious disease. disease. Second, there are are important important conceptual conceptual insights eradication by vaccination of insights about about the the eradication by vaccination of infections infections to to be be gained gained from from studies 994; Grenfell and studies of of the the persistence persistence of of metapopulations metapopulations (Nee, (Nee, 11994; Grenfell and Harwood, 997; Ovaskainen 3 ) . This Harwood, 11997; Ovaskainen and and Grenfell, Grenfell, 200 2003). This chapter chapter therefore therefore explores explores two two main main topics: topics: (i) (i) the the analogies analogies between between the the disciplines disciplines of of ecology ecology and epidemiology and (ii) how meta population and epidemiology at at the the metapopulation-level metapopulation-level and (ii) how metapopulation theory theory at at aa variety variety of of scales scales can can aid aid our our understanding understanding of of epidemiological epidemiological dynamics. dynamics. We We discuss discuss these these issues issues in in the the face face of of aa set set of of detailed detailed models models and and high-resolution disease incidence. high-resolution space-time space-time data data of of disease incidence. Metapopulation-like Metapopulation-like disease disease dynamics dynamics occur occur whenever whenever the the environment, environment, in in this this case case the the population population of of susceptibles, susceptibles, is is sufficiently sufficiently patchy patchy that that isolated isolated clumps clumps of of suitable suitable habitat habitat exist. exist. This This is is always always the the case case at at the the microscale; microscale; each each host host is is an an island island to to be be colonized colonized and and aa resource resource patch patch to to be be depleted. depleted. At At the the macro scale, hosts macroscale, hosts are are usually usually aggregated aggregated in in local local communities communities within within which which transmission transmission is is relatively relatively frequent frequent and and between between which which infection infection spreads spreads at at aa lower rate. rate. Our dominant focus population (macro)scale. lower Our dominant focus is is on on the the meta metapopulation (macro)scale. To To illus illustrate trate the the key key issues, issues, we we first first introduce introduce aa simple simple epidemic epidemic model model and and then then use use this this to to illuminate illuminate the the basic basic processes processes in in the the spatiotemporal spatiotemporal dynamics dynamics of of epidemics. epidemics. Two Two distinct distinct modeling modeling scenarios scenarios are are considered: considered: aa fully fully stochastic stochastic metapopula metapopulation (or community) tion where where the the individual individual level level processes processes within within each each habitat habitat (or community) are are modeled explicitly implicit (Levins-type modeled explicitly and and aa spatially spatially implicit (Levins-type)) metapopulation metapopulation where where habitats formulations habitats are are classified classified into into aa limited limited set set of of discrete discrete classes. classes. Both Both formulations have have associated associated benefits benefits and and allow allow different different insights insights into into the the dynamic dynamic processes processes in population processes in disease disease spread. spread. We We then then revisit revisit how how meta metapopulation processes operate operate at at vari various ous spatial spatial scales scales (individual (individual level, level, local, local, and and regional regional epidemics). epidemics). The The resultant resultant spatiotemporal dynamics dynamics are case studies, spatiotemporal are then then illustrated illustrated through through aa series series of of case studies, which diseases metapopulation which explore explore diseases metapopulation dynamics dynamics at at the the interface interface of of models models and and data. data. We We conclude conclude with with aa section section on on fruitful fruitful areas areas for for future future work. work.
117.2 7.2 THE THE SIR SIR MODEL MODEL FOR FOR EPIDEMIC EPIDEMIC DYNAMICS DYNAMICS We We focus focus here here on on microparasite microparasite infections infections (mainly (mainly viruses viruses and and bacteria), bacteria), where where direct direct reproduction reproduction of of the the pathogen pathogen in in the the host host allows allows us us to to model model dis disease ease dynamics dynamics by by dividing dividing the the host host population population between between compartments, compartments, classified classified by their infection status (Anderson 9 9 1 ) . In by their infection status (Anderson and and May, May, 11991). In contrast, contrast, macroparasitic macroparasitic helminth helminth infections, infections, where where parasite parasite burden burden matters, matters, are are much much harder harder to to model model spatially considered here), analogies have spatially (and (and not not considered here), although although strong strong analogies have been been found found between macroparasite between macroparasite and and metapopulation metapopulation dynamics dynamics (Cornell (Cornell et et ai., al., 2000) 2000).. The The most most studied studied microparasite microparasite system system iiss the the SIR SIR model, model, where where individuals individuals are susceptible (5), are susceptible (S), infected infected (I), or or recovered recovered (R). This This classification classification holds holds analo analogies metapopulation models models in gies to to the the "compartmental" "compartmental" Levins Levins metapopulation in which which patches patches are either occupied occupied or discussed in are classified classified as as either or empty empty (Chapter (Chapter 4). 4). As As discussed in the the next next section, local patch section, the the "reversibility" "reversibility" of of true true metapopulations metapopulations (such (such that that local patch populations then reestablished populations can can become become extinct, extinct, then reestablished by by colonization) colonization) is is aa closer closer match susceptible-infectious-susceptible, such match to to the the SIS SIS dynamics dynamics ((susceptible-infectious-susceptible, such that that
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recovered individuals individuals do not possess sexually recovered do not possess immunity) immunity) associated associated with with many many sexually transmitted 9 9 1 ) . In paradigm, suscep transmitted diseases diseases (Anderson (Anderson and and May, May, 11991). In the the SIR SIR paradigm, susceptible tible individuals individuals can can catch catch the the disease disease from from contact contact with with infected infected individuals; individuals; infected infected individuals individuals then then recover recover at at aa given given rate, rate, after after which which time time they they are are assumed assumed to to be be immune immune to to further further infection. infection. This This leads leads to to the the following following set set of of dif differential equations: equations: ferential
SI dS dS SI == BN - dS B N - - 13 B_--:_as dt dt - mN SI dI dI SI at = ~3-~ gI - dI dI dt = 13 N - gI dR dR dt = gI - dR dt = g I -
((17.1) 1 7. 1 )
dR
N N == S +SI ++R I + R B
d
where where B is is the the birth birth rate, rate, d is is the the natural natural death death rate, rate, 13 13is is the the transmission transmission rate rate between between infected infected and and susceptible susceptible individuals, individuals, and and g is is the the recovery recovery rate. rate. Many Many improvements variations on improvements and and variations on this this underlying underlying framework framework have have been been devel developed diseases and oped successfully successfully to to describe describe the the behavior behavior of of particular particular diseases and hosts hosts (Anderson 9 9 1 ; Grenfell 995; Hudson Hudson et (Anderson and and May, May, 11991; Grenfell and and Dobson, Dobson, 11995; et aI., al., 2002 2002).) . IInn essence, 1 7. 1 ) predicts essence, Eq. Eq. ((17.1) predicts aa stable stable equilibrium equilibrium level level of of susceptibles susceptibles and and infected, infected, which which is is reached reached through through aa series series of of damped damped epidemics. epidemics.
117.3 7.3
g
THE THE SPATIAL SPATIAL DIMENSION DIMENSION Spatial Spatial structure structure and and the the aggregation aggregation of of hosts hosts into into discrete discrete patches patches can can have have dramatic diseases (May dramatic effects effects on on the the dynamics dynamics of of infectious infectious diseases (May and and Anderson, Anderson, 11979; 979; Grenfell 99 8 ) . We Grenfell and and Bolker, Bolker, 11998). We subdivide subdivide these these effects effects into into four four main main groups, groups, which which we we consider consider with with respect respect to to the the dynamics dynamics of of one one large, large, homo homogeneously geneously mixed mixed host host population population versus versus the the dynamics dynamics of of several several smaller, smaller, more more isolated isolated ones. ones.
Isolation and Isolation and Coupling: Coupling: A A Simple Simple Two-Patch Two-Patch Model Model The The most most obvious obvious aspect aspect of of spatial spatial separation separation is is the the isolation isolation of of one one or or more more local local populations. populations. The The degree degree of of isolation isolation is is controlled controlled by by the the coupling coupling between absence of between patches. patches. In In the the absence of coupling, coupling, the the dynamics dynamics in in each each patch patch are are independent, independent, and and as as the the coupling coupling increases, increases, so so does does the the correlation correlation between between them. them. We We generally generally envisage envisage coupling coupling as as the the result result of of the the movement movement of of hosts; hosts; in in such such cases cases it it is is important important to to realize realize that that the the movement movement of of both both susceptibles susceptibles and role. We and infecteds infecteds plays plays an an equal equal role. We also also note note that that two two patches patches can can be be coupled coupled directly directly due due to to the the mixing mixing of of individuals individuals in in aa third third patch patch (e.g., (e.g., people people from from two two outlying outlying towns towns might might meet, meet, and and transmit transmit infection, infection, at at aa nearby nearby large large town). town). As As we we are are concerned concerned primarily primarily with with the the spread spread of of infection infection between between human human communities, communities, we we envisage envisage coupling coupling as as the the result result of of short short duration duration commuter commuter movements. movements. For For other other host host species, species, coupling coupling could could be be generated generated by by
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the permanent permanent movement movement of of hosts hosts or or simply simply the the movement movement of of pathogens pathogens the between local local populations populations (Keeling (Keeling et et al., aI., 2001). 200 1 ) . between A key key question question for for understanding understanding the the ensuing ensuing spatial spatial dynamics dynamics is is how how to to A accurately allow allow for for the the movement movement of of infection. infection. Consider, Consider, first, first, aa metapopulametapopula accurately tion of of just just two two patches patches (Keeling (Keeling and and Rohani, Rohani, 2002). 2002). In In this this model, model, individuals individuals tion commute from from their their home home population population to to the the other other patch, patch, but but return return rapidly rapidly commute ( Sattenspiel and and Dietz, Dietz, 1995). 1 995). We label individuals individuals by by two two subscripts subscripts such such that that (Sattenspiel We label the number number of of susceptibles susceptibles currently currently in in patch patch j, j, whose whose home home is is patch patch i. i. We We Sij isis the Sii commute at at rate rate pi and return return at at rate rate also assume assume that that individuals individuals from from patch patch ii commute also Pi and Ti, independent independent of of their their infectious infectious state. state. If If we we assume assume frequency-dependent frequency-dependent % transmission (de (de Jong Jong et et al., aI., 1995; 1995; McCallum McCallum et et al., aI., 2001), 200 1 ), then then equations equations for for transmission the number number of of susceptibles susceptibles and and infecteds infecteds in in each each patch patch are are given given by by the
dSii
dt
Iii + Iji ~SiiNi i + mji - dSii + "rimij - piSii
= bmii-
dlii
Iii + Iji
dt = ~Sitmii + mji - glii - dIii + "r
- pilii ((17.2) 1 7.2)
dSij = b N i j -
at
Iij + Ijj
f3S#Nij + Njj
_ dSij - "r
+ f)iSii
dIij Iij + Ijj = f3SilNij + Njj - gIij - dIij - $iIij + piIii dt Here, equations equations for for the (R ii and have not been where ii 7= where ~ j.j. Here, the recovered recovered class class (Rii and Ri Rii)j) have not been R == N. given be calculated from the the fact If given explicitly, explicitly, as as they they can can be calculated from fact that that S + R S ++ II + N. If we distribution of individuals to ;;INij = we allow allow the the distribution of individuals to equilibrate, equilibrate, then then N Nii/Nij = T; "ri/Pi. !Pi' Now, summing over all individuals whose home home is Now, summing over all individuals whose is patch patch ii and and assuming assuming that that time relatively short time spent spent away away from from the the home home patch patch is is relatively short compared compared to to the the dis disease dynamics, we get ease dynamics, we get
dSi dSi j] - dSi dt = [(Tii1i + = b b NNi i - - I3Si ~Si[(Yiili + (Tij! (Yijlj]dSi dt dl dlii = j ] - gIi Si [(Tii1i : I3 ~3Si [o'iiIi + + (Tij! o'iilj]gli- - d dIi1i dt dt
((17.3) 1 7. 3 )
where where (T
(Tii (7ii = = (T"II = (T"II
(Yij = (Yji
= =
- 'Y~/i)2 ((11 'Y~/2T iF + + j) Nji ((1 1 iNi + jNj 'Y~liNi - 'Yi)Ni "yi)Ni + + 'Y"yjNi + ((11 - 'Y ~fi)N ((11 - 'Yi i( 1 -- 'Y~lj)j) 'Y~li(1 ~li)~lj l'"Yj -
(( 11
~li)mi + + 'Y ~ljmj jNj -- - 'Yi)Ni
+
+ 'Y ~limi iNi ++ (( 11
((17.4) 1 7.4)
j) Nj - 'Y ~lj)mj
-------
ratio of P' is is the the ratio of commuting commuting to to return return rates rates and and as as such such can can be be cal calwhere where 'Yi ~/i = = � 7-] culated culated from from the the expected expected amount amount of of time time an an individual individual from from patch patch ii spends spends away away from from home home ((= ~/i) ." In In the the much much simplified simplified case case where where the the population population = 11Zi+~/i "YJ sizes sizes and and movement movement patterns patterns are are equal equal in in both both patches, patches,
1 7. 17.
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a··
{7iitl -=
~/2 'Y2 ++ (1 ( 1 --- , y'Y) 2) 2 NN
aij == 2y(1 2'Y( 1 -- y) 'Y) O'ij
( 1 7. 5 ) (17.5)
The factor factor of o f two two in in cr aij0 originates originates because because coupling coupling can can come come from from either either the the The movement of of susceptibles susceptibles or or the the movement movement of of infecteds. infecteds. Quadratic Quadratic terms terms movement occur due due to to two two individuals individuals with with the the same same home home patch patch meeting meeting in in the the away away occur patch. patch. If we we assume assume global global coupling, coupling, such such that that commuter commuter movement movement occurs occurs If equally to to all all other other local local populations populations irrespective irrespective of of distance distance between between them, them, equally patch generalization generalization is is then the the nn patch then ddS· mi o.S1 [ ( 1 -- n{r)Ii _I = na)l·1 ++ ca�(tdS1 = b bNNi -1 - f3Si[(1 r ~ J=-l lljl]J/]m/N -- dSi I-' dt dt
dli o. dli na)l1 ++ ~r~j a�(tJn-_-- IllI jJ]] //N gl1 S [ ( 11 - n~r)Ii == f3Si[( N -- gIi dt I-' 1 dt -
a == 2y(1 2'Y( 1 -- y). 'Y ). o-
li 1 - ddl
(17.6) ( 1 7.6)
-
where y'Y is is again again the the ratio ratio of of the the rate rate of of commuting commuting to to aa given given patch patch to to the the rate rate where of return. The proportion of time spent away from the home patch is now of return. The proportion of time spent away from the home patch is now
ny "1----" 1 n-y ' 1+ +ny"
These models [Eqs. [Eqs. ((17.3)and 1 7.3) and (17.6)] ( 1 7.6)] illustrate illustrate that that even even the the complex complex mechmech These models anistic movement of commuters can generally be expressed as a distributed force anistic movement of commuters can generally be expressed as a distributed force of infection from each infected individual across multiple local populations. of infection from each infected individual across multiple local populations. Thus the the complex patterns of human movements movements can be subsumed into a set of of Thus complex patterns of human can be subsumed into a set parameters parameters a, or, which which specify specify the the relative relative strengths strengths of of within-patch within-patch to to between betweenpatch transmission. These 1 7.3 and 7.6) are those patch transmission. These equations equations ((17.3 and 117.6) are identical identical to to those derived populations is derived when when the the movement movement between between local local populations is permanent permanent immigra immigration (Kot et aI., 11996; 996; Smith tion rather rather than than short-duration short-duration commuter commuter travel travel (Kot et al., Smith et et aI., al., 2002) 2002) and and to to those those formulated formulated when when the the transmission transmission of of infection infection between between dif different local populations 999; Park ferent local populations is is via via wind-borne wind-borne spread spread (Bolker, (Bolker, 11999; Park et et aI., al., 2001 2001).). Therefore, Therefore, the the simple simple and and intuitive intuitive method method of of coupling coupling local local populations populations is applicable applicable to to aa wide wide variety variety of of diseases diseases and and interaction interaction scenarios. scenarios. is The The aforementioned aforementioned framework framework for for studying studying the the dynamics dynamics of of aa disease disease in in aa spatially spatially structured structured population population is is founded founded on on the the premise premise of of deterministic deterministic interactions interactions and and very very rapid rapid movement movement of of commuters commuters back back to to their their home home patch. patch. Now Now we we consider consider how how this this translates translates into into aa more more realistic realistic stochastic stochastic frame flamework, work, where where the the population population is is individual individual based based and and events events are are assumed assumed to to occur occur at at random; random; this this is is often often termed termed demographic demographic stochasticity. stochasticity. In In such such aa 1 7. 3 ) ] , which framework, the the coupled coupled model model [Eq. [Eq. ((17.3)], which has has far far fewer fewer equations equations framework, than 1 7.2)], is than the the full full mechanistic mechanistic model model [Eq. [Eq. ((17.2)], is aa reliable reliable approximation approximation if if the the movement movement rate rate of of individuals individuals between between the the populations populations is is rapid. rapid. However, However, as as the the movement movement rate rate slows slows (e.g., (e.g., if if commuters commuters generally generally spend spend the the entire entire day day or or longer longer away away from from home), home), the the individual individual nature nature of of the the population population plays plays an an ever ever greater greater role. role. If If just just one one individual individual is is infected, infected, then then the the level level of of coupling coupling will will be be influenced 7. 1 shows influenced greatly greatly by by whether whether that that individual individual commutes. commutes. Figure Figure 117.1 shows the the distribution distribution of of cases cases caused caused by by aa single single infectious infectious case case in in their their nonhome nonhome patch. patch. Clearly Clearly the the number number of of cases cases produced produced is is highly highly dependent dependent on on whether whether
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0.25
CJ CJ _
0.2 §0.15
.0 co .0
e Q.
Does not commute Commutes once Commutes twice
0.1
Number of cases in non home patch of a single infectious individual individual causing different different numbers of of second secondFig. 117.1 7.1 Probability of ary cases in their non home patch. If the infectious individual an/cases nonhome individual commutes, commutes, there is a dramatic shift in the the expected number In this scenario, the infectious period is exactly exactly 3 days, days, com in number of cases. cases. In com0 . The muters always spend a full day away from and the basic reproduction from home, and reproduction ratio Ro R0 = = 330. susceptible population population is considered to be very large.
the infectious infectious person person commutes, commutes, although although even even when when they they remain remain in in their their the home home patch patch the the disease disease can can still still spread spread due due to to the the movement movement of of susceptibles. susceptibles. In population models, In stochastic stochastic meta metapopulation models, therefore, therefore, when when the the level level of of infection infection is is low and and the the commuter commuter time time is is of of the the same same order order as as the the infectious infectious period, period, we we must must low be be very very cautious cautious in in our our use use of of approximations approximations to to the the true true mechanistic mechanistic dynamics. dynamics. The The occasional occasional rare rare event, event, when when the the infected infected individual individual commutes, commutes, can can have have large large repercussions repercussions and and leads leads to to aa far far wider wider range range of of outcomes outcomes than than would would be be expected expected from version of simple coupling coupling model 1 7.3)]. from aa stochastic stochastic version of the the simple model [Eq. [Eq. ((17.3)].
Stochastic Stochastic and and Seasonal Seasonal Forcing Forcing The The main main manifestation manifestation of of random random fluctuations fluctuations explored explored in in epidemic epidemic the theory is is the the impact impact of of demographic demographic stochasticity. stochasticity. As As for for conventional conventional metapopu metapopuory lations, major impact demographic stochasticity lations, aa major impact of of demographic stochasticity is is on on the the extinction extinction rate, rate, here populations ((see see next here of of epidemics epidemics in in small small populations next section) section).. However, However, due due to to the the inherent oscillatory nature of inherent oscillatory nature of epidemics, epidemics, stochastic stochastic forcing forcing of of epidemics epidemics can can give rise to regular or issue has give rise to regular or irregular irregular cycles. cycles. This This issue has strong strong parallels parallels with with the the recurrent debate debate in relative impact recurrent in ecology ecology on on the the relative impact of of noise noise and and deterministic deterministic forces 1 ) . In forces on on dynamics dynamics (e.g., (e.g., Bjornstad Bjornstad and and Grenfell, Grenfell, 200 2001). In epidemiology, epidemiology, the the interaction between interaction between deterministic deterministic nonlinearity nonlinearity and and forcing forcing has has been been most most stud studied perturbing forces, ied in in terms terms of of the the perturbing forces, which which may may maintain maintain strong strong recurring recurring epi epidemics demics of of measles measles in in the the prevaccination prevaccination era; era; these these epidemics epidemics are are predicted predicted to to nonseasonal models dampen to to an an equilibrium equilibrium by by simple simple deterministic deterministic nonseasonal models (May (May dampen and 9 9 1 ) . The 1 956, 11957), 957), who and Anderson, Anderson, 11991). The seminal seminal work work here here is is by by Bartlett Bartlett ((1956, who showed that that both both stochastic stochastic forcing forcing or or the the marked marked seasonality seasonality in in transmission transmission due oscilla due to to the the aggregation aggregation of of children children in in schools schools could could excite excite the the measles measles oscillator epidemics. In case of diseases, seasonality tor into into sustained sustained epidemics. In the the case of childhood childhood diseases, seasonality appears role in appears to to play play aa major major role in the the maintenance maintenance of of measles measles cycles cycles (Schenzle, (Schenzle, 11984; 984; Bjornstad 2002; Grenfell aI., 2002) Bjornstad et et aI., al., 2002; Grenfell et et al., 2002)..
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In In general, general, most most observations observations and and stochastic stochastic model model results results agree agree that that the the average average number number of of cases cases in in aa population population scales scales linearly linearly with with population population size size (1 (I ct:.~ N). The The variance variance in in the the number number of of cases, cases, however, however, can can be be approximated approximated by aa power power law, law, with with an an exponent exponent between between 1I and and 2, 2, [var( [var(I) N2% �89:S ~_ a ~ :S ~ 11]] by I) ct:.~ N2a, (Keeling 999; Keeling, (Keeling and and Grenfell, Grenfell, 11999; Keeling, 2000a) 2000a).. This This underlines underlines how how large large popu populations lations have have relatively relatively lower lower standard standard deviations deviations in in the the number number of of cases cases com com- 1 ] and pared pared to to the the mean mean [SD [ S D (( II))/I / I ct:. ~ N N a'~-1] and thus thus behave behave more more like like deterministic deterministic systems. systems. These These subtleties subtleties lead lead to to nontrivial nontrivial consequences consequences of of spatial spatial subdivision subdivision of hosts on epidemic dynamics. Work Work on on whooping whooping cough cough illustrates illustrates the the dramatic dramatic influence influence of of demographic demographic stochasticity stochasticity on on epidemic epidemic dynamics dynamics due due to to the the intricate intricate interaction interaction between between stochasticity and and nonlinearity nonlinearity (Rohani et al., 2000; Keeling Keeling et et aI., al., 200 2001; stochasticity (Rohani et aI., 2000; 1; Rohani 994). A Rohani eett aI., al., 2002; 2002; see see also also Rand Rand and and Wilson, Wilson, 11994). Ass predicted predicted by by stand standard becomes increasingly ard theory, theory, demographic demographic stochasticity stochasticity becomes increasingly important important in in small small populations populations because because one one individual individual in in smaller smaller populations populations is is aa comparatively comparatively larger fraction fraction of of the the entire entire population population and and therefore therefore each stochastic stochastic event event replarger rep resents susceptible and resents aa relative relative larger larger change change to to the the susceptible and infected infected proportions proportions m one infectious infectious individual small village village is likely to one individual in in aa small is likely to infect infect aa greater greater proportion of proportion of the the population population than than one one infectious infectious individual individual in in aa large large city. city. can illustrate the complex of noise on epidemics Thus we we can illustrate the complex roles roles of noise on epidemics by by considering considering the of whooping the stochastic stochastic dynamics dynamics of whooping cough cough across across aa range range of of host host community community sizes (Rohani (Rohani et et aI., al., 2000) 2000).. Small Small model model populations are are seen seen to to display display 4-yr 4-yr sizes cycles close to, to, their cycles driven driven by by stochastic stochastic resonance resonance at, at, or or close their natural natural frequency, frequency, whereas large populations more annual the whereas large populations possess possess more annual dynamics dynamics constrained constrained by by the deterministic attractor (Fig. (Fig. 117.2). deterministic 7.2). We can can extend the concept concept of power-law variances variances to metapopulation with with We extend the of power-law to aa metapopulation n local populations. If the level of coupling coupling between the weak, n populations. If the populations populations is weak, such that dynamics are are almost almost independent, average number of such that the dynamics independent, then then the the average number of cases local populations is the as for large population. population. For cases across across all all local populations is the same same as for one one large For independent variance of of the variances; independent populations, populations, the the variance of the the sum sum is is the the sum sum of the variances; hence, increase in local populations causes aa linear increase hence, an an increase in the the number number of of local populations causes linear increase in the total variance. However, However, a similar similar increase in in the in total variance. the population population size of of one large patch causes than linear linear rise of the the power law. large patch causes aa faster faster than rise due due to to the the scaling scaling of power law. Naively, then, one could be tempted to conjecture that by breaking the the habitat Naively, then, one could be tempted to conjecture that by breaking habitat into multiple multiple (independent) patches, one one would would effect effect aa decrease decrease in in the relative (independent) patches, variability observed observed in in the the aggregate aggregate dynamics. dynamics. This This is is analogous analogous to the statisstatis variability to the tical averaging averaging discussed discussed as as the the "portfolio "portfolio effect" effect" in in community community ecology ecology tical (Tilman, 1999). 1 999). However, However, in in practice, practice, the the significant significant levels levels of of coupling coupling and and the the (Tilman, complex interactions interactions between between nonlinear nonlinear transmission transmission dynamics dynamics and and demodemo complex graphic stochasticity stochasticity mean mean that that no no such such general general assertions assertions are are possible. possible. Wilson Wilson graphic Hassell (1997) ( 1 997) have have shown shown that that such such complexities complexities also also take take place place in in other other and and Hassell host-enemy host-enemy systems.
Extinctions Extinctions A major major effect effect of of demographic stochasticity in in small small populations populations is is the the tenten A demographic stochasticity dency dency for for chance chance extinctions. extinctions. This This behavior behavior is is highlighted highlighted in in Bartlett's Bartlett's classiclassi cal cal work work on on measles, measles, where where the the number number of of fadeouts fadeouts (or ( or localized localized extinctions) extinctions) decreases exponentially exponentially with with population population size size (Bartlett, ( Bartlett, 1957). 1 957). Bartlett Bartlett (1957, ( 1 957, decreases
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1 7. 17.
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identified aa critical critical community community size size (CCS) ( CCS) for for disease disease persistence, persistence, such such 1 960) identified 1960) that above above this this population population size the the disease disease is endemic endemic and and is rarely rarely subject subject to to that stochastic extinctions. extinctions. Interestingly, this this emergent emergent critical critical community community size is stochastic remarkably robust, robust, and and similar similar values values of of around around 300,000 3 00,000 for for measles measles occur occur remarkably for communities communities in in England, England, the the United United States, States, and and isolated isolated islands islands (Bartlett, (Bartlett, for 1 957, 1960; 1 960; Black, Black, 1966). 1 966). The The CCS CCS is is arguably arguably the the best best empirically empirically docudocu 1957, mented extinction threshold threshold in meta population biology biology (Grenfell ( Grenfell and and mented local extinction metapopulation Harwood, 1997; 1 9 97; Keeling and Grenfell, Grenfell, 1997). 1 997). Harwood, Keeling and When considering considering the the regional regional persistence persistence of of an an infectious infectious disease across When populations versus one one large population, population, there there are are conflicting conflicting eleele several small populations ments. Small isolated populations populations exhibit exhibit more more frequent extinctions than than ments. frequent local extinctions populations. However, in a metapopulation metapopulation consisting of of many many small small popupopu large populations. eradication) at at national national or or regional scales requires requires the the lations, extinction extinction (or eradication) concerted collapse collapse of of all local epidemics; in contrast, contrast, regional regional eradication, eradication, concerted where there there is is just just one one large large population, population, only only requires requires aa single single extinction extinction event. event. where population models, coupling enhances persistence through local In classic meta metapopulation persistence through recolonization, but but erodes persistence recolonization, persistence through through synchronizing the local dynam dynamics (Chapter 4). 4). For For epidemic metapopulations, the the relationship relationship between between ics (Chapter epidemic metapopulations, and coupling dis regional persistence and coupling is complex complex and and depends critically on the disease the demography and movement the hosts. hosts. Thus, Thus, it it is far ease parameters parameters and and the demography and movement of of the is far from whether one one large large patch patch or or several several smaller patches have have the from obvious obvious whether smaller patches the greater rate. greater extinction extinction rate. The coupling between local populations on global disease eradieradi The effects effects of coupling much attention attention due to the interesting interesting trade-offs trade-offs that cation have received much due to that arise (Keeling, If the the coupling is very very small, small, then then the the local local populations populations act (Keeling, 2000b). 2000b). If coupling is act independently independently and and there there is is little little or or no no chance chance of of the the disease disease being being reintroduced reintroduced from population; there from another another local local population; there is is no no rescue rescue effect. effect. Thus Thus using using the the lan language populations, the guage associated associated with with Levins Levins meta metapopulations, the local local populations populations have have aa large extinction large extinction rate rate and and aa very very low low probability probability of of colonization. colonization. If If the the coupling coupling is is very very large, large, then then the the local local populations populations act act like like one one large large homogeneously homogeneously mixed mixed population population and and thus thus stochastic stochastic effects effects may may lead lead the the entire entire metapopula metapopulation tion to to extinction. extinction. In In disease disease models, models, heterogeneity heterogeneity plays plays an an important important role role as as low low levels levels of of infection infection allow allow the the susceptible susceptible population population to to recover, recover, which which in in turns cases. Persistence turns promotes promotes future future cases. Persistence is is therefore therefore maximised maximised at at intermedi intermediate ate levels levels of of coupling: there there is is sufficient sufficient coupling to to allow allow recolonization recolonization and bsorb and sufficient sufficient variability variability between between patches patches for for the the metapopulation metapopulation to to aabsorb stochastic stochastic fluctuations. fluctuations. The The global global eradication eradication (extinction) (extinction) of of disease disease metapopulations metapopulations is is obviously obviously aa key key aim aim in in public public health health terms. terms. This This is is generally generally investigated investigated using using com computer puter simulations, simulations, as as analytical analytical techniques techniques have have difficulty difficulty dealing dealing with with the the complexities complexities of of spatial spatial heterogeneities heterogeneities and and the the stochastic stochastic dynamics dynamics that that per permeate 7. 3 illustrates meate the the problem. problem. Figure Figure 117.3 illustrates the the aforementioned aforementioned principles principles using using extinction extinction probabilities probabilities for for aa spatial spatial SIR SIR epidemic epidemic model. model. When When the the coupling at the coupling is is very very low low such such that that recolonization recolonization is is rare, rare, local local extinctions extinctions ((at the local 7.3A) are at local population population level, level, Fig. Fig. 117.3A) are common, common, as as are are global global extinctions extinctions ((at the 7.3B). When the metapopulation metapopulation level, level, Fig. Fig. 117.3B). When the the coupling coupling is is high, high, local local popu populations lations rarely rarely go go extinct. extinct. However, However, because because of of the the synchrony synchrony induced induced by by cou coupling, pling, rescue rescue effects effects are are less less effective effective as as all all epidemic epidemic declines declines are are aligned aligned and and therefore therefore prone prone to to simultaneous simultaneous local local extinctions. extinctions. Hence Hence the the global global extinction extinction
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probability is is again again enhanced. enhanced. There There is is aa clear clear intermediate intermediate minimum minimum {or for probability which disease disease persistence persistence is is the the greatest. greatest. This This compares compares with with the the diminished diminished which persistence of of classic classic metapopulations metapopulations when when embedded embedded in in aa correlated correlated landland persistence (Harrison and and Quinn, Quinn, 1989). 1 9 8 9 ) . It It is is still still an an open open problem problem to to relate relate disdis scape scape (Harrison ease characteristics, coupling to characteristics, host host demography, demography, and and coupling to the the extinction extinction risk risk at at the the metapopulation metapopulation scale for for a wide wide range range of of microparasites microparasites (Keeling, 2000b). 2000b). Changes in in coupling coupling between between populations populations due to social social changes changes and of Changes due to and ease ease of long-distance travel travel have have important important implications implications for for disease extinction and and long-distance disease extinction eradication -this is is aa major major question question for for the the theoretical theoretical epidemiology epidemiology of of the the eradication - this {uture. future.
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Dynamic Dynamic Heterogeneity Heterogeneity In heterogeneity" is In this this context, context, ""heterogeneity" is taken taken to to mean mean the the total total degree degree of of variation variation (or asynchrony) locations. This (or asynchrony) between between epidemic epidemic dynamics dynamics at at different different locations. This includes includes variation locations, as variation due due to to asynchronous asynchronous timing timing of of epidemics epidemics at at different different locations, as well well as as heterogeneities heterogeneities in in local local dynamics dynamics due due to to differences differences in in local local host host demography. demography. Heterogeneities Heterogeneities are are thus thus aa fundamental fundamental difference difference between between spatial spatial and and nonspa nonspatial processes. processes. As As outlined outlined earlier, earlier, heterogeneity heterogeneity is is promoted promoted by by stochasticity stochasticity but but tial is coupling. The is reduced reduced by by coupling. The level level of of heterogeneity heterogeneity further further depends depends on on the the rela relative absolute differences tive and and absolute differences in in host host community community size, size, movement, movement, and and demogra demography, phy, as as well well as as subtle subtle characteristics characteristics of of the the transmission transmission dynamics. dynamics. To To better better understand the understand the causes causes and and consequences consequences of of such such heterogeneity, heterogeneity, we we contrast contrast aa range population models. simplest assumes assumes identical identical demography range of of meta metapopulation models. The The simplest demography within local population, dynamics and within each each local population, deterministic deterministic dynamics and global global coupling, coupling, so so that interaction is same between all patches. that the the interaction is the the same between all patches. Under Under these these simplifying simplifying assumptions, even for low levels assumptions, and and even for very very low levels of of coupling, coupling, we we generally generally observe observe phase phase locking locking where where the the interaction interaction between between patches patches leads leads to to complete complete syn synchronization chronization of of each each local local epidemic epidemic and and zero zero heterogeneity. heterogeneity. When When the the internal internal dynamics dynamics are are stochastic, stochastic, the the spatial spatial dynamics dynamics are are more more complex. complex. Coupling Coupling still still acts acts to to synchronize synchronize the the dynamics dynamics by by homogenizing homogenizing the the level level of of infection infection in in each each local local population. population. In In contrast, contrast, stochasticity stochasticity acts acts to to sep separate arate the the dynamics dynamics as as different different populations populations experience experience different different random random events. events. Figure 7.4 shows Figure 117.4 shows the the correlation correlation in in disease disease incidence incidence between between two two stochastic stochastic local populations coupling. When local populations for for various various levels levels of of coupling. When coupling coupling is is low, low, the the two two populations correlation is populations are are unsynchronized unsynchronized and and the the correlation is zero; zero; however, however, as as cou coupling pling increases, increases, the the stochastic stochastic oscillations oscillations are are increasingly increasingly correlated. correlated. As As seen seen in in Fig. 7.4, coupling Fig. 117.4, coupling has has aa greater greater effect effect for for larger larger populations populations (results (results on on popu populations lations of of more more than than 10,000 10,000 did did not not differ differ significantly), significantly), which which is is primarily primarily due due to aI., 2002). to the the diminished diminished effect effect of of stochasticity stochasticity (Grenfell (Grenfell et et al., 2002). Population Population size size thousand 11 thousand 0 thousand x 1lOthousand
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In presence of dynamics are In the the presence of seasonal seasonal forcing, forcing, space-time space-time dynamics are more more involved. involved. In In general, general, unforced unforced stochastic stochastic epidemics epidemics can can peak peak at at any any time time of of the the year, year, whereas whereas seasonal seasonal forcing forcing usually usually constrains constrains the epidemic epidemic cycle. cycle. Therefore, Therefore, if if seasonality seasonality tends tends to to force force the the epidemics epidemics into into aa rigid rigid annual annual cycle, cycle, populations populations appear appear partially partially or or fully fully synchronized synchronized without without the the need need for for strong strong coupling coupling this echoes echoes the the operation operation of of the the Moran Moran effect effect in in ecology ecology (Moran, (Moran, 11953; this 953; Grenfell 9 9 8 ) . If Grenfell et et aI., al., 11998). If the the epidemic epidemic period period is is multiannual, multiannual, however, however, epi epidemics 99 8 ) , as demics can can become become locked locked out out of of phase phase (Henson (Henson et et aI., al., 11998), as was was the the case case for 950s for the the 2-yr 2-yr epidemics epidemics of of measles measles in in Norwich Norwich and and London London during during the the 11950s ((Grenfell Grenfell et 1 ) , for et aI., al., 200 2001), for which which high high levels levels of of coupling coupling may may be be required required to to regain regain synchrony. synchrony. In In this this latter latter case, case, greater greater levels levels of of stochasticity stochasticity and and weaker weaker attractiveness attractiveness of of the the cyclic cyclic attractor attractor can can also help help synchronization synchronization as as there there is is aa greater greater chance chance that that aa population population will will switch switch phases phases (as indeed indeed happened happened for for the 960s). the Norwich Norwich measles measles epidemics epidemics during during the the 11960s). Two Two other other factors factors influence influence the the synchrony synchrony and and hence hence the the level level of of hetero heterogeneity. local geneity. The The first first is is the the presence presence of of inherent inherent differences differences between between the the local populations, such populations, such as as different different host host reproductive reproductive rates. rates. In In general, general, such such het heterogeneities erogeneities will will act act to to decorrelate decorrelate the the dynamics, dynamics, as as different different populations populations will will obey obey different different underlying underlying models. models. This This was was the the case case for for the the measles measles epi epidemics demics in in Liverpool Liverpool and and Manchester Manchester during during the the prevaccination prevaccination era era when when the the higher birth birth rates rates in in Liverpool Liverpool led led to to annual annual epidemics epidemics whereas whereas the the rest rest of of higher England Finkenstadt et 99 8 ; England and and Wales Wales was was predominantly predominantly biennial biennial ((Finkenst~idt et aI., al., 11998; Grenfell Grenfell eett aI., al., 2002 2002).) . Heterogeneities Heterogeneities iinn the the size size ooff local local host host populations populations may have have contrasting contrasting effects. effects. The The presence of one one large large population population may may act act may presence of to populations; in to synchronize synchronize the the behavior behavior of of many many surrounding surrounding small small populations; in such such scenario of of mainland-island mainland-island epidemic epidemic meta metapopulation, coupling to to the the aa scenario population, coupling large main synchronizing across the whole metapopumetapopu large population population is is aa main synchronizing force force across the whole Grenfell et 1 ; see 7. 1 0 ) . Local, lation ((Grenfell et aI., al., 200 2001; see also Fig. Fig. 117.10). Local, rather rather than than global, global, coupling coupling may may furthermore furthermore lead lead to to epidemic epidemic traveling traveling waves, waves, although although strong strong seasonal com seasonal forcing forcing can can again again counteract counteract this. Such Such epidemic epidemic waves waves are are aa common spatially explicit models of mon feature feature of of many many spatially explicit models of natural-enemy natural-enemy interactions interactions ((rabies, rabies, bubonic bubonic plague, plague, parasitoid-host systems) and parasitoid-host systems) and have have been been confirmed confirmed in ecological and (Nobel, 11974; 974; Grenfell in both both ecological and epidemiological epidemiological systems systems (Nobel, Grenfell et et aI., al., 200 1 ; Ranta 997; Smith 2001; Ranta et et aI., al., 11997; Smith et et aI., al., 2002; 2002; Bjornstad Bjornstad et et aI., al., 2002 2002).) . The The presence presence and and absence absence of of spatial spatial synchrony synchrony can can play play important important roles roles in in the the dynamics dynamics and and persistence persistence of of disease. disease. As As discussed discussed earlier, earlier, heterogeneity heterogeneity can can vastly vastly increase increase the the long-term long-term persistence persistence of of aa disease disease through through local local recolonisa recolonisation tion and and repeated repeated rescue rescue events. events. This This effect effect is is heightened heightened if if there there are are demo demographic or or size size differences differences between between the the populations. populations. Heterogeneities Heterogeneities at at aa graphic smaller scale scale can can also alter the the observed observed aggregate aggregate dynamics. dynamics. Case reports reports are are smaller often often aggregated aggregated at at the the community community or or regional regional level; level; however, however, such such data data may may be be composed composed of of multiple multiple smaller smaller epidemic epidemic within within wards wards or or population population cliques. cliques. As As these these subepidemics subepidemics are are likely likely to to be be somewhat somewhat out out of of phase, phase, the the aggregate aggregate duration epidemic. 7.5 shows picture is is of of aa slower, slower, longer longer duration epidemic. Figure Figure 117.5 shows aa simple simple example example of of this, this, while while each each localized localized epidemic epidemic (gray) (gray) is is of of short short duration, duration, the the aggregate black) is aggregate ((black) is far far longer longer with with aa much much diminished diminished epidemic epidemic peak. peak. Throughout the the examples examples that that follow, follow, we we refer refer continually continually to to the the afore aforeThroughout mentioned mentioned four four basic basic elements elements of of spatially spatially structured structured disease disease dynamics: dynamics: isolation, isolation, stochasticity, stochasticity, extinction, extinction, and and heterogeneity. heterogeneity. We We discuss discuss how how
117. 7.
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metapopulation both full metapopulation models models ((both full stochastic stochastic metapopulations metapopulations and and the the simpler simpler Levins Levins metapopulations) metapopulations) can can be be used used to to represent represent additional additional spatial spatial structure structure and the the insights insights that that can can be be gained from from such such idealized idealized models. models. We We begin begin by by and studying studying the the implications implications of of spatial spatial structure structure at at aa range range of of scales, scales, starting starting with with the individual individual and and working working up up to to the the community community or or even even country country level. level. the General conclusions conclusions of of this exercise exercise are illustrated subsequent case studies. General illustrated in subsequent studies.
117.4 7.4
DISEASE METAPOPULATIONS METAPOPULATIONS AT DIFFERENT SCALES DISEASE AT DIFFERENT SCALES
Individual Level Level Individual The standard SIR can be be explored explored by by considering each individual The standard SIR equations equations can considering each individual host using aa modified form of of the the Levins Levins metapopulation metapopulation formula host as as aa patch patch using modified form formulation. model the the interaction of the the disease disease and tion. The The mechanistic mechanistic approach approach would would model interaction of and the system, leading leading to to models models comparable comparable to to those those used used in in the the the host's host's immune immune system, study of of macroparasites, macroparasites, which which classify classify hosts hosts in in terms terms of of their their burden burden of of parapara study sites (Anderson (Anderson and and May, May, 1991). 1 99 1 ) . However, However, the the more more classical classical models models where where sites hosts are are described described as as unoccupied unoccupied but but suitable suitable habitat habitat (susceptible), ( susceptible), occupied occupied hosts (infected), or or unoccupied unoccupied but but exhausted exhausted habitat habitat (recovered) (recovered) have have aa much much (infected), closer analogy analogy to to standard metapopulation models models for for successional successional habitats. habitats. closer standard metapopulation From this this perspective, perspective, birth birth and and deaths deaths correspond correspond to to the the creation creation and and From destruction of of habitat. Strictly speaking, SIR-type dynamics dynamics cannot cannot correscorres destruction habitat. Strictly speaking, SIR-type pond pond to to "true" Levins Levins metapopulation metapopulation dynamics dynamics at the the individual individual level level because recovered cannot regain infection. However, However, most most sexually sexually because recovered patches patches cannot regain infection. transmitted diseases diseases exhibit exhibit SIS SIS dynamics; dynamics; that that is, is, once once an an infected individual transmitted infected individual recovers, it it is is once once again again susceptible susceptible to to infection. infection. In In such such cases cases there there is is aa direct direct recovers, correspondence between between the the standard standard Levins Levins model model and and the the SIS SIS equations equations correspondence (Ovaskainen ( Ovaskainen and and Grenfell, Grenfell, 2003). 2003 ) .
MATT J. KEELING KEELING ET ET AL. AL. MAI-F J.
4428 28
The The Levins Levins metapopulation metapopulation has has an an extensively extensively developed developed theoretical theoretical armoury that that can can be be applied applied to to the the description description and and understanding understanding of of disease disease armoury dynamics (Chapter ( Chapter 4). 4). Sexually Sexually transmitted transmitted diseases, diseases, which which are are characterized characterized dynamics by aa low low transmission transmission rate rate and and long long infectious infectious period, period, can can be be thought thought of of as as by poor colonizers, colonizers, but but aa persistent persistent species species with with low low local local extinction extinction rates. rates. In In concon poor trast, childhood diseases, diseases, which are short short lived lived and and transmitted transmitted rapidly, rapidly, can can be be trast, childhood which are conceptualized as as good good colonizers colonizers that that exploit exploit the the local local resource resource rapidly, rapidly, drivdriv conceptualized ing themselves themselves extinct. extinct. This This concept concept may may be be extended extended fruitfully to consider consider the the ing fruitfully to competition between between cross-resistant cross-resistant strains strains of of disease disease (Gupta ( Gupta et et al., aI., 1998), 1 99 8 ) , in in competition which case case low low transmission transmission rates (poor colonizing colonizing ability) can be be offset offset by by which rates (poor ability) can good competitive competitive ability ability within within aa patch. patch. good The compartmental compartmental classification classification of of "habitat" " habitat" needs needs not not necessarily oper The necessarily operate at at the the level level of of the the individual. individual. Work Work on on the the 2001 200 1 foot-and-mouth foot-and-mouth epidemic epidemic ate in Great Great Britain Britain considers considers each each farm an epidemic epidemic unit unit ("patch") ( "patch" ) to to be be classiclassi in farm an fied as as susceptible, exposed, infectious, infectious, or or removed (Keeling et et al., aI., 2001; 200 1 ; fied susceptible, exposed, removed (Keeling Ferguson et et aI., al., 2001). Patch "removal" "removal" was in this this case case through through massive Ferguson 200 1 ) . Patch was in culling farm. In In modeling modeling the culling of of all all potential potential host host animals animals on on aa farm. the outbreak, outbreak, the the within-farm favor of Levins-type classification. within-farm epidemic epidemic was was ignored ignored in in favor of this this Levins-type classification. Despite this this great models predicted the course course of Despite great simplification, simplification, these these models predicted the of the the epi epidemic level, justifying demic with with great great accuracy accuracy at at the the regional regional level, justifying the the approximation. approximation.
Within-Community Metapopulations Within-Community Metapopulations Many subdivided into various Many communities, communities, especially especially large large ones, ones, can be subdivided into various weakly interacting components. components. This This subdivision subdivision may may take take place place along along social, weakly interacting social, age, age, or or simply simply spatial spatial boundaries, boundaries, but but inevitably inevitably there there are are many many factors factors that that prevent prevent the the random random mixing mixing of of the the population population and and therefore therefore break break the the assump assumptions population tions underlying underlying standard standard models. models. This This necessitates necessitates the the use use of of meta metapopulationtype type equations equations to to model model the the dynamics dynamics of of these these partially partially separated separated components. components.
Regional Regional Metapopulations Metapopulations The The epidemic epidemic dynamics dynamics at at regional regional or or country country level level begs begs the the use use of of metapopulation metapopulation concepts. concepts. Here Here each each local local population population represents represents aa community, community, and is aa one-to-one at which and hence hence there there is one-to-one correspondence correspondence between between the the scale scale at which the the and the available data. Exploring regional regional dynamics dynamics brings model operates and two two main main challenges: challenges: understanding understanding the the detailed detailed consequences consequences of of demographic demographic heterogeneities heterogeneities between between the the communities communities and and analyzing analyzing the the epidemic epidemic coupling coupling between between communities communities on on real real landscapes. landscapes. The The scientific scientific development development of of the the latter latter issue issue mimics mimics the the succession succession from from the the naive naive to to more more realistic realistic models models of of metapopulation see Chapters 20, and metapopulation theory theory ((see Chapters 4, 4, 5, 5, 20, and 22). 22). The The traditional traditional models models of of identical identical local local populations, populations, with with low low levels levels of of global global coupling, coupling, have have given given way way to to models models with with distance-based distance-based coupling coupling rates. rates. Such Such models models are are slowly slowly being being replaced replaced with with models models embracing embracing heterogeneous heterogeneous patch patch sizes sizes (with (with obvi obvious ous parallels parallels to to the the current current generation generation of of incidence incidence function function models models and and sto stochastic chastic patch patch occupancy occupancy models models as as described described throughout throughout this this book) book).. However, However, for populations, it for aa complete complete understanding understanding of of epidemic epidemic meta metapopulations, it is is becoming becoming increasingly increasingly clear clear that that aa deeper deeper knowledge knowledge of of the the complex complex geometries geometries of of the the "transportation Cliff and "transportation networks" networks" for for the the infections infections is is required required ((Cliff and Haggett, Haggett,
METAPOPULATION DYNAMICS DYNAMICS OF OF INFECTIOUS INFECTIOUS DISEASES DISEASES 117. 7. METAPOPULATION
429 429
This is is likely likely to to provide provide an an exciting exciting area area for for future future research research with with great great 11988). 98 8 ) . This theoretical, empirical, empirical, and and statistical statistical challenges. challenges. theoretical,
117.5 7.5
CASE STUDIES STUDIES CASE
Prevaccination Measles Measles in in England England and and Wales Wales Prevaccination Of parasites, such Of all all infectious infectious diseases, diseases, the the dynamics dynamics of of childhood childhood micro microparasites, such as as measles measles and and whooping whooping cough, cough, are are arguably arguably among among the the best best understood understood with with respect respect to to both both local local and and regional regional dynamics. dynamics. In In particular, particular, the the rich rich data data base base and and the the comparatively comparatively simple simple natural natural history history of of measles measles have have made made this this the the prototypical prototypical system system in in the the study study of of spatiotemporal spatiotemporal dynamics dynamics of of infectious infectious dis disease 9 9 1 ; Bartlett, 957; Cliff 993; Grenfell ease (Anderson (Anderson and and May, May, 11991; Bartlett, 11957; Cliff et et aI., al., 11993; Grenfell and and Harwood, 997; Keeling 997; Grenfell 1 , 2002; Harwood, 11997; Keeling and and Grenfell, Grenfell, 11997; Grenfell et et aI., al., 200 2001, 2002; Keeling Keeling et et aI., al., 2001 2001;; Bj0rnstad Bjornstad et et aI., al., 2002 2002).) . Measles, Measles, along along with with other other child childhood 944. hood infections, infections, was was made made aa notifiable notifiable disease disease in in the the United United Kingdom Kingdom in in 11944. This resulted resulted in in the the collection collection of of weekly weekly reports reports in in 1400 communities communities in in This England England and and Wales Wales through through to to the the present. present. As As such, such, this this is is likely likely to to represent represent the the longest longest and and most most detailed detailed record record of of any any epidemic epidemic metapopulation. metapopulation. Not Not surprisingly, surprisingly, these these data data have have been been studied studied extensively extensively from from epidemiological, epidemiological, mathematical mathematical modeling, modeling, and and time-series time-series analysis analysis perspectives. perspectives. Due Due to to its its very very high basic basic reproductive reproductive ratio, ratio, Ro R0 = 117, most children children were were infected infected with with high 7, most measles (before mass vaccination vaccination campaigns campaigns in 960s) measles (before the the onset onset of of mass in the the late late 11960s) with with an an average average age age of of infection infection around around 4-5 4-5 yr. yr. Before Before mass mass vaccination vaccination was was introduced predominantly biennial biennial introduced in the United United Kingdom, Kingdom, measles displayed displayed predominantly dynamics, major epidemic odd years (Fig. 17.6A) 1 7.6A) (for further further details dynamics, with with a major epidemic in odd details see aI., 2002; Grenfell aI., 2002). 2002 ) . see Bj0rnstad Bjornstad et et al., Grenfell et al., Demographic stochasticity Demographic stochasticity plays aann important important role iinn the dynamics dynamics of measles This arises from the individual of popu measles in small communities. communities. This from the individual nature nature of populations (the that there must be whole whole numbers numbers of of cases) and and the the probaproba lations (the fact fact that there must of events, that transmission transmission of infection in particular bilistic nature nature of events, such such that of infection particular occurs occurs Stochasticity has two on the patterns by chance. chance. Stochasticity two basic effects effects on patterns of of disease behav behavcan push push trajectories trajectories away from the deterministic deterministic attractor ior: it can away from attractor such such that that transient dynamics play aa more more major major role role and and it it can can lead lead to to chance chance extinctions extinctions transient dynamics play due to to the the random random failure failure of chains chains of of transmission transmission (Figs. 17.6B 1 7.6B and and 17.6C). 1 7.6C). due The role role of of patch patch size (host (host population population size) on on epidemic epidemic extinction extinction rates rates is The illustrated wonderfully wonderfully in in the the public public records. records. Extinction Extinction rates rates appear appear to to decay decay illustrated exponentially with with host host population population size size so so that that above above the the critical critical community community exponentially of around around 300,000 300,000 hosts, hosts, extinctions extinctions are are rare rare (Bartlett, (Bartlett, 1957). 1 957). This This pattern pattern size of of size-based size-based extinctions extinctions and and recolonizations recolonizations warrants warrants interpretation interpretation from from the the of metapopulation metapopulation point point of of view. In the the prevaccination prevaccination era childhood childhood diseases, diseases, such such as measles measles or or whooping whooping cough, were were spread spread predominantly predominantly by by school school children children mixing mixing within within the the pripri cough, mary school school environment. environment. In this this respect, respect, the the host host populations populations can can be be thought thought mary of of as subdivided subdivided into into school catchment catchment areas. Considering Considering an an average average primary primary school has has an an intake intake of of around around 150 1 5 0 children children in in each each year, year, then then each each school school school serves aa population population of of around around 10,000; 1 0,000; this this determines our basic basic unit unit of of subsub serves determines our division. A A Levins-type Levins-type metapopulation metapopulation model model (global (global dispersal, dispersal, no no local local division.
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dynamics, etc.), etc . ) , which which splits splits each community into into school school catchment catchment areas areas of of dynamics, each community 1 0,000 people, people, motivates motivates the the following following model model for for the the proportion proportion of of "dis"dis 10,000 eased" local local populations populations (D) (D) in in the the city: city: eased"
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This provides provides easy easy calculation calculation ooff the the long-term long-term distribution distribution ooff disease disease within within This the metapopulation, the = eP1), the metapopulation, the associated associated extinction extinction rates rates ((= eP1), and and the the propor proportion = 1 Po). We tion of of time time the the disease disease should should be be present present ((= - P0). We may may compare compare these these theoretical theoretical predictions predictions with with data data on on measles measles incidence. incidence. Figure Figure 17.7 shows shows the the resultant comparison as population size resultant comparison as the the total total population size and and hence hence the the number number of of local local populations populations increase. increase. It It thus thus appears appears that that Levins-type Levins-type local local patch patch dynam dynamics ics can can reproduce reproduce the the pattern pattern in in data. data. The The Levins Levins approach, approach, which which completely completely ignores ignores the the local local epidemic epidemic dynam dynamics, absence of ics, works works surprisingly surprisingly well well for for predicting predicting the the presence presence or or absence of the the dis disease population sizes. sizes. However, ease across across aa range range of of population However, it it breaks breaks down down if if we we wish wish to to predict predict the the level level of of infection infection within within the the population. population. This This is is for for three three rea reasons. sons. First First the the local local dynamics dynamics of of different different cities cities are are highly highly correlated, correlated, such such that that the the level level of of infection infection is is an an increasing increasing function function of of the the colonization colonization rate, rate, which Fig. 17.8). which in in turn turn is is proportional proportional to to number number of of occupied occupied patches patches ((Fig. Second, Second, in in the the absence absence of of infection, infection, the the level level of of susceptibles susceptibles increases; increases; any any ensuing ensuing epidemic epidemic is is therefore therefore critically critically dependent dependent on on the the local local number number of of sus susceptibles, ceptibles, which which in in turn turn is is dependent dependent on on the the time time since since the the last last epidemic. epidemic. This This induces induces aa level level of of memory memory to to the the local local dynamics dynamics that that breaks breaks with with the the under underlying Markovian Markovian assumption assumption of of the the Levins Levins model. model. Finally, Finally, the the distribution distribution of of lying infection infection (Fig. (Fig. 17.8) does does not not conform conform to to the the Levins-type Levins-type metapopulation metapopulation ideal, ideal, which which assumes assumes that that local local prevalence prevalence should should be be bimodal bimodal and and dominated dominated by by aa zero zero and and aa nonzero nonzero equilibrium equilibrium level. level. For For aa detailed detailed understanding understanding of of the the population population dynamics, dynamics, we we need need to to consider consider aa metapopulation metapopulation with with detailed detailed stochastic Swinton et stochastic dynamics dynamics within within each each patch patch ((Swinton et aI., al., 1998).
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We explore the breakdown breakdown of of the Levins-type Levins-type metapopulation metapopulation model model We explore through comparison with with the full stochastic analogues analogues across across a variety of of scesce through a comparison full stochastic narios. A A range range of of single species models, models, with with aa variety of forms of density density narios. variety of forms of dependence, have have been been explored explored elsewhere elsewhere (Keeling, (Keeling, 2000b). 2000b). These These conform conform to to dependence, Levins-type metapopulation metapopulation behavior behavior when when (a) (a) the the distribution distribution of of population population Levins-type sizes sizes falls falls into into two two distinct distinct classes, classes, extinct extinct and and close close to to carrying carrying capacity, capacity, and and
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(b) (b) the the carrying carrying capacity capacity and and extinction extinction rate rate are are not not significantly significantly affected affected by by the the number of Levins frame number of extinct extinct patches. patches. For For such such single-species single-species systems systems the the Levins framework 7.9A work is is the the ideal ideal tool for for describing the the metapopulation metapopulation dynamics dynamics (Figs. (Figs. 117.9A and 7.9B). However, and 117.9B). However, for for disease disease models models (and (and most most likely likely many many other other enemy-host enemy-host interactions), interactions), aa different different pattern pattern of of regional regional behavior behavior arises arises due due to to the the synchronization synchronization that that occurs occurs as as an an integral integral part part of of the the space-time space-time dynamics dynamics (Chapter 4). epidemics across metapopula (Chapter 4). In In particular, particular, this this synchrony synchrony of of epidemics across the the metapopulation tion will will bias bias the the extinction extinction and and colonization colonization rates rates relative relative to to the the Levins Levins assumption. assumption. Extinctions Extinctions are, are, hence, hence, far far less less common common and and colonizations colonizations more more common 7.9C common than than expected expected when when the the majority majority of of patches patches are are infected infected (Figs. (Figs. 117.9C and 7.9D). In and 117.9D). In general, general, this this leads leads to to two two distinct distinct forms forms of of global global behavior: behavior: per persistent sistent endemic endemic infections infections or or irregular irregular short-duration short-duration epidemics. epidemics. Both Both of of these these states states will will be be stable stable in in the the medium medium to to long long term. term. Which Which type type of of behavior behavior is is observed observed is is critically critically dependent dependent on on the the initial initial conditions. conditions. Spatial Spatial coupling coupling in in epidemic epidemic metapopulations metapopulations consisting consisting of of aa geographic geographic mosaic 9 9 1 , Grenfell mosaic of of cities cities and and villages villages (May (May and and Anderson, Anderson, 11991, Grenfell and and Bolker, Bolker, 11998) 99 8 ) has has represented thorny scientific more than than half half aa cen represented aa thorny scientific question question for for more century. 1 957), in tury. No No simple simple answer answer has has as as yet yet been been found. found. As As ever, ever, Bartlett Bartlett ((1957), in his his study study of of the the scaling scaling of of epidemiological epidemiological coupling, coupling, has has been been seminal seminal in in prompt prompting ing detailed detailed work work in in both both spatial spatial geography geography and and epidemiology. epidemiology. Fifty Fifty years years hence, hence, the the challenge challenge of of understanding understanding epidemic epidemic coupling coupling still still stands. stands. Progress Progress is seasonally forced local is likely likely to to lie lie in in combining combining models models for for the the nonlinear, nonlinear, seasonally forced local dynamics dynamics of of measles measles with with detailed detailed transportation transportation data. data. We We see see two two strands strands of of recent recent work work that that offer offer aa way way forward forward in in the the face face of of this this daunting daunting challenge. challenge.
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First, Finkenstadt First, the the development development of of discrete discrete time time versions versions of of the the SIR SIR model model ((Finkenstfidt and and Grenfell, Grenfell, 2000; 2000; Bjornstad Bjornstad et et aI., al., 2002; 2002; Grenfell Grenfell et et aI., al., 2002) 2002) has has allowed allowed us to to model model the the local local scaling scaling of of dynamics dynamics and and importation importation of of infection infection over over us four population size. four orders orders of of magnitude magnitude in in host population size. Second, Second, time time series analysis analysis of of England England and and Wales Wales data data using using wavelet wavelet phase analysis analysis has has testified testified to to well welldefined defined hierarchical hierarchical traveling traveling waves waves of of infection, infection, moving moving from from large large centers centers to to the 7. 1 0; Grenfell 1). the surrounding surrounding hinterland hinterland (Fig. (Fig. 117.10; Grenfell et et aI., al., 200 2001). These These waves waves echo, echo, oonn aa larger larger spatiotemporal spatiotemporal scale, scale, the the hierarchical hierarchical waves waves detected Cliff et 98 1 ) . Simplistic detected in in earlier geographical geographical work work ((Cliff et aI., al., 11981). Simplistic spatiotem spatiotemporal 1 ) show poral models (Grenfell (Grenfell et et aI., al., 200 2001) show that that the the waves waves arise arise essentially essentially from from "forest " -like dynamics Bak et 990; Rand aI., 11995) 995) in "forest fire fire"-like dynamics ((Bak et aI., al., 11990; Rand et et al., in which which epi epidemic populations ignite demic "sparks" "sparks" of of infection infection from from the the large large core core populations ignite epidemics epidemics in in smaller, smaller, locally locally extinct extinct centers. centers. These These studies studies offers offers aa glimpse glimpse of of an an ultimate ultimate understanding understanding of of the the space-time space-time dynamics dynamics of of measles, measles, but but much much is is yet yet to to be be
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Fig. 117.9 7.9 Continued. SIR disease Continued. (C (C and and D) D) A A SIR disease model model with with measles-like measles-like parameters parameters and and aa between-patch Levins paradigm. between-patch coupling coupling of of f.L I~ = = 0.001 0.001.. Clearly Clearly the the latter latter contrasts contrasts with with the the Levins paradigm.
uncovered. uncovered. In In particular, particular, developing developing models models and and theories theories that that scale scale into into the the vaccination vaccination era era appear appear to to hold hold significant significant challenge. challenge.
Bubonic Bubonic Plague Plague in in the the Middle Middle Ages Ages 3 64, spreading Bubonic plague plague invaded invaded Europe Europe in in 11364, spreading rapidly rapidly north north from from the the ports 974) . For dis ports of of southern southern Italy Italy (Nobel, (Nobel, 11974). For the the next next 300 300 years years or or so the the disease ease ravaged the the towns towns and and cities cities of of Europe causing causing vast vast mortality mortality (Shrewsbury, 970). Bubonic (Shrewsbury, 11970). Bubonic plague plague is is aa disease disease of of rodents rodents that that is is generally generally transmitted transmitted by by fleas; fleas; occasionally occasionally it it spreads spreads to to humans, humans, which which is is when when cases cases are are generally generally first first noticed. noticed. Records Records show show that that although although the the disease disease was was endemic endemic in in Europe Europe as as aa whole whole through through three three centuries, centuries, each each community community displayed isolated isolated epidemics human host host population population cases cases followed displayed epidemics in in the the human followed by by ""disease-free" disease-free" periods. periods. It bubonic plague It has has therefore therefore long long been been thought thought that that bubonic plague exhibited exhibited classic classic metapopulation metapopulation behavior behavior at at the the regional regional scale, scale, with with the the infection infection continually going going extinct extinct and and then then recolonizing recolonizing communities communities (Appleby, 9 8 0 ) . This (Appleby, 11980). This conventional conventional wisdom wisdom contradicts contradicts two two pieces pieces of of
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historical historical evidence. evidence. First, First, there there is is aa fairly fairly regular regular cyclic cyclic nature nature to to human human epi epidemics, demics, which which is is unlikely unlikely to to be be caused caused by by random random imports imports of of infection. infection. Second, Second, even even in in communities communities with with tight tight quarantine quarantine controls, controls, there there is is little little change of epidemics 980). change to to the the pattern pattern of epidemics (Appleby, (Appleby, 11980). Keeling Keeling and and Gilligan Gilligan (2000a,b) (2000a,b) focused focused oonn the the interaction interaction among among rats, rats, fleas, fleas, and and humans humans within within aa metapopulation metapopulation setting. setting. The The life life cycle cycle of of the the plague plague can can be be partitioned partitioned into into distinct distinct stages stages and and follows follows aa general general pattern pattern for for vector vectorborne diseases. borne diseases. Fleas Fleas that that feed feed on on an an infected infected rat rat ingest ingest the the bubonic bubonic plague plague bacteria Yersinia pestis) and bacteria ((Yersinia and become become infectious. infectious. When When an an infected infected rat rat dies, dies, its its fleas, which now infectious, leave to search for host. Usually Usually the fleas, which are are by by now infectious, leave to search for aa new new host. the fleas rats, infect so spread spread the fleas find find other other rats, infect them, them, and and so the disease disease through through the the rodent rodent community. community. Only Only when when the the density density of of rats rats is is low low are are the the fleas fleas forced forced to to feed feed on alternative alternative hosts, hosts, such as humans, humans, and and spark off aa human epidemic. on such as spark off human epidemic. Humans considered aa dead-end host, as transmission from Humans are are considered dead-end host, as transmission from humans humans to to fleas fleas is is rare. rare. Direct Direct transmission transmission between between humans humans is is possible possible if if the the pneumonic pneumonic form form of the such infection, of the disease disease develops, develops, but but due due to to the the rapidity rapidity and and virulence virulence of of such infection,
437 411
117. 7. METAPOPULATION OF IINFECTIOUS NFECTIOUS DISEASES METAPOPULATION DYNAMICS DYNAMICS OF DISEASES
pneumonic pneumonic epidemics epidemics are are small small and and short short lived. lived. These These epidemiological epidemiological obser observations vations can can be be translated translated into into aa mathematical mathematical model: model:
SR F [1 rRSR(1-- ~R ) + RR(1 --p)-- dRSR-- f3R-~R
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where where
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)
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== [[dR ---:it dR + + mR( mR(1l dt dF
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TR
) ] hN - d pF. -- gR gR)]IRNdvF.
== rH (SH + RH ) - dH SH - [3HSHF - aTR), ---:it rH(SH RH)dHSH[3HSHFexp( exp(--aTR), dt dSH dIH
dlH ---:it =- [3HSHF dH + mH)IH f3HSHFexp( e x p (-aT - - a TRR) ) - ((dH mH)IH,, dt
= mHgHIH - dHRH. ---:it dt = mHgHIH- dHRH. dRH
((17.10) 1 7. 1 0 )
where where SSR, and R RR refer ttoo the the number number ooff susceptible, susceptible, infectious, infectious, and and resist resistR, IIR, R, and R refer ant popu ant rats, rats, respectively respectively SSH, and RH R/q are are similar similar quantities quantities for for the the human human popuH, IIH, H, and lation, lation, N N is is the the average average number number of of fleas fleas on on aa rat, rat, and and F is is the the number number of of free-living 17.1 lists free-living infected infected fleas fleas that that are are searching searching for for aa host. host. Table Table 17.1 lists the the meaning and value of other parameters used in meaning and value of all all other parameters used in the the model, model, which which have have been been estimated estimated from from historical historical data data or or experiments experiments (Keeling (Keeling and and Gilligan, Gilligan, 2000b) 2000b).. The The behavior behavior ooff the the theoretical theoretical model model for for this this system system iiss critically critically dependent dependent on on stochasticity stochasticity and and scaling. scaling. For For large large host host populations, populations, aa deterministic deterministic solu solution (as expected tion gives gives rise rise to to aa constant constant level level of of infection infection in in the the rodents rodents (as expected from from most number of human cases. most SIR-type SIR-type models) models) and and aa negligible negligible number of human cases. However, However, when stochastic large epidemics when stochastic effects effects play play aa major major role, role, unusually unusually large epidemics may may drive drive the such low low levels that the forced to the rat rat population population to to such levels that the fleas fleas are are forced to feed feed on on alter alternative occurs. This localized extinction native hosts hosts and and aa human human epidemic epidemic occurs. This results results in in localized extinction of subsequent local build-up of of the the disease. disease. The The subsequent local dynamics dynamics depends depends on on the the build-up of the the susceptible population. Fairly recolonizations of susceptible rat rat population. Fairly rapid rapid recolonizations of infection infection lead lead to to an an endemic population and endemic persistence persistence in in the the rat rat population and few, few, if if any, any, human human cases. cases. In In con contrast, recolonization is population has trast, if if recolonization is rare rare and and hence hence the the susceptible susceptible rat rat population has time time to increase to levels, major spillover in to increase to high high levels, major epidemic epidemic cycles cycles with with resultant resultant spillover in human behaviour is human hosts hosts occur. occur. Thus, Thus, the the epidemic epidemic behaviour is determined determined by by the the mix mixture of local transmission transmission dynamics, dynamics, stochasticity, stochasticity, and and spatial spatial coupling. coupling. ture Good Good evidence evidence suggests suggests that that in in any any large large town town or or city, city, rats rats are are unlikely unlikely to to act act as as aa homogeneously homogeneously mixing mixing host host population, population, and and therefore therefore aa spatially spatially seg segregated metapopulation approach appropriate. Studies regated metapopulation approach may may be be more more appropriate. Studies per performed 1 906) showed formed by by the the Plague Plague Commission Commission in in India India ((1906) showed that that the the spatial spatial
MATT MATT J.I. KEELING KEELINGET ET AL. AL.
438 438
TABLE 117.1 7.1 Parameter Parameter
rrRR p p KR K R ddRR f3[3R R mR m R ggRR I~R ILR a rF dF KF ~F ILF rH dH [3H f3H m H mH gH
Parameters Used in the the Bubonic Plague Model Model
Value
55 0.975 0.975 2500 2500 0.2 0.2 4.7 4.7 20 20 0.02 0.02 0.03 0.03 44 X • 1100-3 -3 20 20 1100 1 . 1 7 mean mean 6.57 3.29 3.29 � --o 111.17 6.57 0.008 0.008 0.045 0.045 0.04 0.04 0.01 0.01 26 26 0.1 0.1
Meaning Meaning Reproductive Reproductive rate rate of of rat rat Probability Probability of of inherited inherited resistance resistance Carrying Carrying capacity capacity of of rat rat Death Death rate rate of of rats rats Transmission Transmission rate rate (Infectious (Infectious period) period) --1l Probability Probability of of recovery recovery Movement Movement rate rate of of rats rats Flea searching searching efficiency efficiency Flea Reproductive Reproductive rate rate of of flea flea Death Death rate rate ooff fleas fleas Carrying Carrying capacity capacity of of flea flea per per rat rat Movement Movement rate rate of of fleas fleas Reproductive Reproductive rate rate of of humans humans Death Death rate rate of of humans humans Transmission Transmission rate rate to to humans humans (Infectious (Infectious period) period)-- l1 Probability Probability of of recovery recovery
spread through the slow due spread of of the the epidemic epidemic through the rodent rodent population population was was extremely extremely slow due to their largely this corresponds to their largely territorial territorial nature; nature; this corresponds well well with with historical historical evidence evidence of cities. Figure 7. 1 1 of slow-moving slow-moving waves waves of of infection infection in in the the large large medieval medieval cities. Figure 117.11 shows population shows the the number number ooff bubonic bubonic plague plague cases cases in in rodents rodents in in aa meta metapopulation model consisting of Persistence of population is model consisting of 25 25 local local populations. populations. Persistence of the the meta metapopulation is due due to to the the local local populations populations that that remain remain close close to to the the endemic endemic state state (e.g., (e.g., cen central tral site site for for the the latter latter part part of of the the simulation), simulation), whereas whereas human human cases cases (and (and thus thus historical historical reports) reports) are are due due to to the the stochastically stochastically driven driven large large epidemics. epidemics. Due Due to to the time time necessary necessary for for the the susceptible susceptible rat rat population population to to recover, recover, these large large epi epithe demics 0-12 yr, demics have have aa period period of of around around 110-12 yr, which which corresponds corresponds remarkably remarkably well well with with the the historical historical observations. observations. As As observed observed earlier, earlier, the the classic classic Levins Levins metapopulation metapopulation does does not not readily readily cap capture ture the the dynamics dynamics of of spatially spatially structured structured epidemics epidemics due due to to the the strong strong correla correlations tions that that often often exist exist between between local local and and global global levels levels of of infection. infection. However, However, for for plague, can be plague, such such correlations correlations are are weak, weak, and and the the local local populations populations can be classified classified into infection and extinc into three three basic basic states: states: endemic endemic (low (low level level of of infection and low low risk risk of of extinction), level of infection and risk of tion), epidemic epidemic (high (high level of infection and high high risk of extinction), extinction), and and extinct extinct ((but but susceptible) susceptible).. The The extinct extinct class class is is further further subdivided subdivided so so as as to to mimic mimic the the gradually 1 2 shows shows aa carica gradually increasing increasing susceptible susceptible rat rat population. population. Figure Figure 17. 17.12 caricature bubonic plague. ture schematic schematic of of the the Levins-type Levins-type model model for for bubonic plague. For For this this type type of of spatiotemporal spatiotemporal dynamics, dynamics, where where the the behavior behavior is is classified classified easily of states, easily into into aa discrete discrete set set of states, the the Levins Levins approach approach provides provides great great improvements computational efficiency improvements in in computational efficiency and and clarity. clarity. The The Levins Levins formulation formulation allows allows us us to to consider consider the the dynamics dynamics at at aa far far larger larger scale scale and and hence hence observe observe the the wave-like spread spread of endemic centers centers (Keeling wave-like of the the epidemics epidemics away away from from the the endemic (Keeling and and Gilligan, models it Gilligan, 2000b) 2000b).. From From these these models it is is clear clear that that the the epidemic epidemic wave wave is is often often short lived lived and and self-extinguishing, of endemic short self-extinguishing, confirming confirming the the importance importance of endemic pop populations in in allowing allowing for for long-term long-term disease disease persistence. persistence. ulations
439 439
NFECTIOUS DISEASES METAPOPULATION DYNAMICS OF IINFECTIOUS DISEASES 117. 7. METAPOPULATION
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Conservation C o n s e r v a t i o n or o r Contamination Contamination An An interesting interesting extension extension to to the the classic classic metapopulation metapopulation models models for for the the popu population 996; lation dynamics dynamics of of endangered endangered species species is is the the inclusion inclusion of of disease disease (Hess, (Hess, 11996; Gog Gog et et aI., al., 2002 2002).) . In In the the absence absence of of infection, infection, increasing increasing the the spatial spatial coupling coupling between between isolated isolated habitats habitats will will increase increase the the level level of of patch patch occupancy occupancy and and decrease decrease the the risk risk of of global global extinction extinction for for one one threatened threatened species. species. Using Using the the Levins Levins metapopulation metapopulation framework, framework, with with coupling SIGMA, SIGMA, the the occupancy occupancy level level x x is is given given by: by:
dx dx
11
((17.11) 1 7. 1 1 )
where where ee is is the the extinction extinction rate rate and and cc is is the the probability probability that that invasion invasion ooff aann empty empty patch patch is is successful. successful. From From this this simple simple model it it is is clear clear that that movement between between largely isolated habitats largely isolated habitats improves improves the the persistence persistence of of the the endangered endangered species. species.
MATT MATT j.J. KEELING KEELING ET ET AL. AL.
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Fig. 1 77.. 1122 Representation of transition states in a structured meta metapopulation Fig. population model (Gyllenberg aI., 11997) 997) of bubonic bubonic plague. plague. Solid arrows represent probabilistic transitions, (Gyllenberg et al., transitions, which occur independently independently of the surrounding surrounding environment. environment. Dashed arrows show transitions which import of infection from from a neighboring neighboring patch. Rates of transition can that only occur due to the import be measured from population model. from the full stochastic meta metapopulation model.
This increased coupling coupling 0" This effect effect occurs occurs for for two two distinct distinct reasons. reasons. Primarily, increased cr leads to likelihood of to a higher higher colonization colonization rate, rate, which which increases the likelihood of rescue rescue events number of occupied local populations populations at equilibrium. equilibrium. Second, large and the number of occupied large and the amounts the dynamics, amounts of movement movement between between patches synchronize synchronize the dynamics, the local populations act habitat, and populations suffer populations act effectively as as one large habitat, and large populations suffer a relatively less extinction stochasticity. The The single-species model extinction risk from from stochasticity. model thus thus reveals no of demographic no benefit of demographic heterogeneities heterogeneities between between local populations populations as one large population (or several tightly tightly coupled populations) populations) shows greatpopulation (or shows the great est persistence. persistence. This result result is echoed echoed by full full stochastic stochastic metapopulation meta population equaequa tions with explicit explicit within-patch dynamics (Keeling, 2000b). tions with within-patch dynamics 2000b). This conclusion conclusion can can be altered altered radically radically in in the the presence presence of of a virulent virulent infecinfec This 1996). tious disease, disease, as coupling coupling also facilitates facilitates the the spread spread of of infection infection (Hess, 1996). tious The resultant resultant cost-benefit cost-benefit trade-off trade-off depends depends on on the the relative relative levels levels of of host host The extinction with and without without the the disease, disease, as well well as the the relative relative colonization colonization extinction with and rates. dynamics of rates. Gog Gog et et al. (2002) (2002) used used the the following following model model to to explore explore the the dynamics of infected infected (I) and and uninfected uninfected (S) habitat: habitat:
dS s S -- cO"OlS r S I S -- ggS S = (rS(1 O"S( l -- II- - SS)) -- eesS dt dt dI dI = (0"r I1((11 -- I1- - S)S) -- eell iI + rSIS + S + (O"OlS + ggS dt dt -
-
( 1 7.12) (17.12)
117. 7.
METAPOPULATION METAPOPULATION DYNAMICS D Y N A M I C S OF OF INFECTIOUS INFECTIOUS DISEASES DISEASES
441 441
where where a (r is is the the rate rate of of movement movement to to and and colonization colonization of of empty empty habitat, habitat, es and and eI e1 are are the the patch patch level level extinction extinction rates, rates, 08 is is the the chance chance that that movement movement leads leads to to the the spread spread of of infection, infection, and and g is is the the import import rate rate of of infection infection from from outside outside the the considered considered population. population. As As this this is is aa model model of of wildlife wildlife disease, disease, the the coupling coupling between between populations populations occurs occurs as as the the random random dispersal dispersal of of organisms organisms rather rather than than the the short-duration short-duration commuter commuter movements movements associated associated with with human human disease disease trans transmission. The mission. The focus focus of of this this model model is is conservation conservation of of an an endangered endangered host, host, and and therefore therefore is is the the reverse reverse of of the the scenarios scenarios discussed discussed earlier earlier where where the the eradication eradication of of infection infection was was the the main main aim. aim. In In agreement agreement with with the the earlier earlier work work of of Hess Hess ((1996), 1 996), this this Levins-like Levins-like model model shows shows that that under under certain certain circumstances circumstances greater greater movement larger a) movement between between patches patches ((larger or) can can lead lead to to aa reduction reduction in in the the number number of of occupied occupied patches patches and and an an increased increased risk risk of of global global extinction extinction to to highlight highlight an an important Fig. 117.13). 7.13). important conservation conservation risk risk ((Fig. IItt iiss informative informative ttoo consider consider aann extreme extreme variation variation ooff this this model. model. Suppose Suppose that that the the disease disease within within an an infected infected patch patch is is severe severe and and widespread widespread so so that that ani animals unable to mals from from infected infected patches patches are are unable to colonize colonize aa new new habitat habitat successfully. successfully. The model model then then can be rewritten as The can be rewritten as
dS dS == as(1 o-S(1 - I I - - S) S) - (ao)lS (ty8)IS - esS esS - gS gS dt dt dl dI == (ao)lS (itS)IS-- ((eI e i -- ees)l s)I- esl esI + + gS dt dt
((17.13) 1 7. 1 3 )
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J i i i i O0 �--J--�----�--L--� 0.2 o0 0.2 0.4 0.6 0.6 0.8 0.8 1 0.4 Movement Movement rate, rate, (J a
Fig. 117.13 7. 1 3 Effects Effectsoonn patch patch occupancy occupancy of of between-patch between-patch coupling coupling a ~ for for the the three three models models are 7.1 2) and and the are shown. shown. Both Both the the model model of of Gog Gog et et a!. al. (2002) ( 2 0 0 2 ) (1 (17.12) the simplified simplified SIR-like SIR-like version version (1 7.1 3) show that increased occupancy in presence (1 7.13) show that increased levels levels of of coupling coupling can can decrease decrease patch patch occupancy in the the presence 1 , e{ 0.2, 1)8 = of of an an infectious infectious disease disease (c (c = = 11,, e = = es = = 0. 0.1, el = = 0.2, = 0.5, 0.5, 9 g = = 0.001 0.001).).
MATT MAI-F ]J.. KEELING KEELING ET ET AL. AL.
442 442
This model then then shares elements with standard SIR models; (TO This model shares many many elements with standard SIR disease disease models; (r8 plays the role of the the transmission �, es plays the role of transmission parameter parameter [3, e s corresponds corresponds to to the the natural natural death rate, and equates to recovery rate. parallel with death rate, and eI ei - es e s equates to the the recovery rate. The The emergent emergent parallel with classical models allows allows us resultant dynamics. dynamics. For classical disease disease models us to to intuit intuit about about the the resultant For example, example, changes changes in in the the movement movement rate rate (T (r correspond correspond to to aa trade-off trade-off between between an an increase increase in in the the birth birth rate rate and and an an increase increase in in disease disease transmission. transmission.
Vaccination, Pulses, Pulses, and Synchrony A issue of population modeling A key key issue of meta metapopulation modeling for for infectious infectious diseases diseases is is to to com compare vaccination strategies likelihood of pare different different vaccination strategies to to optimize optimize the the likelihood of disease disease eradi eradication. cation. As As discussed discussed earlier, earlier, the the global global persistence persistence of of aa disease disease is is determined determined by by two main factors: local extinction two main factors: the the local extinction rate rate and and the the rate rate of of recolonization, recolonization, which population. which in in turn turn is is related related to to the the heterogeneity heterogeneity of of the the meta metapopulation. Figure 7. 1 4 shows Figure 117.14 shows how how these these two two facets facets change change as as the the level level of of vaccination vaccination increases; consider the solid black line, which increases; we we first first consider the solid black line, which corresponds corresponds to to contin continuous vaccination. Below vaccination level uous random random vaccination. Below the the critical critical vaccination level of of 90% 90% the the local local extinction increase with extinction rate rate shows shows only only aa moderate moderate increase with the the level level of of vaccination vaccination so so that that the the expected expected length length of of an an epidemic epidemic decreases decreases slowly. slowly. In In contrast, contrast, the the cor correlation relation between between two two coupled coupled local local populations populations starts starts to to decrease decrease from from the the onset of Therefore, in Levins formulation, moderate levels onset of vaccination. vaccination. Therefore, in the the Levins formulation, moderate levels of of vaccination only cause cause aa small extinction rate, rate, which vaccination only small increase increase in in the the extinction which may may be be counteracted therefore the res counteracted by by the the increase increase in in asynchrony asynchrony and and therefore the increase increase in in rescue most needed. cue effects effects when when they they are are most needed. The The balance balance between between vaccination vaccination increasing increasing the the stochastic stochastic extinction extinction rate rate but but reducing reducing the the synchrony synchrony between between populations populations depends depends on on the the demographic demographic and and epidemiological epidemiological parameters. parameters. Thus Thus while while moderate moderate levels levels of of vaccination vaccination will will always always act act to to reduce reduce the the total total number number of of cases, cases, they they may may surprisingly surprisingly increase persistence of loss of increase the the global global persistence of the the disease disease if if the the loss of synchrony synchrony is is dra dramatic enough. However, as as the the level level of of vaccination vaccination approaches approaches the the critical critical matic enough. However, eradication rapid rise extinctions will eradication threshold, threshold, the the rapid rise in in the the rate rate of of local local extinctions will over overwhelm any any rescue rescue effects effects and and global extinction will will inevitably inevitably follow. follow. whelm global extinction Obviously, be a if as Obviously, vaccination vaccination would would be a much much more more effective effective tool tool if as well well as as reducing reducing the the number number of of cases cases it it could could also also decrease decrease the the global global persistence persistence of of the the disease. disease. In In principle, principle, this this can can be be achieved achieved by by superimposing superimposing periodic periodic "pulses" of of vaccination vaccination on on the the overall overall background background rate. rate. Pulsed Pulsed vaccination vaccination has has "pulses" been aI., 11993), 993), but been proposed proposed to to increase increase the the efficiency efficiency of of vaccination vaccination (Agur (Agur et et al., but it it could could also also have have aa spatial spatial benefit benefit by by "lining "lining up up"" epidemic epidemic troughs troughs and and there therefore (Earn et 9 9 8 ) . The fore reducing reducing rescue rescue effects effects (Earn et aI., al., 11998). The impact impact of of aa simple simple model model for pulse vaccination (in the absence of background vaccination) is shown for pulse vaccination (in the absence of background vaccination) is shown in in 7. 1 4 . The gray in in Fig. Fig. 117.14. The first first observation observation is is that that pulse pulse vaccination vaccination is is associated associated gray also more cases of disease; with aa slightly slightly lower lower local local extinction extinction rate, rate, and and also more cases of the the disease; with this this is is because because in in the the gaps gaps between between the the vaccination vaccination pulses pulses children children that that would would have vaccination have chance of have been been immunized immunized under under continuous continuous vaccination have aa chance of catching catching the pulsing would the infection infection-- in in practice practice though, though, any any pulsing would probably probably be be superim superimposed on constant 'background' realistic. The posed on aa constant 'background' rate, rate, so so that that this this effect effect is is not not realistic. The difference between pulsed pulsed and more dramatic difference between and continuous continuous vaccination vaccination is is more dramatic in in terms of of the the correlation correlation between between epidemics. epidemics. The The significant significant perturbation perturbation terms caused by vaccination campaign caused by aa periodic periodic vaccination campaign acts acts to to synchronize synchronize the the dynamics dynamics
OF INFECTIOUS DISEASES DISEASES 117. 7. METAPOPULATION DYNAMICS OF
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Fig. 11 77.14 of vaccination vaccination on on the the characteristics characteristics of of unforced unforced SIR SIR epidemics. epidemics. Black Black symsym Fig. . 1 4 Effects Effects of to constant constant random random vaccination vaccination at at birth, birth, whereas whereas gray gray symbols symbols correspond correspond to to pulse pulse bols refer to bols refer vaccinating randomly at at regular regular 4-yr 4-yr intervals; intervals; similar similar results results are are achieved achieved for more frequent frequent vaccinating randomly for more yearly pulses. pulses. (A) Change in in the the local local extinction extinction rate (per day) day) of of an an isolated population. yearly (A) Change rate (per isolated population. (B) (B) Change Change in in the the correlation correlation between between two two local local populations populations coupled coupled at at aa level level ~IT == 0.01. 0.01 . (Population size size is is 10,000, 1 0,000, R0 Ro = 11 O, 0, g9 == 10 1 0 days, days, import import rate rate is is 55 per per year.) year.) (Population =
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MATT J.J. KEELING KEELING ET ET AL. AL. MATT
of the the two two populations, populations, thus thus for for pulsed pulsed vaccination vaccination the the correlation correlation remains remains of approximately constant constant for for vaccination vaccination levels levels below below 60%. 60 % . approximately Pulsed vaccination vaccination therefore therefore provides provides aa potentially potentially important important tool tool for for Pulsed increasing local local extinctions, extinctions, without without increasing increasing the the effective effective rescue rescue events, events, and and increasing therefore increasing increasing the the likelihood likelihood of of global global extinctions extinctions (compare (compare to to Levins, Levins, therefore 1 96 9 ) . Results Results shown shown in Fig. 17.14 1 7. 1 4 have have ignored ignored seasonal seasonal forcing, forcing, which which natnat 1969). urally synchronization of urally leads leads to to greater greater synchronization of the the dynamics. dynamics. When When seasonal seasonal forcing forcing important (such ( such as for for most most childhood childhood diseases), diseases) , the the interaction interaction of of backback is important ground vaccination, vaccination, vaccination vaccination pulses pulses and and seasonal seasonal effects effects may may be be very very comcom ground plex; the the precise precise timing timing of of vaccination vaccination could could significantly significantly increase increase the the chance chance plex; of regional regional eradication. eradication. However, However, this this is is very very much much an an open open problem problem for future of for future research. research.
1 7.6 17.6
FUTURE DIRECTIONS DIRECTIONS FUTURE Metapopulation history in the ecological ecological literature literature and Metapopulation theory theory has has aa rich rich history in the and has proved proved itself itself continually continually as as both and an insightful has both an applied applied tool tool and insightful descripdescrip tion of complex world world ((Gilpin Gilpin and Hanski, 1991; 1 99 1 ; Hanski Hanski and Gilpin, tion of the the complex and Hanski, and Gilpin, 1997). The use use of metapopulations 1 997). The of meta populations has has been somewhat somewhat more more limited limited in epiepi demiology to the the more dynamics. However, However, in in demiology due due to more complex complex within-patch within-patch dynamics. recent balance has to be redressed. Several key theoretical theoretical and and recent years this this balance has begun begun to be redressed. issues still still need need to to be be dealt the subject subject to to practical dealt with with successfully successfully to to allow allow the practical issues develop develop further. further.
A better understanding understanding of of how the the epidemiological epidemiological and and demographic demographic 11.. A parameters metapopulation parameters parameters translate translate into into the the Levins-type Levins-type metapopulation parameters of of extinction and and colonization rates. The The ability ability to to translate translate stochastic stochastic extinction simple set population level within-patch population population dynamics dynamics into into aa simple set of of population level within-patch states computational speed states would would lead to to aa vast vast increase in in computational speed and and provide powerful insights insights into into the the spatiotemporal dynamics of of disease disease spread spread powerful spatiotemporal dynamics and and extinction. Although Although moment-closure moment-closure approximations approximations and and quasi quasiequilibrium solutions offer offer aa likely likely approach, they they have have yet yet to to be be applied applied to to realistic realistic seasonally seasonally forced forced dynamics. dynamics. 2. More More detailed detailed simulations simulations of of heterogeneous patches patches with with complex complex connections Chapter 4). connections ((Chapter 4). So So far far the the majority majority of of studies studies have have considered considered equally equally sized sized local local populations populations and and global global coupling. coupling. While While this this is is aa natural natural starting starting point, point, the the real real world world is is far far more more complex, complex, and and developing models models and and intuition intuition for for such such scenarios scenarios will will be be important important developing measures are if spatially spatially targeted targeted control control measures are to to be be applied applied most most effectively. effectively. if 33.. A A range range of of more more powerful powerful statistical statistical and and mathematical mathematical techniques techniques are are also also required required to to deal deal with with coupling. coupling. First, First, there there is is the the complex complex problem problem of how how to to estimate estimate the the coupling coupling between between communities communities from from case case reports. reports. of This This estimation estimation process process is is confounded confounded by by stochasticity, stochasticity, seasonality, seasonality, and and heterogeneities heterogeneities in in demographic demographic rates, rates, although although some some progress progress has has been been Associated with with this this problem problem is is developing developing mathematical mathematical rules for made. Associated the the coupling coupling between between populations populations as as aa function function of of their their separation. separation. In In aa meta population of metapopulation of N patches, patches, there there are are N(N N(N - 1 1)) coupling coupling terms, terms, hence hence in large large systems systems estimating estimating or or even even storing storing all all the the coupling coupling rates rates becomes becomes in -
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problematic problematic so so analytical analytical approximations approximations become become necessary. necessary. Developing Developing gravity Cliff and 9 8 8 ) and gravity ((Cliff and Hagget, Hagget, 11988) and other other formulations formulations of of the the relation relationship ship between between human human movement movement and and disease disease spread spread is is an an important important problem problem for for both both fundamental fundamental population population biology biology and and applied applied epidemi epidemiology. ology. Although Although these these three three problems problems present present formidable formidable challenges, challenges, metapopula metapopulations tions are are likely likely to to see see far far more more use use in in the the future future as as the the degree degree of of realism realism and and resolution resolution required required from from models models increases. increases.
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TOWARD A 18o TOWARD META PO PUL ATION M ETAPO PU LATI ON CONCE PT FOR PL ANTS CONCEPT PLANTS N.J. Ouborg Ouborg and and O. O. Eriksson Eriksson
118.1 8. 1
INTRODUCTION INTRODUCTION The The spatial spatial structure structure of of populations populations and and communities communities has has always always been been an an important e.g., Wright, important component component of of ecological ecological and and evolutionary evolutionary theory theory ((e.g., Wright, 11931; 93 1 ; Andrewartha 954; den Boer, 96 8 ) . Introduction Andrewartha and Birch, Birch, 11954; Boer, 11968). Introduction of the the the theories 967) and metapopula ories of of island island biogeography biogeography (MacArthur (MacArthur and and Wilson Wilson 11967) and metapopulaLevins, 11969) 96 9 ) further tion dynamics dynamics ((Levins, further enhanced enhanced the the use of spatial spatial considerations considerations and and arguments arguments in in explanations explanations of of ecological ecological and and evolutionary evolutionary population concept has phenomena. Over the last several decades, the meta metapopulation become become the the guideline guideline for for our our understanding understanding of of issues issues as as diverse diverse as as large-scale large-scale population population dynamics, dynamics, the the spatial spatial distribution distribution of of species, species, the the dynamics dynamics of of species species interactions, interactions, and and the the effects effects of of habitat habitat fragmentation fragmentation on on biodiversity. biodiversity. The development of metapopulation theory was inspired by the dynamics of of animal animal populations, and and many many of of the the models implicitly implicitly have have the the features features of of aa "model 1 99 6 ) suggested "model animal" animal" as as their their basis. basis. Husband Husband and and Barrett Barrett ((1996) suggested that that this population concept and this may may have have led led to to aa mismatch mismatch between between the the meta metapopulation and the the features of the population biology of plants. Whether or not this is the case, a fact is that despite the patchy spatial structure of plant populations, and despite 970s and 980s of despite the the establishment establishment during during the the 11970s and 11980s of aa population-oriented population-oriented paradigm 977; Silvertown, 987), metapopulation paradigm in in plant plant ecology ecology (Harper, (Harper, 11977; Silvertown, 11987), metapopulation theory theory has has received received comparatively comparatively little little attention attention in in plant plant studies. studies. Two Two reviews reviews
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(Eriksson, 996; Husband 99 6 ) concluded (Eriksson, 11996; Husband and and Barrett, Barrett, 11996) concluded that that there there is is much much indirect indirect evidence evidence for for metapopulation metapopulation dynamics. However, However, only only aa handful handful of of studies studies have have shown shown that that regional regional plant plant populations populations exhibit exhibit features features of of typical typical meta populations, for turnover of metapopulations, for example, example, turnover of populations populations within within aa set set of of suitable suitable sites sites with with colonization colonization rates rates dependent dependent on on distance distance between between suitable suitable sites sites and/or extinction extinction rates rates dependent on local local population population size size (Menges, (Menges, 11990; van and/or dependent on 990; van der 992; Ouborg, 993; McCauley et 995; Quintana der Meijden et et aI., al., 11992; Ouborg, 11993; et aI., al., 11995; QuintanaAscencio and Menges, 11996; 996; Barrett 997; Giles Giles and Ascencio and Menges, Barrett and and Husband, Husband, 11997; and Goudet, Goudet, 11997; 997; Bastin 999; Harrison Harrison et Bastin and and Thomas, Thomas, 11999; et aI., al., 2000) 2000).. The The applicability applicability of of the the metapopulation metapopulation concept concept ttoo plants plants has has even even been been questioned, questioned, both both on on empirical empirical and and on on theoretical theoretical grounds. grounds. Scheiner Scheiner and and Rey ReyBenayas 1 997) analyzed patterns of of aa large Benayas ((1997) analyzed distribution-abundance distribution-abundance patterns large sample sample of plant plant communities communities and and concluded that metapopulation were gen genof concluded that metapopulation models were erally patterns occurring erally unable unable to to predict predict patterns occurring at at larger larger scales scales than than 11 km2• km 2. As As explanations of the population models, Scheiner explanations for for the the failure failure of the examined examined meta metapopulation models, Scheiner and 1 997) suggested and Rey-Benayas Rey-Benayas ((19971 suggested that that assumptions assumptions concerning concerning among-site among-site variation equal migration variation in in quality, quality, site site connectivity connectivity and and equal migration and and extinction extinction rate, rate, do do not not hold. hold. A A problem with with this this kind kind of of test test is is that that species are are lumped lumped together assumed to together and and implicitly implicitly assumed to behave behave similarly similarly with with regard regard to to regional regional dynamics. dynamics. As As discussed discussed later, later, plants plants as as aa group group are are heterogeneous heterogeneous with with regard regard to to their their spatial dynamics, and and therefore therefore we we may may not not expect expect that that distribution distribution patterns species occurring occurring in patterns for for all all species in aa region region will will fit fit into into aa single single metapopu metapopulation lation model. model. The The metapopulation metapopulation concept concept has has also also been been questioned questioned as as aa useful useful tool tool for for understanding understanding regional regional dynamics dynamics of of single single species. species. Freckleton Freckleton and and Watkinson Watkinson (2002) focused focused on on the the methodological methodological difficulties difficulties to to define define suitable suitable ((but unoccu(2002) but unoccu prop pied) patches, patches, to to document document turnover turnover in in sets sets of of local populations, populations, and and to to properly plant erly assess assess dispersal dispersal among patches and and concluded concluded that that in in many many plant populations populations the the assumptions assumptions of of aa metapopulation metapopulation model do do not hold hold and and that that metapopulations metapopulations in in plants may may be be restricted restricted to to relatively relatively few few cases. cases. As As alter alternatives concept, they natives to to the the metapopulation metapopulation concept, they suggested suggested other other concepts, concepts, forming forming aa new new typology of of plant plant regional dynamics, using using "spatially "spatially extended extended popula populations" and and "regional "regional ensembles" ensembles" as as the the main main types. types. The The key key issue issue for for this this alternative alternative typology typology is is the the fraction fraction of of suitable suitable habitat habitat in in aa region. region. If If there there is is aa lot lot of of suitable suitable habitat, habitat, Freckleton Freckleton and and Watkinson Watkinson argued, populations populations will will occur throughout large Thus, local occur continuously continuously throughout large areas. areas. Thus, local processes processes will will domin dominate dynamics. If, (and patches ate the the dynamics. If, however, however, there there is is very very little little suitable suitable habitat habitat (and patches occur populations will occur more more or or less less isolated) isolated),, the the populations will form form regional regional ensembles. ensembles. These These consist consist of of basically basically unconnected unconnected local local populations, populations, in in turn turn leading leading to to aa dominance dominance of local over regional regional processes. processes. Of course, course, extinction extinction is is usually usually considered also in considered aa local local process process also in metapopulation metapopulation models, models, but but strong strong isolation isolation may may imply imply that that aa regional regional turnover turnover is is more more or or less less absent. absent. In In another another review, review, Bullock 2002) suggested that it Bullock et et ai. al. ((2002) suggested that it is is too early early to to determine determine whether "true" "true" plant plant metapopulations metapopulations exist exist because because we we lack lack the the relevant relevant data. data. Nevertheless, Nevertheless, they doubt over they cast cast doubt over the the applicability applicability of of the the concept in in plants, mainly mainly because because local local extinction extinction and and recolonization recolonization phenomena phenomena could could well well be be explained explained by by rather than than regional regional processes. processes. local rather These population concept These critics critics of of the the meta metapopulation concept for for plants plants illustrate illustrate two two prob problems lems that that any any study study of of plant plant metapopulations metapopulations must must deal deal with. with. The The first first type type of of
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problem problem is is habitat habitat related. related. The The distribution distribution of of plants, plants, especially especially at at larger larger spa spatial tial scales, scales, tends tends to to reflect reflect abiotic abiotic conditions, conditions, climate, climate, and and soil. soil. Why Why should should we we expect population models expect that that meta metapopulation models based based on on colonization colonization and and extinction extinction dynamics dynamics dependent dependent on on spatial spatial configuration configuration of of habitats habitats should should add add signifi significantly understanding regional dynamics cantly to to understanding dynamics at at these these scales? scales? This This chapter chapter pro provides vides some some reasons reasons why why the the spatial spatial configuration configuration of of habitats habitats affects affects dynamics dynamics also also when when aa set set of of local local plant plant populations populations do do not not behave behave as as typically typically envisaged envisaged in in metapopulation metapopulation models. Moreover, Moreover, basic basic needs needs of of plants plants (water, (water, nutrients, nutrients, light) light) are are often often quite quite unspecific, unspecific, which which means means that that indirect indirect means means of of assessing assessing habitat habitat quality quality may may prove prove useless. useless. We We will will discuss discuss experimental experimental approaches approaches to to handle handle habitat habitat suitability, suitability, and and we we stress stress the the need need for for considering considering variation variation in in habitat habitat quality quality among among sites. sites. A A second second problem problem relates relates to to features features of of the the plants plants themselves. themselves. Plants Plants pos possess sess aa number number of of life life cycle cycle features features that that make make them them different different from from the the "model "model organism, organism,"" short short lived lived and and mobile, mobile, typically typically envisaged envisaged in in metapopulation metapopulation models. phases may models. Plants Plants often often have have aa long long life life span. span. Vegetative Vegetative life life cycle cycle phases may per persist cases, even Cook 11983). 983). sist for for centuries centuries or, or, in in extreme extreme cases, even millennia millennia (e.g., (e.g., Cook Commonly, Commonly, the the life life span span ooff established established plants plants bbyy far far exceeds exceeds that that of of the the human human observer, observer, making making us us perceive perceive plants plants as as being being "static. "static."" Seeds Seeds may may stay stay dormant dormant in in the the soil soil for for long long periods, periods, making making it it hard hard to to document document true true extinctions extinctions and and to to distinguish distinguish colonization colonization events events from from just just reappearance reappearance of of plants plants that that have have stayed stayed on on the the site. site. Moreover, Moreover, plants plants have have mainly mainly sessile sessile life life stages, stages, but but with with two pollen dispersal two mobile mobile phases: phases: pollen dispersal and and seed seed dispersal. dispersal. For For both both these these mobile mobile phases, phases, the the common common feature feature is is that that pollen pollen and and diaspores diaspores are are not not able able to to direct direct their distance traveled their dispersal, dispersal, neither neither the the distance traveled nor nor the the target target for for deposition. deposition. Long-range Long-range dispersal dispersal is is extremely extremely difficult difficult to to assess, assess, yet yet essential essential for for under understanding 996) and e.g., standing both both gene gene flow flow (e.g., (e.g., Young Young et et aI., al., 11996) and plant plant migration migration ((e.g., Higgins 999; Cain Higgins and and Richardson, Richardson, 11999; Cain et et aI., al., 2000) 2000).. This This chapter chapter discusses discusses how how aa metapopulation metapopulation concept concept iiss useful useful for for aa range range of regional plant populations, despite of issues issues related related to to studies studies of of regional plant populations, despite the the problems problems mentioned 1 ) that mentioned earlier. earlier. The The basic basic tenets tenets for for our our view view are are ((1) that plant plant populations populations are 2 ) that are characteristically characteristically patchy, patchy, at at most most (all) (all) spatial spatial scales, scales, ((2) that despite despite exten extensive spans and apparent ""stasis" stasis" of sive life life spans and an an apparent of many many plant plant populations, populations, viewed viewed over considered in over aa longer longer time time period period than than are are usually usually considered in research research programs, programs, most plants plants possess aa turnover turnover at at aa regional regional scale, scale, ((3) that there there is is convincing convincing most 3 ) that evidence evidence that that patterns patterns of of spatial spatial habitat habitat configuration configuration do do affect affect aa number number of of pollination, herbivory, processes that that influence influence plants plants m pollination, herbivory, diseases, diseases, and and seed seed dispersal (4) that dispersal m and and (4) that the the ongoing ongoing landscape landscape transformation, transformation, including including frag fragmentation natural and and seminatural mentation of of many many natural seminatural habitats, habitats, implies implies that that the the effects effects of are likely likely to of landscape landscape habitat habitat configuration configuration are to increase. increase.
118.2 8.2 THE THE NEED NEED FOR FOR DEFINING DEFINING PLANT PLANT METAPOPULATIONS METAPOPULATIONS The The discrepancy discrepancy between between the the metapopulation metapopulation concept concept as as it it is is prevailing prevailing in in current current models models and and the the specific specific features features of of plant plant populations populations is is perhaps perhaps best best exemplified exemplified by by an an overview overview of of the the discussion discussion on on the the definition definition of of aa metapopu metapopulation. 1 997) defined lation. Hanski Hanski and and Simberloff Simberloff ((1997) defined the the metapopulation metapopulation approach approach as as taking taking into into account account that that "populations "populations are are spatially spatially structured structured into into assemblages assemblages
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of of local local breeding breeding populations populations and and migration migration among among local local populations populations has has some some effect effect on on local local dynamics, dynamics, including including the the possibility possibility of of local local population population reestab reestablishment (p. 6). lishment following following extinction" extinction" (p. 6). The The reviews reviews that that have have been been published published on on metapopulation 996; Eriksson, metapopulation dynamics dynamics in in plants plants (Husband (Husband and and Barrett, Barrett, 11996; Eriksson, 11996; 996; Freckleton Freckleton and and Watkinson, Watkinson, 2002; 2002; Bullock Bullock et et aI., al., 2002) 2002) all all devote devote quite quite some issue. While detail, there con some text text to to the the definition definition issue. While they they vary vary in in detail, there is is general general consensus components of Habitat should sensus about several several components of the the definition. definition. Habitat should be be distrib distributed discontinuously with suitable occupied, occupied, suitable uted discontinuously with aa mix mix of of suitable suitable nonoccupied, nonoccupied, and nonsuitable patches; some and nonsuitable some degree degree of of migration migration should should occur occur between between local populations; and populations; and ultimately ultimately there there should should be be extinction extinction and and (re)colonization (re)colonization at at the the local local patch patch level. level. Freckleton Freckleton and and Watkinson Watkinson (2002) (2002) argued argued that that distribution distribution of of aa habitat habitat is is aa key key element element determining determining whether whether there are are metapopulations metapopulations in in plants. plants. They They claimed that the claimed that the basic basic premise premise of of metapopulation metapopulation theory, theory, that that suitable suitable habitat habitat occurs occurs as as discrete discrete patches patches within within aa matrix matrix of of unsuitable unsuitable habitat, habitat, is is frequently frequently violated some plant plant species exist on on violated in in plants, plants, as as regional regional populations populations of of some species may may exist largely swathes of largely uninterrupted uninterrupted swathes of suitable suitable habitat. habitat. In In addition, addition, Vandermeer Vandermeer and and Carvajal 1 ) showed Carvajal (200 (2001) showed that that the the dynamics dynamics in in the the supposedly supposedly nonsuitable nonsuitable habi habitat tat matrix matrix surrounding surrounding suitable suitable patches patches may may be be of of great great importance importance for for the the overall overall regional regional dynamics, dynamics, thus thus reducing reducing the the importance importance of of the the suitable suitable vs vs nonsuitable nonsuitable distinction. distinction. As As this this is is likely likely to to be be aa process process relevant relevant for for many many plant plant species, species, this this further further emphasizes emphasizes the the need need for for aa revised revised plant plant metapopulation metapopulation definition. definition. It 1 997), in It is is telling telling that that Hanski Hanski and and Simberloff Simberloff ((1997), in an an attempt attempt to to defend defend the the general general applicability applicability of of the the metapopulation metapopulation approach, approach, stress stress that that "" .. ... . empiri empirical metapopulation concept even cal work work has has made made good use use of of the the metapopulation even when when some some tens of of percents percents of of individuals individuals per per generation generation leave leave their their natal natal patch" patch" (p. 9), 9), tens whereas whereas the the hesitation hesitation of of plant plant population population biologists biologists to to embrace embrace the the metapopu metapopulation concept is dispersal distances lation concept is (partly) (partly) based based on on the the very very limited limited dispersal distances that that Husband and 1 996) stated characterize almost almost all all plant plant species. Husband and Barrett Barrett ((1996) stated that that this dispersal makes appropriate for metapopula this restricted restricted dispersal makes plants plants particularly particularly appropriate for metapopulation tion analyses. analyses. It It is, is, however, however, clear clear that that at at least least some some migration migration should should occur occur for for the concept to value, and cases, authors authors place place some some doubt the concept to be be of of value, and in in several several cases, doubt on on whether dispersal whether dispersal in in plants is is not too restricted (e.g., (e.g., Freckleton and and Watkinson, Watkinson, 2002; 2002; Bullock Bullock et et aI., al., 2002 2002).) . Maybe Maybe the the most most prominent prominent feature feature of of metapopulation metapopulation dynamics dynamics iiss the the extinction extinction and and colonization colonization dynamics dynamics at at the the regional regional population population level. level. What What is is accepted accepted as as evidence evidence for for metapopulation metapopulation dynamics dynamics follows follows aa continuum continuum from from the the animal-oriented animal-oriented side, side, where where observations observations of of extinctions extinctions and and recolon recolonizations 997) to izations are are accepted accepted as as evidence evidence (e.g., (e.g., Harrison Harrison and and Taylor, Taylor, 11997) to various various degrees degrees of of relaxation relaxation at at the the plant plant side. side. While While all all authors authors have have some some form form of of the population as the original original Levins Levins meta metapopulation as aa conceptual conceptual starting starting point, point, the the extinc extinction debate when comes to tion criterion criterion is is aa matter matter of of much much debate when it it comes to plant plant dynamics. dynamics. Husband 1 996) stated population concept Husband and and Barrett Barrett ((1996) stated that that "" .. ... . the the meta metapopulation concept has to recognize has been been broadened broadened to recognize that that all all species species have have local local and and regional regional dynamics 1 99 6 ) argued dynamics"" (p. 462 462).) . Eriksson ((1996) argued that that "" . ... . metapopulations metapopulations in in the the strict strict sense s e n s e.. .. , . may may not not occur occur in in all, all, perhaps perhaps not not even even in in most, most, organ organisms" following Hanski isms" (p. (p. 249 249).) . Freckleton Freckleton and and Watkinson Watkinson (2002), (2002), following Hanski and 1 997), relaxed and Simberloff Simberloff ((1997), relaxed the the extinction extinction requirement requirement from from observed observed
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extinction population should extinction to to "" . .. .. . even even the the largest largest local local population should have have aa measur measurable population is source of able risk risk of of extinction extinction (unless (unless the the largest largest population is the the source of aa source-sink This discussion source-sink system) system)"" (p. (p. 420). 420). This discussion around around the the extinction extinction criterion criterion is is not not so so much much motivated motivated by by theoretical theoretical objections, objections, but but rather rather has has aa method methodological base: most ological base: most plants plants have have aa much much slower slower turnover turnover than than most most animals, animals, making making observations observations of of extinctions extinctions and and colonizations colonizations impractical, impractical, if if not not impossible, impossible, in in many many cases. cases. As As aa result, result, there there is is aa clear clear need need for for reconsidering reconsidering metapopulation metapopulation definitions definitions and and concepts. concepts. This This should should be be based based on on aa thorough thorough identification identification of of plant-specific plant-specific problems problems with with the the concept, concept, which which then then may may serve populations. serve as as the the basis basis for for redefining redefining plant plant meta metapopulations.
118.3 8.3
PLANT-SPECIFIC PLANT-SPECIFIC PROBLEMS PROBLEMS WITH W I T H THE THE METAPOPULATION METAPOPULATION CONCEPT CONCEPT
Suitable Suitable but but Unoccupied Unoccupied Habitat Habitat It suitable vs It is is well well known known that that the the simplifying simplifying dichotomy dichotomy suitable vs unsuitable unsuitable habitat does does not not hold hold for for most most organisms. organisms. Habitats Habitats may may have have aa fluctuating fluctuating habitat quality, example, being open for seeds only certain years, quality, for for example, being open for colonization colonization of of seeds only certain years, and and there there may may be be aa more more or or less less continuous continuous variation variation in in abiotic abiotic and and biotic biotic properties determining determining the the suitability suitability of of aa site. site. For For species species inhabiting inhabiting rather rather properties well-defined decomposing wood wood or well-defined substrates, substrates, such such as as decomposing or small small ponds, ponds, the the con concept cept of of unoccupied unoccupied suitable suitable sites sites is is relatively relatively easy easy to to use. use. In In contrast, contrast, for for most most plants plants it it is is very very difficult difficult to to exactly exactly define define the the features features of of aa suitable suitable site site with without species under under study present there. there. However, out having having the the species study present However, assuming assuming that that spa spatially tially delimited delimited sites sites do do vary vary with with regard regard to to quality, quality, defined defined on on the the basis basis of of their their suitability suitability to to harbor harbor populations populations of of aa focal focal species, species, the the methodological methodological problems of of finding finding those those sites sites are are not not an an acceptable acceptable basis for refuting refuting basis for problems metapopulation metapopulation theory. theory. One to overcome the problem of assessing is to to use One method method used used to overcome the problem of assessing site site quality quality is use seed seed sowing sowing or or transplantations transplantations of of the the focal focal species. species. Although Although experimental experimental seed seed sowing sowing has has been been much much used used to to examine examine seed seed limitations limitations in in plant plant distri distributions for aa review, been butions ((for review, see see Turnbull Turnbull et et aI., al., 2000), 2000), this this method method has has been employed occasionally to employed only only occasionally to analyze analyze occupancy occupancy patterns. patterns. For For example, example, Ehrlen Ehrl~n and and Eriksson Eriksson (2000) (2000) estimated estimated the the occupancy occupancy of of seven seven forest forest herb herb species species among among patches patches of of deciduous deciduous and and mixed mixed coniferous-deciduous coniferous-deciduous forests forests by indicated that by sowing sowing seed seed and and transplanting transplanting juveniles. juveniles. Results Results indicated that occupancy occupancy was was related related negatively negatively to to seed seed (diaspore) (diaspore) size; size; large-seeded large-seeded species species may may thus thus be be more more restricted restricted in in their their exploitation exploitation of of available available suitable suitable sites. sites. Interestingly, Interestingly, no no relationship was was found found between between the the actual actual occurrence occurrence of of the the species, species, or or the the relationship success success of of the the sowing, sowing, and and aa number number of of measured measured abiotic abiotic soil soil factors. factors. Thus, Thus, aa study only on factors would would have have yielded study based based only on measuring measuring site site factors yielded misleading misleading results results concerning concerning the the suitability suitability of of the the sites sites and and thus thus the the actual actual occupancy occupancy of of the the species. species. From From aa methodological methodological viewpoint viewpoint there there are are some some important important caveats caveats with with sowing/transplanting experiments that accounted for. sowing/transplanting experiments that must must be be accounted for. First, First, plants plants may may succeed succeed in in recruitment recruitment only only in in certain certain years. years. This This means means that that failure failure to to find not mean mean that altogether unsuitable. find recruitment recruitment may may not that aa site site is is altogether unsuitable.
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Experiments Experiments may may have have to to be be repeated repeated during during several several years. years. Second, Second, aa recorded recorded recruitment population will recruitment may may not not mean mean that that aa population will be be established. established. Plants Plants may may have uveniles, and (Schupp, have differences differences in in the the requirements requirements for for recruits, recruits, jjuveniles, and adults adults (Schupp, 11995; 995; Ohlson Ohlson et 1 ; Eriksson, reliable, et ai., al., 200 2001; Eriksson, 2002 2002).) . Thus, Thus, in in order order to to be be reliable, recruitment recruitment experiments experiments must must be be followed followed several several years years after after the the appearance appearance of the the seedlings. seedlings. Using Using jjuvenile transplants in in combination combination with with seed seed sowing sowing of uvenile transplants is, however, means to reduce the is, however, aa means to reduce the time time needed needed for for the the experiment. experiment. Ehrlen Ehrl~n and and Eriksson (2000) followed the took another Eriksson (2000) followed the recruits recruits for for 4 4 yr; yr; in in fact fact it it took another 33 yr yr before the conclusion was reached that sowing really really resulted resulted in in population population before the conclusion was reached that sowing establishment establishment defined defined as as sown sown plants plants development development to to maturity maturity (Ehrlen (Ehrl~n and and Eriksson, Eriksson, unpublished unpublished results) results).. Assessing Assessing habitat habitat quality quality by by means means of of sowing sowing and least for perennial plants and transplantation transplantation is is therefore therefore time-consuming, time-consuming, at at least for perennial plants that uvenile period. that have have aa long long jjuvenile period. Yet Yet we we suggest suggest that that this this method method is is the the most most appropriate appropriate for for detecting detecting suitable suitable but but unoccupied unoccupied sites sites in in plant plant metapopula metapopulation tion studies. studies.
Long Long Life Life Spans Spans Plants Plants are are tremendously tremendously variable variable with with regard regard to to their their life life span. span. Annuals Annuals may couple of weeks, whereas clonal plants may live live for for aa couple of weeks, whereas clonal plants and and some some trees trees may may live live for oldest documented for centuries. centuries. The The oldest documented life life span span for for trees trees is is more more than than 4000 4000 years 965), whereas years (Currey, (Currey, 11965), whereas the the extreme extreme age age for for some some clonal clonal plants plants is is over over 110,000 0,000 years e.g., Vasek, 98 0 ) . Even years ((e.g., Vasek, 11980). Even if if these these ages ages are are not not representative representative for for plants plants in in general, general, life life spans spans in in the the order order of of aa century century are are common. common. Ehrlen Ehrl~n and and Lehtilii Lehtil~i (2002) (2002) compiled compiled demographic demographic data data for for 71 71 perennial perennial species species and and reana reanalyzed lyzed them them with with use use of of population population matrix matrix models. models. They They found found that that over over half half of of the the species species had had aa projected projected life life span span exceeding exceeding 35 35 years years and and aa quarter quarter of of the the species had span over means that species had aa projected projected life life span over 8800 years. years. This This means that local local popula populations tions may may be be very very persistent, persistent, even even in in cases cases where where the the population population growth growth rate rate is is negative, negative, for for example, example, due due to to lack lack of of recruitment. recruitment. In In many many changing changing habi habitats tats where where the the change change is is experienced experienced as as "deterioration" "deterioration" from from the the viewpoint viewpoint of of some some of of the the inhabiting inhabiting plants, plants, such such as as abandoned abandoned grasslands grasslands and and forests forests undergoing undergoing succession, succession, reproduction reproduction and and recruitment recruitment are are likely likely to to be be the the first first population population processes processes that that are are affected affected negatively. negatively. However, However, populations populations still still persist, and time. Such Such remnant persist, and they they may may do do so so for for extended extended periods periods of of time. remnant popu populations 996) may lations (Eriksson, (Eriksson, 11996) may be be aa characteristic characteristic component component of of the the landscape, landscape, especially especially in in regions regions where where land land use use has has changed changed during during the the last last century century (i.e., (i.e., in in most most parts parts of of the the world world where where humans humans live live).) . Estimates Estimates of of time time to to extinction extinction following land land use use change change in in Scandinavia reveal that that local local populations of following Scandinavia reveal populations of perennial plants plants may persist for periods of 0-100 years Ehrlen, perennial may persist for periods of 550-100 years (Eriksson (Eriksson and and Ehrl~n, 200 1 ) . Thus, Thus, aa fraction occupied" sites regional perspective 2001). fraction of of the the ""occupied" sites in in aa regional perspective may may reflect habitat distribution rather than actual one one present reflect aa historical historical habitat distribution rather than the the actual present in in the the landscape Fig. 118.1). 8.1). landscape today today ((Fig. From From aa conservation conservation perspective, perspective, such such aa time time lag lag in in the the response response ooff species species to to ongoing ongoing habitat habitat changes changes represents represents aa form form of of extinction extinction debt debt (Tilman (Tilman et et ai., al., Hanski and and Ovaskainen, Ovaskainen, 2002 2002).) . If If no no habitats habitats are are available available where where popu popu11994; 994; Hanski lation lation growth growth is is positive, positive, these these remnant remnant populations populations are are slowly slowly moving moving toward toward extinction, species may perceived as extinction, although although the the species may be be perceived as rather rather common common based based on on aa conventional conventional survey. survey. The The expected expected time time to to extinction extinction for for the the regional regional
118. 8. TOWARD FOR PLANTS TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR PLANTS 1 7"24'
1998
453 453 1945
17"24'
58"50'
D
Semi-natural grassland
D
Open 'non-grassland'
Forest
1 000 •
Site
with
m
Filipendula vulgaris
Fig. 118.1 8.1 Distribution lipendula vulgaris, aa plant in aa Distribution of of Fi Filipendula plant typical typical of of seminatural seminatural grasslands, grasslands, in 30 sites recorded in in aa 11998 998 survey, landscape the 30 landscape in in southern southern Sweden. Sweden. Of Of the sites recorded survey, 9 9 sites sites were were found found in 945 landscape in what what is is presently presently forest. forest. If If the the same same sites sites are are mapped mapped on on the the 11945 landscape (as (as revealed revealed by aerial photographs), all but but 11 site in seminatural some sites by aerial photographs), all site is is located located in seminatural grasslands. grasslands. Note Note that that some sites that appear appear to "open nongrasslands" actually occur occur on that to be be located located in in "open nongrasslands" actually on small small grassland grassland frag fragments. 998 map represent remnant (5. Cousins, ments. Most Most likely likely the the forest forest sites sites in in the the 11998 map represent remnant populations populations (S. Cousins, uunpublished npublished result).
population equals equals the the expected expected time time to to extinction extinction for for the the most most persistent persistent local local population. population. However, However, the the likelihood likelihood of of success success after after habitat habitat restoration restoration will will increase increase if if remnant remnant populations populations are are still still present. present. Habitats Habitats dominated dominated by by species species that that develop develop remnant remnant populations populations may may therefore therefore be be the the most most suitable suitable targets targets for for restoration. restoration. If If remnant remnant populations populations generally generally are are common common in in plants, plants, this this has has two two important important implications. implications. First, First, occupancy occupancy patterns patterns of of long-lived long-lived perennial perennial plants plants may may be be far far from from equilibrium equilibrium with with the the present present habitat habitat configuration configuration ((Eriksson Eriksson and 1 ) . This and Ehrlen, Ehrl~n, 200 2001). This means that that estimates estimates of of colonization colonization and and extinction extinction rates rates from from incidence incidence function function models may may yield yield very very misleading misleading results. To To achieve achieve better better estimates estimates of of colonization colonization and and extinction, extinction, aa combin combination ation of of experimental experimental studies studies and and demographic demographic studies studies of of local local populations, populations, assessing assessing the the actual actual growth growth rate rate variation, variation, is is needed. needed. The The turnover turnover of of local local populations populations will will naturally naturally be be very very slow slow in in many many perennial perennial plants. plants. However, However, the the slow dynamics dynamics per se se does does not imply that that there is is no no turnover turnover also also in in regional regional populations populations of of long-lived long-lived plants. plants. The The problem problem is is aa matter of of time time scale, scale, but but given given sufficiently sufficiently long long observation observation series, series, long-lived long-lived plants plants prob probably ably have have regional regional dynamics dynamics not not fundamentally fundamentally different different from from short-lived short-lived 977). organisms (d. (cf. Whittaker Whittaker and and Levin, Levin, 11977). A A second second implication implication is is that that remnant remnant population population systems systems may may function function as as aa temporal temporal source-sink source-sink population. population. Habitats Habitats may may have aa fluctuating quality quality ((from from aa focal persistence of local populations, focal species species viewpoint) viewpoint).. If If the the persistence of local populations, despite despite aa negative negative growth growth rate, rate, is is in in the the magnitude magnitude of of the the period of of habitat quality quality fluctuation, fluctuation, time time periods periods with with aa positive positive population growth growth rate rate compensate compensate for for those those with with population population decline. decline. Thus, Thus, using using aa dichotomized dichotomized simplification simplification unsuitable," colonization of habitats habitats into into "suitable" "suitable" and and ""unsuitable," colonization of of suitable suitable habitats habitats
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may may be be considered considered through through time time as as well well as as through through space. space. This This resembles resembles the the effects seed bank, bank, where where the suitable habitat habitat (e.g., effects of of aa persistent persistent seed the appearance appearance of of suitable (e.g., after after aa disturbance) disturbance) initiates initiates development development of of aa population population of of vegetative vegetative and and reproductive reproductive plants. plants. The The time time lag lag of of plant plant species species response response to to habitat habitat changes changes calls calls for for analyses analyses of based on of population population distributions distributions that that are are based on historical historical data data on on the the popula populations tions themselves themselves and and on on the the landscape. Estimates Estimates of of colonization colonization and and extinc extinction derived from knowledge of previous occupancy tion may may be be derived from combining combining knowledge of previous occupancy patterns, 993; patterns, as as found found in in old old surveys, surveys, with with reinventories reinventories (e.g., (e.g., Ouborg, Ouborg, 11993; Lienert Lienert et et aI., al., 2002; 2002; Lindborg Lindborg and and Ehrlen, Ehrl6n, 2002 2002).) . In In most most cases, cases, such such infor information does does not exist, however, historical effects mation not exist, however, and and considerations considerations of of historical effects can can only only be be inferred inferred indirectly indirectly by by analyzing analyzing present present occupancy occupancy patterns patterns in in relation relation to Fig. 118.1). 8 . 1 ) . Such to known known habitat habitat changes changes ((Fig. Such studies studies provide provide aa direct direct link link between between metapopulation metapopulation studies studies on on the the species species level level and and landscape landscape ecology ecology (Wiens, 997). This emphasis on (Wiens, 11997). This places places aa much much stronger stronger emphasis on the the landscape landscape matrix matrix surrounding the example, habitats habitats that surrounding the target target habitats. habitats. For For example, that are are regarded regarded as as presently should be presently unsuitable unsuitable should be examined examined for for historical historical suitability. suitability. The The histor historical landscape change compared with ical time time frame frame for for landscape change must must then then be be compared with the the time time lag lag of response to landscape changes. of the the local local population population response to the the landscape changes.
Seed Seed Banks Banks In addition addition to long long life spans, spans, many many plant plant species species have long-lived long-lived seed seed banks ((Leck Leck et aI., 11989; 989; Thompson aI., 11997), 997), and et al., Thompson et et al., and for for such such species, species, estimates estimates of of colonization colonization and and extinction extinction are are complicated. complicated. Local Local populations populations (above (above ground) ground) may may vanish, vanish, but but the the species species is is nevertheless nevertheless present present in in the the seed seed bank. bank. D ocumenting local Documenting local population population extinction extinction is is also also therefore therefore difficult difficult if if seed seed bank bank samples samples are are gathered. gathered. Negative Negative findings findings in in seed seed bank bank samples samples are are difficult difficult to to interpret, interpret, and and it it demands demands aa large large sampling sampling effort effort to to safely safely conclude conclude that that aa species seed bank. bank. Estimating species is is in in fact fact missing missing from from the the seed Estimating colonization colonization is is also also complicated complicated by by seed seed banks banks because because of of the the difficulty difficulty of of distinguishing distinguishing between between recruitment from recruitment from seeds seeds arriving arriving at at aa site site through through ordinary ordinary dispersal and and recruitment from the the seed seed bank. bank. recruitment from The these problems The extent extent of of these problems for for aa plant plant metapopulation metapopulation concept concept depends depends on on the the longevity longevity of of seeds seeds in in soil soil and and on on how how often often species species occur occur in in the seed bank not in vegetation (which (which is the seed bank but but not in the the vegetation is in in turn turn likely likely to to reflect reflect the the longevity of seed banks banks in longevity of the the seed seed bank). bank). It It is is common common knowledge knowledge that that seed in many cases do present vegetation, vegetation, although many cases do not not resemble resemble the the present although this this conclusion conclusion may Leck et may partly partly rest rest on on an an insufficient insufficient sampling sampling of of the the seed seed bank bank ((Leck et aI., al., Furthermore, the the potential potential longevity longevity of of many many seeds seeds extends extends over over sev sev11989). 9 8 9 ) . Furthermore, eral Thompson et ai., 11997). 997). eral decades decades and, and, in in some some cases, cases, centuries centuries ((Thompson etal., Although Although such such extended extended longevity longevity of of seeds seeds constitutes constitutes problems problems for for assess assessments ments of of colonization colonization and and extinction extinction in in metapopulation metapopulation studies, studies, the the extent extent Whereas highly of the the problem problem differs differs among among vegetation types. Whereas highly disturbed disturbed habitats, ((e.g., e.g., arable arable fields long-lived seed banks, seeds habitats, fields and and spoil) spoil) contain contain long-lived seed banks, seeds in the the seed seed bank bank of of forests forests and and permanent are generally generally not not long long in permanent grasslands are lived (Thompson et i . , 11998). 9 9 8 ) . For lived (Thompson et aal., For example, example, many many species-rich species-rich grasslands grasslands do ority of do not not generally generally have have the the maj majority of species species represented represented in in the the seed seed banks banks ((Bekker Bekker et 99 7 ) . et aI., al., 11997).
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Only examined the Only aa few few studies studies have have examined the quantitative quantitative relationships relationships between between recruitment bank. Indirect not estimating recruitment from from seed seed rain rain vs vs the the seed seed bank. Indirect studies studies ((not estimating the the resulting resulting recruitment) recruitment) indicate indicate that that contribution contribution of of the the seed seed rain rain may may be be larger 9 8 1 ; Schott larger than than contribution contribution of of the the seed seed bank bank (e.g., (e.g., Rabinowitz, Rabinowitz, 11981; Schott and and Hamburg, 997; Molau Hamburg, 11997; Molau and and Larsson, Larsson, 2000). 2000). However, However, both both seed seed rain rain and and seed seed bank bank may may contribute contribute little little to to regeneration regeneration after after disturbance disturbance (e.g., (e.g., Owen Owen et aI., 2002), et al., 2002), whereas whereas other other studies studies indicate indicate that that the the seed seed bank bank is is important. important. Kalamees 6 % of Kalamees and and Zobel Zobel (2002) (2002) estimated estimated that that 336% of the the regeneration regeneration in in gaps gaps in in species-rich grasslands grasslands came bank. An species-rich came from from the the seed seed bank. An important important finding, finding, how however, below 4 % ) from banks in ever, was was that that rather rather few few recruits recruits ((below 4%) from seed seed banks in permanent permanent grassland grassland belong belong to to species species not not present present in in the the vegetation vegetation (Eriksson (Eriksson and and Eriksson, 997; Kalamees Eriksson, 11997; Kalamees and and Zobel, Zobel, 2002 2002).) . Thus, Thus, at at least least for for some some vegeta vegetation problems for based on tion types, types, the the problems for assessing assessing extinction extinction and and colonization colonization based on aboveground numerous stud aboveground vegetation vegetation seem seem to to be be rather rather limited. limited. Although Although numerous studies described the bank in ies have have described the seed seed bank in different different vegetation vegetation types, types, it it is is obvious obvious that that we basically lack importance we still still basically lack general general quantitative quantitative knowledge knowledge on on the the direct direct importance of banks for need studies addressing how of seed seed banks for regeneration. regeneration. In In particular, particular, we we need studies addressing how frequent present in frequent recruitment recruitment of of species species from from the the seed seed bank bank that that are are not not present in the the vegetation problem with vegetation occurs. occurs. In In order order to to evaluate evaluate the the potential potential problem with such such "pseudo-colonizations, issue will "pseudo-colonizations,"" aa crucial crucial issue will be be to to estimate estimate the the time time scale scale of of the the persistence bank after of conspecific persistence of of the the seed seed bank after the the aboveground aboveground population population of conspecific plants plants has has disappeared. disappeared.
Time Population Turnover Time Lags Lags and and Population Turnover There There are are several several ways ways in in which which one one can can handle handle the the problem problem of of the the time time lag lag of population extinction bank ((after after vege of population extinction resulting resulting from from the the presence presence of of aa seed seed bank vegetative species have presence of tative plants plants of of the the focal focal species have vanished) vanished) or or the the presence of long-lived longqived vegetative example, reprorepro vegetative plant plant individuals individuals in in aa remnant remnant population population where, where, for for example, duction ceased due change. If duction and and recruitment recruitment have have ceased due to to habitat habitat change. If there there are are only only seed seed banks banks or or remnant remnant populations populations left left in in aa region, region, there there will will be be no no dynamics dynamics at unsuitable but at all all at at the the regional regional scale scale unless unless unsuitable but occupied occupied sites sites improve. improve. The The regional population will bank or regional population will go go extinct extinct when when the the last last local local seed seed bank or remnant remnant population local population disappears. disappears. A A more more complex complex situation situation occurs occurs when when there there are are local populations Also, in case, the populations at at still still suitable suitable sites. sites. Also, in this this case, the actual actual pattern pattern of of occu occupancy pancy is is not not in in equilibrium equilibrium with with the the present present habitat habitat configuration configuration as as it it is is lag lagging ging behind behind the the habitat habitat change. change. Thus, Thus, aa key key issue issue is is to to incorporate incorporate the the time time lag lag into population models. occupied into meta metapopulation models. Assuming Assuming an an equilibrium equilibrium fraction fraction of of occupied sites, (or close close to sites, Po, P0, before before habitat habitat change change and and aa new new equilibrium equilibrium (or to equilibrium) equilibrium) fraction fraction of of occupied occupied sites sites after after habitat habitat change, change, Ph Pl, the the time time lag lag T T will will be be equal equal to to the the time time it it takes takes to to move move from from Po P0 to to Pi Pl (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2002 2002).) . T T can be defined either expected time extinction, Te, can be defined either as as the the expected time to to local local extinction, Te, of of the the largest largest seed population or local extinction seed bank bank or or remnant remnant population or as as the the expected expected time time to to local extinction of of the or median) the average average ((or median) sized sized seed seed bank bank or or remnant remnant population population from from the the time time where deteriorate (from species point point of where aa local local site site starts starts to to deteriorate (from aa focal focal species of view) view).. For For seed seed banks, banks, Te Te can can be be estimated estimated from from repeated repeated seed seed samples samples from from aa selection selection of of different different sites. sites. The The decline decline of of vegetative vegetative remnant remnant populations populations may may be be estimated estimated by based on by analyses analyses of of transition transition matrix matrix models models based on aa representative representative sample sample of of populations 1). populations (e.g., (e.g., Eriksson Eriksson and and Ehrlen, Ehrl~n, 200 2001).
N.J. N.J. OUBORG OUBORG AND AND O. O. ERIKSSON ERIKSSON
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Estimates Estimates of of colonization colonization are are also also affected affected by by seed seed banks banks and and remnant remnant populations because both suitable habitat populations because both can can help help bridge bridge periods periods of of suitable habitat condi conditions conditions. If population is tions interrupted interrupted by by unfavorable unfavorable local local conditions. If aa population is able able to to persist conditions, this popu persist through through aa period period of of unfavorable unfavorable conditions, this implies implies that that the the population lation can can be be present present at at aa site site and and start start expanding expanding again again if if conditions conditions improve, colonization through improve, without without aa new new colonization through space. space. While While this this in in principle principle may unusual ability may occur occur for for all all organisms, organisms, plants plants may, may, due due to to their their unusual ability to to with withstand conditions, be stand unfavorable unfavorable conditions, be particularly particularly prone prone to to such such aa "dispersal "dispersal in in time" Eriksson, 11996). 996). time" ((Eriksson, IIn n meta population models, metapopulation models, colonization colonization is is usually usually modeled modeled aass aa function function of some measure of some measure of of habitat habitat connectivity, connectivity, for for example, example, the the total total area area of of suit suitable habitat region weighted between the able habitat in in aa region weighted by by the the distance distance between the focal focal site site and and each Hanski, 11999). 9 9 9 ) . Assuming certain fraction each surrounding surrounding site site ((Hanski, Assuming that that aa certain fraction of of unsuitable but formerly seed banks unsuitable ((but formerly suitable) suitable) sites sites harbors harbors seed banks or or remnant remnant popu populations likelihood that lations and and that that there there is is aa certain certain likelihood that such such aa site site develops develops to to become suitable, new suitable sites ones already become suitable, new suitable sites add add to to the the ones already present, present, being being "colonized" "colonized" already already from from time time zero. zero. Again Again aa key key issue issue is is the the time time it it takes takes before populations go before local local populations go extinct extinct after after aa site site has has changed changed from from suitable suitable to to unsuitable, Te, unsuitable, Te, which which will will set set the the temporal temporal limit limit for for colonization colonization through through time. time. Spatial Spatial relationships relationships for for this this subset subset of of colonizations colonizations are are not not expected expected to habitat configuration. to reflect reflect the the present-day present-day habitat configuration. Rather, Rather, the the occurrence occurrence of of seed banks and remnant populations seed banks and remnant populations in in the the landscape landscape will will reflect reflect the the histor historical ical habitat habitat connectivity connectivity within within the the time time frame frame for for the the persistence persistence of of these these populations. populations. Conceptually, Conceptually, this this way way of of treating treating problems problems with with time time lags lags in in extinction, extinction, habitat connects to models of habitat suitability, suitability, and and colonization colonization connects to models of patch patch dynamics dynamics (Watt, 947; Whittaker 977) and (Watt, 11947; Whittaker and and Levin, Levin, 11977) and "mosaic "mosaic models" models" of of communi communities DeAngelis and 987) in is a patches ties ((DeAngelis and Waterhouse Waterhouse 11987) in which which there there is a turnover turnover of of patches (sites) inhabitant species. species. Given that we use aa (sites) with with regard regard to to their their quality quality for for inhabitant Given that we use strict where populations strict delimitation delimitation of of "suitable "suitable sites" sites" (as (as sites sites where populations potentially potentially have positive population reproduc have aa positive population growth growth or, or, alternatively, alternatively, as as sites sites where where reproduction species is bank or tion is is possible), possible), sites sites where where aa focal focal species is present present only only in in aa seed seed bank or as as aa remnant remnant population population are, are, by by definition, definition, "unsuitable. "unsuitable."" Despite Despite being being unsuit unsuitable, these contribute to overall dynamics able, these sites sites contribute to the the overall dynamics of of the the focal focal species species by by (i) means by (i) delaying delaying regional regional extinction extinction and and by by (ii) (ii) providing providing aa means by which which new new suitable sites suitable sites (re)appear (re)appear already already being being colonized. colonized. Moreover, Moreover, there there is is aa link link to to community resilience Eriksson, 2000); seed banks) community resilience ((Eriksson, 2000); remnant remnant populations populations (and (and seed banks) contribute contribute to to decrease decrease the the return return time time to to community community equilibrium equilibrium (or (or any any quasi quasistable stable state) state) following following disturbance. disturbance.
Source Source Quality Quality Unlike are sessile most of the Unlike animals, animals, plants plants are sessile during during most of their their life life cycle. cycle. In In the mobile of the life cycle, or, for mobile phase phase of the life cycle, seeds seeds ((or, for some some aquatic aquatic plant plant species, species, vege vegetative random spatial tative propagules) propagules) disperse disperse in in aa nondirected, nondirected, random spatial process. process. Once Once established, established, individuals individuals will will have have to to cope cope with with the the local local environment environment and and the the temporal variation these conditions. only have limited temporal variation in in these conditions. To To this this goal, goal, they they only have aa limited number of Seeds can where they will not number of options. options. Seeds can arrive arrive at at sites sites where they will not germinate, germinate, or or will despite the overall suitability will not not establish, establish, despite the overall suitability of of the the patch. patch. For For instance, instance,
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many success many plant plant species species rely rely on on the the presence presence of of gaps gaps in in the the vegetation vegetation for for successful aI., 11995; 995; Moloney 996; ful establishment establishment (e.g., (e.g., Bullock Bullock et etal., Moloney and and Levin, Levin, 11996; Eriksson, 997; Brokaw Eriksson, 11997; Brokaw and and Busing, Busing, 2000 2000).) . A Att the the population population level, level, plants plants can can either either adapt adapt ttoo local local conditions conditions oorr have have enough enough plasticity plasticity to to adapt adapt their their phenotype. phenotype. Local Local adaptation adaptation has has been been demon demonstrated strated in in many many cases, cases, including including local local adaptation adaptation to to heavy heavy metals metals (McNeilly (McNeilly and 968; Antonovics 980), to and Bradshaw, Bradshaw, 11968; Antonovics et et aI., al., 1971; 1971; Pollard, 11980), to competition competition with 979; Burdon, 9 8 0 ) and with other other species species (Turkington (Turkington and and Harper, Harper, 11979; Burdon, 11980) and to to grazing 98 9 ) . Reciprocal grazing pressure pressure (van (van Tienderen, Tienderen, 11989). Reciprocal transplant transplant experiments experiments have have shown shown that that local local adaptation adaptation may may occur occur at at large large spatial spatial scales scales (Schmidt (Schmidt and 9 8 8 ; Jordan, 992), but 00 m and Levin, Levin, 11988; Jordan, 11992), but also also at at small small scales scales of of 1100 m or or less less ((Antonovics, Antonovics, 11976; 976; Schmitt 990; Bell aI., 2002). Schmitt and and Gamble, 11990; Bell et etal., 2002). Metapopulation Metapopulation models, models, when when dealing dealing with with plants, plants, should should take take this this variance variance in in source source quality quality into into account. account. The The extent extent of of genotypic genotypic adaptation adaptation will will be be the the result process. The result of of aa selection selection process. The efficiency efficiency of of this this process process is is influenced influenced strongly strongly by by the the amount amount of of migration. migration. If If migration migration among among populations populations is is com common, aa continuous continuous mix occurs and mon, mix of of genotypes genotypes through through the the regional regional system system occurs and local 1 ) ) to local adaptation adaptation is is unlikely unlikely (although (although not not impossible; impossible; Barton Barton (200 (2001)) to evolve. evolve. However, plants, gene However, if if migration migration rates rates are are low, low, as as will will often often be be the the case case in in plants, gene flow process and flow does does not not disrupt disrupt results results of of the the selective selective process and plants plants become become adapted adapted to to their their local local environments. environments. The The adaptive adaptive differentiation differentiation will will then then be be aa function function of of the the heterogeneity heterogeneity at at the the regional regional scale. scale. A A prominent prominent feature feature of of plants plants is is that that genotypes genotypes often often are are able able to to change change their their phenotype phenotype in in relation relation to to environmental environmental conditions. conditions. Although Although such such phe phenotypic notypic plasticity plasticity does does occur occur in in animals animals as as well, well, it it is is certainly certainly aa prominent prominent fea feature in in plants. plants. Whether Whether plants plants will will cope cope with with environmental environmental heterogeneity heterogeneity via via ture genotypic genotypic adaptation adaptation or or phenotypic phenotypic plasticity, plasticity, that that is, is, whether whether they they will will be be spe specialists cialists or or generalists, generalists, is is aa function function of of the the amount amount of of heterogeneity, heterogeneity, the the fre frequency between various quency of of encountering encountering various various environments, environments, the the dispersal dispersal between various environments, environments, and and the the fitness fitness costs costs of of establishing establishing in in each each local local environment environment ((Gilchrist, Gilchrist, 11995; 995; van 9 97; Reboud 998). van Tienderen, Tienderen, 11997; Reboud and and Bell, Bell, 11998). Thus, Thus, individuals individuals throughout throughout aa regional regional system system will will not not have have the the same same value value for for the the dynamics: dynamics: regional regional systems systems will will be be characterized characterized by by aa variance variance in in source source quality. quality. This This can can be be expressed expressed either either as as aa mean mean source source quality quality value value for for each each combination combination of of source source and and target population population or or as aa variance variance component, component, covering covering the the variance variance in in source source quality quality throughout throughout the the regional regional system, system, that that is is added added to to the the average average colonization colonization rates rates in in models. models. However, However, even even at at the the within-population, within-population, genotypic genotypic level, level, aa significant significant vari variance ance in in source source quality quality may may exist. exist. Ouborg Ouborg et et al. al. (2000) (2000) investigated investigated the the inter interaction action between between the the host host plant plant Silene Silene latifolia latifolia and and its its specific specific pathogen pathogen Microbotryum violaceum. This This host-pathogen host-pathogen system system is is characterized characterized by by metapopulation dynamics (Antonovics 994) . After founding of metapopulation dynamics (Antonovics et et aI., al., 11994). After founding of aa new new development of local population, population, the the development of that that population population will will be be characterized by by aa continuous continuous increase increase in in the the inbreeding inbreeding level. level. Ouborg Ouborg et et al. (2000) (2000) investigated investigated the plant and the effect effect of of inbreeding on on the the interaction interaction between between the the host host plant and the the fun fungal gal pathogen pathogen and and discovered that that although although inbreeding inbreeding on on average average increased increased the the resistance resistance of of the the host, host, there there was was aa strong strong and and significant significant difference difference between between genotypes genotypes in in inbreeding inbreeding effect, effect, both both in in direction direction and and in in magnitude. magnitude. Within Within the the same same population, population, in in some some genotypes, genotypes, inbreeding inbreeding increased increased resistance, resistance, whereas whereas
458 458
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a.
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Fig. 1 88.2 . 2 Regression Regression lines of inbreeding inbreeding level level versus versus percentage infection of Silene Silene alba albainbred lines with with Microbotryum violaceum from the the same field popula violaceum fungus. Lines Lines originate originate from population. The variance between between lines lines reflects the variance in inbreeding effects, both both in magnitude magnitude (slope) (slope) and in direction direction (positive or negative negative slope). After Ouborg Ouborg et al. (2000).
in (Fig. 118.2). 8.2). This in other other genotypes, genotypes, inbreeding inbreeding decreased decreased the the resistance resistance (Fig. This vari variance ance in in inbreeding inbreeding effects effects has has been been found found in in other other plant plant species species and and in in other other traits McCauley et 1 ; Richards, Richards, 2000 consequence of traits (e.g., (e.g., McCauley et aI., al., 200 2001; 2000).) . As As aa consequence of this this variation, regional populations populations will variation, the the effects effects for for the the dynamics dynamics of of local local and and regional will be population. This be very very dependent dependent on on which which genotype genotype founds founds aa new new population. This illus illustrates trates the the intricate, intricate, subtle subtle effects effects of of source source quality quality variation. variation.
Species Species Interactions Interactions Many, Many, if if not not all, all, plant plant species species rely rely on on interactions interactions with with other other species species for for the the completion completion of of their their life life cycle. cycle. The The most most obvious example example is is the the interaction interaction many species have generalist pollinating pollinating animal many plant plant species have with with specialist specialist or or generalist animal species. Plants Plants may may be be pollinated pollinated by by insects, insects, birds, birds, and and mammals. mammals. Without Without the the species. presence presence of of these these pollinators, pollinators, the the seed seed set set will will be be low low or or even even absent. absent. Because Because pollinators pollinators tend tend to to respond respond to to the the abundance abundance of of flowers, flowers, small small plant plant popula populations tions are are often often subjected subjected to to aa reduction reduction in in pollination. pollination. This This leads leads to to aa positive positive density plant densities, ((i.e., i.e., an density effect effect at at low low plant an Allee Allee effect) effect).. Such Such effects effects may may also also be influenced influenced by by patch patch isolation. isolation. For For example, example, Groom Groom ((1998) found that that small small be 1 998) found patches patches of of the the annual annual Clarkia concinna suffered suffered from from reproductive reproductive failure failure due due to to aa lack lack of of pollinators pollinators when when the the patches patches were were more more than than 26 26 m m apart. apart. For For large large patches, threshold occurred. patches, no no isolation isolation threshold occurred. Ample Ample evidence evidence exists exists that that habitat habitat fragmentation fragmentation [which [which in in this this context context can can be be perceived perceived as as the the breaking breaking up up of of previously previously (possibly) (possibly) existing existing metapopulation metapopulation structure] structure] results results in in aa decreased decreased seed aI., 11998; 998; Fisher 997; Luijten seed set set (e.g., (e.g., Oostermeijer Oostermeijer et et al., Fisher and and Matthies, Matthies, 11997; Luijten et aI., 2003) et aI., al., 2000; 2000; Vergeer Vergeer et et al., 2003) or or in in reduced reduced quality quality of of the the resulting resulting seeds seeds as as aa consequence consequence of of increased increased selfing selfing in in local, local, isolated isolated populations populations (Rayman (Rayman et 994; van 993, 11994). 994). Because Because seed et aI., al., 11994; van Treuren Treuren et et aI., al., 11993, seed limited limited recruitment, recruitment, at common (Turnbull at both both local and and regional scales, is is common (Turnbull et et aI., al., 2000), 2000), the the regional dynamics of of plant species species cannot cannot be be understood understood and and modeled com comregional pletely pletely without without taking taking the the local local and and regional regional dynamics dynamics of of their their pollinators pollinators into into account. However, However, the the effects effects on on plant reproduction of of habitat habitat connectivity connectivity account. plant reproduction
118. 8. TOWARD TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR FOR PLANTS PLANTS
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may size and isolation may be more more complex complex than aa simple simple patch size isolation effect effect mediated mediated by by pollinators. Seed Seed predators predators may may also also increase increase their their patch patch incidence incidence when when connectivity connectivity increases, increases, counteracting counteracting positive positive effects effects by by pollinators. Steffan SteffanDewenter aI. (20 0 1 ) found Dewenter et et al. (2001) found such such counterbalancing counterbalancing effects effects of of seed predators predators and and pollinators, resulting resulting in in no no net net effect effect of of habitat habitat connectivity connectivity on on seed seed pro production in Centaurea jacea. Another example of of aa vital vital mutualistic mutualistic relationship relationship is is the the interaction interaction many many plant plant species species have have with with arbuscular arbuscular mycorrhizal mycorrhizal fungi fungi (AMF) (Smith (Smith and and Read, Read, al., 11997). 997). Evidence has been presented (Gange et al., aI., 11993; 993; van der Heijden et aI., 11998) 9 9 8 ) that that successful successful colonization colonization and and establishment establishment of of aa plant plant species species are are only only possible in in the the presence presence of of the the right right species species of of mycorrhiza. mycorrhiza. Moreover, Moreover, the the presence presence or or absence absence of of AMF AMF may may alter the the interaction interaction between plants plants and and herbivore herbivore insects insects (Brown (Brown and and Gange, Gange, 2002 2002).) . Thus, Thus, plant plant metapopu metapopulation lation models models should should take take the the local local and and regional regional dynamics dynamics of of these these fungal fungal mutualists into into account. account. mutualists Plants Plants constitute constitute the the habitat habitat for for herbivores herbivores and and seed seed predators predators as as well well as as pathogens, pathogens, and and plants plants are are used used as as food food for for larger larger mobile mobile herbivores, herbivores, including, including, for example, frugivorous birds and mammals acting as seed dispersers. All these these interactions interactions may may be be potentially potentially important important for for the the plant. plant. If If the the interactors interactors respond respond to to local local population size size and and isolation (which (which seems seems reasonable) reasonable),, the the plant will be affected by the spatial configuration of local populations, even if the by colonization the plants plants themselves themselves ((by colonization and and extinction extinction processes) processes) are are not not affected affected directly directly by by the the underlying underlying habitat habitat configuration. configuration. Evidence Evidence provided provided by trophic by Tscharntke Tscharntke and and co-workers, co-workers, demonstrated demonstrated the the relevance relevance of of multi multitrophic interactions interactions in in assessing the the effects effects of of habitat habitat fragmentation. fragmentation. In In manually Trifolium pratense), isolated isolated patches established islands islands of of red red clover clover ((Trifolium patches were were colonized colonized by by most most of of the the available herbivores but but only only aa few few of of the the available available parasitoid 0% parasitoid species. species. In In isolated isolated patches, patches, herbivores herbivores experienced experienced only only 1199 to to 660% of of the the parasitoid parasitoid attacks attacks compared compared ttoo nonisolated nonisolated patches patches (Kruess (Kruess and and Urtica Tscharntke, 994). In Tscharntke, 11994). In another another experiment, experiment, 32 natural natural stinging stinging nettle nettle ((Urtica dioica) patches size and degree patches of different different size degree of isolation isolation were investigated. investigated. Habitat Habitat fragmentation fragmentation reduced reduced species species richness, richness, but but not not all all species species groups groups were were affected affected to to the the same same degree degree and and in in the the same same way. way. Monophagous insects insects were were most most affected affected by by the the area area of of the the patch, patch, whereas whereas predatory predatory insects insects were were most 9 9 8 ) . In most affected affected by by the the degree degree of of isolation isolation (Zabel (Zabel and and Tscharntke, Tscharntke, 11998). In Vicia sepium) plants experiments experiments with with bush bush vetch vetch ((Vicia plants in in pots, pots, the the overall overall colon colonization ization success success of of insects insects decreased decreased with with increasing increasing distance distance (Kruess (Kruess and and Tscharntke, Tscharntke, 2000 2000).) . Moreover, Moreover, parasitism parasitism on on the the rape rape pollen pollen beetle, beetle, aa pest pest on on Brassica napus, responded responded positively positively to to increased increased habitat habitat connectivity, connectivity, thus thus enhancing 99 9 ) . enhancing seed seed production production (Thies (Thies and and Tscharntke, Tscharntke, 11999). IIn n addition, addition, insects insects that that respond respond ttoo landscape landscape structure structure seem seem ttoo perceive perceive the the landscape spatial scales 997; Steffan landscape at at different different spatial scales (Roland (Roland and and Taylor, Taylor, 11997; SteffanDewenter et implies that Dewenter et aI., al., 2002). 2002). This This implies that analyses analyses of of how how landscape landscape structure structure influences influences the the whole whole range range of of processes processes ultimately ultimately determining determining plant fitness, fitness, pollination, pollination, seed seed production, production, seed seed predation, predation, and and seed seed dispersal dispersal should should account account for for effects effects that that appear appear in in different different spatial spatial scales. scales. Still Still rather rather few few studies studies have have addressed addressed this this complexity complexity in in plant-animal plant-animal interactions interactions in in relation relation to to landscape structure. structure. From From the the studies studies that that are are at at hand, hand, however, however, we we can can conclude conclude that that effects effects of of habitat habitat connectivity connectivity do do occur, occur, but but that that these these are are not not necessarily necessarily
N.J. N.J. OUBORG OUBORG AND AND O. O. ERIKSSON ERIKSSON
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straightforward straightforward positive positive effects effects of of patch patch size size and and negative negative effects effects of of isolation. isolation. The regional regional dynamics dynamics of of plants plants will will be be affected affected through through variable variable responses responses of of The the the next next trophic trophic levels. levels. This This exciting exciting research research field field surely surely deserves deserves more more atten attention in tion in the the future. future.
11 88.4 .4
ESTIMATING E S T I M A T I N G PLANT P L A N T DISPERSAL DISPERSAL The The amount amount of of dispersal dispersal taking taking place place as as aa function function of of distance distance between between local Thomas and local populations populations is is aa core issue for for any any metapopulation metapopulation concept. concept. Thomas and Kunin 1 999) argued Kunin ((1999) argued that that the the type type of of dynamics dynamics that that is is inferred inferred in in studies studies of of regional regional dynamics dynamics may may be be altered altered arbitrarily arbitrarily by by chosing different different spatial spatial scales scales of of study. study. It It is is therefore therefore very very important important to to define define the the appropriate appropriate scale scale for for regional regional studies. studies. Such Such an an aa priori priori definition definition should should be be based based on on estimates estimates of of dispersal dispersal rates rates and and distances. distances. Dispersal Dispersal may may also also be be the the most most important important issue issue underlying underlying the the discussion discussion on on the (e.g., Husband 996; the existence existence of of plant plant metapopulations metapopulations (e.g., Husband and and Barrett, Barrett, 11996; Eriksson, 996; Bullock Eriksson, 11996; Bullock et et ai., al., 2002) 2002).. Obviously, Obviously, the the most most efficient efficient way way to to solve solve this reliable estimates dispersal distances this debate debate is is to to obtain obtain reliable estimates of of plant plant dispersal distances in in aa regional regional context. context. Unfortunately, Unfortunately, quantifying quantifying dispersal, dispersal, especially especially long long distance distance dispersal, dispersal, has has always always been been one one of of the the most most difficult difficult tasks tasks in in plant plant population population biology. biology. An important important topic topic for for the the discussion discussion is interpretation interpretation of of the the plant plant disper dispersal sal distance distance curve. curve. Typically, Typically, such such curves curves are are extremely extremely leptokurtic, leptokurtic, with with the the overwhelming majority seeds dispersing distances (in (in the overwhelming majority of of seeds dispersing over over very very short short distances the order of meter) within local populations order of aa few few meter) within local populations and and only only aa very very small small propor proportion tion dispersing dispersing over over longer longer distances, distances, between between local local populations. populations. Thus, Thus, long longdistance distance dispersal dispersal events events are are rare, rare, but but have have aa great great importance importance for for plant plant migration, migration, probabilities probabilities of of colonization colonization of of suitable but but unoccupied unoccupied habitat, habitat, and for structure in Indeed, studies studies suggest and for the the metapopulation metapopulation structure in general. general. Indeed, suggest that that understanding these understanding these processes processes necessitates necessitates that that the the form form of of the the tail tail of of the the dis disCain et persal curve, curve, resulting resulting from from "chance events," events," is is taken taken into into account account ((Cain et aI., al., Clark, 11998; Higgins and and Richardson, Richardson, 11999; Bullock et et aI., al., 2002). 2002). 11998; 998; Clark, 998; Higgins 999; Bullock Several Several approaches approaches to to studies studies of of dispersal dispersal can can be be found found in in the the literature literature (Table (Table 118.1). 8. 1 ) . Nathan 1 ) mentioned Nathan (200 (2001) mentioned three three categories categories of of approaches. approaches. In In the the first first cat category 1 ) , the egory (movement-redistribution (movement-redistribution methods; methods; Nathan, Nathan, 200 2001), the movement movement of of indi individuals through space is measured directly individuals. In viduals through space is measured directly by by marking marking individuals. In the the strict strict sense of sense of traditional traditional mark-recapture mark-recapture methods, methods, which which were were designed designed for for the the study study of these methods are unsuitable of animal animal dispersal, dispersal, these methods are unsuitable for for the the study study of of dispersal dispersal in in plants plants due due to to the the impracticalities impracticalities of of marking marking and and tracking tracking large large amounts amounts of of small small seeds space. Some Some studies seeds through through potentially potentially large large amounts amounts of of space. studies have have tried tried to to measure measure the the actual actual distance distance over over which which individual individual seeds seeds disperse disperse by by trapping trapping 996; Thiede seeds at at various distances distances from from aa source source (e.g., (e.g., Ruckelshaus, Ruckelshaus, 11996; Thiede and and Augspurger, 996; Bullock and Augspurger, 11996; and Clarke, Clarke, 2000). 2000). However, However, because of of the the logistical logistical problems with with seed seed trapping trapping at at larger distances, distances, these methods almost invariably invariably problems rely dispersal curve, rely on on extrapolation extrapolation to to estimate estimate the the tails tails of of the the dispersal curve, and and therefore therefore the the frequency dispersal. Moreover, frequency and and extent extent of of long long distance distance seed seed dispersal. Moreover, often often these these stud studies ies will will be be performed performed in in situations situations where where aa single single point point source source is is placed placed in in aa habi habitat tat that that is is otherwise otherwise unoccupied unoccupied by by the the focal focal species. Whether Whether the the resulting resulting data data can can be be translated translated without without bias bias to to more more natural natural situations situations is is unclear. unclear. Within Within this this
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118. 8. TOWARD FOR PLANTS TOWARDA METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR PLANTS TABLE 118.1 8.1
An An Overview Overview of of Approaches Approaches to to Studies Studies of of Plant Plant Dispersal Dispersal
Type Type of of estimate estimate
Method Method
Mark -recapture Instantaneous, Mark-recapture Instantaneous, dispersal studies studies of seeds seeds dispersal Seed Seed trap trap experiments experiments Diffusion modeling
Demographic Demographic modeling Indirect molecular techniques techniques Direct molecular techniques techniques
Reliability Reliability of of long-distance long-distance dispersal estimates estimates
Sampling Sampling effort effort
low
Very Very large large
Instantaneous, Instantaneous, low extrapolation, extrapolation, dispersal Prediction, average dispersal
Large Large
Prediction Prediction
Average Average
Unknown Unknown
Historical, Case dispersal dependent dispersal + + dependent establishment establishment Instantaneous, Instantaneous, Average Averageto dispersal dispersal + good establishment establishment
Minor
Average Average
Very Very large large
Refs_ Refs. Platt 1 977); Lee 1 984); Platt and Weis Weis ((1977); Lee ((1984); Nilsson 1 9 9 1 ) ; Johansson Nilsson et al. ((1991); and Nilsson 1 99 3 ) Nilsson ((1993) Ruckelshaus 1 996); Thiede Ruckelshaus ((1996); Thiede and Augspurger 1 996); Bullock Augspurger ((1996); Bullock and Clarke Clarke (2000) (2000) van Dorp 1 996); Greene Dorp et al. ((1996); Greene and Johnson 1 996); Cain Johnson ((1996); Cain et al. ((1998); 1 99 8 ); Jongejans Jongejans and Schippers Schippers ((1999); 1 999); Soons Heil (2002) Soons and Heil (2002) Neubert and Caswell Caswell (2000); (2000); Neubert Bullock Bullock et al. (2002) (2002) d. 1 999) cf. Ouborg Ouborg et al. ((1999) Meagher 1 987); Meagher and Thompson Thompson ((1987); Ashley ((1996); 1 996); Schnabel Dow and Ashley Schnabel et al. ((1998); 1 99 8 ) ; Isagi Isagi et al. (2000) (2000)
same same category category of of approaches, approaches, some some studies studies use use mark-recapture mark-recapture methods methods with with natural 977; Lee, 984; Johansson natural seeds seeds (e.g., (e.g., Platt Platt and and Weis, Weis, 11977; Lee, 11984; Johansson and and Nilsson, Nilsson, 1993) aI., 11991). 9 9 1 ) . In 1993) or or with with artificial artificial seed seed mimics mimics (Nilsson (Nilsson et et al., In addition addition to to the the enor enormous mous effort effort that that has has to to be be put put into into these these type type of of experiments, experiments, both both in in terms terms of of labor labor and and in in the the number number of of seeds seeds to to be be used, used, the the recovery recovery rate rate of of marked marked seeds seeds at unreliable. In at long long distances distances is is very very low, low, making making the the estimations estimations unreliable. In addition, addition, the the general value value of dispersal estimates estimates from experiments is general of dispersal from these these experiments is limited limited because because they process, which they essentially essentially (try (try to) to) measure measure one one realization realization of of aa dispersal dispersal process, which may may change from from situation situation to to situation situation and and from from time time to to time. time. Thus, Thus, it it is is very very difficult difficult change to to reliably reliably estimate estimate long-distance long-distance dispersal dispersal using using this this category category of of approaches approaches (Silvertown, 1991; 1991; Bullock Bullock and and Clarke, Clarke, 2000) 2000).. (Silvertown, A A second second category category ooff approaches approaches uses uses mathematical mathematical modeling modeling ttoo describe describe dis dispersal (Nathan, 200 1 ) . These persal patterns patterns and and infer infer long-distance long-distance dispersal dispersal (Nathan, 2001). These methods methods predict dispersal. Models predict rather rather than than measure measure dispersal. Models that that deal deal with with wind wind dispersal dispersal are are presumably presumably the the most most advanced. advanced. The The basic basic rationale rationale in in these these models models is is that that aero aerodynamic dynamic properties, properties, which which are are measured measured in in wind wind tunnel tunnel experiments experiments (e.g., (e.g., van van Dorp 996; Jongejans 999; Soons 2002) and Dorp et et aI., al., 11996; Jongejans en en Schippers, Schippers, 11999; Soons and and Heil, Heil, 2002) and which which result result in in terminal terminal velocities velocities of of seeds, seeds, are are fed fed into into specific specific aerodynamic aerodynamic models Johnson, 11996; 996; Cain Soons and models (e.g., (e.g., Greene Greene and and Johnson, Cain et et aI., al., 1998; 1998; Soons and Heil, Heil, 2002; 2002; Tackenberg Tackenberg et et aI., al., 2003; 2003; Tackenberg, Tackenberg, 2003 2003),), which which then then transform transform wind wind and and landscape landscape profiles profiles into into distributions distributions of of dispersal dispersal distances. distances. In In general, general, these these methods reliable in reliable in methods are are reliable in the the short short dispersal dispersal range range and and less less reliable in the the biologic biologically ally more more relevant relevant long long dispersal dispersal range. range. It It has has been been argued argued that that the the inaccurate inaccurate description description of of long-distance long-distance dispersal dispersal in in these these models models is is the the consequence consequence of of the the relative Clarke, relative inflexibility inflexibility of of the the mathematical mathematical functions functions used used (Bullock (Bullock and and Clarke,
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00 1 ) . New 000; Nathan 000; Nathan, 2 2000; Nathan and and Muller-Landau, Muller-Landau, 22000; Nathan, 22001). New functions functions have have improvbeen proposed that allow "fatter" tails of the dispersal curve, thereby improv 000 ; ing ing the the fit fit of of the the model to to actual actual dispersal (Nathan and and Muller-Landau, 2 2000; 8 .3). 002; Fig. Bullock Bullock et et aI., al., 22002; Fig. 118.3). Another Another approach approach within within this this category category has has been been proposed proposed by by Neubert Neubert and and 000 ) in 000 ) . Caswell Caswell, 2 Caswell (2 (2000) in analogy analogy to to the the life life table table response response analysis analysis ((Caswell, 2000). Their Their approach approach uses uses aa combination combination ooff matrix matrix projection projection modeling modeling ttoo describe describe and and predict predict local local demography, demography, with with integrodifference integrodifference equations equations to to capture capture dis dispersal. persal. The The method method allows allows calculation calculation of of the the sensitivity sensitivity of of spatial spatial expansion expansion speed speed to to changes changes in in local local demographic demographic parameters parameters and and regional regional dispersal dispersal parameters. Neubert Neubert and and Caswell Caswell ((2000) demonstrated that that the the rate rate of of spatial spatial parameters. 2 000 ) demonstrated expansion expansion is is governed governed by by the the long-distance long-distance component component of of dispersal, dispersal, even even when when long-distance long-distance dispersal dispersal is is rare. rare. This This result result was was confirmed confirmed by by analyses analyses of of demo demographic Calluna vulgaris, graphic and and dispersal dispersal data data for for three three heathland heathland species species ((Calluna vulgaris, Erica 002). Further development cinerea, cinerea, and and Rhinanthus minor) minor) (Bullock (Bullock et et aI., al., 22002). development and and application of of this this approach are are needed to to evaluate evaluate its its usefulness usefulness and practicality practicality in in attempts attempts to to measure measure the the implications of of long-distance dispersal. dispersal. The The third third category category of of approaches approaches is is to to use use molecular molecular markers markers and and popula popula000 ; Ennos, 00 1 ; tion Ouborg et aI., 11999; 999; Cain tion genetic genetic analyses analyses ((Ouborg et al., Cain et et aI., al., 22000; Ennos, 2 2001; 002 ) . With Raybould Raybould et et aI., al., 22002). With the the continuous continuous invent invent ooff new new marker marker techniques techniques and promis and the the increasing increasing automation of of their their application, application, these these methods are are promising. ing. Two Two basic basic methods can can be be followed followed when when applying applying molecular markers markers in in the Ouborg et 999). First, the study study of of dispersal in in plants plants ((Ouborg et aI., al., 11999). First, dispersal dispersal can can be be assessed assessed from from observed observed distributions distributions of of genetic genetic variation variation in in space. space. This This indirect indirect method popu method is is based on on the the quantification quantification of of genetic genetic divergence divergence between between populations lations and and the the interpretation interpretation of of this this divergence divergence in in terms terms of of the the amount amount of of past gene gene flow. flow. Second, Second, direct direct approaches try try to to establish establish the the parent-offspring parent-offspring rela relamolecu tionships between between individuals in in space. The The various approaches approaches using using molecular lar markers markers to to estimate estimate dispersal rates rates are are discussed in in Chapter Chapter 15. 15.
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Fig. Fig, 1 8.3 8 . 3 Three dispersal functions functions demonstrating demonstrating the difference in tail of of the distribution. distribution. The negative exponential - x)] is perhaps the most exponential [y = = exp( exp(-x)] most frequently frequently used; the gaussian (nor (nor2)] has a mal) function -x2)] has a thinner /(1 + function [y = = exp( exp(-x2)] thinner tail, and the cauchy function function [y = = 11/(1 + xx2)] fatter fatter tail, allowing allowing more easy fitting fitting of rare long-distance long-distance dispersal events.
118. 8. TOWARD FOR PLANTS TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR PLANTS
463 463
It population It seems seems that that long long range range dispersal, dispersal, and and its its relevance relevance for for meta metapopulation dynamics, dynamics, will will never never be be quantified quantified easily easily using using aa single single method method (Ouborg (Ouborg et et aI., al., 11999; 999; Raybould Raybould et et aI., al., 2002) 2002).. When When studying studying plant plant dispersal, dispersal, it it seems seems advisable advisable to reasons. to apply apply several several methods methods to to the the same same study study system system for for at at least least two two reasons. First, First, one one method method may may result result in in aa hypothesis hypothesis on on dispersal dispersal and and regional regional connec connection tion between between local local populations populations that that can can be be tested tested independently independently with with aa second second method. method. Second, Second, various various methods methods will will provide provide different different types types of of information information about the the dispersal dispersal process. Direct Direct approaches will will give give estimates estimates of of instantan instantaneous eous dispersal, dispersal, whereas whereas indirect indirect methods methods will will give give estimates estimates of of historical historical gene gene flow. flow. Mark-recapture Mark-recapture estimates estimates will will address address the the seed seed dispersal dispersal component component of of dispersal dispersal only, only, whereas whereas indirect indirect (and (and some some direct) direct) approaches approaches will will estimate estimate the the combined combined results results of of seed seed dispersal dispersal and and subsequent subsequent establishment. establishment. This This overview overview illustrates illustrates the the perhaps perhaps somewhat somewhat depressing depressing complexity complexity of of the the study study of of dispersal; dispersal; it it is is very very likely likely that that this this complexity complexity has has resulted resulted in in the the lack lack of of data data that that is is needed needed to to build build our our arguments arguments on on the the existence existence or or absence absence of of meta populations in metapopulations in plants. plants. However, However, there there are are some some promising promising methods methods that that reduce manageable proportions reduce this this complexity complexity to to manageable proportions by by helping helping us us to to decide decide whether populations do whether at at least least populations do exchange exchange individuals individuals at at all. all. For For instance, instance, Rannala 1 997) presented Rannala and and Mountain Mountain ((1997) presented aa statistical statistical method method for for detecting detecting immi immigration using using multilocus multilocus genotype genotype data data based based on on Bayesian Bayesian statistical statistical inference. inference. In In gration addition, methods have have been addition, other other assignment assignment methods been developed developed where where individuals individuals can can be be assigned assigned to to aa particular particular population; population; any any individual individual assigned assigned to to another another popu population (e.g., Pritchard lation than than where where it it was was sampled sampled indicates indicates aa dispersal dispersal event event (e.g., Pritchard et et aI., al., 2000) 2000).. Some Some authors authors suggest suggest that that the the maximum maximum distance distance seeds seeds disperse, disperse, rather rather than than the the frequency frequency of of dispersal dispersal events, events, determines determines the the regional regional dynamics dynamics (Neubert and aI., 2002) (Neubert and Caswell, Caswell, 2000; 2000; Bullock Bullock et et al., 2002).. If If this this is is true, true, it it perhaps perhaps becomes more assess whether becomes more important important to to assess whether two two populations populations ever ever exchange exchange migrants migrants rather rather than than estimating estimating the the long-distance long-distance dispersal dispersal rate rate in in detail. detail.
118.5 8.5
EXAMPLES EXAMPLES OF OF METAPOPULATION METAPOPULATION STUDIES STUDIES IN IN PLANTS PLANTS An regional population exhibit metapopu An essential essential feature feature of of regional population systems systems that that exhibit metapopulation lation dynamics dynamics is is that that colonization colonization and and extinction extinction processes processes are are related related to to the the configuration configuration of of habitats habitats in in the the region. region. With With configuration, configuration, we we mean mean the the size size distribution distribution of of habitat habitat patches, patches, their their shape, shape, and and the the extent extent to to which which they they are connected to each other habitat corridors corridors or dispersal routes. are connected to each other by by habitat or by by dispersal routes. Effects Effects of of habitat habitat configuration configuration on on regional regional plant plant populations populations can can result result from from different 1 ) Since different mechanisms. mechanisms. ((1) Since seed seed dispersal dispersal from from source source populations populations is is likely distance the likely to to be be related related to to the the distance the likelihood likelihood of of colonization colonization is is expected expected to 2 ) If to decline decline as as habitat habitat patches patches become become more more isolated. isolated. ((2) If the the same same distance distance effects pollen transport, populations may effects occur occur for for pollen transport, isolated isolated populations may suffer suffer from from repro reproductive e.g., in 3) ductive limitations limitations ((e.g., in self-incompatible self-incompatible plants) plants) and and inbreeding. inbreeding. ((3) Interactions Interactions other other than than related related to to pollination, pollination, e.g., e.g., herbivores herbivores and and pathogens, pathogens, may (4) The may reflect reflect landscape landscape structure. structure. (4) The size size and and shape shape of of habitat habitat patches patches may may influence 5 ) The influence their their capacity capacity to to harbor harbor local local populations. populations. ((5) The structure structure of of the the habitat patches may among landscape surrounding surrounding habitat may influence influence the the dispersal among patches, stepping-stone dispersal. issue patches, e.g., e.g., by by "corridors" "corridors" or or by by ""stepping-stone dispersal."" An An issue related hypothesized mechanisms for related to to these these hypothesized for configuration configuration effects effects is is whether whether
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there there are are threshold threshold effects, effects, (e.g., (e.g., if if there there is is aa minimum minimum amount amount of of available available habitat habitat needed needed to to sustain sustain aa persistent persistent regional regional population population or or if if there there is is aa max maximum imum distance distance between between habitat habitat patches patches above above which which no no effective effective dispersal dispersal can population can be be achieved) achieved).. If If such such thresholds exist, exist, the the behavior of of aa local population may may change change drastically drastically if if the the threshold threshold value value is is passed. passed. Evidence Evidence supports supports all all five five mechanisms mechanisms for for habitat habitat configuration configuration effects effects on plants. plants. Distance Distance effects on colonization colonization have been found in temperate temperate woodlands 9 9 8 ; Butaye woodlands (e.g., (e.g., Grashof-Bokdam Grashof-Bokdam and and Geertsema, Geertsema, 11998; Butaye et et aI., al., 200 1 ; Jacquemyn 1 ; Verheyen 1 ) , open 2001; Jacquemyn et et aI., al., 200 2001; Verheyen and and Hermy, Hermy, 200 2001), open grasslands grasslands (e.g., 99 3 ; Fig. 8 .4 ) , serpentine (e.g., Ouborg, Ouborg, 11993; Fig. 118.4), serpentine vegetation vegetation [Harrison [Harrison et et aI., al., (2000), (2000), who who also also found found evidence evidence for for rescue rescue effects effects related related to to distance] distance],, scrub Quintana-Ascencio and 99 6 ) , seasonal scrub vegetation vegetation ((Quintana-Ascencio and Menges, Menges, 11996), seasonal pools pools (Husband 99 8 ) , and Bastin and (Husband and and Barrett, Barrett, 11998), and urban urban vegetation vegetation ((Bastin and Thomas, Thomas, 11999). 99 9 ) . In In practice, practice, it it may may be be very very difficult difficult to to distinguish distinguish effects effects of of size size and and populations, as isolation in in studies studies of of fragmented fragmented plant plant populations, as these these two two landscape landscape features features normally normally change change along along with with each each other. other. Moreover, Moreover, factors factors other other than isolation isolation ((and patch size size)) were also important important for species occupancy occupancy than and patch patterns Bastin and patterns in in some some of of these these studies, for for example, age age of of the the sites sites ((Bastin and Thomas, 999; Jacquemyn 00 1 ) . Small popula Thomas, 11999; Jacquemyn et et aI., al., 22001). Small and and isolated isolated plant plant populations tions have have been been found found to to suffer suffer from from reduced reduced reproduction reproduction (e.g., (e.g., Jennersten, Jennersten, 11988; 9 8 8 ; Groom, 99 8 ; Jules, 99 8 ; Morgan, 999; Cunningham, Groom, 11998; Jules, 11998; Morgan, 11999; Cunningham, 2000 2000)) and and effects ostermeijer et 994; Ouborg Ouborg and effects of of inbreeding inbreeding (e.g., (e.g., O Oostermeijer et aI. al.,, 11994; and van van Treuren, 11994, K~ry et aI., al., 2000; 2000; Richards, 2000; 2000; McCauley McCauley et aI., al., Treuren, 994, 11995; 995; Kery 2 00 1 ) . The 2001). The shape shape of of habitat habitat patches patches affects affects the the edge-interior edge-interior relationship, relationship, which which in in turn turn may may influence influence colonization colonization patterns patterns (e.g., (e.g., Restrepo Restrepo et et aI., al., 11999; 999; Kiviniemi Kiviniemi and and Eriksson, Eriksson, 2002 2002).) . Thus, Thus, although although it it may may well well be be that that the deterioration have the general general effects effects of of habitat habitat decline and and deterioration have aa dominating dominating impact impact on on the the persistence of of plants inhabiting fragmented fragmented landscapes ((Harrison Harrison and 999; Fahrig, 2002 and Bruna, 11999; 2002),) , this evidence suggests that that
2.00 2.00
,------==--"71
go 11.50 .50 X � (D (!)
�E 11.00 .00
-
o 03 C/)
�Oz
0.50 0.50 i J , -"-----"---'---' 0.00 ""'--11.50 .50 2.00 11.00 .00 0.00 2.00 0.50 NOSD NOSD (km) (km) non-extinct non-extinct
Fig. 1 8.4 43 grassland Dutch Rhine, 6 plant species were Fig. 8 . 4 In an analysis of 1143 grassland sites sites along the Dutch Rhine, 116 examined in detail. For most most species, that went went extinct in aa 32-yr had aa examined in detail. For species, populations populations that extinct in 32-yr period period had greater nearest occupied D) than than populations greater distance distance to to nearest occupied sites sites (NOS (NOSD) populations that that remained remained extant. extant. This This illustrates plant populations illustrates the the importance importance of of regional regional processes processes for for the the local local persistence persistence of of plant populations (after 993). (after Ouborg, Ouborg, 11993).
118. 8. TOWARD FOR PLANTS TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR PLANTS
465 465
changes changes in in habitat habitat configuration configuration per se se do do influence influence the regional regional dynamics dynamics of of plants plants in in aa range range of of different different vegetation vegetation types. types. In In contrast, contrast, the the effects effects of of habitat habitat corridors corridors remain remain more more controversial. controversial. Malanson 1 997) suggested Malanson and and Cairns Cairns ((1997) suggested that that the the effects effects of of habitat habitat isolation isolation are are more more dependent dependent on on the the capacity capacity of of source source populations populations to to produce produce diaspores diaspores than than on on the the barrier barrier effect effect of of surrounding surrounding landscapes. landscapes. This This may, may, however, however, be be treated populations in treated conceptually conceptually by by including including the the size size of of the the source source populations in the the def definition of Some authors inition of habitat habitat connectivity connectivity (Moilanen (Moilanen and and Nieminen, Nieminen, 2002 2002).) . Some authors have corridors are have questioned questioned whether whether habitat habitat corridors are effective effective in in enhancing enhancing disper dispersal e.g., van 997; Cain 998). A sal ((e.g., van Dorp Dorp et et al., al., 11997; Cain et et al., al., 11998). A rationale rationale for for this this view view is is that chance events" that long-range long-range dispersal dispersal is is mainly mainly dependent dependent on on unlikely unlikely ""chance events" ((Clark, Clark, 11998; 99 8 ; Higgins 999; Cain Higgins and and Richardson, Richardson, 11999; Cain et et al., 2000; 2000; Bullock Bullock et et al., al., 2002 2002).) . Assuming Assuming this this is is the the case, case, both both range range expansion expansion and and dispersal dispersal among among relatively relatively isolated isolated habitat habitat patches patches may may be be decoupled decoupled effectively effectively from from the the structure landscape surrounding patches, and structure of of the the landscape surrounding habitat habitat patches, and even even the the distance distance among among patches. Despite Despite the the plausibility plausibility of of this this argument, argument, the the studies studies listed listed earlier also beyond earlier indicate indicate that that distance distance does does play play aa role role in in dispersal dispersal also beyond the the local neighborhood. hypothesized cor local neighborhood. Moreover, Moreover, some some evidence evidence supports supports the the hypothesized corridor 996; Burkart, 1 ), hedgerows ridor effects: effects: along along rivers rivers (Johansson (Johansson et et al., 11996; Burkart, 200 2001), hedgerows ((Corbit Corbit et 999), roads roads ((Cousins Cousins and 1 ), and et al., al., 11999), and Eriksson, Eriksson, 200 2001), and railway railway verges verges (Tikka 1). (Tikka et et al., al., 200 2001). Theory Theory suggests suggests that that aann extinction extinction threshold threshold exists exists at at the the lowest lowest amount amount of of habitat Lande, 11987). 987). habitat in in aa region region needed needed for for sustaining sustaining aa regional regional population population ((Lande, Under Under aa set set of of simplifying simplifying assumptions, assumptions, this this amount amount equals equals the the equilibrial equilibrial unoccupied Lawton et 994). A unoccupied fraction fraction of of suitable suitable habitat habitat in in aa landscape landscape ((Lawton et al., al., 11994). A more realistic definition of value for of a more realistic definition of aa threshold threshold value for the the capacity capacity of a certain certain land landscape harbor aa meta population was scape to to harbor metapopulation was suggested suggested by by Hanski Hanski and and Ovaskainen Ovaskainen (2000). (2000). This This measure, measure, the the metapopulation metapopulation capacity, capacity, incorporates incorporates both both aspects aspects of of the the configuration configuration of of habitat habitat patches, patches, i.e., i.e., their their size size and and isolation, isolation, and and dis dispersal persal features features of of the the focal focal species. species. Even Even if if improved improved realism realism makes makes studies studies of of extinction extinction thresholds thresholds more more feasible, feasible, it it is is still still very very difficult difficult to to examine examine extinc extinction empirically (Fahrig, tion thresholds thresholds empirically (Fahrig, 2002 2002).) . One One complicating complicating factor factor is is that that extinction processes for extinction processes for most most plants plants are are subjected subjected to to time time lags lags (Eriksson (Eriksson and and Kiviniemi, 999; Eriksson 1 ; see Kiviniemi, 11999; Eriksson and and Ehrlen, Ehrl~n, 200 2001; see also also Hanski Hanski and and Ovaskainen, Ovaskainen, 2002 2002).) . Nevertheless, Nevertheless, aa few few studies studies provide provide evidence evidence suggesting suggesting that that there there are are threshold threshold effects effects manifested manifested by by aa minimum minimum amount amount of of habitat habitat or or maximum maximum allowed patches. Husband Husband and 1 9 9 8 ) found allowed distance distance among among patches. and Barrett Barrett ((1998) found that that no no populations populations of of the the water water plant plant Eichhornia Eichhornia paniculata paniculata existed existed when when the the density density of below aa certain value. Butaye 1 ) found of potential potential sites sites fell fell below certain value. Butaye et et al., al., (200 (2001) found that that isolation-sensitive isolation-sensitive woodland woodland plants plants did did not not occur occur when when the the distance distance to to source source populations above 200 200 m. herb, Scutellaria populations was was above m. In In aa study study of of an an endemic endemic herb, Scutellaria mon monCruzan (200 (2001) concluded that that effects effects of of habitat habitat configuration configuration appeared appeared 1 ) concluded tana, Cruzan at populations at different different spatial spatial scales. scales. At At aa sampling sampling scale scale of of 22 km, km, small small meta metapopulations had population size had higher higher levels levels of of selfing; selfing; at at aa sampling sampling scale scale of of 88 km, km, meta metapopulation size was was related related to to levels levels of of genetic genetic diversity. diversity. Whether Whether any any of of these these genetic genetic effects effects directly influence influence the populations were, directly the persistence persistence of of the the S. S. montana montana meta metapopulations were, however, however, not not clear. clear. Even there is much to Even though though there is much to be be done done before before well-founded well-founded generaliza generalizations tions can can be be made made on on how how habitat habitat configuration configuration influences influences regional regional plant plant
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populations, populations, evidence evidence suggests suggests that that there there are are configuration configuration effects effects in in many many 8 . 1 ) demonstrated different 2 0 0 3 ; see different kinds kinds of of vegetation. vegetation. Skarpaas Skarpaas ((2003; see Box Box 118.1) demonstrated that population approach that even even though though assumptions assumptions associated with with aa meta metapopulation approach may some) plant populations, the may not not strictly strictly apply apply to to ((some) plant populations, the approach approach may may still still lead population concept lead to to valid valid results. results. A A meta metapopulation concept that that incorporates incorporates time time lags lags ((see see Section 8 . 3 ) and Section 118.3) and allows allows for for slow slow dynamics dynamics is is likely likely to to be be aa most most pro productive ductive approach approach for for future future studies. studies.
BOX 118.1 8.1
Plant Population Population Dynamics in a Fragmented Fragmented Landscape
species, Skarpaas In a comprehensive comprehensive study on spatial dynamics of of two two plant plant species, Skarpaas (2003) - the (2003) tested various modeling modeling approaches. He chose two two contrasting contrasting species speciesoyster plant Mertensia Mertensia maritima (Boraginaceae) and the leafless leafless hawk's hawk's beard Crepis Crepis praemorsa (Asteraceae) (Asteraceae) - - to test whether whether predictions predictions of metapopulation metapopulation models would agree with would with results of a modeling modeling effort effort to incorporate incorporate variance in local dynam dynamics and more realistic dispersal-distance relationships. Both species species are regionally rare and declining declining in northern Europe. M. mertensa mertensa is naturally naturally fragmented fragmented with with suitable sites surrounded surrounded by inhabitable inhabitable unsuitable unsuitable matrix (the (the sea), sea), making the habitat struc strucspecies resemble an ideal metapopulation ture of this species metapopulation structure. In contrast, C. C. pra praemorsa emorsa is recently fragmented fragmented by landscape changes and is inhabiting inhabiting seminatural grasslands, where the distinction distinction between suitable and unsuitable unsuitable habitat habitat may be less less strict. To test for for spatial effects effects on occupancy, logistic regression models were fitted fitted to to either incidence (M. maritima) or extinction extinction and colonization colonization rates (c. (C. praemorsa) praemorsa) using the model
e� )
lOg Iog/i P p l ==[ 3�o 0++ [ 3 �1 1 1I++f 132A 32A p
where p is incidence, extinction, or colonization, colonization, A is patch area, area, and I is isolation of of occupied (when modeling extinction) or unoccupied modeling extinction) unoccupied (when modeling modeling colonization) colonization) patches. (l3i (13jare regression coefficients). coefficients). For M. maritima, maritima, there was no significant significant effect of area area on incidence, but distance to the nearest occupied occupied site or distance to to all occu occupied sites sites significantly significantly affected incidence. For C. C. praemorsa, praemorsa, extinction extinction was not not related to either area area or isolation, whereas colonization was only influenced influenced by isolation meas measures. The fitted regression models were used used in a long-term long-term simulation simulation of metapopula metapopulation dynamics, leading to predictions of occupancy as a function function of time. Skarpaas next next modeled modeled both both the local demography demography of both both species, species, using matrix matrix pro projection models, and the regional dispersal behavior with the use of various dispersal 989; Clarke 999; Nathan et aI., ). In models (e.g., Greene and Johnson, 11989; Clarke et aI., al., 11999; al., 2001 2001). this way he explored the limits of population approach when of a simple meta metapopulation when applied to these species. species. He presented evidence that that the strict dichotomy dichotomy between suitable and nonsuitable nonsuitable habitats did not not apply to both both species. species. In addition, addition, he showed that, con contrary to the assumption of simple meta population models that colonization metapopulation colonization is random, colonization for these species species followed followed an isolation by distance relationship relationship (Skarpaas (Skarpaas and Stabbetorp, 2001 2001;; Skarpaas, Skarpaas, 2003). Despite these deviations from from the basic basic meta population models, he demonstrated metapopulation demonstrated that such models would would still describe the regional dynamics of M. maritima fairly accurately. For C. C. praemorsa, praemorsa, the predictions of metapopulation metapopulation models were not in agreement with with the result of demographic demographic and dis dispersal modeling. populations in plants may modeling. The study demonstrates therefore, that that meta metapopulations exist, even though though not all assumptions strictly apply.
118. 8. TOWARD FOR PLANTS TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR PLANTS
118.6 8.6
467 467
REGIONAL AND REGIONAL DYNAMICS DYNAMICS A N D METAPOPULATION METAPOPULATION ARGUMENTS ARGUMENTS IN IN PLANT PLANT POPULATION POPULATION BIOLOGY BIOLOGY Several Several areas areas in in plant plant population population biology biology have have explicitly explicitly incorporated incorporated regional regional dynamics, population dynamics, dynamics, or or meta metapopulation dynamics, into into their their theories. This This Section Section gives gives three population concon three examples. examples. First, First, dispersal dispersal plays plays aa central central role role in in the the meta metapopulation cept. 1 995) and 1 997) developed cept. Olivieri Olivieri et et al. al. ((1995) and Olivieri Olivieri and and Gouyon Gouyon ((1997) developed aa theory theory for for how how metapopulation metapopulation dynamics dynamics may may affect affect the the evolution evolution of of dispersal dispersal ability. ability. They suggested suggested that two opposing opposing selective selective forces, forces, which which they baptized baptized as as the the meta population effect, metapopulation effect, affect affect dispersal dispersal traits, traits, (i.e., (i.e., fruit fruit and and seed seed traits traits such such as as plumes, bristles, showy fleshy fruits, and spines) spines) that may be interpreted interpreted as adap adaptations tations to to dispersal. dispersal. At At the the local local population population level level there there will will be be selection selection against against dispersal, dispersal, as as seeds seeds may may be be dispersed dispersed to to unsuitable unsuitable sites. sites. At At the the regional regional level, level, how however, ever, there there may may be be selection selection promoting promoting dispersal, dispersal, especially especially in in highly highly dynamic dynamic sit situations uations with with aa high high extinction extinction rate rate of of local local populations. populations. In In this this last last situation, situation, dispersal dispersal will will be be aa means means of of persistence persistence at at the the regional regional scale. scale. Cody 1 99 6 ) found Cody and and Overton ((1996) found that that the the evolution evolution of of dispersal dispersal traits traits may may be be very very rapid rapid in in situations situations where where suitable suitable habitat habitat is is surrounded surrounded by by large large areas of of unsuitable unsuitable habitat. Any Any dispersal dispersal beyond beyond the the borders borders of of the the local local popula population tion will will lead lead to to loss loss of of individuals, individuals, making making the the selection selection pressure pressure against against dis dispersal persal very very high. high. They They measured measured the the dispersal dispersal potential, potential, defined defined as as the the ratio ratio of of pappus pappus to to achene achene volume, volume, of of Lactuca muralis in in mainland mainland and and island island popula populations tions around around Vancouver, Vancouver, Canada. Canada. Newly Newly colonized colonized islands islands had had populations populations with with an an increased increased dispersal dispersal potential, potential, as as only only individuals individuals with with aa good good disper dispersal ability will be able to reach the islands. However, in the years following following col colonization, onization, the the dispersal dispersal potential potential dropped dropped to to levels levels below below that that of of the the mainland mainland populations. populations. Roff 1 ) presented Roff and and Fairbairn Fairbairn (200 (2001) presented evidence evidence that that dispersal dispersal traits traits are are often often correlated correlated genetically to other life life history traits. Thus, evolution evolution driven by the meta population effect metapopulation effect on on dispersal dispersal may may result in in correlated correlated evolution evolution of of other other traits, 996; Olivieri 997). Most traits, such such as as dormancy dormancy (Rees, (Rees, 11996; Olivieri and and Gouyon, Gouyon, 11997). Most evi evidence, Roff and 1 ), dence, however, however, comes comes from from studies studies with with animals animals ((Roff and Fairbank, Fairbank, 200 2001), which is is surprising surprising given given the the suitability suitability of of plants plants to to perform the the large large crossing crossing studies studies necessary necessary to to estimate estimate the the genetic genetic correlative correlative structure. structure. In In conclusion, conclusion, the the balance balance between between opposing opposing forces forces imposed imposed by by local local and and regional regional processes processes may may drive drive evolution evolution of of aa range range of of life life history history traits. traits. This This theory theory is, is, however, however, awaiting awaiting experimental experimental data data to to be be tested. tested. A A second second area area where where metapopulation metapopulation dynamics dynamics enters enters the the theory theory is is the the evo evolutionary lutionary dynamics dynamics of of reproductive reproductive systems. systems. Various Various studies studies present present evidence evidence that that attributes attributes of of the the breeding breeding system system of of plants plants may may be be affected affected by by the the metapopulation 995) of metapopulation effect effect (Olivieri (Olivieri et et aI., al., 11995) of frequent frequent extinction extinction and and colon colonization, Couvet ization, for for example, example, the the frequency frequency of of females females in in local local populations populations ((Couvet et 986), the Husband and et aI., al., 11986), the frequency frequency of of various various flower flower morphs morphs ((Husband and Barrett, Barrett, 11995; 995; Eckert 996), and Eckert et et aI., al., 11996), and the the rate rate of of self-fertilization self-fertilization in in local populations populations (Husband 992). It (Husband and and Barrett, Barrett, 11992). It is is probably probably not not possible to to understand understand the the dynamics dynamics and and evolution evolution of of these these features features without without taking taking regional regional dynamics dynamics into into account. account. A third third example is the evolutionary evolutionary dynamics of plants plants and and their herbivores herbivores or or pathogens. pathogens. Plants Plants in in natural natural populations populations are are generally generally challenged challenged by by aa wide wide
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variety variety of of herbivores herbivores and and pathogens pathogens and and can can be be affected affected strongly strongly by by these these nat natural enemies. enemies. Effects Effects on on individual hosts may may extend to to effects effects on population population size, (Burdon, 11987), 987), as size, dynamics, dynamics, and and population population structure structure (Burdon, as well well as as commun community Dobson and 994; Peters 996), for ity structure structure ((Dobson and Crawley, Crawley, 11994; Peters and and Shaw, Shaw, 11996), for instance, instance, by by altering altering the the relative relative competitive competitive abilities abilities of of species species (e.g., (e.g., Paul Paul and and Ayres, 990; Clay, 990) that Ayres, 11990; Clay, 11990) that cause changes changes in in their their relative relative abundance abundance or or by by affecting rates (Van 99 3 ) and and local local species diver affecting succession succession rates (Van der der Putten Putten et et ai., al., 11993) species diversity Plants have have evolved evolved a variety of mech sity (Packer (Packer and and Clay, Clay, 2000). 2000). Plants a variety of different different mechanisms anisms by by which which they they defend defend themselves themselves against against natural enemies. enemies. Direct Direct defenses of of plants plants include include at at least least three three types types of of mechanisms: mechanisms: avoidance, avoidance, resist resistdefenses ance, ance, and and tolerance. tolerance. Of Of these these mechanisms, mechanisms, the the molecular molecular and and genetic genetic basis basis of of gene for for gene gene (GFG) (GFG) resistance resistance to to pathogens pathogens is is probably probably the the best best documented, documented, gene and mechanism has and this this mechanism has served served as as the the basis basis for for the the majority majority of of models models of of the 992). the evolution evolution of of host-parasite host-parasite interactions interactions (Thompson (Thompson and and Burdon, Burdon, 11992). The The GFG GFG hypothesis hypothesis states states that that for for each each gene gene determining determining resistance resistance (R) (R) in in the the pathogen with host, there there is is aa corresponding corresponding gene gene for for avirulence (Avr) (Avr) in in the the pathogen with which interacts specifically. which it it interacts specifically. An An example example of aa GFG interaction interaction in in aa natural natural system system is is the the interaction interaction between between Linum marginale and and the the rust rust fungus fungus Melampsora lini, endemic endemic to to Australia. Australia. Studies on on the the dynamics of of host resistance resistance types types and and pathogen viru virulence types types in in this this system system have have yielded yielded valuable valuable knowledge knowledge about about the the spatial spatial scale scale at at which which such such GFG GFG interactions interactions occur. occur. The The frequency frequency of of different different races races within within local local pathogen pathogen populations populations in in this this system system appears appears to to be be poorly poorly correl correlated ated with with the the frequency frequency of of different different resistance resistance types types within within the the corresponding corresponding local populations (Jarosz 9 9 1 ) . Stochastic local host host populations (Jarosz and and Burdon, Burdon, 11991). Stochastic processes processes during during population natural selection within population crashes crashes rather rather than than natural within local populations populations appear appear to to be be the the main main cause cause of of large large year-to-year year-to-year variation variation in in the the frequencies frequencies of For instance, host genotypes of R R and and Avr Avr alleles. alleles. For instance, after after aa severe severe epidemic, epidemic, host genotypes resistant pathogen races resistant to to the the pathogen races that that were were present present at at high high frequency frequency during during the the epidemic epidemic surprisingly surprisingly had not not increased increased but but decreased decreased in in frequency, opposing opposing the the view view that that natural natural selection selection in in local local populations populations is is governing governing GFG GFG coevo coevolution 995). The lution (Burdon (Burdon and and Thompson, Thompson, 11995). The authors authors suggested suggested that that individual individual populations populations are are mainly mainly influenced influenced by by genetic genetic drift, drift, extinction, extinction, and and gene gene flow flow among among populations populations within within the the same epidemiological epidemiological region region and and that that GFG GFG coevolution likely to place at population rather coevolution is is likely to take take place at the the meta metapopulation rather than than the the local local population population level. level. In In other other cases, cases, authors authors have have argued argued that that the the stability stability of of host-pathogen possible at host-pathogen interactions interactions is is only only possible at regional regional scale scale levels, levels, whereas whereas at at the one or the local local level, level, one or both both of of the the interactors interactors are are bound bound to to go go extinct extinct (e.g., (e.g., Antonovics ai., 11994; 994; Hess, 996). These Antonovics et etal., Hess, 11996). These examples examples demonstrate demonstrate that that metapopulation plant population metapopulation theory theory forms forms an an inextricable inextricable part part of of plant population biology. biology.
118.7 8.7
CONCLUDING CONCLUDING REMARKS REMARKS Considerable Considerable effort effort has has been been devoted devoted to to defining defining different different types types of of regional regional populations (here used populations (here used in in the the widest widest possible possible sense), sense), for for example, example, source-sink, source-sink, mainland-island, patchy, patchy, and and remnant; remnant; new new suggestions suggestions include "regional "regional mainland-island, ensembles" (Freckleton and ensembles" and and "spatially "spatially extended extended populations" populations" (Freckleton and Watkinson, Watkinson, 2002), metapopulations "in (Eriksson, 11996) 996) or 2002), in in addition addition to to metapopulations "in aa strict strict sense" sense" (Eriksson, or
TOWARD A A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR FOR PLANTS PLANTS 118. 8. TOWARD
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"true Bullock et "true metapopulations" metapopulations" ((Bullock et aI., al., 2002). 2002). This This question question of of classifying classifying dif different ferent types types of of regional regional populations populations would would be be important important if if itit were were shown shown that that different different concepts concepts were were productive productive in in developing developing knowledge knowledge and and insights insights on on the the pattern pattern and and process process of of plant plant regional regional populations. populations. We We agree agree that that it it would would be be erroneous erroneous to to "force" "force" aa diversity diversity of of regional regional dynamics dynamics into into aa narrow narrow set set of of concepts Carey et concepts and and definitions. definitions. For For instance, instance, studies studies on on Vulpia ciliata ((Carey et aI., al., 11995; 995; Watkinson Watkinson et et aI., al., 2000) 2000) suggest suggest that that alternative alternative regional regional models models are are more more suitable suitable than than the the metapopulation metapopulation model. model. However, However, we we are are not not aware aware of of any any alternative alternative conceptual conceptual framework framework that that has has been been even even nearly nearly as as productive productive as as the the metapopulation metapopulation concept concept in in stimulating stimulating studies studies of of regional regional populations. populations. Thus, aa broadening broadening of of the the metapopulation metapopulation concept concept may may be be more more useful useful Thus, than developing developing new new terms terms and and concepts. concepts. Basically, Basically, metapopulation metapopulation models models than focus focus on on colonization colonization and and extinction extinction processes processes and and their their relationship relationship to to the the landscape landscape structure. structure. As As discussed discussed in in this this chapter, chapter, plant plant dynamics dynamics may may be be slow, slow, nonequilibrial, nonequilibrial, determined determined by by landscape landscape history, history, and and it it includes includes plant plant features features that that present present difficulties difficulties to to any any student student working working with with regional regional dynam dynamics: ics: seed seed banks, banks, long long life life spans, spans, and and elusive elusive dispersal dispersal processes. processes. Still, Still, these these difficulties difficulties can can be be incorporated incorporated into into aa conceptual conceptual framework framework based based on on colon colonization/extinction ization/extinction in in relation relation to to landscape landscape structure. structure. The The choice choice of of spatiotem spatiotemporal 999). Typologies poral scale scale is is essential essential (Thomas (Thomas and and Kunin, Kunin, 11999). Typologies of of different different forms forms of of plant regional regional dynamics often often make use of of aa hypothetical hypothetical spectrum spectrum of habitat ((Freckleton Freckleton and of landscapes landscapes with with small small to to large large amounts amounts of of suitable suitable habitat and Watkinson, Watkinson, 2002) 2002) or, or, phrased phrased differently, differently, small small to to large large amounts amounts of of migrants migrants among "true" or among sites sites (Bullock (Bullock et et aI., al., 2002 2002).) . Metapopulations Metapopulations (("true" or "strict" "strict")) are are placed placed at at the the intermediate intermediate portion portion of of this this spectrum. spectrum. However, However, aa broad-sense broad-sense metapopulation concept can can easily incorporate incorporate the the whole spectrum ((Ehrl~n meta population concept whole spectrum Ehrlen and 2003 ) . When amount of suitable habitat and Eriksson, 2003). When there there is is aa small small amount of suitable habitat and and migration also, migration among among sites sites is is small, small, the the temporal temporal scale scale has has to to be be extended; extended; also, slowly plant populations to slowly fluctuating fluctuating regional regional plant populations (where (where colonization colonization appears appears to working in be almost almost nonexistent nonexistent for aa student student working in the the time time scale scale of of an ordinary ordinary research project) project) possess possess regional regional dynamics dynamics likely to to obey obey the the same same colonizacoloniza research tion/extinction dynamics dynamics as more more rapidly fluctuating regional regional populations populations tion/extinction rapidly fluctuating (d. Whittaker Whittaker and and Levin, Levin, 1977). 1 977) . When When there there is is aa large large amount amount of of suitable suitable (cf. habitat and and migration migration among among sites is large, the the spatial spatial scale scale has has to to be extended; extended; habitat plants with with large spatially extended extended populations populations are are also also likely likely to to possess possess a plants large spatially patchiness at at aa larger larger spatial spatial scale scale with with colonization/extinction colonization/extinction dynamics dynamics patchiness similar similar to to more more fine-grained fine-grained populations. populations. Thus, Thus, we we believe believe that that metapopulametapopula tion tion studies studies will will continue continue to to be be the the most most productive productive approach approach to to advance advance the the understanding understanding of of regional regional plant plant population population systems. systems.
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19o LONG-TERM LONG-TERM STUDY OF OF A STUDY PLAN T--PATH PATH OGEN O G EN PLANT META PO PULATION M ETAPO PU LATI O N Janis Antonovics Antonovics
19.1 1 9. 1
INTRODUCTION INTRODUCTION Although there has been a long-standing recognition that that the the numerical and and gene inter gene frequency dynamics dynamics of of natural populations may may be affected by the the inter9 3 1 ; Levins, 1969; 1 969; connectedness of of populations on aa regional scale (Wright, (Wright, 11931; MacArthur and 1 967), it is only since the early 1980s 1 980s that that attention attention has has MacArthur and Wilson, 1967), been given to the explicit study of interconnected interconnected sets of populations populations and and to to the the exploration of of the consequences of spatially explicit models theoretical exploration models for ecological and genetic processes (Silvertown and Antonovics, 2001). 200 1 ) . In the the ecological context that migration context of field field studies, studies, there there is also increasing recognition that populations and local extinction extinction and recolonization are among interconnected interconnected populations rather than than the exception in natural populations (Gilpin ( Gilpin and and Hanski, the rule rather 1991; 1 9 9 1 ; Hanski Hanski and Gilpin, 1996). 1 996). Early metapopulation models assumed simplisimpli fied within population population dynamics driven largely by the effects of colonization, migration, and migration, and extinction (Levins, 1969; 1 969; Caswell, 1978). 1 978). More More recently, with with of increased has been possible to to explore explore the the the advent advent of increased computational power, it has of within within population population dynamics on on spatially extended extended systems and and consequences of multiple interconnected interconnected populations populations (Comins ( Comins et et al., ai., 1992; 1 992; Kareiva, 1994). 1 994). in multiple
Ecology, Genetics, and and Evolution Ecology, of Metapopulations Metapopulations of
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Copyright 2004, 2004, Elsevier, Elsevier, Inc. 0-12-323448-4 0-12-323448-4
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The The major major feature feature to to emerge emerge from from theoretical theoretical studies studies of of spatially spatially explicit explicit systems is that that conclusions conclusions regarding regarding equilibrium equilibrium states states and and dynamics dynamics derived derived systems from from single single populations populations are are changed changed drastically drastically when when interactions interactions among among these populations populations are are included. included. With With regard regard to to ecological ecological dynamics, dynamics, the the bestbest these known known conclusion conclusion is that that systems systems that that show show locally locally unstable unstable population population dynamics dynamics (such ( such as as would would lead lead to to extinction) extinction) can can be be stabilized stabilized readily readily when when extended extended spatially spatially (Comins (Comins et et al., aI., 1992; 1 992; Antonovics Antonovics et et al., aI., 1994; 1 994; Molofsky Molofsky et et al., aI., 2001). 200 1 ) . Conversely, Conversely, it it has has been been shown shown that that changes changes in in connectedness connectedness of of populations populations can, can, in in and and of of itself, drastically drastically influence influence the the overall overall prevalence prevalence of of a species, without without changes changes in in the the local local dynamics dynamics (Carter (Carter and and Prince, Prince, 1988; 1988; May May and and Anderson, Anderson, 1990; 1 990; Hanski, Hanski, 1991). 1 99 1 ) . With With regard regard to t o evolutionary evolutionary dynamics, dynamics, genetic genetic change change within within populations populations can can also be be stabilized and and allelic allelic diversity can extended periods can be be maintained maintained for for extended periods in spatially spatially explicit explicit models models (Frank, (Frank, 1991). 1 99 1 ) . Many Many of of the the statistics of of among among population population differentiation differentiation are are also explicit consideration consideration of also altered altered by explicit of extinction extinction and and colonization colonization processes processes (McCauley, (McCauley, 1993). 1 99 3 ) . Metapopulation Metapopulation structure structure can can enhance enhance the the importance importance of of or group selection by by altering the local frequency frequency of of phenotypes phenotypes kin group selection altering the kin selection or (Gilpin, ( Gilpin, 1975; 1 9 75 ; McCauley McCauley and and Taylor, 1997; 1 997; O'Keefe O 'Keefe and and Antonovics, Antonovics, 2002). 2002) . Short-term off a metapopulation off population Short-term studies o metapopulation can lead to estimates o population turnover turnover and and can be used to to parameterize parameterize models that that can can be used as "surro"surro gates" gates" for experimental experimental studies (Antonovics (Antonovics et al., aI., 1998). 1 99 8 ) . Even single season studies of metapopulations provide useful data for for assessing distance metapopulations can provide habitat occupancy to dependence and and size size dependence of habitat occupancy and and as as aa guide guide to conservation decisions (Hanski, (Hanski, 1991). 1 9 9 1 ) . However, whether whether a metapopulation metapopulation is itself stable can only be determined if there data on the state of there are historical data the the system system at at some some point point in in the the past past or or by long-term studies studies..
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Fig. 19.1 19.1 Diagrammatic Diagrammatic map map of of the the census censusarea areashowing showing the the roads roadsalong along which which populations populations Fig. were censused censused (gray) (gray) and and position position of of roadside roadside segments segments that that had had healthy healthy (small (small dots) dots) or or dis diswere eased (large (large dots) dots) populations populations at at some some time time during during the the census. census. Note Note that that the the scale scale results results in in an an eased apparent apparent overlap overlap of of populations populations that that are are often often separated. separated. Dotted Dotted lines linesseparate separatethe the four four "repli "replicate" cate" areas areasidentified identified in in the the analyses. analyses.Populations Populations were were not not censused censusedalong along Route Route460, 460, which which isis aa major major highway; highway; however, however, only only rarely rarely was was the the occasional occasional plant plant seen seen along along this this highway. highway.
119. 9. PLANT-PATHOGEN PLANT-PATHOGENMETAPOPULATION METAPOPULATION
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We We have have been been studying studying anther-smut anther-smut disease disease caused caused by by the the fungal fungal pathogen pathogen
Microbotryum Microbotryum violaceum violaceum ((= Ustilago violacea) in in several several hundred hundred popula popula14 yr tions tions of of the the plant plant Silene latifolia ((= S. alba) for for 14 yr in in the the region region of of =
=
Mountain 9 . 1 ) . This Mountain Lake Lake Biological Biological Station Station in in western western Virginia Virginia (Fig. (Fig. 119.1). This chap chapter ter reports reports the the results results of of these these studies studies and and discusses discusses the the factors factors that that contribute contribute to the the overall overall stability stability of of this this metapopulation metapopulation system. system. to
11 9.2 9.2
THE STUDY THE STUDY SYSTEM SYSTEM
Life Life Cycles Cycles of of Host Host and and Pathogen Pathogen
Silene latifolia, or or white white campion, campion, is is aa short-lived short-lived perennial perennial herb herb native native to to Europe Europe commonly commonly found found in in ruderal ruderal habitats habitats throughout throughout the the northern northern regions regions of of the the United United States States and and in in upland upland areas areas farther farther south. south. Infection Infection by by M. violaceum results results in in the the plant plant producing producing anthers anthers that that release release fungal fungal spores spores rather than than pollen. pollen. In In S. latifolia, which which is is dioecious, dioecious, in in addition addition to to infecting infecting rather the anthers in males, the pathogen induces induces the produce sta the anthers in males, the pathogen the female female flowers flowers to to produce stamens bear diseased diseased anthers. ovary is aborted and sterile, although mens that that bear anthers. The The ovary is aborted and sterile, although it it is visible as disease therefore is still still visible as aa rudimentary rudimentary structure. structure. The The disease therefore has has aa large large fit fitness and diseased ness effect effect by by sterilizing sterilizing the the host, host, and diseased plants plants are are identified identified easily easily in in the their dark-smutted slow" dis the field field by by their dark-smutted anthers. anthers. Anther Anther smut smut is is aa relatively relatively ""slow" disease period. The ease with with aa long long latent latent period. The pathogen pathogen does does not not convert convert existing existing flow flowers state, but into very young developing ers into into aa smutted smutted state, but grows grows into very young developing flower flower buds, buds, which which are are then then converted converted into into smutted smutted flowers. flowers. This This process process generally generally takes takes no no less less than than 33 weeks, weeks, and and sometimes sometimes more more than than 66 weeks weeks from from initial initial infection. infection. Because Because S. latifolia in in Virginia Virginia flowers flowers from from mid-May mid-May until until early early October, October, there between one one and there are are probably probably between and three three fungal fungal generations generations per per flowering flowering season, average life span of season, depending depending when when infection infection first first takes takes place. place. The The average life span of aa plant that flowers is see later) plants may plant that flowers is ca. ca. 2 2 yr yr ((see later).. Initially, Initially, infected infected plants may be be par partially tially diseased, diseased, but but the the disease disease soon soon becomes becomes systemic. systemic. The The disease disease persists persists between between seasons seasons inside inside the the overwintering overwintering rosette rosette of of the the host host plant. plant. The The disease disease is is transmitted transmitted when when pollinators pollinators move move from from flower flower to to flower. flower. Because Because pollinators pollinators adjust adjust flight flight distances distances to to compensate compensate for for plant plant density, density, transmission on the the frequency and not not the transmission at at moderate moderate plant plant densities densities depends depends on frequency and the density pollin density of of infectious infectious individuals, individuals, whereas whereas at at very very high high densities, densities, when when pollinators ators become become limiting limiting per per capita, capita, transmission transmission rates rates decline decline (Alexander (Alexander and and Antonovics, 992; Antonovics aI., 11995). 995). Although Antonovics, 11992; Antonovics et et al., Although the the pathogen pathogen is is actually actually vector vector transmitted, transmitted, the the frequency-dependent frequency-dependent nature nature of of the the transmission transmission and and the the expression expression of of the the disease disease in in the the sexual sexual organs organs of of the the adult adult plants plants result result in in strong strong parallels parallels between between the the biology biology of of this this host-pathogen host-pathogen system system and and other other sexually 995; Lockhart 99 6 ) . sexually transmitted transmitted diseases diseases (Kaltz (Kaltz and and Schmid, Schmid, 11995; Lockhart et et aI., al., 11996). There There iiss substantial substantial genetic genetic variation variation iinn S. latifolia for for disease disease resistance, resistance, and and most most populations populations are are aa mixture mixture of of genotypes genotypes that that range range from from being being almost almost completely 9 89; Alexander completely resistant resistant to to completely completely susceptible susceptible (Alexander, (Alexander, 11989; Alexander et aI., 11993; 993; Biere 995). Although et al., Biere and and Antonovics, Antonovics, 11995). Although resistance resistance has has aa high high her heritability aI., 11993), 99 3 ) , the itability (Alexander (Alexander et et al., the precise precise genetics genetics underlying underlying the the resistance resistance is not not known. known. Additionally, Additionally, large large fitness fitness costs costs are are associated associated with with resistance resistance in in is the the absence absence of of the the disease. disease. More More resistant resistant plants plants flower flower later later in in the the season season and and
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produce fewer fewer flowers flowers (Alexander, (Alexander, 1989; 1 989; Biere Biere and and Antonovics, Antonovics, 1995). 1 995). produce Unexpectedly, the the fungus fungus appears appears to to be be relatively relatively uniform uniform with with regard regard to to its its Unexpectedly, pathogenicity, and and therefore therefore this this host-pathogen host-pathogen system system does does not not follow follow the the pathogenicity, classical gene-for-gene gene-for-gene scenario scenario (Jarosz (Jarosz and and Burdon, Burdon, 1991). 1 99 1 ). Whether Whether this this is is classical because the the disease disease has has been been recently recently introduced introduced into into the the United United States States from from because Europe and and has has gone gone through through aa bottleneck bottleneck is is not not known known (see ( see Section Section 19.4). 1 9.4) . Europe All M. violaceum violaceum oonn S. latifolia latifolia iiss host host specific specific in in the evidence evidence indicates indicates that that M. All the the United United States, States, although although aa recent recent host host shift shift to to S. vulgaris vulgaris has has been been observed observed the (Antonovics et et al., aI., 2002). 2002) . We have found found anther anther smuts smuts on on two two other other native native (Antonovics We have virginica and and S. caroliniana). caroliniana). However, However, species in in the the southeast southeast United United States States (S. virginica species these anther anther smuts smuts are are phylogenetically phylogenetically and and chromosomally chromosomally quite quite distinct distinct from from these the one one on on S. latifolia latifolia (Perlin, (Perlin, 1996; 1 996; Perlin Perlin et et al., aI., 1997). 1 997). There There is is no no evidence evidence the for any any cross-species cross-species transmission transmission with with the the native native species. species. Moreover, Moreover, for virginica in in this this Antonovics et et al. ai. (1995b) ( 1 995b) showed showed that that the the anther anther smut smut on on S. virginica Antonovics area is is isolated isolated reproductively reproductively from from the the anther anther smut smut on on S. latifolia. latifolia. area
Study and Census Census Methods Study Area Area and Methods
latifolia is is a is largely confined to to area, S. latifolia In the the study study area, a ruderal ruderal species species that that is largely confined In roadsides. us to access to to many roadsides. Its Its roadside roadside distribution distribution allows allows us to gain gain rapid rapid access many popu populations over large area (25 km km from north to to south and 30 30 km km from east to to lations over aa large area (25 from north south and from east west), while at the same same time time being being confident that we we are are missing missing very very few west), while at the confident that few populations. populations. Because the is distributed in patches patches of of differing spacing, Because the plant plant is distributed in differing sizes sizes and and spacing, which may coalesce coalesce or separate due colonization and extinction events, we which may or separate due to to colonization and extinction events, we do population in themselves but do not not define define aa population in terms terms of of the the patches patches themselves but count count numbers numbers of of diseased diseased and and healthy healthy individuals individuals within within contiguous contiguous 40-m 40-m segments segments of 994). Therefore, Therefore, in terms, we of roadsides roadsides (Antonovics (Antonovics et et aI., al., 11994). in formal formal terms, we collect collect data data on on aa one-dimensional one-dimensional grid grid system system at at aa local local scale, scale, but but at at aa larger larger scale scale the the topology area. Distances topology of of this this grid grid follows follows the the pattern pattern of of the the roads roads in in the the area. Distances on on curves curves are are estimated estimated on on the the right-hand right-hand side side of of the the road road in in the the direction direction that that the the census (unusual trees, census is is being being made. made. Local Local landmarks landmarks (unusual trees, driveways, driveways, telephone telephone poles, etc poles, etc.). ) are are used used to to demarcate demarcate each each segment. segment. We diseased and We have have counted counted the the number number of of diseased and healthy healthy individuals individuals within within each 9 88. The each roadside roadside segment segment since since 11988. The main main census census is is done done prior prior to to seed seed dispersal dispersal in in June, June, and and aa recensus recensus in in August August is is restricted restricted to to checking checking aa much much smaller smaller subset subset of of the the populations populations that that have have been been recorded recorded as as extinct extinct or or that that have been been recorded recorded as as having having lost lost the the disease. disease. Although Although we we make make no no attempt attempt have to to map map individuals individuals within within aa segment segment to to aa precise precise location, location, we we note note the the loca location (approximate (approximate distance distance from from start start of of grid grid unit unit and and distance distance from from edge edge of of tion road) of of either either healthy healthy or or diseased diseased individuals individuals when when there there are are very very few few in in aa road) grid unit; unit; this this helps helps us us relocate relocate those those individuals individuals in in subsequent subsequent censuses censuses and/or and/or grid confirm confirm their their absence. absence. Our Our census census is is therefore therefore simple simple and and rapid; rapid; field field work work can can be be completed completed by by three three crews crews of of two two to to three three people people in in less less than than 11 week. week. The one-dimensional one-dimensional grid grid units units of of 40 40 m m include include perhaps perhaps one one or or two, two, but but not not The many, "genetic "genetic neighborhoods" neighborhoods" (i.e., (i.e., areas areas within within which which genetic genetic exchange exchange is is many, essentially essentially random) random) as as estimated estimated from from spore, spore, pollen, pollen, and and seed seed dispersal dispersal distances 990). They distances (Alexander, (Alexander, 11990). They may may include include several several distinct distinct patches patches of of S. latifolia and and sometimes sometimes these these patches patches are are contiguous contiguous between between grid grid units. units.
PLANT-PATHOGEN METAPOPULATION METAPOPULATION 119. 9. PLANT-PATHOGEN
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However, However, rarely rarely is is there there aa continuous continuous "swath" "swath" of of S. latifolia latifolia that that spans spans more more than two two grid grid units. units. By By analyzing analyzing field field data data based based on on pooling pooling two, two, four, four, or or than eight adjacent adjacent segments, segments, we we have have shown shown that that the the patterns patterns of of disease disease incidence incidence eight are remarkably remarkably robust robust over over aa grid grid scale scale of of 40-160 40-160 m m (Antonovics (Antonovics et et aI., al., 11998). are 99 8 ) . Throughout Throughout the the study study w wee take take pains pains not not ttoo disturb disturb the the system system by by our our own own activities activities during during the the census. census. Because Because the the flowers flowers usually usually close close before before midday, midday, we :30 and and 111:00 1 :00 A.M., we census census between between 55:30 A.M., during during which which time time we we can can determine determine the disease disease status status visually visually without without touching touching the the plants plants or or trampling trampling on on the the the sites. sites. In In this this way way we we avoid avoid becoming becoming disease disease dispersal dispersal agents agents ourselves. ourselves. Not Not all all populations populations of of S. latifolia latifolia occur occur at at roadsides. roadsides. Some Some populations populations also also occur on on waste waste ground ground or or along along field field edges edges away away from from the the actual actual road. road. There There occur are relatively relatively few few so-called so-called "off-road "off-road sites" sites" (on (on average average 5.9 5.9 % % of of occupied occupied grid grid are units number of plants in these sites units in in any any 11 yr) yr) but but the the number of diseased diseased and and healthy healthy plants in these sites was was also also recorded. recorded. Sketch Sketch maps maps were were used used to to identify identify these these sites. sites. However, However, because very variable, they were because the the area area of of these these sites sites is is very variable, they were not not included included in in the the following analyses. following analyses.
General General Characteristics Characteristics of of the the Silene-Microbotryum Silene-Microbotryum Metapopulation Metapopulation It is is clear clear that that our our system system does does not not fit fit the the simple simple conceptualization conceptualization of of It metapopulations 1 96 9 ) (uniform metapopulations presented presented by by Levins Levins ((1969) (uniform populations, populations, no no distance distance dependence, dependence, instantaneous instantaneous within within population population dynamics) dynamics).. The The popu populations are are very very different different in in size, size, dispersal dispersal is is limited, limited, and and within within population population lations dynamics dynamics is is important important relative relative to to the the annual annual time time scale scale of of the the study. study. Moreover, Moreover, as in is not not possible as in many many plant plant metapopulations, metapopulations, it it is possible to to define define "suitable "suitable habi habidiscontinuity ((see see Chapter Chapter 18 1 8 for tats" of tats" of S. latifolia latifolia by by aa clear clear environmental environmental discontinuity for a the metapopulation as applied applied to to plants). plants) . Previous a discussion discussion of of the metapopulation concept concept as Previous studies shown that the host host and and pathogen studies have have shown that colonization colonization and and extinction extinction of of the pathogen populations frequent (Antonovics 994, 11998; 99 8 ; Thrall Thrall and and populations are are frequent (Antonovics et et aI., al., 11994, Antonovics, 995; see Antonovics, 11995; see also also results) results).. These These colonization colonization events events enhance enhance the the degree of genetic populations (McCauley (McCauley et 995). degree of genetic differentiation differentiation among among populations et aI., al., 11995). with the the disease disease The growth rate rate of of healthy healthy populations populations is is density density dependent, dependent, with The growth having negative effect et al., aI., 1998). 1 99 8 ) . In In having aa negative effect on on population population growth growth (Antonovics (Antonovics et particular, high high levels levels of disease shift population growth from positive positive to to particular, of disease shift population growth rates rates from negative values (Antonovics et et al., The impact of the disease on on popunegative values (Antonovics aI., 11998). 99 8 ) . The impact of the disease popu lation declining population lation extinction extinction is is gradual; gradual; the the disease disease results results in in aa declining population growth rate, rate, and and aa small small population size in in turn turn presages presages an an increased increased probprob growth population size ability of extinction. extinction. Using models, Antonovics Antonovics (1999) ( 1 999) showed showed ability of Using simulations simulations models, that that the the presence presence of of the the pathogen pathogen can can more more than than halve halve the the number number of of occupied occupied segments in segments in the the metapopulation metapopulation as as aa whole. whole.
19.3 1 9.3 LONG-TERM LONG-TERM TRENDS TRENDS Analysis In the the first first year, year, 1988, 1 9 8 8 , data data were were gathered gathered on on aa 0.1-mile O . I -mile (ca. (ca. four four grid grid units) units) In scale, and and no no recensus recensus was was carried carried out. out. Therefore, although data data from from this this first first scale, Therefore, although year were were valuable valuable in in indicating indicating the the high high rate rate of of turnover turnover in in the the populations populations year
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JANIS ANTONOVICS JANIS
and were were aa stimulus stimulus for for embarking embarking on on the the study, study, we we did did not not use use these these data data and directly in in the the statistical statistical analysis analysis of of population population trends trends presented presented here. here. directly To assess assess whether whether any any long-term long-term changes changes were were general general to to the the census census as as aa To whole, we we divided divided the the census census region region into into four four areas areas representing representing different different valval whole, ley systems systems and and separated separated by by high high elevation elevation areas areas (Fig. (Fig. 19.1). 1 9 . 1 ). Area Area 11 was was the the ley valley region region of of Big Big Stony Stony Creek; Creek; Area Area 22 was was the the lowland lowland area area and and foothills foothills in in valley the New New River River Valley; Valley; Area was Clover Clover Hollow Hollow and and Route 700 up up to to the the the Area 33 was Route 700 Biological Area 44 was Station; and and Area was Maggie Maggie Valley Valley and and the the area area east east of of Biological Station; Simmondsville, Route Route 42. 42. Simmondsville, The scope scope of of the the census census changed somewhat with circumstances. Thus Thus The changed somewhat with circumstances. access to to one one section section of of the the census census was was denied denied by by the the land land managers managers in in 1994 1 994 access ( excluded nine units either either side side of of the the road). road ) . In In Area Area 4, 4, diseased diseased sites sites (excluded nine grid grid units present in in the the most most northerly northerly region region were were the the result result of of artificial artificial introduction introduction present of the the disease disease from from aa spore spore dispersal dispersal experiment; experiment; this this area area was was therefore therefore not not of included in in analyses analyses of of disease disease parameters. parameters. In In 1998, 1 99 8 , we we included included aa new new secsec included tion south of Area Area 4 4 when when this this became became the the focus focus of related demographic demographic tion south of of related studies. Disease was was present in this this region. region. However, However, in in order order to to avoid avoid pospos studies. Disease present in sible confounding effects, analyses analyses of trends were only sible confounding effects, of the the long-term long-term trends were based based only on data data from from the the roadside roadside segments that were were censused censused throughout throughout the the on segments that whole study period. whole study period.
Host Host and and Pathogen Pathogen Occurrence Occurrence At regional level, level, the of aa species species can can be be assessed in terms terms of of At aa regional the abundance abundance of assessed in the number number of populations as as well well as as the average number number of of individuals individuals within the of populations the average within populations. Because the census was based on populations. Because the census was based on aa grid grid system system of of roadside roadside seg segments, measured the of ments, we we measured the number number of of host host populations populations in in terms terms of of number number of segments population size segments occupied occupied and and population size in in terms terms of of the the number number of of individuals individuals within analyses did within each each segment. segment. The The number number of of grid grid segments segments used used in in the the analyses did not not change, and regional abundance change, and therefore therefore the the former former is is aa measure measure of of regional abundance and and the the latter measure of (at the latter is is aa measure of local local abundance abundance (at the segment segment scale). scale). We We measured measured disease abundance regional disease abundance as as the the fraction fraction of of segments segments occupied occupied by by S. latifolia latifolia regional that were were diseased (we refer this as disease incidence") local abun that diseased (we refer to to this as ""disease incidence") and and the the local abundance dance as as the the fraction fraction of of individuals individuals that that were were diseased diseased within within each each occupied occupied segment segment (we (we refer refer to to this this as as "disease "disease prevalence" prevalence").). (Fig. 119.2A) 9.2A) did The fraction fraction of of segments segments occupied occupied by by S. latifolia latifolia (Fig. did not not change change The significantly there was year interac significantly overall overall (P (P < < 0.27), 0.27), but but there was aa significant significant area* area '" year interaction 1 9, tion (P (P < < 0.0001 0.0001).) . In In Area Area 11 the the occupancy occupancy declined declined significantly significantly (P (P < < 0.00 0.0019, bb = 0.003 8 ) , while increased in = -0.0038), while it it increased in Areas Areas 2 2 and and 33 (P (P < < 0.015, 0.015, bb = = 0.0029 0.0029 and and P P< < 0.0089, 0.0089, b = = 0.0020). 0.0020). There There was was no no significant significant change change in in Area Area 4. 4. 9.2B) declined The The average average number number of of S. latifolia latiflolia within within each each segment segment (Fig. (Fig. 119.2B) declined markedly 1, b = markedly overall overall (P (P < < 0.000 0.0001, = - 00.01 . 0 1 112, 2 , 10glO log10 scale). scale). The The decline decline occurred occurred in 8 8 ), but in all all four four areas, areas, significantly significantly so so in in Areas Areas 1-3 1-3 (P (P all all < < 0.00 0.0088), but not not in in Area Area 4 .13). 4 (P (P < < 00.13). The The fraction fraction ooff S. latifolia latifolia segments segments that that were were diseased, diseased, oorr "disease "disease inci incidence, 9.3A) declined 1; regression dence,"" (Fig (Fig 119.3A) declined significantly significantly overall overall (P (P < < 0.000 0.0001; regression coefficient coefficient b = = - 00. .00115533,, arcsin arcsin square square root root transformed transformed data data).) . Although Although dis disease ease incidence incidence declined declined in in Areas Areas 1-3, 1-3, the the rate rate of of decline decline differed differed among among the the areas site interaction, 1 ) . Area areas (year" (year'~site interaction, P P< < 0.000 0.0001). Area 33 was was particularly particularly interesting interesting in 995 in that that it it showed showed an an initial initial increase increase in in disease disease incidence, incidence, peaking peaking in in 11995
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Fig. (A) Fig. 1 1 99.2 .2 (A) Fraction Fraction of of roadside r o a d s i d e segments s e g m e n t s occupied o c c u p i e d by by S. S. latifolia latifolia aand n d (S) (B) average a v e r a g e num number ber of of S. latifolia latifolia individuals individuals within within each each occupied occupied segment s e g m e n t in in each each year year for for the the four four areas areas of of the metapopulation. the metapopulation.
when 5 % of when nearly nearly 335% of the the populations populations were were diseased, diseased, followed followed by by aa rapid rapid decline. decline. Three Three subareas subareas were were identified identified within within this this area area on on the the basis basis of of sepa separation ration by by long long runs runs of of unoccupied unoccupied segments. segments. All All three three subareas subareas showed showed aa sim sim1 990s and ilar ilar pattern pattern with with disease disease incidence incidence peaking peaking III in the the midmid-1990s and then then declining Fig. 119.4). 9.4) . declining ((Fig.
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Fig. (A) Fraction of diseased (disease F | g . 11 99.3 .3 (A) Fraction of S. latifolia latifolia populations populations that that were were diseased (disease incidence) incidence) and and (8) (B) fraction fraction of of individuals individuals that that were were diseased diseased (disease (disease prevalence) prevalence) within within each each diseased diseased popula population tion for for three three areas areas of of the the metapopulation. metapopulation.
The The fraction fraction of of individuals individuals that that were were diseased, diseased, or or "disease "disease prevalence," prevalence," within Fig. 119.3B) 9.3B) increased within each each diseased diseased population population ((Fig. increased significantly significantly overall overall (P < 5 1 , arcsin < 0.0042, 0.0042, b = = 0.00 0.0051, arcsin square square root root transformed transformed data data).) . All All areas areas showed " year interaction showed an an increase increase in in disease disease prevalence prevalence and and the the area area':year interaction
479 419
PLANT-PATHOGEN METAPOPULATION METAPOPULATION 119. 9. PLANT-PATHOGEN
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Fig. Disease incidence and disease three different different sections sections of Fig. 11 9.4 0.4 Disease incidence and disease prevalence prevalence in in three of Area Area 3. 3. Incidence Incidence is is the the fraction fraction of of roadside roadside segments segments occupied occupied by 5. S. latifolia latifolia that that contained contained at at least least one one diseased diseased plant, plant, and and prevalence prevalence describes describes the the fraction fraction of of plants plants within within each each diseased diseased segment segment that that were were diseased. diseased.
JANIS JANIS ANTONOVICS ANTONOVICS
480 480
approached (P < . 06 1 ) . The approached significance significance (P < 00.061). The increase increase was was individually individually signifi significant only (P < .0 1 1 ) . The absolute number number of dis cant only in in Area Area 1I (P < 0.0005, 0.0005, b = - 00.011). The absolute of diseased plants significantly (P (P < 0.123) eased plants per per segment segment decreased decreased significantly < 0.0037, 0.0037, b = = -0.123) and year interaction and the the area" area*year interaction was was not not significant significant (P < < 0.21 0.21).) . Within Within the the subareas subareas ooff Area Area 33 disease disease prevalence prevalence was was positively positively but but nonsignifi nonsignificantly incidence in cantly correlated correlated with with incidence in two two subareas subareas ((rr = 0.19, 0.19, 0.38; 0.38; P < < 0.51, 0.51, 0.17), 0.17), while while in in the the other other area area they they were were negatively negatively and and nearly nearly significantly significantly correlated correlated (r = 0.50, P < (r -0.50, < 0.069). 0.069).
Host Host Colonization Colonization and and Extinction Extinction A A host host colonization colonization was was identified identified as as the the presence presence of of aa population population in in aa roadside segment after year when when no plants were were seen seen in roadside segment after aa year no plants in that that segment; segment; the the host colonization measure that host colonization rate rate is is therefore therefore aa compound compound measure that includes includes recruit recruitments plants that remained vegetative ments from from the the seed seed pool, pool, recruitment recruitment of of plants that had had remained vegetative for whole year, immigration from for aa whole year, and and immigration from other other sites. sites. We We calculated calculated the the colonization colonization rates rates of of the the host host S. latifolia latifolia as as the the number number of of new new populations populations at at time time t per per existing existing population population at at time time t -- 11.. This This "per "per capita" colonization rate capita" colonization rate does does not not take take into into account account the the number number of of empty empty segments available these were 1 989: segments available for for colonization, colonization, as as these were extremely extremely numerous numerous ((1989: 645 1 , 11990-2002: 990-2002: 6 6 1 6-6694) and 6451, 6616-6694) and did did not not vary vary appreciably appreciably with with changes changes in in host per unoccupied host occupancy. occupancy. Calculations Calculations on on aa ""per unoccupied segment" segment" basis basis (i.e., (i.e., equi equivalent "c" in 969) valent to to Levins' Levins' "c" in the the canonical canonical metapopulation metapopulation model, model, Levins, Levins, 11969) did did not not change change the the results results appreciably. appreciably. We We included included both both healthy healthy and and diseased diseased populations as populations as sources sources because because the the latter latter also also produced produced seed seed (except (except in in the the very case where where there 00 % disease very rare rare case there was was 1100% disease of of females females and/or and/or males). males). Results Results (Fig. 9 .5A) showed showed that healthy populations (Fig. 119.5A) that the the colonization colonization rates rates of of healthy populations (b = 0.0041, P < declined of the declined over over the the time time period period of the study study (b - -0.0041, < 0.040) 0.040) and and that that the the rate rate of of decline decline was was not not significantly significantly different different in in the the different different areas areas ((area*year area*year interaction interaction P < < 0.44). 0.44). Host Host extinction extinction was was identified identified as as the the absence absence of of aa population population in in aa roadside roadside segment segment after after aa year year when when plants plants had had been been seen seen in in that that segment segment the the previous previous year. apparent" host year. Strictly Strictly speaking, speaking, it it is is an an ""apparent" host extinction extinction rate rate because because it it refers refers to does not preclude the to the the absence absence of of flowering flowering individuals individuals and and does not preclude the persistence persistence of individuals or bank. Generally, of the the population population as as vegetative vegetative individuals or in in the the seed seed bank. Generally, most most plants plants flower flower every every year, year, except except for for very very small small individuals. individuals. When When vegetative vegetative plants were seen, the plants were occasionally occasionally seen, the population population was was not not recorded recorded as as extinct; extinct; however, because vegetative individuals are however, plants plants may may have have been been missed missed because vegetative individuals are not not very (Fig. 119.5B) 9.5B) showed very conspicuous. conspicuous. Results Results (Fig. showed that that the the extinction extinction rates rates of of the the host period of host tended tended to to decline decline over over the the time time period of the the study, study, but but this this decline decline was was not 0 . 0 1 7, P < decline was not significant significant (b = = -0.017, < 0.076 0.076).) . The The rate rate of of decline was not not signifi significantly areas (area ':· year interaction interaction P < 1). cantly different different in in the the different different areas (area*year < 0.2 0.21).
Disease Disease Colonization Colonization and and Extinction Extinction A A disease disease colonization colonization event event was was identified identified as as the the presence presence of of the the disease disease in in aa population of disease had population of S. latifolia after after aa year year when when no no disease had been been seen seen in in that that popu population lation the the previous previous year. year. Disease Disease colonization colonization is is most most probably probably by by immigration, immigration,
119. 9.
PLANT-PATHOGEN PLANT-PATHOGEN METAPOPULATION METAPOPULATION
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Fig. (A) and (B) in each each year F i g . 11 99.5 .$ (A) Colonization Colonization rate rate and (B) extinction extinction rate rate of of S. latifolia latifolia in year for for the the four four areas areas of of the the metapopulation. metapopulation. Colonization Colonization rate rate is is measured measured as as the the number number of of new new populations populations in in the in aa given given year year per per existing existing population population in the previous previous year. year. Extinction Extinction rate rate is is measured measured as as the the number of number of populations populations that that went w e n t extinct extinct in in aa given given year year as as aa fraction fraction of of the the number number of of popu populations lations in in the the previous previous year. year.
482 482
JANIS ANTONOVICS ANTONOVICS JANIS
but or aa plant but the the persistence persistence of of the the disease disease in in aa vegetative vegetative plant plant ((or plant that that was was not not flowering flowering at at the the time time of of census) census) cannot cannot be be precluded. precluded. Across Across season season soil-borne soil-borne transmission transmission and and vertical vertical transmission transmission of of the the disease disease have have never never been been observed. observed. We We calculated calculated the the colonization colonization rate rate of of the the disease disease as as the the number number of of newly newly diseased diseased populations populations at at time time t per per existing existing population population at at time time t - 1i divided divided by by the the number number of of healthy healthy populations populations available available for for colonization colonization in in an an area area (i.e., (i.e., Levins' Levins' "c" "c").). Results 9.6A) showed Results (Fig. (Fig. 119.6A) showed that that the the colonization colonization rates rates of of disease disease declined declined over over the the time time period period of of the the study study (b = = - 00.025, .025, P < < 0.029 0.029)) and and that that the the rate rate of of decline year inter decline was was not not significantly significantly different different in in the the different different areas areas (area" (area*year interaction P < < 0.44). 0.44). action Disease Disease extinction extinction was was identified identified as as the the absence absence of of disease disease in in aa population population that been diseased apparent" extinc that had had been diseased in in the the previous previous year. year. Again Again this this is is an an ""apparent" extinction because the tion rate rate because the disease disease may may have have persisted persisted in in nonflowering nonflowering individuals. individuals. Results 9.6B) showed Results (Fig. (Fig. 119.6B) showed that that the the extinction extinction rate rate of of the the disease disease did did not not (b = .46) and change change over over the the time time period period of of the the study study (b = 0.0094, 0.0094, P < < 00.46) and that that the the extinction extinction rate rate was was not not significantly significantly different different in in the the different different areas areas (area*year (area*year interaction P < < 0.20). 0.20). The The correlation correlation between between disease disease extinction extinction and and colon coloninteraction ization ization rate rate was was not not significant. significant.
Disease Transmission Disease Transmission Rates Rates Disease transmission rates calculated using Disease transmission rates were were calculated using populations populations where where disease disease had had been been present present in in two two successive successive time time intervals intervals so so as as not not to to confound confound the the estimates colonization or likelihood estimates with with disease disease colonization or extinction extinction rates. rates. Maximum Maximum likelihood methods (5) and disease transmission methods were were used used to to estimate estimate the the survival survival rate rate (S) and disease transmission 13 ) for rate ((13) for each each year year by by fitting fitting the the following following model model to to the the data data (and (and mini minirate mizing mizing the the sum sum of of squares squares of of the the log log of of predicted predicted minus minus the the log log of of observed): observed):
Yt+l = S(Yt +
x,(1 -exp(-~3Yt/Nt))
((19.1) 19.1)
where where Xt X t iiss the the number number ooff healthy healthy plants plants in in year year t, t, Yt, Yt, Yt+ Yt+l1 iiss the the number number of of diseased diseased plants plants in in year year tt and and tt + + 11,, and and Nt Nt = = Xt Xt + + Yt. Yr. Note Note that that the the param parameter eter 13 13 represents represents aa within within season season transmission transmission coefficient coefficient (assuming (assuming no no sum summer mortality) mortality) and and 5 S represents represents overwinter overwinter survival. survival. Equivalent Equivalent analyses analyses were were mer also PROC NUN 999) and also carried carried out out using using PROC NLIN in in SAS SAS (SAS (SAS Institute, Institute, 11999) and gave gave identical identical results. results. The The frequency-dependent frequency-dependent transmission transmission model model always always resulted resulted in in aa better better fit fit - I3 Yt)]; than than the the density-dependent density-dependent model model [where [where force force of of infection infection = = 11 - exp( exp(-IBYt)]; the the latter latter also also frequently frequently produced produced unrealistic unrealistic estimates estimates of of 5S (equal (equal to to or or close close to ). A to 11). A good good fit fit was was also also obtained obtained with with aa model model where where the the force force of of infection infection was exp( - I3 Y/N/Nt),, aa model was = = 11 - exp([3Yt/Nt*Nt) model form form appropriate appropriate for for vector-based vector-based trans transmission, but mission, but because because the the relative relative values values of of 5S and and 13 [3 did did not not differ differ much much between between models, we present present the the results results of of the the more more familiar familiar frequency-dependent frequency-dependent model. model. models, we There There was was aa strong strong colinearity colinearity in in the the estimates estimates of of 5 S and and 13, [3, such such that that high high estimates estimates of of 5S were were correlated correlated with with low low estimates estimates of of 13 [3 and and vice vice versa. versa. We We there therefore fore standardized standardized the the survival survival rate rate by by taking taking the the average average over over all all years years and and including this this average average in in the the model model to to estimate estimate 13. 13. Therefore, Therefore, this this estimate estimate in in including effect effect represents represents an an overall overall "cross-season" "cross-season" transmission transmission coefficient coefficient that that is is aa com compound pound of of the the survival survival rate rate of of diseased diseased plants plants and and the the within within season season transmission. transmission.
119. 9.
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Within Fig. F i g . 11 99.7 .7 Within population population disease disease transmission transmission rates rates per per year year for for three three areas areas of of the the metapopulation metapopulation with with diseased diseased populations populations (see (see text text for for details details of of estimation)" estimation).
Results Results showed showed that that the the transmission transmission rates rates of of the the disease disease within within popula populations Fig. 119.7) 9.7) did tions ((Fig. did not not change change significantly significantly over over the the time time period period of of the the study study .015; P .44 ) . Nor area inter (b = 0 0.015; P< < 0 0.44). Nor was was there there any any evidence evidence for for aa year* year*area inter(b = action . 9 3 ) ; regressions action (P (P < < 00.93); regressions for for each each area area were were slightly slightly positive positive but but did did not approach significance, not individually individually approach significance, even even when when an an outlier outlier was was removed removed (1' < 0.56-0 0.56-0.82). (P < .82). There a s no There w was no significant significant relationship relationship between between disease disease transmission transmission rates rates within within populations populations and and disease disease colonization colonization rate rate (correlation (correlation coefficient, coefficient, r = . 8 3 ) . When outlier was removed ((1989 1 9 8 9 estimates), = - 00. . 007, 7, P P < < 0 0.83). When an an outlier was removed estimates), the the (r = . 2 9 ) but (P < . 3 6 ) . The relationship was relationship was positive positive (r = 00.29) but still still not not significant significant (P < 0 0.36). The relationship relationship between between disease disease transmission transmission rates rates within within populations populations and and dis disease extinction (P < ease extinction rate rate was was negative negative (( - 00.26) . 2 6 ) but but not not significant significant (P < 0.40) 0.40) and and was 9 8 9 estimate was essentially essentially unchanged unchanged when when the the 11989 estimate was was removed. removed. The The same same trends trends (greater (greater colonization colonization and and lower lower extinction extinction when when the the disease disease trans transmission mission rate rate was was higher) higher) were were obtained obtained when when the the analysis analysis was was carried carried out out for for each each area area individually, individually, but but these these trends trends were were not not significant. significant.
Weather Data Weather Data Prior 997, weather Prior to to 11997, weather data data at at Mountain Mountain Lake Lake Biological Biological Station Station were were gath gathered ered manually manually and and were were obtained obtained from from the the National National Climate Climate Data Data Center. Center. In In 11994, 994, aa new weather station acquisition was new weather station with with automatic automatic data data acquisition was installed installed and and run correspondence between run by by the the station. station. There There was was aa close close correspondence between weather weather data data (monthly (monthly mean mean temperature, temperature, highest highest temperature, temperature, lowest lowest temperature, temperature, and and
119. 9. PLANT-,PATHOGEN PLANT-PATHOGEN METAPOPULATION METAPOPULATION
485 485
precipitation) precipitation) during during the the 2 2 to to 33 yr yr period period when when both both types types of of data data were were being being gathered. gathered. Therefore Therefore the the two two types types of of data data were were averaged averaged during during this this overlap overlap period 9 8 8 to period and and were were used used to to span span the the period period 11988 to the the present. present. We We investigated investigated aa specific specific set set of of weather weather variables variables that that we we thought thought might might be be related related to to host host and and pathogen pathogen colonization colonization and and extinction, extinction, as as well well as as to to within within population Based on natural history history observations population disease disease transmission. transmission. Based on our our natural observations of of field summers would would decrease field experiments, experiments, we we hypothesized hypothesized that that hot hot dry dry summers decrease dis disease ease transmission transmission and and hence hence disease disease colonization. colonization. We We also also hypothesized hypothesized that that cold cold winters winters and/or and/or unusually unusually cold cold weather weather in in early early spring spring would would increase increase host host extinction extinction rates. rates. For For each each year year of of the the census, census, for for the the summer summer (June, (June, July, July, and and August), August), we we calculated calculated precipitation precipitation and and mean mean daily daily maximum maximum and and minimum minimum temperatures; temperatures; for for the the winter winter (December, (December, January, January, and and February) February) we we calculated calculated mean mean daily daily maximum maximum and and minimum minimum temperatures. temperatures. We We also also calculated calculated the the minimum minimum temperature temperature in in March, March, as as this this represents represents the the incidence incidence of of unusually unusually cold weather weather in in the the early early spring. spring. cold Over Over the the period period of of the the census, census, there there was was aa significant significant decrease decrease in in summer summer (r = 3 ) and daily daily maximum maximum temperatures temperatures (r =-0 .0.57, 57, P < < 0.03 0.033) and an an increase increase in in (r = summer summer and and winter winter minimum minimum temperatures temperatures (r = 0.76, 0.76, P < < 0.0015; 0.0015; r = = 0.56, 0.56, P< < 0.045 0.045).) . Analysis Analysis of of weather weather data data at at Mountain Mountain Lake Lake Biological Biological Station Station from 11972 to 2001 2001 showed showed aa gradual but nonsignificant nonsignificant increase in mean, mean, from 972 to gradual but increase in maximum, 1 , 0.023, 8°C maximum, and and minimum minimum summer summer temperatures temperatures (0.02 (0.021, 0.023, and and 0.01 0.018~ per per year, year, respectively) respectively);; the the decrease decrease in in summer summer maximum maximum temperatures temperatures since since was therefore therefore contrary contrary to to the the longer longer term term trend. trend. Summer Summer precipitation precipitation 11988 98 8 was did did not not change change systematically systematically with with year, year, but but was was correlated correlated negatively negatively with with maximum (r = 3 ) . No maximum summer summer temperatures temperatures (r = - 00.60, .60, P < < 0.02 0.023). No other other weather weather relationships relationships showed showed aa significant significant change change with with year. year. With were not not correlated With aa few few exceptions, exceptions, the the population population parameters parameters were correlated with with weather Host extinction was negatively winter minimum weather data. data. Host extinction was negatively correlated correlated with with winter minimum (r = temperatures .0042) , and disease colonization colonization rate but not temperatures (r = - 00.76, .76, P < < 00.0042), and disease rate ((but not transmission transmission rate) rate) was was significantly significantly negatively negatively correlated correlated with with summer summer mean mean minimum 1 ). A minimum temperature temperature (r = = - 00. .660, 0, P < < 0.03 0.031). A Bonferroni Bonferroni correction correction of of the the P< 30 correlations < 0.05 0.05 criterion criterion for for significance significance (given (given that that 30 correlations were were estimated) estimated) 1 7. Under results results in in aa value value of of P < < 0.00 0.0017. Under this this criterion criterion none none of of the the aforemen aforementioned relationships be deemed deemed significant. significant. tioned relationships would would be Examination prevalence in Examination of of the the change change in in incidence incidence and and prevalence in Area Area 33 where where incidence was was initially initially low low and and then then peaked peaked in in the the midmid-1990s Fig. 119.4) incidence 1 990s ((see see Fig. 9.4) showed showed no no obvious obvious or or even even suggestive suggestive relationship relationship with with the the weather weather variables. variables.
11 9.4 9.4
DISCUSSION DISCUSSION This This study study provides provides clear clear evidence evidence that that the the Silene-Microbotryum Silene-Microbotryummetapopu metapopulation 9 8 8 is "global sta lation that that we we have have been been studying studying since since 11988 is not not in in aa state state of of "global stability. bility."" This This result result came came very very much much aa surprise surprise with with regard regard to to our our ongoing ongoing impressions populations. Indeed, impressions of of the the populations. Indeed, analysis analysis of of the the 1155 yr yr of of data data was was stimu stimulated lated by by an an assessment assessment of of whether whether it it was was "worthwhile" "worthwhile" continuing continuing with with the the census, given needed to our census, given the the resources resources and and effort effort needed to carry carry it it out out every every year year ((our attempts several years attempts several years ago ago to to get get funding funding for for the the study study were were unsuccessful! unsuccessful!).). Year-by-year not give Year-by-year observations observations did did not give us us the the sense sense that that diseased diseased populations populations
486 4 86
JANIS JANIS ANTONOVICS ANTONOVICS
were declining declining in in frequency, frequency, as as every every year year there there were were always always reports reports of of both both were disease extinctions extinctions and and colonizations. colonizations. disease issues are are raised raised by by these these data. data. First, First, what what is is the the proximal proximal cause cause of of Several Several issues the decline? decline? In In particular, particular, is is it it driven driven by by changes changes in in the the external external environment environment the or is is it it intrinsic intrinsic to to the the disease disease dynamics? dynamics? Second, Second, if if it it is is the the latter, latter, is is the the instainsta or bility related related to to the the fact fact that that both both the the host host and and the the disease disease are are relatively relatively recent recent bility introductions into into the the United United States? States? Finally, Finally, is is the the system system moving moving toward toward introductions some eventual eventual equilibrium equilibrium with with host-pathogen host-pathogen coexistence coexistence or or will will the the outout some come be be disease disease extinction? extinction? come It is is well well known known from the crop crop literature that variation variation in in weather weather can can It from the literature that greatly influence the the prevalence prevalence of of disease. disease. However, However, in in the the weather weather data data we we greatly influence analyzed, only only 22 out out of of aa possible possible 30 30 correlations correlations were were significant. significant. While While the the analyzed, decrease in in host-extinction host-extinction rate rate with with increasing increasing winter winter minimum minimum temperatures temperatures decrease is hard hard to to interpret interpret causally, causally, the the increase increase in in disease disease colonization colonization rate rate with with is decreasing summer summer minimum minimum temperatures temperatures is is consistent consistent with with our our own own obserobser decreasing vations that is highest low temperatures temperatures and high vations that disease disease transmission transmission is highest at at low and high humidity (Alexander (Alexander et 1 99 3 ) . These These low low temperatures temperatures are are most most likely likely to humidity et aI., al., 1993). to occur night, which is also also the the period moth visitation visitation occur during during the the night, which is period of of greatest greatest moth (Altizer aI., 1998) 1 9 9 8 ) and therefore the the period period most most likely likely for the long long distance distance (Altizer et et al., and therefore for the transport of spores. transport of spores. Other unrelated to to the the weather weather may also have Other environmental environmental changes changes unrelated may also have had had an relative importance importance is is hard hard to to judge. judge. In In Area 1, elimelim an effect, effect, although although their their relative Area 1, ination of heavily diseased diseased off-road in 1995 1 995 by by extensive reland ination of several several heavily off-road sites sites in extensive relandscaping by a local lime-manufacturing may have have reduced reduced the the scaping by a local lime-manufacturing company company may available disease sources. sources. In In one one part part of Area 2, 2, road road widening widening in in 1990 1 990 elimelim available disease of Area inated five 995 it another two inated five of of six six diseased diseased sites sites and and in in 11995 it eliminated eliminated another two diseased diseased sites sites nearby. nearby. However, However, it it is is doubtful doubtful that that this this had had aa cascading cascading effect effect elsewhere elsewhere in area. The whole region also subject subject to in the the area. The whole region of of the the census census was was also to early early spring spring spraying control gypsy 9 9 8 ) . However, spraying to to control gypsy moth moth (Sharov (Sharov and and Liebhold, Liebhold, 11998). However, because the spraying in this because much much of of the spraying in this area area has has been been with with male male mating mating pheromone whose effect likely to specific to pheromone whose effect is is likely to be be specific to gypsy gypsy moths moths (and (and which which have reached epidemic levels in census area) overall impact have not not reached epidemic levels in the the census area),, the the overall impact on on moth (which are also disease probably been been small. moth pollinators pollinators (which are also disease vectors) vectors) has has probably small. An An alternative alternative explanation explanation for for the the disease disease decline decline is is that that it it is is intrinsic intrinsic to to the the dynamics of host pathogen pathogen system dynamics of the the system system as as aa whole. whole. In In aa simulation simulation of of this this host system (Antonovics 998), the (Antonovics et et aI., al., 11998), the disease disease could could only only be be sustained sustained in in little little over over 50% 50% of of the the runs. runs. We We have have not not reparameterized reparameterized or or reevaluated reevaluated this this model model based based on on more best estimates more recent recent data, data, but but it it is is nonetheless nonetheless interesting interesting that that our our ""best estimates"" based based on on values values from from the the earlier earlier part part ooff this this census census and and from from experimental experimental stud studpredicted that population. ies often often predicted that the the disease disease would would be be lost lost from from the the meta metapopulation. ies Moreover, Moreover, as as the the disease disease was was lost, lost, the the prevalence prevalence of of the the disease disease within within the the remaining remaining populations populations increased, increased, as as we we have have observed observed in in this this study. study. This This is is largely largely because because newly newly founded founded populations populations with with low low levels levels of of disease disease were were no no longer longer being being produced. produced. The The decreasing decreasing disease disease incidence, incidence, the the increasing increasing preva prevalence disease colonization lence within within populations, populations, and and the the declining declining of of disease colonization rate rate observed observed here population. here are are all all consistent consistent with with gradual gradual disease disease extinction extinction in in the the meta metapopulation. In In this this region region of of Virginia Virginia there there is is extensive extensive genetic genetic variation variation in in the the host, host, yet yet no no detectable detectable variation variation in in the the infectiousness infectiousness of of the the pathogen. pathogen. Thus Thus Antonovics Antonovics et 1 998) showed et al. al. ((1998) showed that that if if the the simulation simulation is is carried carried out out with with aa genetically genetically
119. 9. PLANT-PATHOGEN PLANT-PATHOGEN METAPOPULATION METAPOPULATION
487 481
uniform uniform host host population population with with aa resistance resistance that that is is intermediate intermediate between between that that of of the the most most susceptible susceptible and and most most resistant resistant genotypes, genotypes, with with an an exponential exponential 13[3 = 2.00, would persist persist about 2.00, and and aa survival survival of of 0.50, 0.50, then then the the metapopulation metapopulation would about 90% 90% of of the the time. time. (Analysis (Analysis of of census census data data gave gave an an average average value value of of 13 [3 over over all all years years of of 2.89 2.89 and and an an average average survival survival of of 0.55, 0.55, remarkably remarkably close close to to values values used used in in the the ear earlier lier simulations.) simulations.) However, However, when when the the simulation simulation was was carried carried out out with with aa geneti genetically variable host host population, less frequent cally variable population, persistence persistence was was much much less frequent (ca. (ca. 40%). 40%). Introduction Introduction of of the the disease disease into into aa population population led led to to aa rapid rapid local local spread spread of of the the resistance gene populations that resistance gene and and the the generation generation of of resistant resistant populations that were were not not colon colonized ized readily readily by by the the disease. disease. Populations Populations only only become become readily readily available available for for colon colonization ization by by the the disease disease when when the the gene gene for for susceptibility susceptibility increased increased because because of of the the cost % ; Biere 995). cost of of resistance resistance (estimated (estimated to to be be about about 25 25%; Biere and and Antonovics, Antonovics, 11995). In In experimental experimental populations populations of of S. latifolia latifolia where where individuals individuals are are not not replaced replaced over over successive successive years, years, disease disease transmission transmission showed showed an an extremely extremely rapid almost zero individ rapid decline decline to to almost zero within within 2 2 yr, yr, due due to to the the fact fact that that the the only only individuals remaining resistant families uals remaining healthy healthy were were from from genetically genetically resistant families (Alexander, (Alexander, 11989; 989; Alexander 995). Disease Alexander et et aI., al., 11995). Disease prevalence prevalence also also dropped dropped rapidly rapidly in in experimental populations started resistant genotypes experimental populations started with with progeny progeny of of resistant genotypes but but not not in (Thrall and in populations populations started started with with progeny progeny of of susceptible susceptible genotypes genotypes (Thrall and Jarosz, 11994a,b). 994a,b). Moreover, Jarosz, Moreover, detailed detailed demographic demographic studies studies of of extant extant diseased diseased populations 99 0 ) . It populations have have shown shown low low transmission transmission rates rates (Alexander, (Alexander, 11990). It is is there therefore disease colonization fore possible possible that that the the decline decline in in disease colonization rates rates may may be be due due to to an an increased disease resistance resistance in increased level level of of disease in the the metapopulation metapopulation as as aa whole. whole. It It is is relevant relevant to to place place our our metapopulation metapopulation in in aa broader broader geographical geographical and and historical historical context, context, as as this this may may help help with with the the interpretation interpretation of of the the local local changes. changes. In In aa survey survey of of over over aa thousand thousand herbarium herbarium specimens specimens of of S. latifolia latifolia in in the the eastern eastern United United States, States, there there was was no no evidence evidence that that the the plant plant had had been been col collected 9 1 4 (Antonovics aI., 2003 lected south south of of the the Pennsylvania Pennsylvania line line before before 11914 (Antonovics et et al., 2003),), apart 89 6 on apart from from aa collection collection made made in in 11896 on the the Biltmore Biltmore estate estate in in North North Carolina. House was 8 95, and Carolina. Biltmore Biltmore House was opened opened in in 11895, and it it is is likely likely that that the the estate estate imported meadows. The imported seeds seeds from from New New England England for for hay hay or or for for the the meadows. The first first record Virginia was was in 924, and and it not until 93 0s that that collections record in in Virginia in 11924, it was was not until the the 11930s collections in in Virginia Virginia became became frequent. frequent. The The first first record record we we could could find find for for Giles Giles County, County, where of the the metapopulation meta population is is located, 9 3 8 . Therefore, where the the majority majority of located, was was 11938. Therefore, the the weight weight of of the the evidence evidence is is that that the the host host plant plant has has only only been been in in the the Mountain Mountain Lake 80 years. years. Lake area area for for perhaps perhaps less less than than 80 The disease is unknown. Previously, The history history of of the the disease is completely completely unknown. Previously, M. M. vio violaceum laceum had had been been noted noted on on S. caroliniana caroliniana in in Virginia Virginia and and New New York York State State and and 9 8 9 ) , but on on several several species species of of Silene Silene in in the the western western United United States States (Farr (Farr et et aI., al., 11989), but there there is is no no record record of of it it on on S. latifolia, latifolia, even even though though other other fungal fungal diseases diseases are are recorded Farr et aI., 11989). 98 9 ) . None recorded for for this this species species in in the the United United States States ((Farr et al., None of of the the herbarium herbarium specimens specimens we we examined examined were were diseased diseased so so they they did did not not help help resolve resolve the the question question of of the the disease disease origins. origins. The The current current distribution distribution of of S. latifolia latifolia and and M. M. violaceum violaceum in in the the eastern eastern United United States States was was studied studied by by A. A. M. M. Jarosz Jarosz and and E. E. Lyons Lyons (personal (personal communication) communication).. They They found found that that the the disease disease was was largely largely confined (where 116% 6% confined to to the the ridge ridge and and valley valley system system of of western western Virginia Virginia (where ooff 1102 02 populations populations were were diseased). diseased). Further Further iinn the the northeast, northeast, they they only only found found 11 diseased (in Pennsylvania) 69, except diseased population population (in Pennsylvania) out out of of 1169, except for for 33 diseased diseased populations populations on on Nantucket Nantucket Island, Island, Massachusetts. Massachusetts. Diseased Diseased plants plants have have been been
488 488
JANIS JANIS ANTONOVICS
known 9 80s (T. known from from Nantucket Nantucket Island Island since since the the early early 11980s (T. Meagher, Meagher, personal personal communication). communication). In In the the north north central central United United States, States, aa single single diseased diseased plant plant was 8 7 populations sampled. The reason for was found found out out of of 3387 populations sampled. The reason for the the absence absence of of the the disease from unknown. In disease from more more northern northern latitudes latitudes is is unknown. In field field experiments experiments along along aa latitudinal latitudinal gradient, gradient, A. A. M. M. Jarosz Jarosz and and E. E. Lyons Lyons (personal (personal communication) communication) showed showed that that northern northern populations populations were were susceptible susceptible to to disease disease in in their their local local areas, areas, but but that that they they were were also also somewhat somewhat more more resistant resistant than than plants plants derived derived from seeds of relatively susceptible susceptible parent parent from Mountain Lake from seeds of aa relatively from Mountain Lake that that was was used pollination" with used as as aa contro!' control. Artificial Artificial hand hand ""pollination" with spores spores produced produced aa higher higher incidence incidence of of disease disease than than open open visitation, visitation, suggesting suggesting aa shortage shortage of of pollinators pollinators may may limit limit disease disease transmission. transmission. Given Given that that the the host host has has moved moved into into this this part part of of Virginia Virginia only only recently recently and and that that the disease is the disease is near near the the southern southern edge edge of of the the current current range range of of S. latifolia, latifolia, yet yet is is found sporadically in plausible that found sporadically in its its former former range, range, it it is is plausible that we we may may be be seeing seeing the the movement disease "front" movement of of aa disease "front" that that is is following following the the host host as as it it colonizes colonizes new new areas. areas. The The movement movement of of this this disease disease front front may may be be driven driven by by the the evolution evolution of of more more resistant resistant populations populations in in the the wake wake of of the the disease. disease. The The spread spread of of this this disease disease in the the United United States States may may therefore therefore be be analogous analogous to to the the spread spread of of many many other other epi epiin demics. demics. In In animal animal populations, populations, "waves" "waves" of of disease disease spread spread are are often often driven driven by by the the development development of of immunity immunity in in the the wake wake of of the the epidemic, epidemic, but but aa genetic genetic component component to this this immunity immunity has has also been posited posited frequently. frequently. In In the the present present metapopula metapopulato also been tion, major driving tion, this this genetic genetic component component may may be be the the major driving force. force. However, However, the the issue issue of of whether whether the the changes changes we we are are observing observing are are due due to to climatic climatic and and management management changes or or to to intrinsic genetic and and demographic factors cannot be determined determined by by changes intrinsic genetic demographic factors cannot be descriptive or require further descriptive or simulation simulation studies studies alone, alone, but but will will require further experiments experiments and and more studies of individual populations. more directed directed field field studies of individual populations.
20
META M ETAPO PO PULATION PU LATIO N DYNAMICS IN IN DYNAMICS CHANGING CH ANGING ENVIRONMENTS ENVIRONMENTS:: BUTTERFLY PONSES BUTTERF kY RES RESPONSES TO H ABITAT AND AND TO HABITAT CLI MATE CHAN GE CLIM ATE C H ANGE Chris D. Thomas Thomas and Chris and Ilkka Ilkka Hanski Hanski
20.1 20. 1
INTRODUCTION INTRODUCTION A A major major criticism criticism of of the the applications applications of of metapopulation metapopulation models models in in conserconser vation has has been been that that real real metapopulations metapopulations rarely rarely conform conform to to the the assumptions assumptions vation of classic classic theory theory (Harrison, (Harrison, 1991; 1 99 1 ; Harrison Harrison and and Taylor, Taylor, 1997). 1 997). In In metapopumetapopu of lation theory, theory, it it is is usually usually assumed assumed that that the the extinction of aa particular particular local local lation extinction of population generates one more more patch patch of of empty habitat that that is is subsequently subsequently population generates one empty habitat available for for colonization colonization and and that that each each new new colonization colonization removes removes aa previously previously available empty patch patch that that is is no no longer longer available available for for colonization colonization (Chapter (Chapter 4). 4 ) . This This empty assumption is is the the basis basis of of the the stochastic stochastic quasiequilibrium quasiequilibrium between between colonizacoloniza assumption tions and and extinctions. extinctions. An An unusual unusual number number of of extinctions extinctions in in one one generation generation tions
Ecology, Genetics, and Evolution Ecology, of Metapopulations Metapopulations
4489 89
2004, Elsevier, Elsevier, Inc. Copyright 2004, 0-12-323448-4 0-12-323448-4
CHRIS D. THOMAS HAN SKI CHRIS D. THOMAS AND AND ILKKA ILKKA HANSKI
490 490
would would likely likely be be followed followed by by an an excess excess of of colonization colonization events events in in subsequent subsequent generations, generations, and and an an unusual unusual number number of of colonization colonization events events would would likely likely be be followed by generated by followed by an an excess excess of of extinctions. extinctions. However, However, if if extinctions extinctions are are generated by habitat follow an habitat deterioration deterioration and and if if colonizations colonizations follow an improvement improvement in in environ environmental mental conditions, conditions, then then this this feedback feedback is is broken broken and and there there is is no no logical logical reason reason why population should why aa meta metapopulation should exist exist in in any any kind kind of of equilibrium equilibrium (Thomas, (Thomas, 11994a,b). 994a,b) . In case, the In this this case, the metapopulation metapopulation dynamics dynamics of of an an organism organism will will be be superimposed (or fail superimposed upon, upon, and and track track (or fail to to track), track), the the dynamic dynamic distribution distribution of of suitable suitable habitat. habitat. At glance, this At first first glance, this criticism criticism seems seems to to be be extremely extremely serious serious because because most most of population models of the the practical practical applications applications of of meta metapopulation models relate relate to to habitat habitat that populations at that is is changing, changing, in in which which context context meta metapopulations at equilibrium equilibrium might might be real issue issue is be expected expected to to be be particularly particularly rare. rare. However, However, the the real is how how fast fast species tracking changing population species are are tracking changing environments environments and and whether whether meta metapopulation models models can can provide provide insight insight into into these these processes. processes. If If tracking tracking is is fast, fast, the the prob problem lem is is in in understanding understanding and and predicting predicting how how the the environment environment is is changing; changing; if if tracking tracking is is slow, slow, there there is is additionally additionally the the problem problem of of transient transient metapopu metapopulation dynamics responding to lation dynamics responding to the the changing changing environment. environment. This This chapter chapter reviews application of population concepts concepts and reviews the the application of meta metapopulation and models models to to situations situations where environment is persistence of metapopulation where the the environment is changing changing or or the the persistence of the the metapopulation is precarious because because the species occurs close to is precarious the species occurs close to the the extinction extinction threshold threshold ((Chapter Chapter 4). 4). Butterfly populations represent Butterfly meta metapopulations represent excellent excellent systems systems with with which which to to assess assess long-term long-term and and nonequilibrium nonequilibrium dynamics dynamics because because the the quality quality of of historical historical information information on on their their distributions distributions allows allows us us to to be be confident confident whether whether popu populations expanding or declining. Box lations are are expanding or declining. Box 20. 20.11 presents presents aa brief brief history history of of butterfly metapopulation studies. butterfly metapopulation studies. In In some some cases, cases, results results of of past past mapping mapping of of distributions allow us observed changes. distributions allow us to to test test model model predictions predictions against against observed changes. Furthermore, Furthermore, knowledge knowledge of of the the often often quite quite specific specific habitat habitat requirements requirements of of many habitat networks, many butterfly butterfly species species allows allows us us to to define define habitat networks, and and changes changes in in the the structure structure of of such such networks, networks, independently independently of of the the distribution distribution of of the species. We population models models can the species. We find find that that meta metapopulation can have have great great predictive predictive power power in in nonequilibrium nonequilibrium systems systems and and that that they they can can be be particularly particularly useful useful in in enhancing enhancing our our understanding understanding of of the the responses responses of of species species to to landscape landscape and and climate climate change. change.
BOX 20.1
Brief History of Butterfly Metapopulatlon studies
Butterfly biologists developed the concept of "open" and "closed" population structures in the 1 9605 and 1 970s (Ehrlich, 1 961 , 1 965, 1 984; Ehrlich et aI., 1 975; Thomas, 1 984), following in the steps of E.B. Ford who, in the 1 9 30s and 1 9405, observed the sedentary behavior of many butterflies, confining most individuals to their natal habitat patch. The notion of fairly discrete and often small local populations paved the way to considerations of meta populations, or assemblages of such local populations. The first full-fledged butterfly metapopulation study was due to Harrison et al. (1 988), who demonstrated a mainland-island metapopulation structure in the
20. 20. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS ENVIRONMENTS
checkers pot butterfly Euphydryas edith a in California. The first study of the Glanville fritillary in Finland produced evidence for a classic metapopulation, and Hanski et aL (1 994) concluded that "the Melitaea cinxia metapopulation . . . p rovides a contrasting example to the Euphydryas editha metapopulation reported by Harrison et aL (1 988). Unlike the latter case, there is no large "mainland" population in the M. cinxia meta population, and its long-term persistence appears to depend on genuine extinction colonization dynamics." Studies of several British butterflies fou nd that a lmost all local breeding populations occur within a dispersal range of other local populations of the same species, which finding suggested, along with direct evidence of colonizations and extinctions, that metapopulation dynamics were likely to be commonplace (Thomas et aL, 1 992; Thomas and Harrison, 1 992; Thomas a nd Jones, 1 993; Thomas, 1 994a,b). From this point onward, metapopulation studies on butterflies have taken place mostly in Europe. Extinction-colonization dynamics have been researched intensively in several species, but most notably in M. cinxia (Hanski et aL, 1 994, 1 995a,b, 1 996; Kuussaari et aL, 1 998; Hanski, 1 999b; N ieminen et aI., 2004), Plebejus argus (Thomas, 1 99 1 ; Jordan o et aL, 1 992; Thomas and Harrison, 1 992; Brookes et aL, 1 99 7; Lewis et aI., 1 997; Thomas et aI., 1 999a, 2002a), Hesperia comma (Thomas et aL, 1 986, 2001 a; Thomas and Jones, 1 993; Hill et aI., 1 996; Wilson and Thomas, 2002) and Proclossiana eunomia (Baguette and Neve, 1 994; Neve et aI., 1 996a; Petit et aI., 2001 ; Sawchik et aL, 2002; Schtickzelle et aI., 2002) in Europe and E. edith a (Harrison et aL, 1 988; H arrison, 1 989; Thomas et aI., 1 996; Boughton, 1 999; McLaughlin et a I ., 2002) in North America. Studies of these and tens of other species (e.g., Warren, 1 987, 1 994; Settele et aL, 1 996; Gutierrez et aI., 1 999, 2001 ; K nutson et aI., 1 999; Mousson et aI., 1 999; Shahabuddin and Terborgh, 1 999; Baguette et aI., 2002; Bergman and Landin, 2001 ; Bulman, 2001 ; Nekola and Kraft, 2002; Wahlberg et aI., 2002a,b; Wilson et aL, 2002) have shown great variation in metapopulation structure and that dynamics a re nearly as variable within as among species. This latter conclusion u nderscores the pivotal role of landscape structure in influencing spatial dynamics. Nonetheless, these studies have confirmed that the general metapopulation notion provides valuable insight i nto the dynamics and distribution of many, although not a l l, butterfly species at the landscape level. The metapopulation approach can be applied to virtually a l l species that were formerly considered to have "closed" population structures (Thomas, 1 984). Since the mid-1 990s, the emphasis on butterfly meta population studies has been in adding further details and evaluating how robust and useful the approach is under different circumstances. Studies have examined the validity of the major assumptions and processes of metapopulation dynamics, incorporated multispecies patterns and dynamics into the common framework (Lei and Hanski, 1 997; van Nouhuys and Hanski, 1 999, 2002), investigated the evolutionary and genetic dynamics of metapopulations (Neve et aI., 1 996b, 2000; Singer and Thomas, 1 996; Brookes et aI., 1 997; Saccheri et aI., 1 998; Thomas et aI., 1 998; Barascud et aL, 1 999; Keyghobadi et aI., 1 999; Kuussaari et aL, 2000; Nieminen et aL, 200 1 ; Saccheri and Brakefield, 2002), and applied meta population a pproaches at increasingly large scales in relation to conservation and climate change, as described in the main text. Research on butterflies has played an important, and in some cases pivotal, role in the development of the science of meta population biology and i n the application of the metapopu lation approach to conservation. Many of the studies cited here and in this chapter were at least partially motivated by conservation concerns, and this pattern is likely to continue.
491 491
4492 92
CHRIS CHRIS D. D. THOMAS THOMAS AND AND ILKKA ILKKA HANSKI HAN SKI
20.2 HABITAT HABITAT FRAGMENTATION FRAGMENTATION 20.2 Habitat loss loss typically typically results results in in ffragmentation smaller and and more more scattered scattered Habitat r a g m e n t a t i o n- smaller fragments of of habitat habitat than than existed existed formerly. formerly. In In many many cases, cases, the the habitat habitat does does not not fragments immediately all all become become unsuitable, unsuitable, and and populations populations may may be be found, found, for for some some immediately time at at least, least, in in patches patches of of habitat habitat that that have have recently recently become become small, small, more more time isolated, or or both. both. At At equilibrium, equilibrium, many many of of these these patches patches would would be be expected expected to to isolated, be unoccupied unoccupied most most of of the the time time (even (even though though they they may may contain contain perfectly perfectly be suitable habitat) habitat) because because the the rate rate of of extinction extinction of of small small populations populations is is likely likely to to suitable be high high and and the the recolonization recolonization rate rate of of isolated isolated patches patches will will be be low low (Hanski, (Hanski, be 1 994, 1999b; 1 99 9b; Chapter 4 ) . However, However, during during and and immediately immediately following following aa period period 1994, Chapter 4). of fragmentation fragmentation aa species species occupying occupying the the remaining remaining fragments fragments may may show show aa of period of of decline, decline, during during which which the the rate rate of of local local extinction extinction exceeds exceeds the the rate rate of of period recolonization. In In such such situations, situations, the the potential potential contributions contributions of of metapopulametapopula recolonization. tion models models are are to to help help understand understand the the timescale timescale of of decline, decline, the the spatial spatial pattern pattern tion of decline, decline, and and whether whether aa species species will will decline decline to to aa reduced reduced metapopulation metapopulation size size of (restricted distribution) distribution) or or become become completely completely extinct extinct from from the the areas areas where where (restricted fragmentation has has taken taken place. place. Such Such insights insights can can be be extremely extremely important important fragmentation because they they may may provide provide an an understanding understanding of of why why some species continue continue because to decline decline long long after after the the damage damage to to the the environment environment has has taken taken place. place. The The to following two examples lag behind behind habitat loss. following two examples illustrate illustrate how how species species may may lag habitat loss.
Melifaea and the Speed of Metapopulation Decline Decline Melitaea cinxia cinxia and the Speed of Metapopulatton The distribution distribution of of the the Glanville Glanville fritillary, Melitaea cinxia, in northern and The Melitaea cinxia, northern and western western Europe Europe has has become become greatly greatly reduced reduced over over the the past past decades, decades, and and the the species 995; species has has gone gone regionally regionally extinct extinct in in many many areas areas (Hanski (Hanski and and Kuussaari, Kuussaari, 11995; Maes 999; van 999). It Maes and and van van Dyck, Dyck, 11999; van Swaay Swaay and and Warren, Warren, 11999). It is is apparent apparent that that habitat habitat loss loss is is the the primary primary or or even even the the only only significant significant cause cause of of the the decline. decline. 970s (Marttila In In Finland, Finland, M. cinxia went went extinct extinct in in the the mainland mainland in in the the 11970s (Marttila et 990), and land Islands Islands in et ai., al., 11990), and it it now now occurs occurs only only in in the the A Aland in Southwest Southwest Finland Finland (Hanski 995). Luckily (Hanski and and Kuussaari, Kuussaari, 11995). Luckily for for this this butterfly butterfly and and many many other other species species of of insects insects and and plants, plants, land land use use practices practices have have changed changed less less drastically drastically in land Islands in the the A Aland Islands than than in in most most other other parts parts of of northern northern Europe. Europe. Dry Dry meadows meadows with with Plantago lanceolata and and Veronica spicata, the the two two host host plants plants of land, partly of M. cinxia, still still abound abound in in A Aland, partly because because the the general general topography topography with with numerous numerous small small granite granite outcrops outcrops prevents prevents large-scale large-scale agricultural agricultural intensi intensification. fication. At At present, present, the the suitable suitable habitat habitat covers covers ca. 66 km2, km 2, which which is is 0.6% 0.6 % of of the the total total land land area area (Nieminen (Nieminen et et ai., al., 2004) 2004).. Nonetheless, Nonetheless, substantial substantial habitat habitat loss loss has land in has occurred occurred in in parts parts of of A Aland in recent recent decades, decades, as as the the following following example example shows, shows, with with adverse adverse consequences consequences for for the the occurrence occurrence of of the the butterfly. butterfly. Figure land Islands, Figure 20.1 20.1aa shows shows one one network network of of habitat habitat patches patches in in the the A ,~land Islands, with 9 92. Thanks 1 995) detailed with 42 42 patches patches in in 11992. Thanks to to Hering's Hering's ((1995) detailed analysis analysis of of old old aerial aerial photographs photographs and and interviews interviews of of local local people, people, we we know know that that 20 20 yr yr previ previously ously there there had had been been 55 55 distinct distinct patches patches in in this this network, network, and and nearly nearly three three times times more more habitat habitat for for M. cinxia. In In this this case, case, the the area area of of suitable suitable habitat habitat had had declined 1 996) used declined largely largely because because of of reduced reduced grazing grazing pressure. pressure. Hanski Hanski et et ai. al. ((1996) used the 994; Chapter ) , parameterized the incidence incidence function function model model (Hanski, (Hanski, 11994; Chapter 44), parameterized previ previously ously for for M. cinxia, cinxia, to to assess assess the the likely likely metapopulation metapopulation dynamic dynamic consequences consequences
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Aland Islands Fig. 20.1 (a) The extent of suitable habitat for Melitaea cinxia in one part of the 992 the/~land Islands in 11992 (shaded) and 20 yr previously (Hering, 11995). 995), (b) Metapopulation dynamics of M. cinxia in the landscape shown in (a), (a), modeled with the IFM, IFM. The size size of the metapopulation is is shown, shown. (Top) The number of patches and the amount of habitat during the simulation, including the observed reduction in habitat over the 20-yr period documented in (a), 0 replicate pre (a). (Bottom) The equilibrium metapopulation size size (thick line) and 110 predicted trajectories trajectories before, before, during, and after the period of of habitat loss loss (thin lines), lines). (c) Similar results results for aa hypo hypothetical scenario scenario of further loss loss of 50% of the area area of each each of the remaining patches in (b) (from (from Hanski Hanski et aI., al., 11996; 996; Hanski, 999b), Hanski, 11999b).
of of this this particular particular scenario scenario of of habitat habitat loss. loss. Initially, Initially, the the landscape landscape had had so so many many well-connected 1 a ) that well-connected habitat habitat patches patches (Fig. (Fig. 20. 20.1a) that the the butterfly butterfly was was able able to to occupy occupy most most of of them them most most of of the the time. time. The The modeling modeling assumed assumed that that habitat habitat was was lost lost gradually gradually over over aa period period of of 20 20 yr, yr, and and the the equilibrium equilibrium state state of of the the
CHRIS CHRIS D. D. THOMAS THOMAS AND ILKKA ILKKAHANSKI HANSKI
494 494
metapopulation corresponding to to the the prevailing prevailing amount amount of of habitat habitat was was calcu calcumeta population corresponding lated for for each each year. year. This This equilibrium equilibrium is is shown shown by by aa thick thick line line in in Fig. Fig. 20. 20.1b lated 1b ((Bottom). Bottom). Next, Next, 1100 replicate replicate simulations simulations were were run run for for 500 500 yr, yr, with with the the 20-yr 20-yr period period of of habitat habitat loss loss starting starting in in year year 300. 300. The The result result shows shows that that butterfly butterfly dynamics tracked tracked the the changing changing environment environment very very closly: closly: the the predicted predicted dynamics metapopulation metapopulation size size is is very very close close to to the the equilibrium equilibrium during during and and following following the the period of of environmental environmental change, change, with with aa lag lag of of only only aa few few years years (Fig. (Fig. 20. 20.1b). period 1b). this particular particular case case there there was was very very little little time time lag lag iinn the the dynamics dynamics ooff IIn n this the the butterfly butterfly iinn aa changing changing environment, environment, but but iitt would would bbee wrong wrong ttoo assume assume that the the same same result result applies applies to to all all butterflies butterflies and and to to all all environmental environmental changes. changes. that As aa matter matter of of fact, fact, and and as as Fig. Fig. 20. 20.1c clearly shows, shows, even even the the same same species species can can As 1 c clearly respond very very differently differently under under different different environmental environmental conditions. conditions. Figure Figure respond 20.1c is is aa continuation continuation of of the the example example in in Fig. Fig. 20. 20.1b: we now now assume that that 20.1c 1 b: we 992 loses each one one of of the the remaining remaining 42 42 patches patches that that existed existed in in 11992 loses 50% 50% of of each area in in the the next next 20 20 yr. yr. This This amount amount of of habitat habitat loss loss is is drastic drastic enough enough in in its area this particular particular case case for for the the equilibrium equilibrium metapopulation metapopulation size size to to move move all all the the this way to to zero (Fig. (Fig. 220.1c). However, the the model-predicted model-predicted final final extinction extinction of of way 0 . 1 c ) . However, the took aa very time, with the butterfly butterfly metapopulation metapopulation took very long long time, with aa median median time time to to extinction Fig. 20.1c). extinction of of more more than than 200 200 yr yr ((Fig. 20.1c). The The striking striking contrast contrast between between the the two understood in two examples examples in in Fig. Fig. 20.1 20.1 can can be be understood in light light of of the the general general theory theory developed by by Ovaskainen Ovaskainen and and Hanski ((2002) and outlined outlined in in Chapter Chapter 4 4 2002) and ((Section Section 4.4). length of delay in population response 4.4). The The length of the the time time delay in meta metapopulation response to to environmental environmental change change is is expected expected to to be be especially especially long long in in cases cases where where the the new new equilibrium or any equilibrium following following habitat habitat loss loss ((or any other other perturbation perturbation)) is is close close to to the the extinction threshold m as as it it evidently evidently is is in in Fig. Fig. 20. 20.1c. In reality, reality, metapopulaextinction 1c. In metapopula tion extinction extinction is is hastened hastened by by temporal temporal variation variation in in environmental conditions tion environmental conditions ( regional stochasticity; Chapter 4) 4) even in the the absence absence of of further environ (regional stochasticity; Chapter even in further environmental eventual extinction mental changes, and and the eventual extinction would would probably probably take take less time than qualitative result than in the example example in Fig. 20.1c. 20. 1c. Nonetheless, Nonetheless, the the qualitative result and and message remain the the same.
Euphydryas aurinia aurinia and and Euphydryas
Extinction Debt Extinction Debt
The marsh marsh fritillary, Euphydryas aurinia, has declined throughout The has declined throughout Europe, Europe, especially in northern Europe (van Swaay and and Warren, especially northern Europe Warren, 1999). 1 999). In northwest northwest Europe, Europe, E. aurinia is restricted to to rough rough and and mainly moist moist pastures, pastures, where where its host host plant plant Devil's bit bit scabious, scabious, Succisa pratensis, pratensis, grows grows among among tall vegetation. The butterfly has The butterfly has been eliminated completely from from eastern eastern parts parts of of Britain, where agriculture agriculture is is most most intensive, intensive, and and it it only only survives survives as as metapopulations metapopulations of of where small local local populations populations in in the the west. west. Even Even here, here, the the original original extent extent of of suitable suitable small habitat habitat has has been reduced reduced and and fragmented fragmented greatly. was studied studied in in aa 625-km 625-km22 area area in in southwest southwest England England Euphydryas aurinia was in the the county county of of Dorset Dorset in 1981 1 9 8 1 (Warren, (Warren, 1994) 1 9 94) and and again again in 1998-2000 1 998-2000 (Bulman, ( Bulman, 2001). 200 1 ) . In this this landscape, landscape, the the butterfly butterfly shows shows typical "metapopulation "metapopulation patterns." patterns." It It was was found found in only 14 out out of of the the 123 123 habitat habitat patches patches that that were were delimited in in 1998-2000, 1 998-2000, occupying occupying large large patches patches of of high high quality quality (tall (tall vegetavegeta delimited tion) tion) that that contained contained large large quantities quantities of of larval larval food food plants plants and and patches patches that that were were closely connected connected to to other other patches patches also also supporting supporting local local populations. populations. Between Between closely the the two two time time periods, periods, from from 1981 1 9 8 1 to to 1998-2000, 1 998-2000, 10 1 0 populations populations became became
20. 20.
METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS ENVIRONMENTS METAPOPULATION
495 495
extinct, mostly mostly in in small small patches, patches, and and four four empty empty patches patches were were colonized, colonized, three three extinct, of which which were were very very close close to to existing existing populations populations (Bulman, ( Bulman, 2001). 200 1 ) . of Bulman (2001) (20 0 1 ) fitted fitted the the incidence incidence function function model model (IFM; (IFM; Hanski, Hanski, 1994; 1 994; Bulman Chapter 4) 4) to to data data from from the the part part of of the the Dorset Dorset system system that that had had the the highest highest Chapter E. aurinia. aurinia. As As the the patch density density and and that that was was most most heavily heavily occupied occupied by by E. patch butterfly had had not not declined declined in in this this area area between between 1981 1 9 8 1 and and 1998-2000, 1 9 98-2000, it it butterfly was reasonable reasonable to to assume assume in in parameter parameter estimation estimation that that the the metapopulation metapopulation was in this this region region was was at at quasiequilibrium quasiequilibrium (Bulman, ( Bulman, 2001; 200 1 ; see see Chapter Chapter 55 for for the the in of the the equilibrium equilibrium assumption). assumption). The The model model with with parameter parameter values values importance importance of thus estimated was then then applied applied to to 10 10 independent independent patch patch networks networks within within thus estimated was of 16 16 km km22 scattered across Britain, Britain, as as well well as as to to two two other other networks networks aareas r e a s of scattered across in in Dorset. Dorset. Half Half of of the the 12 12 networks networks were were centered centered on on individual individual E. E. aurinia aurinia populations known known to to have have become become extinct extinct in in the the last last 15 15 yr yr and and where where it it was was populations known that that the the butterfly butterfly was was extinct from the the entire 1 6-km22 square. square. The The other known extinct from entire 16-km other E. aurinia aurinia populations, populations, located located nonrandomly nonrandomly half were were centered on surviving surviving E. half centered on in to be populated by by E. E. aurinia best aareas in areas areas still still considered considered to be strongly strongly populated aurinia (the (the best reas available within region). Networks with surviving metapopulations concon available within each each region). Networks with surviving metapopulations tained more larger habitat habitat patches were better better connected connected to to one one tained more and and larger patches and and were another patches in in the in which which the the butterfly butterfly had another than than patches the networks networks in had gone gone extinct extinct (Table 1 ; Bulman, Bulman, 200 1). (Table 20. 20.1; 2001). Metapopulation dynamics in each each 16-km 1 6-km22 square were predicted predicted by by the the IFM IFM Metapopulation dynamics in square were with the the initial initial condition condition of of all all patches patches occupied. All of of the the extinct extinct networks networks with occupied. All were predicted to become become extinct, extinct, with with median median times times to to extinction extinction between between were predicted to 1 1 and and 26 26 yr, yr, depending depending on the network. network. This This implies implies that that not not all such 11 on the all such metapopulations would would have have become become extinct extinct within within a 1 5-yr period period and and that that metapopulations a 15-yr some metapopulations might borrowed time. some metapopulations might still still be be surviving surviving for for longer longer on on borrowed time. The populations in The latter latter is is illustrated illustrated by by the the analysis analysis of of the the six six surviving surviving meta metapopulations in
TABLE population Capacity in 116-km 6-km22 Areas TABLE 20. 2 0 . 11 Habitat Availability and Meta Metapopulation Areas Where Euphydryas Has either either Survived Become Extinct Extinct at the Landscape Euphydryas aurinia aurinia Has Survived or or Become at the Landscape levelG level a Network Network
Survived Survived
North Wales Wales Mid Wales Wales Southwest Southwest A A Southwest Southwest BB Cumbria Dorset Dorset
Extinct Extinct
North Wales Wales Mid Wales Wales Southwest Southwest A A Southwest Southwest BB Cumbria Cumbria Dorset
No. of of patches
Total patch area (ha)
Metapopulatlon Metapopulation capacity (AM) (AM)
Median Median time time to to extinction extinction (yr) (yr)
15 88 117 7 15 15 6 1188 7 14 5 33 2 4
1115 15 41 33 1116 16 14 8800 110 0 20 9 9 88 17 66
5.3 3.8 2.8 4.7 1.5 4.0 0.6 1.0 2.0 11.2 .2 1.2 0.4
>200 1116 16 50 >200 24 130 130 15 21 21 17 22 26 1111
aa Simulated population model, Simulatedtimes times to to extinction extinction are are given givenusing usingthe incidence incidence function function meta metapopulation model, starting starting
simulations 1 ) and simulations with with all all patches patches occupied occupied (see (seetext text for for further further details) details).. From From Bulman Bulman (200 (2001) and C. C. Bulman Bulman et et al. al. (unpublished (unpublishedresults). results).
CHRIS SKI CHRIS D. D. THOMAS THOMAS AND AND ILKKA ILKKA HAN HANSKI
496
Bulman's 1 ) study, Bulman's (200 (2001) study, four four of of which which had had predicted predicted median median times times to to extinction extinction between 3 0 yr between 24 24 and and 1130 yr (the (the remaining remaining two two metapopulations metapopulations survived survived the the entire entire duration 1 ) . As duration of of the the simulations; simulations; Table Table 20. 20.1). As the the surviving surviving metapopulations metapopulations were chosen chosen to to be be in in the the most most favorable favorable landscapes landscapes within within each each region, region, it it were is unlikely unlikely that that these these meta metapopulations would be be rescued rescued by by immigrants immigrants from from is populations would other landscapes landscapes still still containing containing E. aurinia. There There appears appears to to be be aa substantial substantial other extinction debt debt in in this this species, species, which which is is likely likely to to decline decline for for many many decades decades extinction into the the future future even even if if there there are are no no further further habitat habitat losses losses (Bulman, (Bulman, 200 2001). If the the into 1 ) . If Dorset Dorset system, system, for for which which the the model model was was parameterized, parameterized, is is actually actually in in overall overall decline, even even these these projections projections are are too too optimistic. optimistic. decline, The The projected projected times times to to extinction extinction should should not not be be interpreted interpreted too too literally, literally, as as the the model model was was parameterized parameterized with with aa limited limited amount amount of of information information and and no no regional regional stochasticity stochasticity was was taken taken into into account. account. However, However, the the modeling modeling exercise exercise has helped helped reveal reveal why why this this butterfly butterfly appears appears to to be be becoming becoming extinct, extinct, region region has after after region, region, even even when when the the remaining remaining populations populations fall fall within within protected protected areas. areas. Modeling Modeling results results suggest suggest that that protection protection of of all all remaining remaining populations populations and and habi habitat tat patches patches in in some some regions regions may may not not be be enough, enough, whereas whereas the the butterfly butterfly is is pre predicted 00 dicted to to survive survive indefinitely indefinitely in in the the two two largest largest networks networks that that contain contain over over 1100 ha of of habitat. habitat. Despite the non nonequilibrium nature of of the the system, metapopulaha Despite the equilibrium nature system, aa metapopula tion approach approach has has provided provided insight insight into into the the recent recent decline decline and and has has identified identified tion minimum viable network for the the long-term long-term conservation conservation of of the the species. species. minimum viable network goals goals for
Extinction Debt D e b t and and Conservation Conservation Extinction The The concept concept of of extinction extinction debt debt is is usually usually applied applied to to communities communities and and in in the the context (Tilman et et a!., 1 994; Hanski 2002 ). context of of species species diversity diversity (Tilman al., 1994; Hanski and and Ovaskainen, Ovaskainen, 2002). Species-area are performed performed to estimate the the numbers numbers of species Species-area calculations calculations are to estimate of species that might become extinct habitat loss loss (Brooks (Brooks and and that might eventually eventually become extinct following following habitat Balmford, 1996; 1 996; Brooks a!., 1997; 1 997; Cowlishaw, 1 999). These Balmford, Brooks et et al., Cowlishaw, 1999). These calculations calculations have extinction pprocess r o c e s s- and have provided provided insight insight into into the the extinction and into into human human impacts impacts on but it an approach a on biodiversitybiodiversity - but it is is an approach without without hope, hope, as as it it does does not not provide provide a practical conservation action. action. However, practical way way forward forward for for conservation However, aa metapopulation metapopulation approach provides aa way way forward forward even the initial initial prognosis prognosis may may be be approach provides even though though the equally pessimistic. Each will have habitat requirements, requirements, which which means Each species species will have slightly slightly different different habitat means that individual species have have somewhat different habitat habitat networks even in in the the that individual species somewhat different networks even same Guti&rez et same fragmented fragmented landscape landscape (e.g., (e.g., Gutierrez et al., a!., 2001; 200 1 ; Thomas Thomas et et al., a!., 2001b). 200 1 b ) . Species will will also also differ differ in in local local population population densities densities and and dispersal dispersal abilities, abilities, Species and and hence hence local local extinction extinction and and colonization colonization rates rates will will differ. differ. Using Using aa singlesingle species species approach, approach, it it is is possible possible to to identify identify which which areas areas of of the the fragmented fragmented landscape are are likely likely to to be be most most important important for for particular particular species species and and potentially potentially landscape to assess assess whether whether aa given given species species will will eventually eventually decline decline to to extinction extinction or or become become to restricted restricted to to some some limited limited area. area. The The theory theory described described in in Chapter Chapter 44 has has the the potential to to achieve achieve this this for for species species whose whose environments environments are are highly highly fragmented. fragmented. potential If the the prognosis prognosis is is metapopulation metapopulation extinction, extinction, the the metapopulation metapopulation approach approach If can extinction threshcan be be used used (1) ( 1 ) to to identify identify which which landscapes landscapes are are closest closest to to the the extinction thresh old old and and (2) (2) to to identify identify how how extinction extinction and and colonization colonization rates rates could could be be altered, altered, via management management of of landscape landscape structure, structure, to to ensure ensure that that extinction extinction does does not not actuactu via ally ally take take place. place. In In other other words, words, the the theory theory provides provides means means of of targeting targeting conserconser-
20. 20. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS ENVIRONMENTS
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vation of E. aurinia, discussed this action might be be vation action. action. In In the the case case of discussed earlier, earlier, this action might ensuring within focal focal regions have the right ensuring that that all all grasslands grasslands within regions are are grazed grazed to to have the right vegetation that management host plant and that that vegetation height, height, that management increases increases host plant densities, densities, and habitat all meas habitat areas areas are are increased increased by by the the restoration restoration of of adjacent adjacent habitat h a b i t a t- all measures ures to to reduce reduce extinction extinction rates. rates. Similarly, Similarly, increasing increasing habitat habitat quality, quality, restoring restoring new possibly providing new habitats, habitats, and and possibly providing stepping-stone stepping-stone habitats habitats to to connect connect semi semiisolated isolated patch patch networks networks could could all all increase increase colonization colonization rates. rates. By By targeting targeting these these actions actions within within the the most most favorable favorable existing existing landscapes, landscapes, rather rather than than investing investing effort species is effort where where the the species is already already doomed doomed to to extinction, extinction, real real long-term long-term success success may possible. Most may be be possible. Most conservation conservation applications applications of of metapopulation metapopulation theory theory have have stressed stressed the the need need to to increase increase habitat habitat areas areas and and minimize minimize patch patch isolation, isolation, but but often often these these particular particular suggestions suggestions are are impractical. impractical. For For example, example, changing changing the the spatial spatial locations locations of of habitat habitat patches patches is is not not usually usually an an option. option. It It is is important important to to realize, realize, however, however, that that any any actions actions that that reduce reduce extinction extinction and and increase increase colo colonization population approach" nization rates rates are are equally equally valid valid applications applications of of the the "meta "metapopulation approach" to to conservation. conservation. For For example, example, manipulation manipulation of of habitat habitat quality quality and and the the geom geometry etry of of the the landscape landscape are are equally equally legitimate legitimate means means of of altering altering extinction extinction and and colonization aI., 2001b; 200 1 b; Box Box 20.2). 20.2) . colonization rates rates (Thomas, (Thomas, 1994a; 1994a; Thomas Thomas et et al.,
BOX 20.2 Reconciling Habitat and Metapopulatlon Approaches in Butterfly Biology
Many simplifications were made during the early development of the metapopu la tion paradigm in butterfly biology. One of these relates to the emphasis on the spatial configuration (geometry) of suitable habitat in the landscape: what are the areas of habitat patches and how isolated they are from each other? Metapopulation studies appeared to pay less attention to the role of variation in habitat quality, which had pre viously been recognized as a major determinant of butterfly distributions (Thomas, 1 984). However, this perception is somewhat misleading, as even the earliest b utterfly metapopulation studies took account of habitat quality. For example habitat quality thresholds were used to define habitat patches (Harrison et aL, 1 988), and variation in habitat quality was widely recognized as the driving force behind extinction and colon ization dynamics within many metapopulations (Warren, 1 98 7; Thomas, 1 994a,b, 1 996; Hanski, 1 999b; Wahlberg et aL, 2002a). Variation in habitat quality also underlies source-sink dynam i cs w ithin butterf ly metapopulations (Thomas et aL, 1 996) and may infl uence migration among patches (Box 20.3). The pe rception that the metapopulation approach" is somehow an alternative to the "habitat approach" has nonetheless persisted. Most recently, attempts have been made to tease apart the relative importance of variation in habitat q u ality and the spatial arrangement of habitats (Dennis and Eales, 1 999; Tho m as et aI., 2001 b; Fleishman et aI., 2002). However, this is not very satisfactory because the m etapopulation and habitat approaches operate at different levels of a hierarchy. At the metapopulation level, we are primarily interested in the probability of extinction of local populations. Habitat quality habitat type and patch size all contribute to that probability The term habitat quality is itself a "black box" simplifying complex interactions among species as well as responses to the physical environment (e.g., Hochberg et aL, 1 992; Jordano et aL, 1 992). A habitat quality approach is often a useful abstraction to summarize the conse quences of multiple interactions within (usually) single landscape elements, just as a
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CHRIS CHRIS D. D. THOMAS THOMAS AND AND ILKKA ILKKA HANSKI HANSKI
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metapopulation approach is a useful abstraction to summarize the behavior of populations in many such elements. Ultimately, the issue is not whether habitat quality matters, but how we deal with it in population biology and conservation. Most butterflies whose meta population biology has been studied are habitat specialists. In all cases where a metapopulation approach is deemed appropriate, the "first cut" is between habitat (patches) and nonhabitat (matrix). This is a given. After this first cut, the question is whether vari ation in habitat quality is great enough within patches, or within the matrix (Sutcliffe and Thomas, 1 996; Ricketts� 2001 ; Keyghobadi et aI., 2002), for it to be necessary to incorporate habitat variation within the meta population level of analysis. This can only be answered in specific terms; that is, whether and how variation in habitat quality should be treated in a particular landscape and for a particular species (Hanski . et aI., 2004). The example on Hesperia comma (Sections 20.5 and 20.6) illustrates one specific case. The truly critical issue; and the one on which the relevance of the metapopulation approach rests, is whether population connectivity makes a difference to the distri bution and spatial dynamics of the species. In this context, it is unfortunate that most empirical studies continue to use a simplistic measure of connectivity distance to the nearest population (or even distance to the nearest habitat patch) - which measure �
lacks power. Instead, one should use a connectivity measure that takes into account the distances to and sizes of all neighboring . populations (Hanski, 1 999b; Moilanen and Nieminen, 2002). Such a measure of connectivity is also an integral part of the i ncidence function metapopulation model (Hanski, 1 994, 1 999b), which has been applied exten sively in the research reviewed in this chapter. '
20.3 20.3
PRECARIOUS PRECARIOUS METAPOPULATION METAPOPULATION PERSISTENCE PERSISTENCE Some Some landscapes landscapes have have ample ample habitat habitat that that ensure ensure metapopulation metapopulation persist persistence ence and and high high patch patch occupancy, occupancy, whereas whereas other other landscapes landscapes have have very very little little habi habitat tat and and no no chance chance for for metapopulation metapopulation persistence. persistence. Yet Yet other other landscapes landscapes contain contain intermediate intermediate amounts amounts of of habitat habitat that that may may permit permit periodic, periodic, but but not not permanent permanent occupancy, occupancy, in in which which case case metapopulations metapopulations may may flip flip back back and and forth forth between and absence that recolonization between presence presence and absence (provided, (provided, of of course, course, that recolonization from from outside outside is is possible possible following following extinction). extinction). It It is is easy easy to to misinterpret misinterpret the the dynam dynamics systems, especially studies cover cover too too small region and too ics of of such such systems, especially when when studies small aa region and too short long-term and short aa time time to to encompass encompass the the long-term and large-scale large-scale dynamics dynamics of of the the sys system. serious implication tem. The The most most serious implication is is that that researchers researchers might might study study aa patch patch net network work that that is is currently currently empty empty and and conclude conclude erroneously erroneously that that it it is is of of no no consequence consequence for for conservation conservation or or the the same same network network when when it it is is well well occupied occupied and and conclude conclude that that it it is is sufficient sufficient for for long-term long-term persistence. persistence. The The following following examples examples illustrate illustrate that that such such precarious precarious metapopulation metapopulation persistence persistence may may be be commonplace. commonplace.
Arida Aricia agestis agestis in in
North Wales North Wales
The tis, is The brown b r o w n argus argus butterfly, butterfly, A. ages agestis, is a a specialist specialist on on common c o m m o n rock rock rose rose Helianthemum nummularium nummularium plants plants in in north north Wales, Wales, where where the the plant plant is is
METAPOPULATION DYNAMICS IN CHANGING CHANGING ENVIRONMENTS 20. METAPOPULATION
4499 99
restricted to to limestone limestone grasslands grasslands and and crags. crags. Therefore, Therefore, both both the the plant plant and and restricted the butterfly butterfly share share aa very very patchy patchy distribution distribution in in north north Wales Wales (Fig. (Fig. 20.2). 20.2 ) . the Within aa 600-km 600-km22 area, area, habitat habitat patchiness patchiness at a t aa coarse coarse scale scale is i s determined determined by by Within the distribution distribution of of limestone limestone outcrops, outcrops, and and at at aa finer finer scale scale by by the the distribution distribution the of traditional traditional flower-rich meadows and and crags crags (Wilson (Wilson et et al., ai., 2002). 2002). of flower-rich meadows The butterfly butterfly shows shows the the usual usual metapopulation metapopulation patterns. patterns. It It is is most most likely likely The to be be present present in in Helianthemum-containing Helianthemum-containing patches patches that that are are large large and and close close to together; some colonizations colonizations and and extinctions have been been observed, observed, and and individindivid together; some extinctions have uals have have been been recorded recorded moving moving between between habitat habitat patches patches (Wilson (Wilson and and Thomas, Thomas, uals 2002; Wilson Wilson et et al., ai., 2002). 2002 ) . Peripheral Peripheral populations populations tend tend to to contain contain only only aa subsub 2002; set of of the the genetic present within core areas, areas, suggesting suggesting colonization colonization set genetic variation variation present within core by relatively small numbers numbers of of individuals (1. Wynne Wynne et et al. ai. unpublished unpublished result). result) . by relatively small individuals (I. Wilson eett aal.i . (2002) (2002) defined defined groups groups of of meadows meadows aass semi-independent semi-independent networks networks Wilson (SINs) of of habitat habitat if if they they were were separated separated from from other other such such groups groups by by 3 3 km km or or (SINs) more of of unsuitable unsuitable habitats. habitats. Because Because movements movements over over distances distances greater greater than than more
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Fig. 20.2 2 0 . 2 Distribution of occupied (solid) and empty (open) limestone grassland habitat for Aricia tis in north Wales 997 (northwest part) and 11999 999 (other areas). Aricia ages agestis Wales in 11997 areas). Circles Circles denote individual habitat patches (all (all containing the host plant Helianthemum Helianthemum nummularium) in the
main figure; squares squares represent semi-independent networks (SINs), (SINs), separated by 33 km or more of of unsuitable habitats from from other other networks (inset). Simulated median times to extinction extinction of SINs 25 yr (medium-sized squares), SINs are are >200 yr (large squares), squares), 25-1 25-125 squares), and <25 yr (small squares) (R. (R. ].J. Wilson, Wilson, unpublished result). result). Modified from from Wilson et al. (2002).
500 500
CHRIS CHRIS D. THOMAS THOMAS AND ILKKA ILKKAHANSKI HANSKI
33 km km are are rare, rare, it it is is reasonable reasonable to to interpret interpret patch patch occupancy occupancy as as aa function function of of habitat extrapolating from habitat availability availability within within each each network network ((extrapolating from mark-recapture mark-recapture study; % of estimated to move the .6-km distance study; 0.0004 0.0004% of butterflies butterflies were were estimated to move the 33.6-km distance separating habitat networks separating the the two two habitat networks nearest nearest to to one one another; another; Wilson Wilson and and Thomas, Thomas, 2002). 2002). Metapopulations Metapopulations in in two two of of these these networks networks have have been been observed observed to become extinct 15 yr, to become extinct in in the the last last 15 yr, even even though though there there is is no no indication indication that that the the habitat Simulations using habitat has has deteriorated deteriorated recently recently in in either either network. network. Simulations using the the IFM IFM revealed harbor extinction-prone revealed that that most most of of the the semi-independent semi-independent networks networks harbor extinction-prone metapopulations, metapopulations, with with just just the the three three largest largest networks networks exhibiting exhibiting consistent consistent long-term aI., 2002, long-term persistence persistence (Wilson (Wilson et et al., 2002, unpublished unpublished result) result).. The The fourth fourth most persistent network, network, as predicted by most persistent as predicted by the the IFM IFM simulations, simulations, was was one one of of the become extinct (the southernmost the SINs SINs that that was was observed observed to to become extinct in in reality reality (the southernmost network habitat patch network in in Fig. Fig. 20.2) 20.2).. Furthermore, Furthermore, habitat patch networks networks were were more more likely likely to close together. to be be occupied occupied if if they they were were large large and and close together. Crucially, Crucially, extinction extinctionprone networks occupied if prone networks were were significantly significantly more more likely likely to to be be occupied if they they were were close close to other, other, usually usually larger larger networks networks (Wilson (Wilson et et aI., al., 2002 2002).) . to The The most most likely likely interpretation interpretation of of these these results results iiss that that most most of of the the habitat habitat net networks A. agestis in too small consistent occupancy and works for for A. in north north Wales Wales are are too small for for consistent occupancy and that smaller networks occupied periodically, that many many of of the the smaller networks are are occupied periodically, become become extinct, extinct, and more persistent and then then again again recolonized recolonized from from the the more persistent networks. networks. By By taking taking aa metapopulation metapopulation approach, approach, it it was was possible possible to to identify identify the the key key areas areas that that were were likely Durham likely to to be be responsible responsible for for persistence persistence at at the the regional regional scale. scale. In In county county Durham in populations in northern northern England, England, there there may may be be no no robustly robustly surviving surviving meta metapopulations of of the the related related Aricia artaxerxes, and and its its regional regional persistence persistence may may depend depend on on the 999b) the dynamics dynamics of of aa "megapopulation" "megapopulation" (Hanski, (Hanski, 11999b) - - aa "metapopulation "metapopulation of of metapopulations. metapopulations."" Here Here each each semi-independent semi-independent metapopulation metapopulation is is extinction extinction prone, migrants. prone, but but metapopulations metapopulations are are recolonized recolonized by by rare rare long-distance long-distance migrants. How stable such dynamics might case is open question, How stable such dynamics might be be in in this this case is an an open question, and and the the entire (Wilson et entire system system may may be be slowly slowly going going extinct extinct (Wilson et aI., al., 2002). 2002).
Melitaea Melitaea cinxia cinxia in i n Aland Aland Melitaea cinxia occupies occupies aa large large network network of of ca. ca. 4000 4000 habitat habitat patches patches in in the the A land Islands Aland Islands in in southwest southwest Finland, Finland, including including very very small small patches patches and and patches patches of marginal marginal quality quality (Nieminen (Nieminen et et al., 2004). The The patches patches have have been been divided divided of aI., 2004). into (Hanski et aI., 11996b). 996b). Within into semi-independent semi-independent patch patch networks networks (Hanski et al., Within aa SIN, SIN, patches patches are are located located so so close close to to each each other other that that butterfly butterfly movements movements are are quite quite frequent, frequent, and and aa large large fraction fraction of of butterflies, butterflies, of of the the order order of of half half of of all all the the individuals, individuals, move move from from one one patch patch to to another another during during their their lifetime lifetime (Hanski (Hanski et al., 2000). In In contrast, contrast, different different SINs SINs are are separated separated by by dispersal dispersal barriers barriers or or et aI., 2000). are are so so isolated isolated from from each each other other that that butterfly butterfly movements movements are are infrequent. infrequent. Some Some migration migration among among SINs SINs still still occurs, occurs, making making it it possible possible that that aa SIN SIN from from which which aa butterfly butterfly metapopulation metapopulation has has gone gone extinct extinct will will become become recolonized. recolonized. Figure 993-2001 that Figure 20.3 20.3 shows shows the the number number of of years years in in the the period period 11993-2001 that 53 53 SINs SINs were were occupied occupied (the (the smallest smallest SINs SINs mostly mostly consisting consisting of of just just aa single single habitat population habitat patch patch have have been been omitted) omitted).. The The horizontal horizontal axis axis gives gives the the meta metapopulation capacity Ovaskainen, 2000), capacity of of the the network network (Hanski (Hanski and and Ovaskainen, 2000), which which is is aa proper proper measure "size" of measure of of the the "size" of the the network, network, integrating integrating the the effects effects patch patch number, number, patch Chapter 4). the patch sizes, sizes, and and their their connectivities connectivities ((Chapter 4). The The vertical vertical axis axis gives gives the
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in the A land Islands. patterns of network occupancy for Melitaea cinxia in Fig. 20.3 20.3 Large-scale Large-scale patterns ~,land Each point point represents represents a separate semi-independent semi-independent network network (SIN). The number number of years that that a particular SIN was occupied out of 9 yr is shown as a function function of the size size of the SIN (measured by meta population capacity) and its connectivity metapopulations in the surrounding metapopulation connectivity to other metapopulations surrounding SINs SINs (calculation (calculation explained in the text). The size size of the the dot dot is proportional proportional to to the number number of occupied (smallest of years years that that the the network network was was occupied (smallest dot, dot, network network always always unoccupied). unoccupied). Figure Figure prepared by O. Ovaskainen, Ovaskainen, data from from I.I. Hanski (unpublished). (unpublished).
connectivity connectivity of of each each network network using using an an exponential exponential dispersal dispersal kernel kernel with with the the same same parameter parameter value value as as in in the the M. cinxia examples examples in in Chapter Chapter 4, 4, metapopu metapopulation lation capacity capacity as as aa measure measure of of network network size, size, and and P PxA calculated calculated on on the the basis basis of of the the observed observed occupancy occupancy states states of of the the patches patches as as aa measure measure of of SIN SIN occupancy occupancy of P ((see see Chapter Chapter 4 4 for for the the definition definition of Px). apparent that that SINs SINs with with large large A l . ItIt isis apparent metapopulation capacities tended to be occupied more regularly than SINs metapopulation capacities tended to be occupied more regularly than SINs with small metapopulation note that with small metapopulation capacities, capacities, but but note that the the connectivity connectivity of of the the SIN SIN also makes aa difference: small metapopulation tended to also makes difference: SINs SINs with with small metapopulation capacities capacities tended to be populations be occupied occupied more more frequently frequently if if they they were were well well connected connected to to meta metapopulations in in the the neighboring neighboring SINs, SINs, exactly exactly the the same same patterns patterns as as seen seen in in A. agestis. The The effects effects of of metapopulation metapopulation capacity capacity and and SIN SIN connectivity connectivity were were both both significant significant in 993-200 1 in aa logistic logistic regression regression contrasting contrasting SINs SINs that that were were never never occupied occupied in in 11993-2001 versus versus those those that that were were occupied occupied during during at at least least 1i yr yr (for (for both both effects effects P p = = 0.01; 0.01; I. Hanski, Hanski, unpublished unpublished result) result).. I.
Plebejus Plebejus argus argus iinn North North Wales Wales The The silver-studded silver-studded blue blue butterfly, butterfly, P. argus caernensis, occurs occurs within within aa restricted restricted area of of 35 km2 km 2 in in the the Creuddyn Creuddyn Peninsula in in north north Wales. Wales. The The subspecies subspecies area (or "race" (or "race")) caernensis, endemic endemic to to the the peninsula, peninsula, has has been been described described based based on on its its morphology (unusually especially blue females) and habitat requirerequire morphology (unusually small, small, with with especially blue females) and habitat ments patches of ments (medium-height (medium-height turf turf with with patches of bare bare ground, ground, on on southerly southerly facing, facing, species-rich grasslands and species-rich limestone limestone grasslands and crags), crags), and and on on its its interactions interactions with with host host plants plants (especially (especially Helianthemum Helianthemum species species and and Lotus Lotus corniculatus) and and mutualistic mutualistic ants (Thomas, 1985a,b; 992; Jordano Jordano ants (Lasius alienus) (Thomas, 1985a,b; Jordano Jordano and and Thomas, Thomas, 11992; et 992; Thomas aI., 11999b). 999b). Such implies et aI., al., 11992; Thomas et et al., Such aa level level of of local local specialization specialization implies
CHRIS D. THOMAS THOMAS AND AND ILKKA ILKKA HAN SKI CHRIS D. HANSKI
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that that the the butterfly butterfly has has persisted persisted in in the the peninsula peninsula for for aa long long time, time, probably probably for for several 1 ,000 or several thousand thousand to to 111,000 or so so years. years. Sporadic Sporadic information information on on the the occurrence occurrence of of the the butterfly butterfly is is available available for for the 00 yr, the last last 1100 yr, and and more more detailed detailed information information on on its its distribution distribution has has been been obtained 970-1971, 11983, 983, 11990, 990, and 996-1 999 obtained in in surveys surveys carried carried out out in in 11970-1971, and 11996-1999 (Thomas (Thomas et et aI., al., 2002a) 2002a).. The The dynamics dynamics of of the the butterfly butterfly and and its its habitat habitat are are relatively relatively slow, slow, and and hence hence only only aa few few colonization colonization and and extinction extinction events events are are observed in periods, potentially potentially leading observed in short short study study periods, leading to to the the incorrect incorrect conclusion conclusion that 970-1 971, 1188 out that the the distribution distribution is is stable. stable. However, However, since since 11970-1971, out of of the the 20 20 habitat habitat patches patches that that have have been been occupied occupied at at any any time time have have shown shown popula population tion turnover, turnover, with with 1166 further further patches patches remaining remaining unoccupied; unoccupied; at at the the timescale timescale of (Fig. 2004). of decades, decades, the the system system has has functioned functioned as as aa metapopulation metapopulation (Fig. 20.4). The The long-term landscape actually long-term study study has has shown shown that that the the landscape actually consists consists of of two two parts parts for metapopulation on for the the butterfly, butterfly, including including aa core core metapopulation on Great Great Orme's Orme's Head Head to to the the northwest northwest of of the the dashed dashed lines lines in in Fig. Fig. 2004 20.4 and and aa more more widely widely scattered scattered patch patch network network in in the the rest rest of of the the peninsula. peninsula. The The Great Great Orme's Orme's Head Head core core area contains more patches are located much area contains more habitat, habitat, and and the the patches are located much closer closer together, together, than those rest of than those in in the the rest of the the landscape. landscape. In In the the core core area, area, aa high high percentage percentage of of
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Fig. 20.4 patches for Fig. 2 0 . 4 Distribution Distribution of occupied (black) and empty empty (open) habitat habitat patches for P/ebejus Plebejus caernensis on the Crueddyn Peninsula in north Wales since 11970. 970. The solid line is the argus caernensis coast. Only two 970, both two patches of habitat have remained constantly constantly populated populated since 11970, both on Great Orme's Head, which population (to which forms forms the core of the meta metapopulation (to the northwest northwest of of the dashed line). The marginal landscape to to the southeast southeast of Great Orme's Orme's Head has been colonized sporadically, but the the butterfly this area: 997 populations sporadically, but butterfly does does not not persist persist in in this area: X X shows shows 11997 populations that that became 999. Modified al. (2002a). became extinct extinct by by 11999. Modified from from Thomas Thomas et et al. (2002a).
20. 20.
METAPOPULATION CHANGING ENVIRONMENTS METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING ENVIRONMENTS
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the the suitable suitable habitat habitat is is inhabited inhabited (Fig. (Fig. 2004 20.4).) . In In the the rest rest of of the the landscape, landscape, local local populations populations have have been been established established in in the the more more scattered scattered patches patches on on at at least least three occasions during 00 yr: 940s, after 9 7 1 , and three occasions during the the last last 1100 yr: in in the the 11940s, after 11971, and again again in in the 990s. Each populations in the mid-late mid-late 11990s. Each time time the the populations in the the peripheral peripheral areas areas became became extinct extinct again again soon soon after after coloniozation, coloniozation, apparently apparently due due to to stochastic stochastic processes. processes. A 1988) A comparable comparable metapopulation metapopulation structure structure was was found found by by Harrison Harrison et et al. al. ((1988) iin n aa study pot butterfly, study ooff the the Bay Bay checkers checkerspot butterfly, Euphydryas editha, iinn California, California, where where aa large large core core patch patch apparently apparently maintained maintained aa persistent persistent population, population, whereas whereas small, small, marginal marginal populations populations were were ephemeral. ephemeral. Thomas al. (2002a) Thomas et et al. (2002a) used used the the IFM IFM to to model model the the dynamics dynamics of of P. argus as aa whole and also in the the marginal caernensis caernensis in in the the peninsula peninsula as whole and also separately separately in marginal areas. Their conclusion was areas. Their first first conclusion was that that the the longer longer the the data data series series available available to to parameterize parameterize the the model, model, the the greater greater the the amount amount of of population population turnover turnover predicted. predicted. Short-term Short-term studies studies may may give give the the impression impression of of stability; stability; relatively relatively long-term long-term empirical empirical studies studies are are required required to to obtain obtain reliable reliable insights insights into into large largescale scale and and long-term long-term spatial spatial dynamics. dynamics. The The large-scale large-scale and and long-term long-term study study of land Islands Islands provides provides further examples (Hanski, 999b; of M. cinxia in in the the A ,~land further examples (Hanski, 11999b; Ovaskainen Ovaskainen and and Hanski, Hanski, 2003b; 2003b; Nieminen Nieminen et et aI., al., 2004). 2004). The The second second conclusion conclusion is is that, that, in in the the case case of of P. P. argus caernensis, there there is is aa core part part of usually fully and expected core of the the distribution distribution that that is is usually fully occupied occupied and expected to to be be persistent, persistent, unless unless the the habitat habitat deteriorates, deteriorates, and and that that there there is is aa marginal marginal landscape landscape where where habitat habitat patches patches are are smaller smaller and and relatively relatively isolated isolated from from one one another another and and where where habitat habitat is is occupied occupied only only sporadically. sporadically. This This marginal marginal landscape landscape was was rarely rarely occupied for occupied for more more than than aa few few years years at at aa time time during during the the 20th 20th century century because because the the populations populations always always happened happened to to become become extinct extinct before before they they had had had had time time to to grow grow to to aa large large size. size. However, However, if if populations populations would would manage manage to to grow grow to to the the local local carrying carrying capacity, capacity, there there is is no no obvious obvious reason reason why why they they could could not not sur survive landscape. Thomas vive for for decades decades in in the the marginal marginal landscape. Thomas et et al. al. (2002a) (2002a) inferred inferred that that the median time 30 yr landscape starting the median time to to extinction extinction was was 30 yr in in the the marginal marginal landscape starting with observed in 997, with 00 with the the pattern pattern of of patch patch occupancy occupancy observed in 11997, with seven seven of of 1100 simulation runs surviving surviving for more than than 100 Starting with all patches simulation runs for more 100 yr. yr. Starting with all patches occu occupied, 59 yr pied, the the median median time time to to extinction extinction was was 59 yr in in the the marginal marginal landscape. landscape.
Sporadic Sporadic Metapopulation Metapopulation Occurrence Occurrence and and Conservation Conservation In just discussed, In each each of of the the cases cases just discussed, and and also also in in the the American American pika pika at at Bodie Bodie in (Moilanen et 998; Chapter 1 ) , parts landscape were in California California (Moilanen et aI., al., 11998; Chapter 2 21), parts of of the the landscape were occupied only occupied only periodically, periodically, and and long-term long-term persistence persistence was was dependent dependent on on aa combination core areas the combination of of extinction-resistant extinction-resistant core areas permitting permitting recolonization recolonization of of the more marginal landscapes, rescue and among semi more marginal landscapes, or or reciprocal reciprocal rescue and recolonization recolonization among semiindependent patch patch networks. networks. These These examples actually actually represent represent aa substantial substantial independent fraction of of all all metapopulation metapopulation studies studies for for which which long-term long-term and/or and/or large-scale large-scale fraction data probably not data are are available, available, and and they they are are probably not unusual. unusual. Therefore, Therefore, we we contend contend that that aa high high proportion proportion of of habitat habitat specialist specialist species species with with restricted restricted distributions distributions will will exhibit exhibit unstable unstable dynamics dynamics in in at at least least parts parts of of their their distributions, distributions, even even if cases, metapopulation if there there is is no no further further habitat habitat degradation. degradation. In In many many cases, metapopulation dynamics dynamics without without regional regional stochasticity stochasticity are are sufficient sufficient to to explain explain these these patterns, patterns, but cases they causing but in in other other cases they are are likely likely to to be be driven driven by by regional regional stochasticity stochasticity causing the the extinction extinction of of even even relatively relatively large large metapopulations. metapopulations.
CHRIS D. THOMAS CHRIS D. THOMAS AND AND ILKKA ILKKA HANSKI HANSKI
504 504
The implications implications for for conservation conservation are are that that planning planning may may need need to to encompass encompass The even even larger larger regions regions than than currently currently considered, considered, even even when when aa metapopulation metapopulation approach approach is is being being employed. employed. There There are are often often likely likely to to be be parts parts of of species' species' distri distributions relatively safe parts that butions that that are are relatively safe from from extinction, extinction, but but other other parts that are are not. not. The population capacities, The former former may may occur occur in in landscapes landscapes with with high high meta metapopulation capacities, with with many close together, regions where many large large patches patches clustered clustered close together, or or they they may may be be regions where semi semiindependent metapopulations are close together independent metapopulations are located located sufficiently sufficiently close together to to allow allow some some migration migration and and gene gene flow. flow. To To allow allow the the conservation conservation of of specialist specialist species species in in such such areas, areas, it it is is clearly clearly imperative imperative first first to to identify identify them them correctly correctly and and then then target target them them for for conservation. conservation. It It would would be be tragic tragic to to lose lose the the few few remaining remaining core core areas areas to to ongoing ongoing habitat habitat loss loss while while conservation conservation measures measures were were targeted targeted at at areas areas where where eventual eventual metapopulation metapopulation decline decline was was most most likely, likely, in in marginal marginal networks networks where persistence is where long-term long-term persistence is unlikely unlikely without without migration migration from from outside. outside.
20.4 20.4
INTRODUCTIONS OF SPECIES INTRODUCTIONS OF SPECIES Successful Successful introductions introductions of of species species to to empty empty habitat habitat beyond beyond their their normal normal dispersal dispersal range range provide provide experimental experimental evidence evidence for for the the presence presence of of suitable suitable but but unoccupied unoccupied habitat. habitat. Such Such introductions introductions also also allow allow us us to to test test whether whether metapopu metapopulation models habitat networks. lation models can can be be used used to to predict predict the the invasion invasion of of empty empty habitat networks. Although Although some some introductions introductions fail fail instantly, instantly, usually usually for for unknown unknown reasons, reasons, many many butterfly 9 8 1 ; Oates butterfly introductions introductions have have been been successful successful (Holdren (Holdren and and Ehrlich, Ehrlich, 11981; Oates and 990; Warren, 992; Neve aI., 11996b; 996b; Hanski and Warren, Warren, 11990; Warren, 11992; N~ve et et al., Hanski et et aI., al., 2004 2004).) . Even Even then, then, the the timescale timescale ooff success success has has often often been been limited, limited, with with populations populations surviving surviving only only for for aa few few years years before before they they die die out out again. again. In In many many cases, cases, the the area area of of habitat habitat to to which which the the population population was was introduced introduced has has been been small, small, or or the the habitat habitat has has changed changed following following introduction, introduction, and and the the population population became became extinct Oates extinct before before it it could could spread spread to to other other potential potential habitats habitats in in the the region region ((Oates and 990). This and Warren, Warren, 11990). This is is metapopulation metapopulation failure: failure: some some suitable suitable patches patches of of habitat habitat are are present present in in the the landscape, landscape, but but the the rate rate of of colonization colonization of of new new patches, patches, starting starting from from the the site site of of introduction, introduction, was was lower lower than than the the rate rate of of extinction. population approach extinction. A A meta metapopulation approach can can help help understand understand which which introduc introductions metapopulations will tions will will be be successful successful and and help help predict predict how how introduced introduced metapopulations will spread. spread. Fortunately, Fortunately, aa large large body body of of empirical empirical information information and and mechanistic mechanistic understanding understanding already already exists exists of of migration migration and and colonization colonization of of butterflies butterflies in in metapopulations metapopulations (Box (Box 20.3 20.3).) .
Introductions I n t r o d u c t i o n s of of Plebejus Plebejus argus argus in in North North Wales Wales As As described described previously, previously, P. P. argus argus caernensis caernensis was was restricted restricted to to limestone limestone grassland grassland on on one one peninsula peninsula in in north north Wales. Wales. Other Other outcrops outcrops of of limestone limestone in in the the same region were same region were unoccupied unoccupied by by the the butterfly, butterfly, despite despite the the fact fact that that the the habitat habitat looked 942, looked superficially superficially similar similar and and potentially potentially suitable suitable for for the the butterfly. butterfly. In In 11942, 90 90 females females were were introduced introduced to to one one patch patch of of grassland grassland in in this this previously previously unoccupied area, unoccupied area, from from which which they they spread spread to to occupy occupy 1177 out out of of 20 20 patches patches in in 11990, 990, establishing establishing aa metapopulation metapopulation with with roughly roughly 90,000 90,000 adult adult butterflies butterflies (Thomas, 985b; Thomas Thomas and 992) . Using (Thomas, 11985b; and Harrison, Harrison, 11992). Using aa metapopulation metapopulation model model parameterized parameterized for for the the original original distribution distribution of of P. argus argus caernensis, caernensis, Hanski Hanski and and
20. 20. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS ENVIRONMENTS
BOX 20.3
505 505
Butterfly Studies on Migration and Colonization
One strong theme of research in butterfly metapopulation biology has been m igra tion within metapopulations. Mark-release-recapture projects on butterflies started with Dowdeswell et al. (1 940), followed by Ehrlich's (1 961 ; Ehrlich and Davidson, 1 961 ) important studies on the spatial structure of Euphydryas editha populations in California. Many multi population mark-release-recapture studies have been conducted on European butterflies in the 1 990s (e.g., Baguette and Neve, 1 994; Hanski et aI., 1 994; Hill et aI., 1 996; Neve et aI., 1 996; Thomas and Wilson, 2002), and new methods of data analyses have been developed (Hanski et aI., 2000, Ovaskainen, 2003) and applied (Ricketts, 200 1 ; Petit et aI., 2001 ; Wah lberg et aI., 2002b; Schtickzelle et aI., 2002). Researchers have attempted to identify the factors that are responsible for variation in the observed rates of migration, such as habitat patch areas and distances between patches, the quality and quantity of larval and adult resources within patches, popula tion density, the sex ratio, the type of boundary surrounding habitat patches, and the nature of the matrix between habitat patches (e.g., Thomas and Singer, 1 987; Kuussaari et aI., 1 996; Hill et aI., 1 996; Sutcliffe and Thomas, 1 996; Sutcliffe et aI., 1 997; Schultz, 1 998; Haddad, 1 999a; Haddad and Baum, 1 999; Roland et aI., 2000; Thomas et aI., 2000; Ricketts, 200 1 ; Fownes and Roland, 2002; Matter and Roland, 2002; Wilson and Thomas, 2002). Because al most all of these variables operate through the behavioral responses of individuals, this has led to the development of individual-based dispersal models, such as random wal k simulations of the dispersal of individuals in habitat patches (Schultz and Crone, 2001 ; R. Setchfield, manuscript in preparation) and through ecological corridors (Haddad, 1 999b). Ovaskainen (2003) has developed a diffusion model that allows quantitative analysis of individual movements in hetero geneous landscapes, including the cost of migration in terms of mortality in the matrix (see also Hanski et aI., 2000). Evolutionary studies have shown that migration rates, behaviors, and flight mor phologies may evolve in response to the structure of the landscape (Thomas et a I ., 1 998; Hanski and Singer, 2001 ; Heino and Hanski, 2001 , Hanski et aI., 2002), including fac tors such as the host plant composition of i ndividual habitat patches (Hanski and Singer, 2001 ; Hanski et aI., 2002; Hanski and Heino, 2003). In Melitaea cinxia, females in newly established populations are more dispersive than females in older popUlations (Hanski et aI., 2002), suggesting that extinction-colonization dynamics select for an increased migration rate at the meta population level. However, the actual behavioral mechanisms that butterflies use to navigate and select habitat patches within metapopulations are largely unknown. Field experiments with individual butterflies removed from their natural habitat patches revealed apparently deliberate search behaviors and the poten tial of individuals to return preferentially to their patch of origin (Conradt et a I ., 2000, 200 1 ) . The behavioral means by which migrant individuals locate new patches of habitat requires further research.
Thomas ((1994) 1 994) predicted should spread spread successfully Thomas predicted that that the the introduction introduction should successfully and and establish aa persistent persistent metapopulation metapopulation in in the the Dulas Dulas Valley. Valley. The The metapopulation metapopulation establish still still thrives thrives in in the the Dulas Dulas Valley, Valley, 60 60 yr yr after after the the original original introduction. introduction. Given Given the the success success of of the the metapopulation metapopulation approach approach to to predict predict the the butterfly'S butterfly's eventual eventual extinction from from aa landscape landscape with with limited limited habitat habitat availability availability and and its its successful successful extinction invasion basis to assess other areas invasion of of an an empty empty habitat habitat network, network, there there is is the the basis to assess other areas of of limestone for for their their potential potential for for introductions. introductions.
506 506
CHRIS LKKA HANSKI CHRIS D. D. THOMAS THOMAS AND AND IILKKA HANSKI
Introductions Introductions of of Melitaea Melitaea cinxia cinxia to to Unoccupied Unoccupied Patches Patches and Patch and Patch Networks Networks
Melitaea cinxia has has been introduced since early 11990s 990s to 10 pre been introduced since the the early to some some 10 previously unoccupied land Islands Islands and mainland Finland viously unoccupied sites sites within within the the A Aland and in in mainland Finland (M. Kuussaari (M. Kuussaari and and I. I. Hanski, Hanski, unpublished unpublished results) results).. All All but but two two of of these these intro introductions occurred occurred at isolated and poorly ductions at sites sites that that were were either either completely completely isolated and poorly connected to habitat patches; introductions to connected to other other (generally (generally small) small) habitat patches; the the introductions to single single patches patches and and sparse sparse habitat habitat networks networks all all failed. failed. Of Of the the two two successes, successes, one one site site in in mainland mainland Finland Finland (Fagervik) (Fagervik) has has persisted persisted for for at at least least 66 yr, yr, but but unfortun unfortunately been surveyed surveyed systematically. case there ately the the populations populations have have not not been systematically. In In this this case there are several suitable sites for usual migration are several suitable sites for the the butterfly butterfly within within its its usual migration distance, distance, up to to 3-4 3-4 km km (Hanski, (Hanski, 11999b; van Nouhuys Nouhuys and and Hanski, 2002 2002).) . up 999b; van The 991, The other other introduction introduction that that turned turned out out ttoo bbee successful successful took took place place iinn 11991, when when 7722 larval larval groups groups were were translocated translocated ttoo 1100 small small meadows meadows on on the the island island of 20 km land Island. of Sottunga, Sottunga, located located 20 km east east from from the the main main A Aland Island. Sottunga Sottunga is is 4 4 km km long 20 suitable suitable meadows long and and 22 km km wide; wide; it it had had some some 20 meadows for for M. M. cinxia cinxia when when surveyed 9 9 1 , but populations of surveyed in in June June 11991, but no no populations of M. M. cinxia. cinxia. The The metapopulation metapopulation established in 9 9 1 has 12 yr, established in 11991 has persisted persisted for for the the past past 12 yr, although although it it went went through through aa bottleneck meadows in 999 (Fig. bottleneck of of just just three three occupied occupied meadows in 11999 (Fig. 20.5 20.5).) . It It is is noteworthy none of created in 9 9 1 has has noteworthy that that none of the the original original local local populations populations created in 11991 survived, hence this this introduction survived, and and hence introduction provides provides an an experimental experimental demonstration demonstration of classic metapopulation metapopulation dynamics action. The of classic dynamics in in action. The metapopulation metapopulation capacity capacity ((Chapter Chapter 4) of network is based on of the the Sottunga Sottunga network is so so large large that, that, based on the the occurrence occurrence 1 00
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Year Fig. 20.5 20.5 Experimental Experimental metapopulation metapopulation of of Melitaea cinxia cinxia on on the the island island of of Sottunga Sottunga in in the the A land Islands. population was established in a previously empty network of ca. 20 small ~,land Islands. The meta metapopulation patches in 11991 991 (72 larval groups placed in 1100 meadows). Observed changes in metapopulation metapopulation size size in in terms terms of of the the number number of of larval larval groups groups and and the the number number of of occupied occupied meadows, meadows, as as well well as as the the number number of of turnover turnover events events (extinctions (extinctions and and colonizations) colonizations) between between consecutive consecutive years, years, are are shown. shown. None original populations 991 has None of of the the original populations that that were were established established in in 11991 has survived survived until until present, present, and and hence population has survived in a stochastic hence this meta metapopulation stochastic balance between local local extinctions and recolonizations recolonizations of of empty empty meadows meadows as as predicted predicted by by the the theory theory (from (from Nieminen Nieminen et et aI., al., 2004). 2004).
20. 20. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS ENVIRONMENTS
507 507
of mainland A land, persistence persistence in of the the butterfly butterfly in in patch patch networks networks on on mainland .~land, in Sottunga Sottunga is is expected expected (Hanski (Hanski et et aI., al., 2004). 2004).
20.5 20.5
METAPOPULATIONS METAPOPULATIONS RESPONDING RESPONDING TO TO CLIMATE CLIMATE CHANGE CHANGE For For thermally thermally sensitive sensitive species species such such as as butterflies, butterflies, climate climate warming warming is is predicted predicted to to alter alter the the quantity, quantity, quality, quality, and and distribution distribution of of suitable suitable habitats habitats in in This section section describes describes how how climatic climatic improvement improvement can can increase increase aa landscape. This the close to the density density of of suitable suitable habitat habitat patches patches in in landscapes landscapes close to northern northern (poleward) boundaries, permitting population invasion invasion of (poleward) range range boundaries, permitting meta metapopulation of pre previously population viously empty empty habitat habitat networks. networks. We We also also evaluate evaluate whether whether meta metapopulation and and invasion invasion models models accurately accurately predict predict the the expansion expansion at at range range margins. margins. At At southern southern boundaries, boundaries, deteriorating deteriorating conditions conditions may may lead lead to to reduced reduced habitat habitat availability, availability, such such that that landscapes landscapes where where species species persisted persisted in in the the past past may may no no longer longer allow allow metapopulations metapopulations to to persist. persist. Range Range contractions contractions at at southern southern mar marCalifornia gins gins have have been been demonstrated demonstrated for for Euphydryas Euphydryas editha editha in in Mexico Mexico and and California (Parmesan, 99 6 ) and number of species in Parmesan et (Parmesan, 11996) and for for aa number of species in Europe Europe ((Parmesan et aI., al., 11999), 99 9 ) , but but no no quantitative quantitative modeling modeling of of southern southern range range contractions contractions has has yet yet been attempted. been attempted.
Multispecies Multispecies Patterns Patterns A qualitative qualitative test test of of the the role role of of habitat habitat availability availability in in range range expansion expansion A comes comes from from aa cross-species cross-species comparison comparison of of range range changes changes at at the the northern northern boundaries of 1 ) evaluated boundaries of butterfly butterfly distributions distributions in in Britain. Britain. Warren Warren et et ai. al. (200 (2001) evaluated the species, all the changes changes in in the the distribution distribution of of 46 46 southerly southerly butterfly butterfly species, all of of which which were become more widespread and were predicted predicted to to become more widespread and expand expand their their distributions distributions because because of of ameliorating ameliorating climatic climatic conditions. conditions. In In practice, practice, however, however, almost almost all all of of the of the habitat habitat specialists specialists and and half half of of the the habitat habitat generalists generalists declined declined in in terms terms of their 970-1 982 and 995-1 999 due their distribution distribution between between 11970-1982 and 11995-1999 due to to habitat habitat losses. losses. The The low low habitat habitat availability availability for for habitat habitat specialists specialists means means that that most most British British landscapes expansion that landscapes fail fail to to permit permit metapopulation metapopulation expansion that would would otherwise otherwise be be expected expected to to occur occur under under aa warmer warmer climate. climate.
Hesperia through Fragments Fragments of Hesperia comma comma Expanding Expanding through of Chalk Chalk Grassland Grassland The The means means by by which which climate climate warming warming can can increase increase habitat habitat availability availability is is clearly clearly seen seen in in the the silver-spotted silver-spotted skipper, skipper, H. H. comma. comma. Like Like many many other other poikilo poikilothermic 999b), this thermic animals animals at at their their northern northern range range margin margin (Thomas (Thomas et et aI., al., 11999b), this butterfly 982 to butterfly was was restricted restricted in in 11982 to unusually unusually warm warm habitats: habitats: southerly southerly facing facing hillsides, where it laid its hillsides, where it laid its eggs eggs on on the the grass grass Festuca Festuca ovina ovina in in warm warm hollows hollows close close to bare bare ground ground (Thomas (Thomas et et al., However, by by 2000 H. H. comma comma had had shown shown to aI., 11986). 98 6 ) . However, aa significant significant widening widening of of the the range range of of habitats habitats used, used, spreading spreading into into north-facing north-facing habitats habitats and and laying laying eggs eggs in in taller taller and and more more continuous continuous vegetation vegetation than than had aI., 200 1a; Z. unpublished had been been used used previously previously (Thomas (Thomas et et al., 2001a; Z. Davies Davies et et aI., al., unpublished result) result).. The The species species now now occupies occupies habitats habitats that that would would have have been been regarded regarded as cool for 982. In as too too cool for the the species species in in 11982. In the the case case of of H. H. comma, comma, availability availability of of aa
508
CHRIS CHRIS D. D. THOMAS THOMAS AND AND ILKKA ILKKA HANSKI HANSKI
suitable suitable habitat habitat has has also also increased increased due due to to increased increased grazing grazing by by expanding expanding rab rabbit 982 (a (a deliberate bit populations populations and and livestock livestock since since 11982 deliberate conservation conservation measure) measure).. Habitat Habitat management management and and climate-induced climate-induced habitat habitat changes changes have have approxi approximately mately doubled doubled the the quantity quantity of of suitable suitable habitat habitat for for this this species species in in one one part part of of its its distribution distribution in in Britain, Britain, in in the the South South Downs Downs hills hills in in southern southern England, England, aa region region been confined hillside in where where H. H . comma c o m m a had had been confined to to aa single single large large hillside in the the late late 1970s. 1970s. The The incidence incidence function function model model was was parameterized parameterized for for H. H . comma c o m m a using using data data from applied to from aa different different part part of of southern southern England England and and was was then then applied to predict predict the the changes distribution in application represents changes in in distribution in the the South South Downs. Downs. This This application represents aa gen genuinely independent test power of model. When uinely independent test of of the the predictive predictive power of the the model. When model model sim simulations ulations were were run run assuming assuming the the definition definition of of suitable suitable habitat habitat that that was was valid valid in in 11982, 982, the about 5 the model model predicted predicted an an expansion expansion of of about 5 km km in in 1188 yr yr (the (the distance distance from 10 populations from the the source source population population to to the the furthest furthest 10 populations in in surviving surviving meta populations; the some simulation simulation runs). metapopulations; the metapopulation metapopulation went went extinct extinct in in some runs). However, However, assuming assuming the the relaxed relaxed definition definition of of habitat habitat as as demonstrated demonstrated empiri empirically expansion was cally in in 2000, 2000, expansion was much much faster, faster, with with an an average average expansion expansion of of 14.3 14.3 km, km, and and now now none none of of the the 100 100 simulation simulation runs runs resulted resulted in in extinction extinction (Fig. (Fig. 20.6). 20.6). The 1 6.4 km) The observed observed expansion expansion after after 1188 yr yr ((16.4 km) was was not not significantly significantly different different from expansion predicted from this this latter latter prediction, prediction, but but was was further further than than the the expansion predicted in in any any simulation run 982 definition habitat. The reduced need simulation run using using the the 11982 definition of of habitat. The reduced need for for hot, hot, south-facing south-facing hillsides hillsides means means that that there there are are now now more more and and larger larger patches patches of of habitat habitat available available in in the the landscape, landscape, and and hence hence shorter shorter distances distances between between the the patches. patches. Shorter Shorter distances distances led led to to more more numerous numerous colonization colonization events; events; more more new new populations colonists, which populations increased increased the the number number of of potential potential colonists, which led led to to aa positive positive feedback feedback between between increased increased habitat habitat availability availability and and rate rate of of expansion. expansion. An An approximate approximate doubling doubling of of the the availability availability of of habitat habitat in in this this landscape landscape resulted resulted in in aa threefold expansion rate. threefold increase increase in in the the predicted predicted expansion rate. These These studies studies have have been been extended extended to to other other parts parts of of the the British British distribution distribution of of H. H . comma c o m m a with with similar similar results results (R. (R. J. J. Wilson Wilson et et aI., al., unpublished unpublished result). result). Differences in in expansion expansion rates rates in in different different landscapes landscapes were were predictable from dif difDifferences predictable from ferences ferences in in the the level level of of habitat habitat availability. availability. Observed Observed expansion expansion distances distances were were closely correlated with distances predicted 00 simulation closely correlated with mean mean expansion expansion distances predicted from from 1100 simulation separate meta populations, P = runs each landscape runs for for each landscape (r22 = = 0.95, 0.95, n n = = 55 separate metapopulations, = 0.001; 0.001; R. R. J. J. Wilson Wilson et et aI., al., unpublished unpublished result). result). The The prediction prediction was was accurate accurate in in four four out out of of five five landscapes landscapes studied. studied. In In the the fifth, fifth, the the model model predicted predicted faster faster expansion expansion than than was 95% confi was observed, observed, with with the the observed observed expansion expansion falling falling just just outside outside the the 95% confidence case, however, dence limits limits of of IFM-predicted IFM-predicted expansion. expansion. In In this this case, however, the the average average habi habitat tat quality quality (host (host plant plant density) density) was was lower lower than than in in the the other other landscapes. landscapes. When When reduced habitat was taken reduced habitat quality quality was taken into into account account by by reducing reducing effective effective habitat habitat areas (see later), later), the areas in in the the IFM IFM (see the errant errant landscape landscape fell fell back back into into line. line. These These results results give possible to give genuine genuine hope hope that that it it may may be be possible to manage manage metapopulation metapopulation recoveries recoveries and responses to level. and predict predict responses to climate climate change change at at aa landscape landscape level.
Scaling Scaling up up from from Metapopulations Metapopulations to to Geographical Geographical Distributions Distributions The The expansion expansion of of H. H . comma c o m m a can can be be regarded regarded as as an an example example of of conventional conventional metapopulation dynamics: metapopulation dynamics: increased increased regional regional habitat habitat availability availability led led to to expan expansion sion rates rates up up to to 1100 km km per per decade decade in in habitat habitat patch patch networks networks of of up up to to aa few few
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Fig and observed observed (c) Fig 20.6 2 0 . 6 Simulated Simulated (a, (a, b) b) and (c) expansion expansion of of Hesperia comma comma from from aa single single large large population 982 in the South Downs hills hills of southern England (solid population (triangle) that existed in 11982 line is the south coast of 982; of England). Circles are habitat patches that that were unoccupied unoccupied in 11982; those and those in those that that had had been been colonized colonized by by 2000 2000 are are show show in in black black and those that that were were still still empty empty in 2000 using the incidence function function model 2000 are shown as open circles in (c). Simulated expansion using shows patches patches that were colonized colonized in more more than (black) and less less than (gray) 50% 50% of simulation simulation runs: open circles were not colonized in any simulation 00). In (a), the 11982 982 defin simulation run (out of of 1100). definition of habitat was used, and expansion expansion was predicted predicted to be limited. limited. By 2000 2000 (b), the range of habitats habitats available available to to H. comma had had increased increased (principally (principally as as aa result result of of climate climate warming), warming), roughly roughly doubling doubling the availability of of habitat in the landscape. Simulated Simulated expansion in the the expanded expanded habitat habitat network network (b) (b) was was much much faster faster than than in in (a), (a), and and not not significantly significantly different different from from the observed Modified from (2001 a). the observed expansion expansion (c). (c). Modified from Thomas Thomas et et al. al. (2001a).
hundred patches. patches. Other response to hundred Other species species that that are are expanding expanding northward northward in in response to climate change habitats available climate change tend tend to to have have aa broader broader range range of of habitats available to to them, them, and and in cases this habitat patches. in many many cases this means means that that it it is is difficult difficult to to delimit delimit distinct distinct habitat patches. However, However, increased increased colonization colonization potential potential and and larger larger population population sizes sizes at at newly newly colonized colonized sites sites should should increase increase with with habitat habitat availability availability even even when when the the habitat habitat is is not colonization rate not fragmented fragmented into into distinct distinct patches. patches. Similarly, Similarly, the the colonization rate is is expected expected to depend depend on on the the distances distances to to currently currently populated populated areas, areas, whether whether they they occur occur in in to discrete discrete patches patches or or not. not. Thus Thus range range dynamics dynamics of of species species that that do do not not form form recog recognizable metapopulations may nizable metapopulations may nonetheless nonetheless show show metapopulation-like metapopulation-like features. features. To resolution grid-based grid-based model To this this end, end, we we adopted adopted aa 200-m 200-m resolution model (MIGRATE) (MIGRATE) to to simulate simulate range range expansion expansion in in butterflies butterflies that that do do not not always always live live in in distinct distinct habi habitat aI., 1999). tat patches patches (Hill (Hill et et al., 1999). The The model model focuses focuses on on colonizations colonizations only only because because
CHRIS LKKA HAN SKI CHRIS D. D. THOMAS THOMAS AND AND IILKKA HANSKI
5 5 11 00
(a) it is is generally generally difficult difficult to to identify identify extinctions extinctions when when the the landscape landscape and and hence hence (a) it population population is is not not highly highly fragmented, fragmented, (b) (b) there there have have been been very very few few observed observed extinctions, extinctions, relative relative to to the the number number of of colonizations, colonizations, in in the the expanding expanding species species since 970, and (c) the since 11970, and (c) the computational computational time time associated associated with with modeling modeling extinctions extinctions as as well well as as colonizations colonizations would would have have been been prohibitative prohibitative at at 200-m 200-m grid grid resolution resolution across across the the whole whole of of Britain. Britain. The was initially The model model was initially run run on on the the speckled speckled wood, wood, Pararge Parargeaegeria, aegeria, which which occurs woodland and occurs in in woodland and scrub scrub in in the the northern northern parts parts of of its its geographic geographic dis distribution tribution in in Britain. Britain. Although Although habitat habitat quality quality inevitably inevitably varies, varies, almost almost all all types types of of woodland woodland and and scrub scrub contain contain suitable suitable breeding breeding areas areas and and host host plants plants (grasses), possible to (grasses), making making it it possible to identify identify potential potential habitat habitat with with satellite satellite images. images. The wood The model model was was run run for for two two parts parts of of Britain: Britain: for for Yorkshire, Yorkshire, where where the the woodland % , and land cover cover was was estimated estimated to to be be 2.72 2.72%, and for for an an area area around around Inverness Inverness in Scotland, where . 5 8 % (Hill 1 ) . In in Scotland, where woodland woodland cover cover was was 33.58% (Hill et et ai., al., 200 2001). In both both areas, the expansion of commenced around 970, hence areas, the actual actual expansion of P. aegeria aegeria commenced around 11970, hence simulation 9 70 simulation and and empirical empirical results results were were compared compared for for the the period period between between 11970 and 2000. and 2000. The The MIGRATE MIGRATE model model was was ran ran at at 200 200 m m resolution, resolution, but but the the fit fit was was assessed assessed at at 55 km km resolution, resolution, at at which which scale scale the the field field data data had had been been collated collated (Hill (Hill et et ai., al., 2001 2001).) . Most Most parameters parameters of of the the MIGRATE MIGRATE model model were were obtained obtained from from inde independent pendent field field data data (e.g., (e.g., population population density, density, population population growth growth rate), rate), but but some some parameters parameters were were estimated estimated by by maximizing maximizing the the fit fit between between the the observed observed and and the the modeled modeled expansion expansion in in Yorkshire. Yorkshire. The The latter latter involved involved adjusting adjusting the the habitat habitat threshold, threshold, permitting permitting successful successful colonization colonization of of aa grid grid square; square; the the need need to to do do this this may may stem stem from from the the lack lack of of an an extinction extinction component component in in the the MIGRATE MIGRATE modei. 53 occurrences model. In In Yorkshire, Yorkshire, the the model model predicted predicted 53 occurrences in in the the 5-km 5-km grid grid squares, 49 recorded squares, compared compared to to 49 recorded occurrences. occurrences. More More interesting interesting was was the the comparison comparison between between model model prediction prediction and and empirical empirical observation observation for for the the region region around around Inverness Inverness using using the the parameter parameter values values estimated estimated in in Yorkshire. Yorkshire. The The only only extra extra element element was was the the imposition imposition of of an an upper upper elevational elevational threshold threshold for for the the occurrence does not above 200 200 m. occurrence of of the the butterfly, butterfly, as as it it does not occur occur above m. The The model model predicted 1 9 occurrences predicted 1119 occurrences around around Inverness, Inverness, while while 121 121 occurrences occurrences were were observed 1 ) . This observed (Fig. (Fig. 20.7; 20.7; Hill Hill et et ai., al., 200 2001). This result result strongly strongly suggests suggests that that habitat habitat availability availability is is an an important important determinant determinant of of the the rate rate of of expansion expansion in in species species responding responding to to climate climate change change and and that that metapopulation-like metapopulation-like models models can can be be used to to predict predict distributional distributional changes changes in in regions regions where where range range expansions expansions are are used taking taking place. place. The The model model has has subsequently subsequently been been applied applied to to two two other other related related species, Pyronia Pyronia tithonus tithonus and and Aphantopus Aphantopus hyperantus, hyperantus, as as well well as as to to P. P. aegeria aegeria species, across the whole of Britain. that differences across the whole of Britain. Results Results of of these these analyses analyses show show that differences in in the of expansion three species can be the rates rates of expansion of of the the three species can be predicted predicted from from differences differences in in habitat and population population parameters habitat availability availability and parameters (especially (especially population population density; density; s. S. Willis Willis et et ai., al., unpublished unpublished result) result)..
20.6 20.6
CONCLUSIONS CONCLUSIONS Contrary Contrary to to initial initial concerns concerns about about the the applicability applicability of of the the metapopulation metapopulation approach to to fragmented fragmented populations populations in in changing changing environments, environments, these these situa situaapproach tions actually actually provide provide some some of of the the strongest strongest evidence evidence for for the the importance importance of of tions
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20.7 (a) Distribution nverness in Scotland: small circle circle represents Fig. 20.7 Distribution of woodland woodland around around IInverness represents 5-1 0% woodland, woodland, medium-sized 1 -25%, and >25% woodland 5-10% medium-sized circle circle 111-25%, and large large circle circle >25% woodland cover. cover. Woodland above 200 200 m level) asl 0-km intervals. Woodland above m elevation elevation (above (above sea sea level) asl was was excluded. excluded. Lines Lines at at 110-km intervals. (b) Observed distribution 970 (striped squares) and 11995-1999 995-1 999 (triangles) 5-km grid distribution in 11970 (triangles) at a 5-km resolution. resolution. Simulated Simulated distribution distribution is is based based on on five five runs runs of of MIGRATE MIGRATE showing showing areas areas normally normally (dark gray; in �4 and sometimes <4 runs) runs) colonized. (dark gray; colonized colonized in ___4 runs) runs) and sometimes (pale (pale gray; gray; colonized colonized <4 colonized. Simulations 970. Simulations were were seeded seeded in in the the six six grid grid squares squares occupied occupied in in 11970.
metapopulation metapopulation dynamics dynamics and and for for the the predictive predictive power power of of metapopulation metapopulation models. Applied Applied carefully, carefully, quantitative quantitative predictions predictions are are likely likely to to be be sufficiently sufficiently population models accurate to to use use in in practical conservation. conservation. Simple meta metapopulation models do do not not encompass encompass all all of of the the reality of of natural natural systems, systems, such such as as complex complex variation variation in in habitat habitat quality, quality, behavior-driven behavior-driven dispersal, dispersal, and and so so forth. forth. These These are are important important areas areas for for further further research. research. However, However, provided provided that that one one has has an an adequate adequate knowledge see later) knowledge of of the the habitat habitat requirements requirements ((see later) and and dispersal capacity capacity of of the population models seem the focal focal species, species, simple simple pattern-based pattern-based meta metapopulation seem to to capture capture enough enough of of the the dynamics dynamics to to provide provide both both insight insight and and practical practical guidance guidance for for conservation conservation and and management. management. For For example, example, we we can can manage manage for for expansion in in restoration restoration ecology ecology by by targeting targeting conservation conservation management management (increasing (increasing habitat habitat
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CHRIS N D ILKKA CHRIS D. D. THOMAS THOMAS A AND ILKKA HANSKI HANSKI
quality quality and and quantity) quantity) to to increase increase metapopulation metapopulation capacity capacity in in areas areas currently currently predicted only just below the predicted to to be be only just below the threshold threshold for for invasion. invasion. Even Even if if not not all all of of the the details details of of the the dispersal dispersal process process are are known known exactly, exactly, the the recommendations recommendations are are likely likely to to be be quite quite robust. robust. The The same same approach approach allows allows us us to to make make the the transition transition from from landscape-level landscape-level dynamics population dynamics to to geographic geographic distributions distributions and and range range dynamics. dynamics. A A meta metapopulation approach approach helps helps understand understand why why some some species species are are expanding expanding at at their their northern northern range range margins margins as as the the climate climate warms, warms, but but others others are are not, not, and and helps helps us us under understand landscapes are stand why why some some landscapes are permanently permanently populated populated by by aa species species while while others population others are are occupied occupied only only sporadically. sporadically. More More impressively, impressively, meta metapopulation models models make models and and spatially spatially explicit explicit colonization colonization models make accurate accurate predictions predictions of of differences expansion in landscapes and differences in in the the rates rates of of expansion in different different landscapes and in in different different species (in the species introductions species (in the context context of of species introductions and and climate climate change). change). The The pre predictive models to dictive power power is is sufficient sufficient for for the the current current generation generation of of models to be be used used in in the development of environmental policy. the development of environmental policy. There also important There are are also important limits limits to to the the predictive predictive power power of of the the current current models. models. Some Some of of these these relate relate to to shortcomings shortcomings in in the the simplest simplest (and (and hence hence most most practical) practical) models. The models. The most most obvious obvious concern concern has has been been the the need need to to incorporate incorporate variation variation in in habitat quality population studies. need is habitat quality into into meta metapopulation studies. The The primary primary need is to to be be able able to to dis distinguish between suitable requires detailed tinguish between suitable and and unsuitable unsuitable habitat, habitat, which which often often requires detailed studies question. Because studies of of the the ecology ecology of of the the species species in in question. Because apparently apparently suitable suitable empty researchers have empty habitat habitat may may not not actually actually be be suitable, suitable, researchers have to to be be careful careful while while mapping mapping patch patch networks. networks. Whenever Whenever possible, possible, statistical statistical habitat habitat quality quality threshold threshold models should applied, and should be models should be be applied, and the the suitability suitability of of empty empty habitat habitat should be checked checked by by experimental experimental introductions introductions and and through through long-term long-term observations observations that that permit permit the the natural natural colonization colonization of of empty empty habitats habitats to to be be observed observed (Harrison (Harrison et et a!., al., 1988; 1988; Harrison, 989). However, Harrison, 11989). However, clear clear separation separation of of suitable suitable habitat habitat from from unsuitable unsuitable areas is areas is only only possible possible in in some some cases. cases. In In the the silver-studded silver-studded blue blue butterfly butterfly (Plebejus argus), itit is is easy easy (after (after much much research) research) to to distinguish distinguish between between suitable suitable and and unsuit unsuitable habitat areas of able habitat in in areas of limestone limestone grassland, grassland, but but much much more more difficult difficult in in heathland heathland vegetation, continuously (Thomas, vegetation, where where habitat habitat quality quality appears appears to to vary vary more more continuously (Thomas, 11985b; 985b; Thomas 992). In Thomas and and Harrison, Harrison, 11992). In heathland, heathland, some some unoccupied unoccupied habitat habitat is is clearly clearly suitable, suitable, as as evidenced evidenced by by successful successful introductions, introductions, but but other other patches patches of of habitat possible to habitat seem seem to to be be of of intermediate intermediate quality. quality. Consequently, Consequently, it it has has been been possible to apply apply patch patch occupancy occupancy metapopulation metapopulation models models (Chapters (Chapters 4 4 and and 55)) to to P. argus where where it it inhabits inhabits limestone limestone grasslands grasslands (Hanski (Hanski and and Thomas, Thomas, 1994; 1994; Thomas Thomas et et a!., al., 2002a), 2002a), but but not not in in areas areas of of heathland, heathland, even even though though extinction-colonization extinction-colonization dynamics species in dynamics are are even even more more important important to to the the persistence persistence of of the the species in succes successional heathland (Thomas and Harrison, 11992; 992; Lewis sional heathland habitats habitats than than on on limestone limestone (Thomas and Harrison, Lewis et 997). This et a!., al., 11997). This is is likely likely to to be be aa widespread widespread phenomenon, phenomenon, with with only only some some species, clear distinction species, and and only only some some regions regions in in many many species, species, showing showing aa clear distinction between habitat and habitat. Identifying between habitat and non nonhabitat. Identifying suitable suitable habitat habitat is is an an essential essential first first step, account of result step, and and aa failure failure to to take take account of variation variation in in habitat habitat quality quality is is likely likely to to result in inadequate or unrealistic metapopulation studies (Thomas, 996; Dennis in inadequate or unrealistic metapopulation studies (Thomas, 11996; Dennis and and Eales, 11999; Thomas et et a!., al., 200 2001b; Fleishman et et a!., al., 2002) 2002).. Eales, 999; Thomas 1 b; Fleishman Once Once "suitable" "suitable" habitat habitat has has been been identified, identified, habitat habitat quality quality will will still still vary, vary, and may represent and some some occupied occupied patches patches may represent population population sinks sinks (Thomas (Thomas et et a!., al., 11996; 996; Thomas Thomas and Kunin, 11999). 999). One simple and approach is and Kunin, One very very simple and practical practical approach is to to modify the the true true areas of of habitat patches by by their quality quality (Hanski, (Hanski, 11994; modify 994;
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Moilanen 99 8 ) . For For example, Moilanen and and Hanski, Hanski, 11998). example, in in the the study study of of H. H. ccomma o m m a we we observed observed that that range range expansion expansion was was slower slower than than predicted predicted in in one one patch patch network network (outside (outside the the 95% 9 5 % confidence confidence limits limits of of simulated simulated expansion). expansion). In In these these networks, networks, habitat host plant habitat patches patches vary vary greatly greatly in in host plant density, density, largely largely due due to to the the manage management al. (unpublished ment histories histories of of the the sites. sites. Therefore, Therefore, R. R. J. J. Wilson Wilson et et al. (unpublished result) result) adjusted adjusted each each patch patch area area by by the the density density of of host host plant plant so so that that patches patches with with low low densities densities were were treated treated as as if if they they were were smaller smaller patches patches than than their their actual actual physical physical dimensions would dimensions would indicate, indicate, and and patches patches with with high high plant plant densities densities were were treated treated would have as larger larger than than their their physical physical size. size. Thereby Thereby low-quality low-quality patches patches would have as higher physical area higher rates rates of of extinction extinction than than other other patches patches of of the the same same physical area and and would would contribute contribute less less to to the the colonization colonization of of empty empty habitat habitat patches. patches. The The result result of of this this adjustment adjustment was was to to predict predict the the dynamics dynamics of of H. H. comma c o m m a accurately accurately for for the the deviating Section 20. 5 ) . This deviating patch patch network network as as well well as as for for the the other other networks networks ((Section 20.5). This example example relates relates to to host host plant plant density, density, but but it it would would be be equally equally simple simple to to apply apply the the approach approach to to any any other other measure measure of of relative relative patch patch quality. quality. Of Of course, course, this this sort sort of of approximation approximation will will not not work work in in all all situations, situations, and and it it may may conflict conflict with with the the need Chapter 4). need to to scale scale immigration immigration and and emigration emigration rates rates with with true true patch patch area area ((Chapter 4). Nonetheless, Nonetheless, we we think think that, that, in in many many or or even even most most cases, cases, relatively relatively simple simple adjustments models are adjustments to to existing existing models are likely likely to to provide provide greater greater insight insight into into the the key key dynamics populations than dynamics of of meta metapopulations than more more complicated complicated models models that that simultan simultaneously eously take take into into account account of of aa wide wide range range of of habitat habitat quality quality variables. variables. A A more more fundamental fundamental problem problem is is that that distribution distribution of of aa suitable suitable habitat habitat does does not remain remain constant constant through through time, time, particularly particularly in in species species that that track track the the distri distrinot bution 9 9 1 ; Thomas, 994a,b). In modern bution of of successional successional habitats habitats (Harrison, (Harrison, 11991; Thomas, 11994a,b). In modern landscapes, (due to habitat losses) landscapes, shrinking shrinking (due to habitat losses) and and expanding expanding (e.g., (e.g., due due to to climate change) patch cases in point. However, studies climate change) patch networks networks are are particular particular cases in point. However, studies reviewed models can reviewed in in this this chapter chapter demonstrate demonstrate that that metapopulation metapopulation models can be be applied successfully applied successfully to to these these systems, systems, provided provided that that one one knows k n o w s how how the the patch patch network network has has changed changed (documented (documented fragmentation) fragmentation) or or can can predict predict these these changes changes (based on, e.g., (based on, e.g., changing changing climate climate or or successional successional dynamics) dynamics).. In In the the situations situations examined, predictions 20- to examined, predictions were were reasonably reasonably accurate accurate over over 20to 30-yr 30-yr periods periods ((or or generations, generations, for for these these butterflies), butterflies), allowing allowing managers managers to to predict predict species species responses responses to to given given environmental environmental changes. changes. Whether Whether these these predictions predictions can can be be extended extended over over longer longer periods periods of of time time is is another another question. question. Today, Today, we we know know that that various various environmental environmental changes changes have have taken taken place place and and we we can can predict predict the the responses responses of of species species to to these these changes changes with with some some degree degree of of accuracy, accuracy, including including extinction extinction debts debts and and distributions distributions lagging lagging behind climate timescale of behind climate change. change. However, However, the the longer longer the the timescale of projection, projection, the the more more important important it it will will become become to to be be able able to to predict predict changes changes to to the the environment environment itself, itself, and and this this is is of of course course outside outside the the scope scope of of metapopu metapopulation lation biology. biology. Metapopulation Metapopulation projections projections can can be be applied applied to to scenarios scenarios of of hypothetical hypothetical environmental environmental changes, changes, but but they they can can never never be be better better than than the the environmental environmental projections projections on on which which they they are are based. based. The population The second second issue issue relating relating to to long-term long-term prediction prediction is is that that meta metapopulation processes processes are are stochastic, stochastic, which which will will limit limit predictability. predictability. Events Events that that are are rare rare may may nonetheless nonetheless be be critical critical determinants determinants of of large-scale large-scale metapopulation metapopulation dynamics, dynamics, and and no no models models can can ever ever predict predict particular particular realizations realizations of of such such processes processes accur accurately. ately. Models Models may may be be able able to to predict predict the the possibility possibility of of such such events, events, but but never never exactly "correct" predictions exactly when when and and where where they they will will occur. occur. Therefore, Therefore, even even "correct" predictions
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may may not not be be of of great great practical practical use. use. This This is is best best illustrated illustrated by by two two examples. examples. Metapopulation Metapopulation models models can can include include aa wide wide variety variety of of dispersal dispersal functions functions (see (see Box Box 4.3), 4.3), some some with with more more and and some some with with less less long-distance long-distance migration, migration, but but no no model model will will ever ever be be able able to to predict predict exactly exactly when when and and where where aa new new long-distance long-distance Section 20.5) is aa colonization colonization will will take take place. place. P. P. aegeria aegeria colonizing colonizing Inverness Inverness ((Section 20.5) is case was about about 100 nearest P. aegeria case in in point. point. Inverness Inverness was 100 km km from from the the nearest aegeria popula population 960s. Once tion when when it it was was colonized colonized in in the the late late 11960s. Once Inverness Inverness had had been been col colonized, possible to onized, however, however, it it was was possible to predict predict subsequent subsequent spread spread away away from from this this population focus population focus for for another another 50+ 50+ km, km, based based on on the the frequency frequency of of more more usual colonization events ((Hill al., 200 2001). Hill et aI., 1). Second, Second, rare rare climatic climatic events events may may wipe wipe out out large large and and vigorous vigorous metapopula metapopulations tions in in episodes episodes of of what what are are essentially essentially extremes extremes of of regional regional stochasticity. stochasticity. In In E. editha, an an exceptional exceptional summer summer frost frost killed killed off off all all populations populations in in forest forest clearings clearings within population (Thomas et 996), and within one one meta metapopulation et aI., al., 11996), and large-scale extinctions extinctions have also been caused by 980). If have also been caused by summer summer droughts droughts (Ehrlich (Ehrlich et et aI., al., 11980). If the the dis distribution tribution of of stochastic stochastic events events was was known, known, the the metapopulation metapopulation consequences consequences could be could be predicted predicted statistically, statistically, although although not not of of course course in in terms terms of of exact exact timing. timing. In In practice, practice, the the situation situation is is worse worse because because we we do do not not know know the the distribution distribution of of the the relevant relevant stochastic stochastic perturbations perturbations m the the longer longer we we observe observe any any system, system, the the more more we we know know of of ever ever rarer rarer events, events, and and these these may may be be the the ones ones that that eventually eventually cause cause the the demise demise of of the the population. population. Rare Rare events events in in space space and and time time do do not not under undermine metapopulation approach, approach, but mine the the overall overall utility utility of of the the metapopulation but they they limit limit the the spa spaand temporal temporal scales over which which it is practical practical to to make make population tial and population predictions. predictions. In application of metapopulation models models to In summary, summary, the the application of metapopulation to environments environments that that are are changing changing seems seems daunting. daunting. The The environments environments occupied occupied by by virtually virtually all all rare 00 yr, rare and and threatened threatened species species have have changed changed over over the the past past 1100 yr, and and in in most most cases cases are are likely likely to to continue continue to to do do so so in in the the future future as as aa result result of of continuing continuing land land use use and and climatic climatic changes. changes. Yet, Yet, the the application application of of metapopulation metapopulation models models to to butterfly butterfly distributions distributions and and dynamics dynamics has has survived survived this this test test well: well: butterfly butterfly responses accurately over responses to to environmental environmental changes changes have have been been predicted predicted accurately over periods periods of of up up to to 30 yr yr (generations). (generations). Road Road testing testing with with appropriate appropriate species species has population approach robust not has shown shown the the meta metapopulation approach to to be be sufficiently sufficiently robust not only only to to aid aid the the theoretical theoretical understanding understanding of of large-scale large-scale dynamics, dynamics, but but also also to to guide guide the the management populations and populations in management of of populations and meta metapopulations in practice practice over over time time periods periods of of several several decades. decades. Rare Rare stochastic stochastic events events may may make make longer longer term term predictions predictions less less certain, certain, but but high high predictive predictive power power over over several several decades decades is is likely likely to to be be useful useful for for most most management management decisions. decisions. As As aa final final caveat, caveat, we we do do not not claim claim that that all all of of the the processes processes relevant relevant to to the the regional regional persistence persistence and and dynamics dynamics of of species species are are encompassed encompassed within within simple simple metapopulation metapopulation models models such such as as IFM. population approaches IFM. Rather, Rather, meta metapopulation approaches capture capture aa sufficient sufficient amount amount of of the the dynamics dynamics of of many many species species to to provide provide both both useful useful insight insight and and practical practical tools tools for for conservation conservation planning. planning.
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INFERRING IN FERRIN G PATTERN AND PROCESS IN AND IN SMALL SMALL M M AAMMAL MMAL META PO PULATIONS M ETAPO PU LATI O N S:: INSIG H TS FROM INSIGHTS FROM ECOLOGIC AL AND AND ECOLOGICAL GENETIC ATA G EN ETIC D DATA Xavier Lambin, Jon Jon Aars, Stuart Stuart B. Piertney, and and Sandra Telfer
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INTRODUCTION INTRODUCTION The population paradigm The meta metapopulation paradigm is is increasingly increasingly being being used used to to describe describe the the structure structure and and dynamics of mammalian populations, populations, as it is with with many other other taxonomic groups. Previous Previous reviews reviews have, have, however, however, highlighted highlighted that that meta metapopulation population terminology terminology in in mammals is is used used to to define define aa number number of of different different population 99 1 ; Harrison and Taylor, 11997) 997) and observed population structures structures (Harrison, (Harrison, 11991; that that case studies studies of of classical classical metapopulation metapopulation structures structures are are rare rare (Elmhagen (Elmhagen and Angerbjorn, 1 ) . Most frequently, Angerbjorn, 200 2001). frequently, the the observed observed pattern pattern is is of of aa "population "population of of populations," populations," or or aa suite suite of of local local populations, populations, that that is is not not always always discrete discrete but but where where limited limited dispersal dispersal gives gives rise rise to to discontinuous discontinuous spatial structure structure
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(e.g., Szacki, 11999; 999; Entwistle Entwistle et aI., al., 2000; 2000; Gaggiotti Gaggiotti et aI., al., 2002). 2002). Metapopu Metapopustructure is also invoked frequently frequently to describe those those sets of of populations populations lation structure where where each each population population has has become become effectively effectively isolated isolated with with no no potential potential for for the the rescue effect, and so are ultimately destined 97 1 ) . The destined for extinction extinction (Brown, (Brown, 11971). aforementioned aforementioned structures structures represent represent the the extremes of aa continuum continuum of habitat habitat fragmentation. population turnover fragmentation. Determining Determining whether whether the processes of population turnover and and dispersal between patches patches have a strong strong influence on on overall persistence in a habitat habitat with with a given degree of of fragmentation fragmentation requires information information on the dispersal dispersal behavior behavior of of individuals, individuals, as well well as as on on the the processes processes responsible responsible for for the the extinction extinction of local populations. populations. Much Much progress progress has been made made in understanding understanding the the process process of of dispersal with with small small mammals, but but extinction extinction processes remain poorly poorly understood. understood. processes Relatively few studies to to date date have have compellingly characterized characterized metapopula metapopulation tion processes in small mammals. This This begs the question question whether whether metapopu metapopulation processes processes are not not being observed because studies studies focus at at inappropriate inappropriate temporal scales or whether whether small mammal populations populations do not not spatial or temporal inherently show such processes. processes. A short short life span, specific habitat habitat requirements ((or or at at least preferences) preferences),, and and local organization into social units or or habitat habitatdefined aggregations may intuitively suggest that that small mammals will form meta population structures. metapopulation structures. Conversely, however, mammals may fail to to form form metapopulations 1 ) species with specific habitat metapopulations because because ((1) habitat requirements requirements may have sufficiently sufficiently high high mobility mobility relative to to the the grain of landscapes landscapes that that the the dynamics of local populations populations is typically synchronized; (2) the frequency of population population turnover turnover may be much much reduced by effective and and targeted targeted immi immigration populations ((Stacey Stacey et aI., 1 997); or or (3) local gration from from adjacent adjacent local populations al., 1997); population size in small mammals may normally be so large or local populapopulation popula tions tions so well connected connected as to make make extinction through through demographic demographic stochas stochasticity and and recolonization recolonization dynamics insignificant, except except when when larger larger scale dynamics processes bring about regional declines. Two separate reviews in Two separate Hanski 1 997), of largely the same empirical studies Hanski and and Gilpin ((1997), studies of of small mammals, reach consensus whether meta population structure mammals, failed failed to to reach consensus on on whether metapopulation structure should be considered considered the norm should norm in small mammals. Stacey et al. ai. ((1997) 1 997) wrote wrote ""Many Many diverse species of small mammals may be predisposed predisposed towards towards metapopulations metapopulations because they they show show spatial spatial population population structure structure as as aa result of metapopulations, local sociality or or habitat habitat fragmentation. fragmentation. In such rescue-effect metapopulations, populations independently of low to populations fluctuate fluctuate fairly independently of one another, another, yet exchange exchange low moderate that metapopulation metapopulation structures moderate numbers numbers of immigrants immigrants such that structures have an an important population important stabilizing effect at at the the regional level even without without population turnover. " However, Harrison 1 997) stated turnover." Harrison and and Taylor ((1997) stated that that they knew knew of of no no good where metapopulation important stabilizing good example example where metapopulation structure structure has has an an important stabilizing without population turnover. This lack of consen effect at at the the regional regional level without population turnover. consensus sus reflected reflected the the absence absence of of any any compelling compelling empirical empirical study study of of small small mammal mammal metapopulations available at both the processes of metapopulations at that that time that that documented documented both dispersal population dispersal and and population population turnover. Thus Thus deciding whether whether meta metapopulation structure structure even existed among among small mammals and and deriving generalities as to to their thus timely to their frequency frequency was was a matter matter of of judgment. judgment. It is thus to ask whether whether new evidence has come to to light to to resolve the apparent apparent contradiction contradiction between ai. ((1997) 1 997) and 1 997) . This chapter first Stacey et et al. and Harrison Harrison and and Taylor Taylor ((1997). first briefly briefly reevaluates population structures three other reevaluates the the evidence evidence of of meta metapopulation structures for for three other small
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mammal species considered considered ttoo conform conform most most closely ttoo the meta metapopulation mammal population paradigm Cynomys ludovicianus), American paradigm m black-tailed black-tailed prairie prairie dogs ((Cynomys American pika ((Ochotona Ochotona princeps), princeps), and and field voles (Microtus (Microtus agrestis). agrestis). Next, Next, the chapter chapter introduces introduces new new studies studies of water water voles (Arvicola terrestris) terrestris) that that contain strong strong evidence of meta population processes and metapopulation and progresses our understanding understanding of metapopulation processes. metapopulation processes. Throughout, Throughout, ecological and and genetic data data are con considered that that can be used to identify metapopulation metapopulation patterns patterns in small mam mammals. We also ask whether whether common common features features of small mammal mammal dispersal dispersal and and of the the processes processes responsible responsible for for extinction extinction of small mammal mammal local populations populations result in a distinct distinct mammalian mammalian perspective of the the metapopulation metapopulation paradigm. paradigm.
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CLASSICAL CLASSICAL SMALL SMALL MAMMAL M A M M A L METAPOPULATIONS METAPOPULATIONS A small number number of well-documented well-documented studies have proven proven highly influential influential in suggesting and subsequently subsequently understanding understanding metapopulation metapopulation processes in small mammals. mammals. With increasingly detailed study incorporating incorporating both both ecological and and genetic genetic data, data, these investigations investigations also highlight highlight common common problems problems associ associated with with understanding understanding mammalian mammalian metapopulation metapopulation dynamics, dynamics, namely namely that that it not necessarily straightforward straightforward to to infer infer process process from from pattern. pattern. is not
American Pika at A m e r i c a n Pika at Bodie Bodie Pikas Pikas are are small small lagomorphs lagomorphs closely closely associated associated with with patchily patchily distributed distributed rocky outcrops. outcrops. In their their usual high-altitude high-altitude habitat, habitat, patches patches are typically large and 00 individuals. and populations populations semicontinuous semicontinuous comprising comprising up up to to 1100 individuals. Smith ((1974) 1 974) reported reported that that such such populations populations rarely experience extinctions. extinctions. At At aa single single lower lower altitude altitude site, the Bodie mining area (ca. 12 km km 22)) in California, California, popula population tion turnover turnover does occur, and this this set of populations populations has been presented presented as the best example of a classical meta population in small mammals metapopulation mammals (Moilanen (Moilanen et ai., al., 11998). 99 8 ) . Evidence shows shows that that migration migration maintains maintains genetic diversity and and rescues local populations Smith and 997; Stacey et ai., 997; populations from from extinction extinction ((Smith and Gilpin, 11997; al., 11997; Peacock 1 ). Pikas defend Peacock and and Ray, 200 2001). defend individual individual territories, territories, such that that upon upon weaning weaning all juveniles must must either inherit inherit or disperse disperse from from their their natal natal territory. unusually inflexible link between the size of a patch patch and and the number There is an unusually number of individuals individuals it can accommodate accommodate and acquiring acquiring an individual individual territory is a key transition transition in pika life life history. Vagility is reduced by low tolerance tolerance to daytime daytime high temperature temperature at Bodie which lies at the lower lower altitudinal limit of the species (Smith, 11974). 974). Unless a territory territory is vacant, vacant, residents aggressively inhibit immi immigration and the immigration rate rate is thus thus density dependent. In addition, addition, immi imminext to grants settle settle preferentially preferentially next to opposite-sex opposite-sex conspecifics. conspecifics. Adjacent Adjacent territories territories are are typically typically occupied occupied by by opposite opposite sex sex individuals individuals and, and, in in the the study study of Smith and 1 984), territory disappeared were always and Ivins ((1984), territory owners owners that that disappeared replaced replaced by members members of the same sex. Most Most young young pikas are philopatric philopatric and and settle settle in vacant vacant territories territories in the natal natal patch. patch. Dispersal between between patches patches also occurs, although although at low frequency, and and tends to be directed directed toward toward the closest patch 997; Peacock and 1). patch with with vacancies vacancies (Peacock and and Smith, 11997; and Ray, 200 2001). Repeated Repeated surveys surveys ooff patch patch occupancy occupancy bbyy pikas pikas at at Bodie Bodie spanning spanning 2200 years years reveal a sharp sharp decline in the southern southern part part of the patch patch network network and and a more more
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part (Smith, 11974; 974; Smith and constant rate of patch patch occupancy in its northern northern part 997). Moilanen Smith and 1 99 8 ) used these data Gilpin, 11997). and Hanski Hanski ((1998) data to para parameterize Hanski's 1 994) incidence function Hanski's ((1994) function model (IFM) assuming a classical meta population structure metapopulation structure and and including including the the rescue rescue effect effect (Chapter (Chapter 4). 4). The The parameterized parameterized model successfully successfully predicted predicted observed observed spatial spatial variation variation in in the the extent extent of of the the decline, decline, suggesting suggesting that that the the dynamics dynamics is is consistent consistent with with that that of of aa meta population in metapopulation in aa constant constant environment. environment. When When low low levels of of region-wide region-wide environmental stochasticity, implemented as a synchronized variation in the size of of all patches, patches, hence, indirectly, indirectly, as as variation variation in in extinction extinction probability, probability, were were included, model iterations iterations suggested that that the southern southern part part of the metapopu metapopulation was extinction extinction prone. Model simulations thus thus showed showed that that the observed spatially spatially correlated correlated pattern pattern in in patch patch occupancy occupancy in in pika pika at at Bodie may may have have arisen arisen from from extinction-colonization extinction-colonization dynamics dynamics without without the the need need to to assume assume that that any any spatially spatially correlated correlated processes processes affected affected the the local local dynamics dynamics and population population extinction 99 8 ) . extinction in in aa subset subset of of patches patches (Moilanen (Moilanen et et a!., al., 11998). contrast ttoo the interpretation interpretation that that the Bodie pika population population persists IIn n stark contrast extinction-recolonization (Moilanen (Moilanen et a!., al., 11998), in a quasistable balance of extinction-recolonization 998), Clinchy Clinchy et et a!. al. (2002) (2002) suggested suggested that that the the observed observed dynamics dynamics of of pika pika at at Bodie Bodie could could equally equally well well be be explained explained by by spatially spatially correlated correlated extinction extinction caused caused by by predation predation in in aa slowly slowly declining declining fragmented fragmented population. population. Predation Predation by by mustelids mustelids death and and local population population extinction extinction (Smith and and Gilpin, is the main cause of death 11997). 997). With With such small and and mobile predators, predators, individuals in small local populations populations are expected to be exposed to a higher higher risk of predation predation mortality than populations. In addition, addition, the fate of adjacent adjacent than individuals in large local populations. local populations potentially simultaneously exploited by a predator predator may become spatially correlated. correlated. In support support of their their argument, argument, Clinchy et a!. al. (2002 (2002)) presented simulations simulations that that assumed assumed spatially spatially correlated correlated extinctions extinctions caused by by a mustelid in the Bodie patch network, network, but did not include dispersal or other demographic processes. The simulations suggested that that more more isolated patches patches became became both both less likely likely to to be be occupied and and more more likely likely to to go go extinct extinct under under the the influence influence of of spatially spatially correlated extinction. extinction. Similar Similar patterns patterns of of patch patch occupancy occupancy are observed at Bodie and, more generally, predicted predicted by distance-dependent distance-dependent dispersal and a!. (2002) interpreted and the rescue effect. Clinchy et al. interpreted their simula simulations tions as showing showing that that recolonization recolonization of vacant vacant rock piles and and the the rescue rescue effect played only minor minor roles in the overall pika dynamics at Bodie. They noted that that recolonized recolonized patches were significantly more likely to go extinct than patches patches that that were occupied in both of the preceding preceding surveys. Clinchy et a!. al. (2002) interpreted interpreted this this pattern pattern as evidence that that extinctions extinctions are a direct result of recolonization. recolonization. That That colonizations colonizations often often fail is not not implausible for for pika because of their low fecundity fecundity and and the requirement requirement for for colonists to accumulate accumulate sufficient sufficient plant plant material material in in hay hay piles before before winter. winter. The a!. ((1998) 1 99 8 ) and a!. (2002) The contrast contrast between Moilanen Moilanen et al. and Clinchy et al. interpretations interpretations of of the the same same data data shows shows that that it it is is difficult difficult to to infer infer process process from from pattern. pattern. Neither Neither model captures all patterns patterns in the data data and, in the real world, world, processes may mask each other. For instance, behavioral evidence of the rescue effect, such as dispersal directed at low-density patches where where opposite opposite sex relatives relatives are present, present, may not not translate translate into into a detectable detectable impact on extinction probability probability when when predator-induced predator-induced extinctions extinctions dominate. dominate. In In addition, addition, little little is known known about about the dispersal behavior behavior of pika and how how it accords with with the
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relationships used iin n the IFM. There are, however, more serious flaws with the Clinchy et al. (2002) argument. It implausibly assumes a mustelid capable of causing a 20-yr declining trend in pika but not overall extinction and causing correlated extinction extinction in the southern half of the network without without any notice noticeable able influence on its northern northern part. It also fails to consider the likely relation relationprobability of extinction extinction and patch quality and/or and/or size that that may ship between probability account for the observed relationship relationship between extinction extinction and recolonization. recolonization. inconsistent with with the observa observaThe assumed insignificance of dispersal is also inconsistent tion of Peacock and Ray (200 1 ) that (2001) that pika in Bodie had had a level of heterozygosity as high as that that observed in a more continuous continuous habitat. habitat. This level of genetic diversity is consistent consistent with a pattern of extinction-recolonization extinction-recolonization within Bodie combined combined with rare immigration immigration from from distant distant continuous continuous populations populations (Peacock and Ray, 200 2001). (Peacock 1).
Black-Tailed Prairie Prairie Dogs Dogs Black-Tailed According to Stacey et ai. 1 997), studies of black-tailed al. ((1997), black-tailed prairie dogs dogs provide provide detailed and metapopulations with a strong rescue and convincing evidence that that metapopulations effect may be common common among among social mammals. mammals. Black-tailed Black-tailed prairie dog live socially in towns towns (local populations) populations) of extended extended families. Early descriptions descriptions suggest that that black-tailed prairie dog towns towns stretched stretched over extensive areas of land and thousands of individuals. and consisted of tens of thousands individuals. As a result result of human human distur disturbance and and the destruction destruction of their habitat, habitat, prairie prairie dogs over most of their range now live live in small, scattered, and and ephemeral local populations populations (Halpin, 11987; now 987; Lomolino 1 ; Antolin Lomolino and Smith, 200 2001; Antolin et aI., al., 2002). 2002). The The decline has been been most most pronounced United States where where an an introduced introduced pathogen, pronounced in the western western United pathogen, Yersinia pestis, the to which which prairie prairie dogs are highly the agent of sylvatic plague to susceptible, 1 930s (Antolin et al., aI., 2002). susceptible, has become established starting starting in the 1930s 2002). Black-tailed studied most Black-tailed prairie prairie dogs have been studied most intensively in plague-free Wind Dakota, where colonies Wind Cave National National Park, South South Dakota, colonies are still large and Franklin, 1988; Hoogland, 1995). 1 995 ). Behavioral studies of (e.g., Garrett Garrett and 1988; Hoogland, (Garrett and dispersal in this area show very strong strong conspecific conspecific attraction attraction (Garrett 1 98 8 ) . Trajectories Franklin, 1988). Trajectories of radio-tracked dispersers were meandering meandering when away became directed when in sight of when away from from towns, towns, but but became directed when of towns. towns. All intertown 1 988) immigrated immigrated intertown dispersers dispersers in the study study by Garrett Garrett and and Franklin Franklin ((1988) into town. This was despite being the the target outright hostility hostility by into existing existing town. target of outright residents that that resulted resulted directly or or indirectly indirectly in the the death death of of some some newly residents established dispersers. Such behavior behavior by dispersers dispersers would would not not be conducive conducive established to to recolonization. recolonization. Studies of black-tailed an area area where plague is established established show show Studies of black-tailed prairie prairie dogs in an where plague that towns towns have have become become fragmented fragmented and and go extinct asynchronously, coinciding coinciding that extinct asynchronously, with plague epizootics epizootics decimating towns. Other Other causes causes of of extinction extinction include include with decimating towns. stochastic factors as well as persecution. Prairie dogs appear appear to to now now persist persist in stochastic factors persecution. Prairie a metapopulation-like metapopulation-like state state (Cully and and Williams, Williams, 2001; 2001; Roach Roach et al., aI., 2001; 200 1 ; Antolin eett al., aI., 2002). 2002). Despite Despite their their strong strong conspecific conspecific attraction, attraction, dispersing dispersing Antolin prairie dogs aggregated in and and recolonized recolonized an an empty empty habitat habitat following following a poisonpoison prairie ing campaign campaign (Cincotta (Cincotta et et al., aI., 1987). 1 987). Colonization Colonization of of towns towns was was evident in ing evident in plague-infected areas, areas, and and genetic genetic data data also also demonstrated demonstrated a substantial substantial dispersal dispersal plague-infected between existing existing towns towns (see also also Halpin, Halpin, 1987), 1 9 8 7), consistent consistent with with the the pattern pattern of of between
XAVIER LAMBIN LAMBIN ET ET AL. XAVIER
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attraction documented more saturated saturated environment environment (Garrett (Garrett and and conspecific attraction documented in a more Franklin, 1988). 1 98 8 ) . Franklin, Plague is i s epizootic epizootic not not enzootic enzootic in i n prairie prairie dogs dogs due due to to their their high high suscepsuscep Plague tibility and and is maintained maintained within within a community community of of more more resistant resistant rodent rodent and and tibility carnivore species whose whose fleas fleas reinfect reinfect prairie prairie dogs dogs (Biggins (Biggins and and Kosoy, 2001; 200 1 ; carnivore and Williams, Williams, 2001). 200 1 ) . During During epizootics epizootics in prairie prairie dogs, dogs, however, direct direct Cully and transmission between between prairie dogs during during their their social encounters encounters takes takes place place transmission (Antolin et al., aI., 2002). 2002). Cully and and Williams Williams (2001) (200 1 ) hypothesized that the the transtrans (Antolin hypothesized that plague is facilitated the social cohesiveness of of colonies such that, that, mission of plague facilitated by the though all local populations, populations, regardless regardless of of size, are are vulnerable vulnerable to to extincextinc even though tion from from plague, large towns towns may may be more more susceptible, susceptible, a pattern pattern opposite opposite that that tion demographic stochasticity. The The pattern pattern of of extinction extinction reported caused by demographic reported in Roach et al. aI. (2001) (20 0 1 ) is consistent consistent with with this. this. Roach Plague progressed only gradually over 3 years years in three adjacent colonies colonies Plague progressed gradually over three adjacent spanning km (Cully and and Williams, Williams, 2001). 2 00 1 ) . Whether Whether this this pattern pattern of of slow slow spanning 30 km spatial moderately spatially spatial spread spread and and moderately spatially correlated correlated extinction extinction was was caused caused by prairie dogs more complex process of amplification amplification dispersal by by infective infective prairie dogs or or a more of and dispersal of the disease in the environment, environment, infection infection and dispersal by resistant hosts and not known known and their their fleas fleas and and reinfection of of adjacent prairie prairie dogs dogs towns towns are not ((Cully Cully and 1 ; Biggins and and Kosoy, 2001). 200 1 ) . Without Without a better better underunder and Williams, 200 2001; standing populations, it not standing of of the the role of of pathogens pathogens in prairie prairie dogs dogs meta metapopulations, it is is not possible to to establish whether whether long-term long-term coexistence with with plague is possible for highly social prairie prairie dogs. dogs.
Field Archipelago Field Voles V o l e s in in Baltic Baltic A rchipelago A A 6-yr study of field voles (M. agrestis) agrestis) living on islands islands in the the Tvarminne Tv~irminne Archipelago of the southeast of Finland's mainland (Pokki, 11981; 9 8 1 ; Crone et aI., al., 2001) mammal population population closely 200 1 ) represents a compelling example of a small mammal approximating population structure. approximating a meta metapopulation structure. Patterns Patterns of occupancy, changes changes in abundance, abundance, and withinwithin- and between-island dispersal by field voles were monitored hal to larger monitored on 71 islands ranging in size from from treeless skerries «(< 11 ha) islands ((>10 > 1 0 ha) hal with with heterogeneous heterogeneous vegetation including old fields, forested area, and less suitable heath. Levels of fragmentation fragmentation have not been modified by anthropogenic influences, although the invasion by American mink mink (Mustela (Mustela vison) vison) over the last 50 years may have had some impact impact on vole dynamics ((Banks Banks et aI., al., unpublished unpublished results). On treeless skerries, skerries, vole abund abundance mirrored the seasonal development development of the vegetation. Vole populations grew grew during the early summer flush of vegetation growth, growth, and voles on skerries had higher maturation maturation rates, rates, litter size, and mean densities than on larger larger islands islands during during this this period. period. However, However, skerry skerry populations populations invariably invariably declined in late summer summer when when favored plant species were were largely consumed or or wilted. These declines resulted from both higher mortality and higher emigration movement to other islands. On larger islands, populations peaked in autumn and voles responded to seasonal crowding by intraisland movements from preferred old-field and meadows habitat habitat to suboptimal heath and forests, with relatively low rates of interisland movements. Despite the proximity of the mainland (less (less than 3 km from from the outermost islands), there there was little evidence that it was the primary source of colonists for unoccupied islands and
21.. SMALL SMALLMAMMAL MAMMAL METAPOPULATIONS METAPOPULATIONS 21
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that the the archipelago was part part of an island-mainland metapopulation ((Crone Crone al., 200 2001). et ai., 1). Extinctions Extinctions and and recolonizations recolonizations ooff local local island island populations populations were were common common and affected all island size categories. categories. Extinction rates rates were were similar for large large and medium medium islands, islands, but but populations populations on on small small islands islands turned turned over over at at aa much much higher higher rate than on larger islands. Despite the high extinction extinction probability on skerries, seasonal extinction extinction was not fully deterministic. deterministic. The median period of occupancy for skerries was 2 yr. Only 1i of 113 3 large islands monitored was occupied for all 6 yr. It is suggested that two mechanisms were important for local population population extinction: demographic stochasticity caused by low population population sizes on all islands (mean population size on skerries in May was <9 <9 voles, with with mean peak size of 34.9 in July) and environmental variability, driven driven by annual variation variation in population population density and rainfall, reducing carrying capacity on skerries (seasonal reduction in food availability). Interannual variation in rainfall must also have affected affected the the development development of of vegetation vegetation on on all all islands islands simultaneously simultaneously and and is is therefore therefore a spatially correlated contributor contributor to to extinction (Banks et ai. al. unpub unpublished results). The relationship relationship between island extinction probability and island size neatly matched Crone et ai., 1 ) , as expected matched the exponential function function ((Crone al., 200 2001), when 993; Ray, 200 1). when demographic stochasticity is the main influence (Lande, 11993; 2001). feature ooff the archipelago meta metapopulation that a substantial substantial A distinctive feature population was that fraction of extinctions were caused by deteriorating environmental conditions, both high mortality and and emigration emigration (Pokki, 11981). leading to both 9 8 1 ). fraction of voles from from skerries than than from from large islands islands A much higher fraction dispersed dispersed between islands. Field voles on large large islands dispersed dispersed within-island within-island to suboptimal habitats habitats in late summer. As suboptimal habitats habitats were were absent absent from small islands, on small islands attained densities at from islands, field vole populations populations on islands attained densities at least twice as high as typical peak densities in productive peak densities productive mainland mainland habitat habitat (Myllymfiki, al., 1992). escaped overcrowding and (Myllymaki, 1977; 1 977; Agrell et ai., 1 992 ). Field voles escaped overcrowding and a deteriorating across open open water. deteriorating environment environment on on small islands by swimming swimming across Immigration was disproportionately disproportionately directed toward toward larger islands. islands. Unlike mainland dispersal (Sandell ( Sandell et ai., mainland field voles that that have strongly strongly sex-biased dispersal al., 11990, 990, 11991), 99 1 ), there was no significant the was no significant sex sex bias bias among among dispersers dispersers in the Archipelago, nor was was dispersal restricted Dispersal Archipelago, nor restricted to to immature immature individuals. Dispersal was thus causing colonization. colonization. Contrary Contrary to the the standard standard assumption assumption was thus effective in causing of population models models that of meta metapopulation that there there is either either no, or or a negative, negative, relationship relationship between population extinction probability and and its production production of of between a local local population extinction probability emigrants, responded to to deteriorating deteriorating conditions conditions by by dispersing. dispersing. Per emigrants, field voles responded may have have been been positively related related to to extinction rate (Crone ( Crone capita emigration emigration may extinction rate et al., ai., 2001). 200 1 ) . the Tvfirminne Tvarminne metapopulation metapopulation may b described aass a combination combination of of Overall, the bee described persistent, low-emigration local populations populations and and ephemeral high-emigration persistent, populations. It meets all criteria to to be considered considered a classical metapopumetapopu sublocal populations. per capita capita dispersal of of voles inhabiting inhabiting small islands increased increased the the lation. A high per importance of of small islands in the the metapopulation. The The contrast contrast in disdis relative importance persal contributed to persal dynamics between large large and and small islands contributed to maintaining maintaining asynchrony in vole dynamics between of different different sizes in the the face face of of a asynchrony between islands of emi synchronizing climatic influence. influence. The The important important inference inference that that per capita emisynchronizing gration gration was was related related positively to to extinction extinction rate rate stems stems from from fitting fitting a modified modified incidence incidence function function model model to to occupancy occupancy data data (Crone ( Crone et et al., ai., 2001). 200 1 ) . There There are, are,
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XAVIER LAMBIN LAMBIN ET ET AL. AL. XAVIER
however, strong strong correlations correlations among among parameters parameters when when fitting fitting the the IFM, IFM, and and future future however, analyses should should verify this insight. insight. In In addition, addition, despite despite evidence of of life history history analyses in a similar island island archipelago archipelago in in Sweden Sweden (Ebenhard, (Ebenhard, 1990), 1 990), whether whether evolution in tendency has has been been subjected subjected to to different different selection pressures pressures than than on on dispersal tendency the mainland is not not known known but but would would provide provide a useful useful test test of of model model predictions predictions the (e.g., Gandon Gandon and and Michalakis, Michalakis, 1999). 1 999). (e.g.,
2211. .3 3
WATER VOLE METAPOPULATIONS W ATER V OLE M ETAPOPULATIONS Given Given the the lack of of consensus consensus on on the the very existence existence of of metapopulation metapopulation strucstruc tures in in small mammals, mammals, we first first describe how how water water voles approximate, approximate, and and tures sometimes deviate from, from, a classical metapopulation metapopulation structure. structure. Our Our studies studies of of sometimes metapopulation dynamics dynamics and and dispersal dispersal in water include both both naturally naturally metapopulation water voles include and anthropogenically anthropogenically fragmented fragmented populations, populations, broken broken up up over over a range range of of and and encompass encompass variation variation in patch timescales, and patch size, size, levels of of dispersal, dispersal, and and population isolation. By considering the dynamics of the the same species population isolation. considering the dynamics of species in a range situations, we control control for for potentially potentially confounding factors range of ecological situations, confounding factors such structure and and inherent inherent dispersal ability. This section describes such as social structure describes in how ecological and and genetic have helped helped characterize characterize in detail detail how genetic approaches approaches have metapopulations and shows that that similar processes operate contrasting these metapopulations and shows operate in contrasting habitats. habitats.
Water W a t e r Vole V o l e Habitats Habitats Water Water voles voles are are exceptionally exceptionally large large microtines, microtines, with with aa mass mass reaching reaching 300 300 gg in in Britain. Britain. British populations populations are are confined confined to to waterway waterway edges, occupying aa range range of of habitats habitats from from upland upland streams to to agricultural ditches and and wide wide rivers. rivers. Reproduction Reproduction in in the the year year of of birth birth is is much much less less frequent frequent than than in in smaller smaller microtines. Over the last 50 yr, water water voles have suffered suffered a catastrophic catastrophic decline decline caused caused primarily by by Britain's Britain's invasion by by the the American American mink, mink, an an generalist predator predator able to enter enter its burrow burrow systems. Estimates of efficient generalist the 8 % of 950s the magnitude magnitude of of the the decline decline vary vary geographically, geographically, but but reach reach 998% of the the 11950s population Strachan et aI., population level in some some regions ((Strachan al., 2000 2000).). The distribution distribution of water water voles voles along along waterways waterways is is generally generally fragmented, fragmented, irrespective irrespective of of the the effects effects of Stoddart, 11970; 970; Lawton roffe, 11991; 9 9 1 ; Aars of mink ((Stoddart, Lawton and and Wood Woodroffe, Aars et et aI., al., 2001; 2001; Telfer 1 ), although Telfer et et aI., al., 200 2001), although large large and and more more continuous continuous populations populations can can occur 999). occur along along productive productive lowland lowland rivers rivers (Macdonald (Macdonald and and Strachan, Strachan, 11999). Fragmented populations (Lawton Fragmented populations populations are are thought thought to to function function as as meta metapopulations (Lawton and 9 9 1 ; Aars 1 ; Telfer 1). and Woodroffe, Woodroffe, 11991; Aars et et aI., al., 200 2001; Telfer et et aI., al., 200 2001). We We have have intensively intensively studied studied several several networks networks of of populations populations from from both both low low productivity upland upland moors and higher productivity lowland farmland areas within Scotland (Fig. 1 . 1 ): ((1) 1 ) four networks in the mountainous far northwest (Fig. 221.1): of of Scotland Scotland (Upland (Upland Assynt Assynt hereafter), hereafter), which which lie lie to to the the north north of of the the American American mink mink invasion invasion front. front. It It is is reasonable reasonable to to assume assume that that these these populations populations fluctuate fluctuate around around their their long-term long-term equilibrium equilibrium state: state: (2) (2) Two Two networks networks in in the the mountains mountains of of northeast where mink northeast Scotland Scotland (Upland (Upland Grampians) Grampians)where mink are are present present but but distributed distributed patchily 998). Water patchily (Lambin (Lambin et et aI., al., 11998). Water voles voles were were almost almost completely completely extirpated extirpated from 999, leaving from one one of of these these blocks blocks in in aa localized localized mink mink advance advance in in 11999, leaving the the second second
221. 1 . SMALL MAMMAL METAPOPULATIONS
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Figure . 1 Maps study areas Assynt and and Maps showing showing water water vole vole metapopulation metapopulation study areas in in Assynt Figure 21 21.1 Grampians the lowland lowland Keithfield Keithfield Burn Burn and Island study study Grampians Mountains Mountains of of upland upland Scotland Scotland and and the and Island delineated by areas. Survey blocks blocks are delineated by boxes. boxes. Areas Areas shaded shaded gray are are mountains, mountains, those those filled filled are are lochs. In In the the lowland Burn area, thick lines lines denote denote sections occupied in Iochs. lowland Keithfield Keithfield Burn area, thick sections of of waterways waterways occupied in by water arrow indicates the the downstream downstream flow of the Note at least 11 yr by water voles. The arrow flow of the River River Ythan. Ythan. Note that the four four areas are not not drawn drawn to that the to scale.
block 20 km to to the the East unaffected: ((3) lowland farmland 20 km 3 ) One One network network in a lowland farmland area of northeast northeast Scotland Scotland (Lowlands). This area area was first colonized by mink up to of was first to 30 yr ago, but but clusters of fragmented populations populations of water water voles still persist in upper tributaries tributaries that that are reached reached only occasionally by mink from from the main main river (Telfer (Telfer et al., aI., 2001): 200 1 ): (4) Island populations populations in the the Sound of of Jura, Jura, southwest southwest coast of of Scotland, where where water water voles are are not not associated with with waterways waterways but but live fosfos sorially as in continental continental Europe Europe (Telfer (Telfer etal., et aI., 2003a). 2003a). The The past past history history of founder founder effect) effect) of of water water voles on on these these islands is unknown, unknown, but but (e.g., degree of population assignment and and genetic infer negligible migration between population genetic analyses infer migration between island groups so they may (Telfer et al., may be distinct distinct populations populations (Telfer aI., 2003a). 2003a). Whether or moorland, Whether in farmland farmland or moorland, water water voles voles select narrow narrow watercourses watercourses with with slow-flowing slow-flowing water water and and banks banks suitable suitable for for burrowing burrowing and and with with abundant abundant vegetation (Aars et al., aI., 2001; 200 1 ; Telfer et al., aI., 2001). 200 1 ) . These are often often in headwaters headwaters vegetation These are of tributaries tributaries such such that that local populations populations in in adjacent adjacent watersheds watersheds may be in in close of proximity proximity (Fig. 21.2). 2 1 .2). The The distribution distribution of of water water vole vole habitat habitat patches patches differs differs between between areas. areas. In upland upland areas, areas, suitable suitable habitat habitat exists exists as as distinct distinct patches patches of of grass (Aars et grass surrounded surrounded by wholly wholly unsuitable unsuitable heather heather moorland moorland (Aars et al., aI., 2001). 200 1 ) . The off patches The boundaries boundaries o patches in in the the productive productive lowlands lowlands are are less less well well defined defined and and the the probability probability of of occupancy occupancy of of a section section of of waterways waterways is only only predictable predictable from from a combination combination of of habitat habitat features features (Lawton ( Lawton and and Woodroffe, Woodroffe, 1991; 1 9 9 1 ; Macdonald Macdonald and and Strachan, Strachan, 1999; 1 999; Telfer Telfer et et al., aI., 2001). 200 1 ). Changes Changes in in water water vole vole distribution distribution may centered on may consist consist of of both both expansions expansions and and contractions contractions of of populations populations centered on
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Figure .2 CChanges in colony colony size size for for the the four four blocks blocks in in the the Upland on the the Figure 21 21.2 h a n g e s in U p l a n d Assynt Assynt area area are are on first those from from the the Upland Grampians are on the second second line, line, and and those those for for the the lowland lowland first line, line, those Upland G r a m p i a n s are on the area are third line. Circle sizes reflect colony and values values on on the the axis distance in area are on on the the third line. Circle sizes reflect colony size, size, and axis are are distance in kilometers. kilometers.
the most recolonization of most suitable suitable habitat in addition to the extinction extinction or recolonization of distinct sections of distinct sections of waterways. waterways.
Population In Water P o p u l a t i o n Turnover T u r n o v e r in W a t e r Vole V o l e Metapopulatlon Metapopulation Water vole populations populations in both both lowlands lowlands and uplands have a fragmented distribution with significant turnover between years (Figs. 2 1 .2 and 21.3). 21.2 21.3). The lowland network included 22 local populations 996 and 11999, 999, occu populations between 11996 occuwaterways. Assuming this represents the length of suitable habi habipying 21 km of waterways. tat, yearly occupancy rates ranged from 57 to 67% (Fig. 21 .2; see also Telfer 21.2; Teller et 1 ) . Water 7% of et al., al., 200 2001). Water voles voles were were lost lost from from between between 1155 and and 117% of waterways waterways each year with similar lengths being recolonized (Telfer (Teller et al., 2001 2001).) . The proportion of suitable patches occupied in the uplands is more variable both spatially and between between years. Observed occupancy occupancy per survey survey block in Upland 62% 21.3). Assynt ranged from 0 to 62 % (Fig. 2 1 . 3). The Upland Grampians patch network 1 7-53 % , Fig. network had had low low occupancy occupancy and and recolonization recolonization rates rates ((17-53%, Fig. 21.3). 21.3). Water 5 surveyed Water voles voles were were present present on on 55 of of 115 surveyed islands islands in in southwest southwest Scotland Scotland (Table 1 . 1 ) . No 996, but (Table 221.1). No extinction extinction has has been been observed observed since since 11996, but two two presently presently unoccupied Corbet et 970). unoccupied islands islands held held water water voles voles 40 40 years years ago ago ((Corbet et al., al., 11970). Local populations populations are are larger larger in in lowlands lowlands than than uplands (Table 221.1). Local 1 . 1 ) . Only 113% 3 % of of 76 76 upland local populations populations had had more more than than 22 overwintered overwintered animals animals trapped, % of trapped, compared compared with with 81 81% of 26 26 lowlands lowlands populations. The The largest largest local local population in in the the lowlands lowlands had had only only 1188 animals animals in in spring spring and and was was therefore therefore
21 . SMALL SMALL MAMMAL METAPOPULATIONS METAPOPULATIONS 21.
5525 25
0 . 75 0.75
Q)
�
0. 5 0.5
> u cc
8
0o 0.25 0 .25
O +------.---r--�=---_,�--� 2001 1 999 2000 1999 2000 2002 Year I
I
~
I
Figure survey blocks blocks in Upland Figure 2 2 11..33 Occupancy rate of habitat patches patches in four upland survey Assynt (filled blocks) and Upland Grampians (empty circle). (filled symbols symbols for for the four survey survey blocks)
TABLE 1.1 TABLE 2 21.1
Area Area Lowland Lowland area area Upland A Upland A Assynt, B Assynt, B four four survey survey C C D blocks blocks D Upland Upland Grampians Grampians Islands Islands
Summary of Summary of Demographic Demographic Properties Properties of of Water Water Vole Vole PopulationsG Populations a Range in annual annual Range median median distance distance from population population from to to closest other other population population (km) (km)
Median Median and and range range of of population population sizes (adults (adults in spring) spring) (N) (N)
Number Number of of patches
Mean Mean distance to distance to closest patch patch (km) (SE) (km)
44 (26) (26) Range: Range: 2-18 2-18
m
m
11 (30) (30)
20 20 20 20 19 19 29 29 47 47
0.58 0.58 (0.09) (0.09) 0.65 0.65 (0.08) (0.08) 0.60 0.60 (0.08) (0.08) 0.55 0.55 (0.04) (0.04) 0.58 0.58 (0.04) (0.04)
0.62 (0.08) (0.08) to to 0.62 0.70 (0.06) 0.46 0.46 to to 0.85 0.85 0.61 0.61 to to 2.18 2.18 1.41 1.41 ttoo 2.33b 2.336 0.60 0.60 to to 1.76 1.76 0.67 to 0.67 to 1.70 1.70
> > 12 12
0.31 0.31 (0.02)d (0.02) d
0.2 0.2
Range: Range: 1-5 1-5
(46) 11 (46) Range: Range: 1-5 1-5
285 (5) (5) 285 Range: Range: 50-621c 50-621 c
aa
Although Although water water voles voles populations populations in in the the lowlands lowlands are are fragmented, fragmented, discrete discrete patches patches of of suitable suitable habitat habitat are are difficult difficult to to identify identify (see (see text) text) and and therefore therefore patch patch details details are are not not presented. presented. On On the the islands, islands, water water voles voles live live fossorially. fossorially. The The prevalence prevalence of of mink mink differs differs between between areas. areas. Mink Mink have have been been resident resident in in the the lowland lowland area area for for approximately approximately 30 30 yr yr but but are are found found predominantly predominantly along along larger larger waterways, waterways, with with some some upper upper tributaries tributaries only incursions. In only being being subjected subjected to to occasional occasional incursions. In Upland Upland Assynt, Assynt, mink mink are are absent. absent. The The Grampians Grampians uplands uplands are ected to are at at the the edge edge of of the the invasion invasion and and are are subj subjected to occasional occasional invaders. invaders. Mink Mink are are long-term long-term residents residents of of the islands. Values the west west coast coast and and are are caught caught occasionally occasionally on on the the islands. Values are are reported reported for for the the four four survey survey blocks blocks in in Upland Upland Assynt. Assynt. bb Distance Distance to to the the closest closest population population in in 2002 2002 for for the the single single occupied occupied patch patch in in survey survey block block C C was was to to aa local local popu population lation in in survey survey block block D. D. c Population Population sizes sizes on on the the islands islands estimated estimated from from transects transects of of signs, signs, calibrated calibrated with with trapping trapping data data (see (see Telfer Telfer et et aI., al., 2003). 2003). d a Mean Mean distance distance between between islands islands calculated calculated for for the the 12 12 islands islands in in the the main main group, group, excluding excluding Jura Jura and and 22 islands islands surveyed surveyed 10 10 km km to to the the northeast. northeast. c
526 526
XAVIER XAVIERLAMBIN LAMBINET ETAL. AL.
not not immune to to stochastic extinction. extinction. In contrast, each island island population population contained contained several hundred hundred water water voles (Telfer et aI., al., 2003a). 2003a). Thus Thus water water voles occur in fragmented fragmented populations in a range of habitats. habitats. On On the mainland, water populations experience water vole vole populations experience frequent frequent turnover, turnover, especially especially in in the the uplands, uplands, indicating that population processes are important that meta metapopulation important for for regional regional dynamics. Similar patterns patterns of of distribution distribution have have been been found found in in the the closely related related species, species, the the southern southern water water vole, Arvicola sapidus, with with occupancy occupancy being being related related to to both local local attributes attributes (abundant (abundant grass cover) cover) and and landscape landscape attributes attributes (distance (distance to closest known Fedriani et aI., known pond pond holding holding water water voles) ((Fedriani al., 2002). 2002). True extinctions extinctions and recolonizations, recolonizations, as opposed opposed to to expansions expansions and con contractions tractions of populations, populations, were were relatively rare in the lowlands lowlands with with only 9 extinc extinctions and 1100 recolonizations observed over 4 yr. In contrast, extinction and and recolonization recolonization processes were frequent frequent in the uplands with 75 and and 68 extinc extinctions and recolonizations, 999 and 2002. recolonizations, respectively, between 11999 2002. Consequently, the life span of local populations populations was longer in lowland than than upland popula populasections of lowland lowland waterways occupied by tions. Fifty-nine percent of the 22 sections water % of water voles voles were were at at least least partially partially occupied occupied in in all all 4 4 yr. yr. Only Only 55% of 98 98 upland upland patches 999 and patches inhabited inhabited between between 11999 and 2002 held held water water voles voles in in all all 4 4 yr. yr. Unlike Unlike the Bodie pika meta population ((Clinchy Clinchy et aI., metapopulation al., 2002), 2002), there there was was no evidence that that most most recolonizations recolonizations were were ephemeral and and ultimately unsuccessful. unsuccessful. Seven Seven out out of 1100 recolonizations recolonizations persisted into a second year in the lowlands lowlands and and 84% 84% (n = -- 36) in the uplands. uplands. The extinction rate of newly colonized patches in the 1 ) does not differ uplands 1 6 % yeac uplands ((16 year -1) differ from the rate rate of extinction of longer lived patches patches (34% (34% year-I). year -1). The The distance distance between between local local populations and and their their nearest nearest neighbor, a measure of connectivity in meta populations, was more variable and metapopulations, tended populations. In tended to to be be higher in in upland upland than than lowland meta metapopulations. In the the lowlands lowlands the mean distance from each population to its nearest nearest neighbor ranged from 0 .62 to 0.70 1 . 1 ) . In contrast, 0.62 0.70 km (Table 2 21.1). contrast, changes in patch occupancy in the uplands uplands resulted resulted in in much variation variation in in the the distance distance between between local local populations populations between 1 . 1 ) (range between blocks blocks and and years years (Table (Table 2 21.1) (range of of annual annual block block level level means: 0.46-2.33 0.46-2.33 km). km). The The highest highest values values of of isolation were were observed observed in in the the absence absence of of mink, mink, suggesting suggesting that that water water voles, voles, in in their their pristine pristine environment, environment, may may persist persist with with substantial turnover, despite being distributed distributed very sparsely in some years.
Correlates In Water C o r r e l a t e s of o f Extinction E x t i n c t i o n in W a t e r Vole V o l e Metapopulatlons Metapopulations Even though though patterns of of occupancy can be suggestive suggestive of of mechanisms oper operating in fragmented populations, populations, examining patterns of changes in occupancy is more revealing of the importance of meta population processes. Having metapopulation Having docu documented similarities and differences in patterns patterns of occupancy, occupancy, we next ask whether whether similar similar processes influence influence the the distribution distribution of of lowland, lowland, upland, upland, and and island water water voles using logistic models. Analyses for the lowlands and islands have been published previously (Telfer et aI., 1 ) and al., 2003, 2003, 200 2001) and we only sum summarize the main results here. Analyses for the lowland lowland metapopulation con consider sider the the occupancy occupancy status status of of 200-m sections sections of of waterway waterway in in relation relation to to habitat habitat characteristics characteristics and measures measures of population population size and and isolation in the the preceding aI., 200 1 ) . In preceding year year (Telfer (Telfer et et al., 2001). In three three analyses analyses investigating investigating patterns patterns of of patch patch occupancy, extinction, extinction, and and recolonization recolonization in the upland upland metapopula metapopulations, the effects of survey block, year, and and isolation isolation in the preceding year were
21 SMALL MAMMAL 21.. SMALL MAMMAL METAPOPULATIONS METAPOPULATIONS
527 $21
explored. explored. The The effect effect of of local local population population size size on on extinction extinction probability probability was was also also considered. considered. Population Population size size was was taken taken as as the the number number of of voles voles trapped trapped over over aa 1 ) and single 4-day 4-day period period in in June-September June-September in in the the uplands uplands (Aars et et aI., al., 200 2001) and as aI., 200 1 ) . In as the the length length of of aa population population in in lowlands lowlands (Telfer (Telfer et et al., 2001). In order order to to quantify quantify connectivity, connectivity, we we assumed assumed that that the the number number of of immigrants immigrants aa patch patch receives receives depends depends on on both both the the sizes sizes of of the the populations populations in in the the surrounding surrounding area area and populations [e.g., [e.g., Hanski, 1 994) with and distance distance to to those those populations Hanski, ((1994) with a (x = = lid 1/d in in Telfer Telfer et 1 ) ] . Occupancy of islands was related to the et al. al. (200 (2001)]. Occupancy of islands was related to to island island size, size, distance distance to the nearest occupied occupied island, island, and and distance to to the the mainland. mainland. The The relationship relationship between occupancy occupancy and and population population size changed changed in the the lowlands, lowlands, following following the the partial partial invasion invasion of of the the network network by by American American mink mink in in the 998. Before populations were the summer summer of of 11998. Before mink mink invaded invaded the the area, area, large large populations were more likely to to persist than than small ones, possibly possibly reflecting demographic demographic sto stochasticity aI., 200 1 ). The chasticity (Telfer (Telfer et et al., 2001). The lack lack of of any any detectable detectable influence influence of of popula population tion size size thereafter thereafter may may have have been been aa reflection reflection of of the the ability ability of of mink mink to to decimate decimate even the the largest water water vole population. population. Extinction Extinction events events also became 999 with became correlated correlated spatially in 11999 with pairs pairs of of extinction extinction events less than than 500 500 m m apart apart more more frequent frequent than than expected expected by by chance. chance. Occupancy Occupancy of of habitat habitat patches patches in in the the uplands uplands could could not not be be predicted predicted accord according ing to to measures measures of of connectivity connectivity within within block. block. Instead, Instead, patch patch occupancy occupancy fluc fluctuated asynchronously tuated asynchronously between between blocks and and years, with with different different blocks having having high high occupancy rates rates in in different different years, but but block-wide synchrony. synchrony. Our Our isol isolation ation measures measures reflect reflect the the number number of of voles voles within within aa relatively relatively local local area, area, as as connectivity connectivity measures measures only only considered considered populations populations within within aa distance distance of of 66 km km and and values values of of d up up to to 2.5 2.5 km. km. No No measure measure of of isolation isolation had had aa significant significant effect, effect, suggesting suggesting that, that, unlike unlike in in the the lowlands, lowlands, local populations populations are are not not clustered clustered within within blocks. blocks. Extinction Extinction probability probability was was also variable variable at at the the scale scale of of block in each each year year but, but, as as in in the the lowlands, lowlands, small populations populations were were more more likely likely to to go go extinct than large large populations, populations, suggestive suggestive of aa role of of demographic demographic stochastic stochasticity. ity. There There was was no no detectable detectable influence influence of of connectivity connectivity at at any any of of the the scales scales considered considered that that would would have have reflected reflected aa rescue rescue effect. effect. As extinction probabilities, As with with occupancy occupancy and and extinction probabilities, recolonization recolonization rates rates varied varied with with year year and and block. Patterns Patterns of of patch patch occupancy are are clearly clearly dominated dominated by by fac factors tors that that result result in in spatially correlated correlated extinctions and and recolonizations recolonizations at at about about the scale of a block block or higher. From From an an analysis analysis of distribution distribution patterns, patterns, there there is no no evidence evidence of of interdependency interdependency of of populations populations over over aa smaller smaller spatial spatial scale. scale. Upland Upland water water voles voles in in an an area area little little influenced influenced by by humans humans or or mink mink thus thus persist persist in in highly highly fragmented fragmented environments, environments, with with high high rates rates of of turnover, turnover, despite despite spatially spatially correlated correlated extinctions extinctions at at the the scale scale of of 25-30 25-30 km km 22.• Local Local climate climate is is unlikely unlikely to to vary vary at at such such aa small small spatial spatial scale scale and and be be responsible for for the the block block level level variation variation in occupancy, extinction, extinction, and and recolonization recolonization rates. One One possibility is that that dis dispersal block" statistically persal is is so so widespread widespread that that ""block" statistically captures captures its its influence influence better better than than the the relatively small-scale small-scale connectivity connectivity measures considered (edge (edge effects effects preclude preclude consideration consideration of of larger larger scale dispersal dispersal coefficients). coefficients). Alternatively, Alternatively, mobile predators may mobile native native predators may be be causing causing population population turnover turnover to to be be correlated correlated spatially. spatially. Indeed, Indeed, the the very very low low biomass biomass of of alternative alternative prey prey available to to mustelids mustelids present present in in the the area area (Mustela nivalis vulgaris, Mustela erminea, and and Lutra Lutra lutra) may magnify their impact impact on each local water water vole population population and and cause them to to exploit large large areas. areas.
XAVIER AL. XAVIERLAMBIN LAMBINET ETAL.
528 528
No bserved on No extinction extinction was was oobserved on the the islands islands over over 4 4 yr; yr; however, however, large large islands islands close close to to other other occupied occupied islands islands were were more more likely likely to to support support water water vole vole popula populations. tions. Occupancy Occupancy was was not not related related to to distance distance from from the the mainland mainland (Telfer (Telfer et et aI., al., 2003a). 00 m 2003a). Voles Voles on on occupied occupied islands islands 50 50 to to 1100 m from from each each other other had had low low genetic genetic differentiation separated by Section 2 1 .5). This differentiation unlike those on on islands separated by 11 km km ((Section 21.5). This suggests suggests that that dispersal dispersal between between islands islands occurs occurs over over short distances, distances, despite despite the the fast metapopu fast tidal tidal current current in in the the area. Island Island archipelagoes may may function function as as aa metapopulation over timescales, but, but, unlike vole populations lation over evolutionary evolutionary timescales, unlike field field vole populations in in the the population processes Baltic, meta metapopulation processes are are unlikely to to influence influence the the dynamics dynamics over over ecological timescales. timescales.
Dispersal Dispersal in in Water W a t e r Vole V o l e Metapopulations Metapopulations Dispersal Dispersal is is obviously obviously paramount paramount to to the the persistence persistence of of fragmented fragmented popula populations tions with with turnover. turnover. Precise Precise and and unbiased unbiased estimates estimates of of dispersal dispersal rates rates and and dis distances, tances, especially especially for for rare, rare, long-distance long-distance events, events, are are required required to to predict predict the the effects effects of of dispersal dispersal on on population population dynamics dynamics and and genetic genetic composition. composition. Obtaining Obtaining such such estimates estimates is is extremely extremely difficult difficult (Ims (Ims and and Yoccoz, Yoccoz, 1997). 1997). If If these these estimates estimates are are to to be be used used in in predicting predicting metapopulation metapopulation persistence, persistence, char characterizing acterizing density density dependence dependence in in emigration emigration and and immigration immigration is is also also essential. essential. Given populations, even Given the the spatial spatial scale scale involved involved in in water water vole vole meta metapopulations, even intensive intensive standard CMR) techniques would standard capture-marking-recapture capture-marking-recapture ((CMR)techniques would underestimate underestimate the the rates rates and and scales scales of of dispersal. We We overcame overcame this this difficulty difficulty through through combining CMR CMR with with microsatellite microsatellite genotyping genotyping to to identify identify parents parents and and offspring offspring in in differ different ent populations populations (Telfer (Telfer et et aI., al., 2003b). 2003b). Voles Voles sampled sampled during monthly live live trapping of 21 local populations over 2 yr in the lowlands lowlands area area were were genotyped genotyped trapping and and assigned to pools pools of putative putative parents. parents. A similar approach approach was used in the uplands, although although local local populations were were trapped trapped only only once once each each year year (J. (J. Aars Aars et et aI., al., unpublished unpublished results). results). The The parentage parentage assignment assignment approach approach does does not not require require frequent proportions of frequent sampling, and and similar similar proportions of juveniles juveniles were were assigned assigned success successfully to at least one parent 8%, n parent in the two two areas areas (lowlands: 668%, n = - 675; uplands: 67%, 67%, n n = = 253). 253). The The percentages percentages of of assigned assigned juveniles juveniles identified identified as as dispersers dispersers were were also similar in in both the the lowlands lowlands and and the the uplands [lowland females females 13.6% 13.6% ((nn = 14); upland n == 242); = 2 214); upland females females 12.8% 12.8% ((nn -= 78); 78); lowland lowland males males 14.9% 14.9% ((n 242); n == 92)] upland 7.4% ((n upland males males 117.4% 92)].. The The proportion proportion of of overwintered overwintered individuals individuals that that contribution of dispersal to to regional regional have dispersed is the best reflection of the contribution dynamics dynamics as as this this cohort cohort produces produces most most recruits. recruits. In In the the lowland lowland metapopulation, metapopulation, we 9 % of had we estimated estimated that that 33% 33% of of males males and and 119% of females females present present in in spring spring 1998 1998 had dispersed dispersed between between local local populations. Comparable Comparable estimates estimates based based on on CMR CMR alone alone were were 12 and and 7%, 7%, respectively, respectively, reflecting reflecting that that most most water water vole vole dispersal dispersal takes takes place place early early in in life, life, before before reproduction. reproduction. A A single single instance instance of of breeding breeding dis dispersal persal by by an an adult adult female female water water vole vole in in lowland lowland populations populations is is much much cited cited (Stoddart, 970). Despite the (Stoddart, 11970). the intensive intensive trapping trapping in in our our lowland study, study, no no adult females ((nn -= 2 1 1 ) and % of adult n == 265) 211) and only 11% adult males ((n 265) were were trapped trapped in different different populations populations in in the the lowlands. lowlands. Breeding Breeding dispersal dispersal was was detected detected more more frequently frequently in in the the uplands, uplands, despite despite aa less less intensive intensive sampling regime. regime. Nine Nine percent percent of n == 123) % of n == 1132) 32) were of adult adult males males ((n 123) and and 2 2% of adult adult females females ((n were caught caught in in more more n == 123). The average distance moved between than than one population population ((n between popu populations by adult 1km (SE = adult males was 0.6 0.61km = 0.08). 0.08).
21 . 21.
SMALL MAMMAL MAMMAL METAPOPULATIONS METAPOPULATIONS SMALL
5529 29
The high high rates rates of of successful successful natal natal dispersal dispersal rates rates in in water water voles voles have have the the The potential to to influence influence regional regional dynamics dynamics in in both both the the lowlands lowlands and and the the uplands. uplands. potential If immigration immigration rates rates are are negatively density density dependent, dependent, as as with with other other small small mammam If mal species species (see ( see later), later), dispersal dispersal could could effectively effectively reduce reduce extinction extinction rates rates mal through the the "rescue "rescue effect." effect. " We used used juveniles j uveniles assigned to to parents parents in in the the lowlow through land study study area area to to investigate how how the the density of of breeding breeding individuals individuals and and isolisol land ation influenced influenced the proportion of of immigrants immigrants in local populations populations in summer summer ation the proportion after members of of the the spring-born spring-born cohorts cohorts had had dispersed. dispersed. Increasing Increasing immigraimmigra after members tion in well-connected well-connected populations populations is the the process process assumed to to underlie underlie the the resres tion cue effect. We fitted logistic models models with with connectivity connectivity measures measures encompassing encompassing cue We fitted different scales and and selected the best best models models following pro different selected the following backward backward stepwise stepwise procedure. Immigration rates rates in males males increased increased significantly significantly with with decreasing isol cedure. decreasing isolation, with with d' d' == 1.2 1 .2 km km yielding the the lowest lowest deviance. deviance. The The proportion proportion of of male male ation, immigrants populations also decreased decreased significantly with with increasing increasing immigrants in local populations densities of of breeding breeding adults. adults. Although Although immigration immigration rates rates in females also also densities decreased there was was no no detectable of isolation decreased with with increasing density, density, there detectable effect effect of [odds ratio (95% 0.708 [odds ratio (95% CL) for for density density controlling controlling for for distance; distance; males: 0.708 (0.658-0.762), 0. 8 1 0 (0.761-0.861)]. (0.76 1-0.86 1 )]. However, underlying the the expectexpect (0.658-0.762), females: 0.810 However, underlying that connectivity influences immigration rates rates is the assumption assumption that that concon ation that measure of number of This nectivity is a good good measure of the number of dispersers dispersers reaching reaching a site. This may not not be be the the case case if if emigration rates are are density density dependent. dependent. may emigration or or immigration immigration rates meta populations with with small small local populations, In metapopulations populations, dispersal dispersal should should contribute to metapopulation dispersers are are attracted attracted to to contribute more more to metapopulation growth growth if dispersers opposite-sex conspecifics than if random with with respect respect to to the the opposite-sex conspecifics than if settlement settlement were were random populations with with no no potential potential for repro presence of of mates. mates. Single-sex local local populations for repropresence duction duction would would be be most most likely likely where where overall overall occupancy occupancy is is low low and and local local popu populations are small, such as in pika pika at at Bodie and and water water voles in the the uplands. uplands. lations There There was was strong strong evidence of nonrandom nonrandom settlement in the the uplands uplands with with only 88 single-sex Binomial single-sex upland upland populations populations out out of of 54 54 with with only only two two adults adults ((Binomial test .0 1 ) . Experimental test P < < 00.01). Experimental translocation translocation and and radio radio tracking tracking of juvenile juvenile water water voles voles into into good good habitat habitat sites, sites, which which were were either either occupied occupied or or vacant, vacant, show show how how well well the the settlement settlement behavior behavior of of dispersing dispersing water water voles voles is is suited suited to to locating Fisher et locating conspecifics conspecifics at at low low density density ((Fisher et al. unpublished unpublished results results).). Immigration Immigration by by voles voles translocated translocated to to lowland-occupied lowland-occupied sites sites was was not not inhib inhibited. ited. Voles Voles translocated translocated to to sites sites that that remained remained vacant vacant for for the the subsequent subsequent week week abandoned abandoned the the sites sites and and moved moved to to new new sites sites either either overland overland or or by by water. water. In In two two cases cases out out of of seven, seven, an an opposite-sex opposite-sex natural immigrant immigrant arrived arrived to oin the to jjoin the translocated translocated vole vole in in aa previously vacant vacant site. site. Voles Voles remained remained in in the step the transient transient phase phase of of dispersal dispersal for for many many days days and and often often followed followed aa ""stepping stone" stone" trajectory, trajectory, stopping stopping for for several several days days at at successive successive sites. sites. Predation Predation mortality during dispersal was, however, high (observed mortality of dis dispersers - l ) relative persers 4% 4% day day -1) relative to to daily daily disappearance disappearance rate rate estimates estimates obtained obtained by by capture-recapture capture-recapture for for voles voles from from the the same-age same-age cohorts cohorts in in the the summer summer of of two two separate 1 997: 11.23% .2 3 % ((95% 9 5 % CI: 0.41-1 .28); 11998: 99 8 : 00.7% . 7 % ((95% 9 5 % CI: separate years [[1997: 0.41-1.28); 00.48-2.68)] .48-2 . 6 8 ) ] (Telfer (Telfer and and Lambin Lambin unpublished unpublished results) results).. not con conIIff water voles disperse until they find a suitable mate and are not strained strained by by their their inherent inherent mobility, mobility, dispersal dispersal distances distances are are predicted predicted to to increase increase with with decreasing decreasing density density of of voles voles at at the the local local scale scale (density (density of of voles voles within within popu popuand at at the the regional regional scale scale (density (density of of populations populations within within an an area). area). The The lations) and
XAVIER XAVIER LAMBIN LAMBIN ET ET AL. AL.
530 530
mean .3 9 km (95% mean dispersal distance distance from a fitted negative exponential exponential was 11.39 (95% CI = = 11.19-1.64 lowlands (Telfer et ai., al., 2003b) 2003b) and 40% 40% longer . 1 9-1 .64 km) in the lowlands .77-2 . 1 8 km)] in the uplands. [[1.96 1 .96 km (95% (95% CI = = 11.77-2.18 uplands. Similar rates rates of success successful parentage parentage assignment in both both areas indicate indicate that that rates of successful dis dispersal are comparable comparable and that that longer dispersal was not not accompanied accompanied by low immigration unconstrained immigration rates. Water Water vole dispersal thus thus appears relatively unconstrained by the distance distance between between local populations populations and between between habitat habitat patches. Regardless Regardless of of habitat habitat type, type, directed directed dispersal by by water water voles voles creates creates aa degree degree of demographic demographic interdependence interdependence between between local populations populations over hitherto hitherto unsuspected large spatial spatial scales. As dispersal is mostly an attribute attribute of subadults, subadults, this interdependency interdependency is loose and and does not not entirely preclude preclude extinc extinctions caused by demographic demographic stochasticity. stochasticity. It is sufficient sufficient to allow for for metapopulation metapopulation persistence, persistence, despite the synchronizing influence of predators predators acting over the scale of several local populations. populations.
21.4 21 .4 TOWARD TOWARD A A METAPOPULATION METAPOPULATION PARADIGM PARADIGM IN IN SMALL SMALL MAMMALS? MAMMALS?
Metapopulation M e t a p o p u l a t i o n Processes Processes in in Nonclassical Nonclassical Metapopulation Metapopulation Structures Small Mammals Structures in in Small Mammals The The increasing increasing use of of metapopulation metapopulation concepts concepts in mammals mammals (Elmhagen (Elmhagen and 00 1 ) must and Angerbjorn, Angerbjorn, 22001) must no doubt doubt reflect reflect the perception perception by authors authors that that mammals show such structures structures because of of their their patchy patchy distribution, distribution, habitat habitat mammals show fragmentation, fragmentation, and and social structure structure or that that invoking invoking the regional regional dynamics dynamics underpin metapopulation processes that that underpin metapopulation adds adds to our our understanding understanding even when when ecological processes processes do not not operate operate on both both local and and regional regional scales. Indeed, Indeed, in the following following examples, examples, regional regional dynamics dynamics akin to metapopula metapopulation tion processes processes operate, operate, but their their influence influence on long-term long-term dynamics dynamics is limited limited either dominant synchronizing mis either because because of a dominant synchronizing influence influence or or because because of a mismatch behavior and match between between the dispersal behavior and the scale of habitat habitat patchiness patchiness and and disturbance. disturbance. Numerous Numerous small rodent rodent populations populations fluctuate fluctuate violently between between years, sometimes populations, periods sometimes with with regular regular periodicity. In cyclic or irruptive irruptive populations, periods of high density may may be separated separated by 1i or more more years when when numbers numbers are so low low as to be almost almost undetectable undetectable at a landscape landscape scale. Because of their their scarcity, lit little is known known about about the spatial structure structure of such populations, populations, but but it seems likely that that they they form form temporary temporary metapopulations metapopulations in in the the low low phase phase of of population population fluctuations, populations in refuge habitats fluctuations, with with small local populations habitats subjected subjected to extinction-recolonization ai., 11996). 996). The extinction-recolonization dynamics dynamics (e.g., Lima et al., The dynamics dynamics of tundra tundra voles (Microtus (Microtus oeconomus), oeconomus), inhabiting inhabiting mires and and bogs scattered scattered in the the taiga taiga in southeast southeast Norway, Norway, supports supports this conjecture conjecture (R.A. Ims, personal personal com communication). munication). Clearly, any metapopulation metapopulation structure structure is a transient transient state state occur occurring only for a limited fraction of the spatially correlated correlated fluctuations fluctuations of regional populations. populations. As regional abundance abundance increases, so would would the contri contribution bution of dispersal, first linking extinction-prone extinction-prone local populations, populations, then resulting in effective rescue effect, and finally binding local populations populations in a synchronized 991). synchronized patchy population population (see (see Harrison, Harrison, 11991).
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Populations of of three three species of of shrews shrews (Sorex (Sorex araneus, araneus, S. minutus, minutus, and and Populations caecutiens) on on islands islands in Finnish Finnish lakes lakes experience experience turnovers. turnovers. Colonization Colonization S. caecutiens) depends on on distance from the the mainland, mainland, and and extinction extinction probability probability relates relates to to depends distance from body size and and competitive competitive ability (Hanski, (Hanski, 1986; 1 986; Peltonen Peltonen and and Hanski, Hanski, 1991). 1991). body These These islands islands are excellent excellent model model systems for for elucidating elucidating the the processes processes of o f disdis persal, colonization, and extinction extinction also operating metapopulations. persal, colonization, and operating in classical metapopulations. archetypal "mainland "mainland island metapopulations," metapopulations," a dominant dominant influence on on the the As archetypal occupancy shrews, however, is their their dynamics dynamics on on the the mainland, mainland, occupancy of islands by shrews, which is loosely linked linked to to that that of of microtine (Henttonen, 1985). 1 985). which microtine rodents rodents (Henttonen, ( 1 997) argued argued that that spatial spatial structures structures caused caused by sociality and and Stacey et al. (1997) would predispose predispose mammals mammals to to metapopulation metapopulation persistence. Indeed, Indeed, group group living would the fates of social groups groups may may be independent independent and and influenced heavily by demodemo the fates of and environmental environmental stochasticity. New New groups groups may may be founded, graphic and founded, usually graphic of existing groups (e.g., Rood, Rood, 1986). 1 986). Group Group territory territory dynamics, how by fission of existing groups dynamics, howrecolonization of of habitat habitat patches metapopulation as territerri ever, differs differs from from recolonization patches in a metapopulation not inherited as whole entities following the disappearance disappearance of of a group, tories are are not structure, such as traditional traditional unless they are centered on a long-lived physical structure, burrows (e.g., Angerbjorn, 2001). 200 1 ). If new dens or burrows (e.g., Arctic foxes; Elmhagen and and Angerbjorn, groups groups are formed formed by the gradual fission of otherwise socially cohesive groups, groups, the habitat may hinder hinder colonization, the requirement requirement to enter enter aa matrix matrix of unsuitable habitat colonization, despite despite the the obvious obvious physical ability to do do so, as appears appears the case with with Samoango monkeys fragmented habitats habitats in South African monkeys (Cercopithecus (Cercopithecus mitis) in recently fragmented forest (Lawes et aI., 2000) . In this scenario, once patches are disjoint, the spatial al., 2000). configuration configuration of the habitat habitat has no no significant significant consequences for for the dynamics of group group territories. population theory for managing previously continuous Invoking meta metapopulation continuous but now now fragmented mammalian populations populations is increasingly common. Observing populations populations of the same species, such as water water voles and field voles persisting both in highly fragmented and more continuous environment, environment, suggests that that ecological and behavioral traits are sufficiently flexible. Some degree of local adaptation adaptation of life history traits has taken place in natural natural field vole metapopu metapopulations (Ebenhard, 11990). 990). Likewise, the increase of dispersal distance with with degree of fragmentation fragmentation in water water voles may reflect adaptation adaptation or plasticity and lack lack of of constraint constraint on on dispersal. dispersal. Predicting Predicting whether whether species species such such as as black-tailed black-tailed prairie dogs in farmland farmland or arboreal arboreal marsupials in recently recently fragmented forest patches ((Lindenmayer Lindenmayer and Possingham, 11996) 996) possess the traits required for persistence population remains persistence in in aa meta metapopulation remains difficult. difficult. The The predictors predictors of of extinction extinction used population approach may be used to rank habitat degra used within the meta metapopulation degradation scenarios, but in the absence of empirical evidence of patch recoloniza recolonization tion through through an an unsuitable unsuitable habitat habitat matrix matrix (e.g., (e.g., Lindenmayer Lindenmayer and and Possingham, Possingham, 11996), 996), the power of the metapopulation metapopulation approach is limited. For instance, traits traits such as conspecific attraction attraction by dispersers displayed in continuous colonies ((Garrett Garrett and Franklin, 11988; 988; Weddell, 11991) 99 1 ) may contribute to a "res "rescue effect" but could also limit the colonization of empty patches regardless of the 99 1 ). In the inherent inherent mobility mobility of of the the species species (Ray (Ray et et aI., al., 11991). In fact, fact, based based on on the the aforementioned review, it appears less likely that mobility per se, rather than the the details details of dispersal behavior, would constrain persistence after fragmenta fragmentation. Even where meta population processes operate, ill-adapted dispersal may metapopulation cause cause aa mismatch mismatch between between the the frequency frequency of of extinction extinction and and recolonization recolonization such such
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that that there there may may be be no no stable stable equilibrium, equilibrium, the the so-called so-called "extinction "extinction debt" debt" (Hanski, 11999). 999). Even then, managing such systems as metapopulations metapopulations is a legitimate, if temporary, option, although long-term persistence obviously requires balancing of colonization and extinction extinction rates. We are, however, not aware aware of any mammalian mammalian metapopulation metapopulation created by human's activity having conditions for persistence. In contrast, it may be easier been shown to meet the conditions to use meta population parameters to predict the persistence of natural metapopulation meta populations that metapopulations that are subject to further further fragmentation. fragmentation. Examples of the number of populations populations connected latter include lowland water voles where the number by by dispersal is reduced by habitat habitat destruction destruction or mink invasion or Vancouver Island marmots marmots (Marmota (Marmota vancouverensis) vancouverensis) following the creation of low lowquality habitat patches through through logging that may reduce the colonization colonization of productive distant distant patches patches (Bryant, 1996; Bryant Bryant and Janz, 11996). more productive 996).
Metapopulation M e t a p o p u l a t i o n Persistence Persistence despite d e s p i t e Spatially Spatially Correlated C o r r e l a t e d Extinction Extinction That That spatial correlation correlation in local population population extinction extinction is is detrimental detrimental to to metapopulation population metapopulation persistence is a long-established long-established lesson of meta metapopulation Chapter 44).) . However, in all four four instances instances given ear earbiology (Hanski, 11999; 999; Chapter extinction processes were were correlated correlated spatially. The causes of such correla correlalier, extinction either abiotic (climatic) or biotic (predation (predation and and pathogens, pathogens, and and tion were either dispersal) probabilities of dispersal).. These unexpected unexpected patterns patterns include probabilities of extinction extinction being independent population size, as exemplified independent of population exemplified in the following following paragraph. paragraph. Not Not surprisingly, given the the small spatial spatial scale involved, involved, abiotic abiotic climatic fac factors have a synchronizing synchronizing influence influence on disjoint small mammal mammal populations. populations. Eastern common voles (Microtus rossiomeridionalis) per Eastern common rossiomeridionalis) in high arctic arctic Svalbard persist in a metapopulation-like state along a strip of cliff cliff up to 10 metapopulation-like state 1 0 km km long. Vole populations have high growth populations growth rates rates and and reach reach extremely high densities in some years numerous patches patches are colonized but but this can be followed years during during which numerous followed by catastrophic but but irregular irregular population and Yoccoz, 11999). 999). catastrophic population crashes (Ims and Population independent of prevailing prevailing population population size Population crashes occurred occurred independent size and and result from from icing of of soil and following rain events in early winter, which result and grass following rain events renders food inaccessible. Icing events are correlated correlated spatially over altitudinal altitudinal renders food inaccessible. but asynchrony and persistence persistence of the meta population is bands, but asynchrony in extinction extinction and metapopulation made gradient in altitude altitude of habitat made possible by the gradient habitat patches patches such that that precipitaprecipita tion falls as snow snow and allows persistence on on some habitat habitat patches patches (Ims (Ims and and Yoccoz, 1999). 1 999). Heterogeneity Heterogeneity in aspect and and associated associated variation variation in duration duration of Yoccoz, contributes to to maintaining maintaining asynchrony asynchrony within within metapopulations metapopulations snow cover also contributes Australia despite of despite a of rock-dwelling rock-dwelling pygmy possums possums (Burramys (Burramys parvus) parvus) in Australia variation in climate (Broome, (Broome, 2001). 2001 ). In the the two two previous examples, examples, region-scale variation topography makes the the scale of of abiotic abiotic correlations correlations small relative relative to to the the local topography total population population and and allows allows for for metapopulation metapopulation persistence. persistence. Predators of of small mammals mammals operate operate on on larger larger spatial spatial scale than than their their prey prey Predators such such that that predation predation may may result result in a pattern pattern of of spatially spatially correlated correlated local local population extinctions (Clinchy direct evidence evidence that population extinctions (Clinchy et et al., aI., 2002). 2002). There There is no no direct that mustelid predation predation causes causes the the observed observed spatially spatially correlated patterns of of extincextinc mustelid correlated patterns tions tions either either in in the the pika pika at at Bodie Bodie (Smith ( Smith and and Gilpin, Gilpin, 1997) 1 997) or or in in the the water water vole vole Upland areas; areas; however, however, in in both both instances, instances, they they are are the the most most likely likely candicandi in in Upland dates. The The change change from from a spatially spatially uncorrelated uncorrelated pattern pattern to to aa more more aggregated aggregated dates.
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pattern pattern of extinction during a short-lived mink invasion in the lowland area water vole population population amounts amounts to more direct evidence of the synchronizing synchronizing impact of predation. The scale of correlated correlated extinction in the lowland (500 m) that observed in upland upland areas (25-50 (25-50 km22)) where only is much smaller than that native mustelids occur. Note that for both pika and water voles, there was also evidence of demographic demographic stochasticity stochasticity within within colonies contributing contributing to extinctions. extinctions. The drastic decline of one survey block in Upland Upland Assynt from 2000 1 .2) is not unlike the regional decline observed in the 2000 to 2001 (Fig. 221.2) southern part part of the Bodie mining area and, as suggested by Smith and Gilpin, ((1997), 1 997), it is quite quite possible that that extinction-recolonization extinction-recolonization stochasticity stochasticity slowed recovery. Although data data are only fragmentary, fragmentary, the putative putative role of plague in causing colony colony extinction extinction in black-tailed black-tailed prairie dogs dogs raises the interesting interesting possibility possibility that that the pattern pattern of density dependence dependence in mortality mortality and and colony colony extinction extinction may reveal the process involved. involved. Predation Predation mortality mortality from from small mobile mobile predators predators such as mustelids mustelids is expected to be negatively density dependent, whereas whereas the impact impact of a pathogen pathogen such as plague plague that that requires dependent, large or transmission may cause positive or dense host host populations populations for transmission positive or step stepdensity-dependent mortality. mortality. wise density-dependent In all case studies considered considered here, small mammal mammal populations populations persisted persisted as meta populations in naturally metapopulations naturally patchy patchy habitats, habitats, despite being being exposed exposed to moder moderspatially correlated correlated extinctions. extinctions. Thus, Thus, not not only can spatially correlated correlated ately spatially extinction result in patterns patterns resembling resembling those caused caused by meta metapopulation extinction population processes ((Clinchy Clinchy et al., aI., 2002), 2002), but but empirical empirical evidence suggests that that small mammals population persistence mammals adapted adapted to meta metapopulation persistence may exist despite correlated correlated extinctions. extinctions. The scale of extinction extinction due to to predation predation is typically larger than than that that sufficiently small to metapopulaof a single patch patch but but still sufficiently to allow persistence persistence as a metapopula water voles in Upland tion. In the extreme, exemplified exemplified by water Upland Assynt (e.g., Fig. 21.3), 21 .3), regional between population population networks networks instead instead of within regional asynchrony asynchrony exists between of within networks. Such persistence possible if dispersal more effective than than networks. persistence is only possible dispersal is more predicted if it were random. random. predicted
Targeted Mammal Metapopulations Metapopulations Targeted Dispersal in Small Mammal A general rodents general pattern pattern emerges emerges from from recent recent studies studies with with microtine microtine rodents highlighting they are flexible flexible dispersers dispersers and density-dependent highlighting that that they and that that density-dependent immigration may fragimmigration may ensure ensure frequent frequent rescue rescue and and colonization colonization in a highly highly frag mented habitat. habitat. Mechanisms Mechanisms potentially potentially important important in metapopulations metapopulations are are mented well experimental studies well illustrated illustrated by experimental studies with with root root voles voles in in patchy patchy populapopula tions. spring or tions. Most Most dispersal dispersal by root root voles voles takes takes place place in spring or early early summer summer aI., 1999), 1 999), when when densities densities are are at at their their seasonal seasonal low. Female Female effective effective (Aars et al., dispersal is very very restricted restricted when when densities densities across across local local populations populations vary little, little, dispersal whereas most most males males leave their their natal natal population population (Aars and and Ims, Ims, 1999). 1 999). As whereas with other other vole vole and and many many mammal mammal species, species, root root voles voles appear appear to to emigrate emigrate so with as to contact with relatives (Wolff, to avoid avoid close close contact with opposite-sex opposite-sex close close relatives (Wolff, 1992; 1 992; Lambin, 1994; 1 994; Gundersen Gundersen and and Andreassen, Andreassen, 1998). 1 99 8 ) . At At low low density, density, animals animals Lambin, are more more likely likely to to immigrate immigrate into into patches patches with with few members of of the the same same sex sex are few members and and more more members members of of the the opposite opposite sex sex (Andreassen (Andreassen and and Ims, Ims, 2001). 200 1 ) . Negative Negative density-dependent female female immigration immigration combined combined with with a rigid rigid male male dispersal dispersal density-dependent pattern can lead lead to to local local sex sex ratio ratio bias bias variation, variation, and and thus thus to to significant significant male male pattern can
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variation in in reproductive reproductive success success between between demes demes (Aars (Aars and and Ims, Ims, 2000). 2000 ). As As variation with water water voles, voles, root root vole vole dispersal dispersal was was costly costly in in terms terms of of survival survival (Aars (Aars with et al., aI., 1999; 1 999; Ims Ims and and Andreassen, Andreassen, 2000). 2000 ). Over Over the the relatively relatively short short spatial spatial et scale involved involved in in the the experimental experimental enclosures, enclosures, emigration emigration from from aa patch patch was was scale influenced heavily heavily by by the the presence presence of of aa vacant vacant habitat habitat nearby, nearby, indicating indicating that that influenced voles made made exploratory exploratory movement movement followed followed by by immigration immigration into into low-density low-density voles patches (Gundersen ( Gundersen et et al., aI., 2001). 200 1 ) . The The patterns patterns of of opposite-sex opposite-sex attraction attraction and and patches the stepping-stone stepping-stone dispersal dispersal patterns patterns by by pikas pikas and and water water voles voles suggest suggest that that the similar patch patch sampling sampling processes processes may may also also operate operate in in metapopulations, meta populations, similar despite larger larger distances distances between between patches. patches. Animals Animals emigrating emigrating to to escape escape despite unfavorable conditions conditions (crowding, (crowding, presence presence of of relative) relative) are are likely likely to to settle settle unfavorable where conditions conditions are are more more favorable. favorable. Despite Despite the the general general pattern pattern of of negative negative where density-dependent immigration, immigration, emigration emigration was was the the main main proximate proximate cause cause of of density-dependent extinction of of small small experimental experimental populations populations (Andreassen (Andreassen and and Ims, Ims, 2001), 200 1 ) , as as extinction with field field voles on on small small islands islands in in the the Baltic Baltic archipelago archipelago metapopulation meta population and and with some upland upland water water vole populations. populations. Thus, Thus, contrary contrary to to the the standard standard assumpassump some tion of of metapopulation meta population models models that that there there is either either no no or or a negative negative relationrelation tion ship between between local local population population extinction extinction probability probability and and production production of of ship emigrants, small mammals mammals may may respond respond to to deteriorating deteriorating conditions conditions by disdis emigrants, persing rather rather than than dying dying passively. As a result, result, per per capita capita emigration emigration may may be be persing related positively to to the the extinction extinction rate Crone et aI., 2001). 200 1 ) . There was also related rate ((Crone et al., There was of the the so-called ""social social fence effect" effect" ((Hestbeck, Hestbeck, 1982) 1 9 82) in root root voles, evidence of whereby almost completely ceased ceased when when densities moder whereby immigration immigration almost densities were were moderoften in late late summer summer (Aars et 1 999). Patterns Patterns of of low rates of of ate or or high, often et aI., al., 1999). low rates migration at at high densities densities seem to to be a common common trait mammals ((see, see, migration trait in small mammals 998; Lambin aI., 200 1 ; Lin 200 1 ) . e.g., Blackburn Blackburn et et aI., al., 11998; Lambin et et al., 2001; Lin and and Batzli, 2001). Overall, the aforementioned aforementioned features ooff dispersal, including stepping-stone dispersal, dispersal, active active patch patch selection (sometimes (sometimes preceded preceded by by sampling), sampling), conspecific conspecific attraction, attraction, and negative density-dependent density-dependent immigration, immigration, should should result in in aa strong strong rescue effect. effect. These These complications complications may may also also cause aa mismatch mismatch between between the the effective effective impact impact of of dispersal dispersal and and that that predicted from from its its frequency frequency and and scale, scale, as as was was the the case case for for both both pikas pikas and and water water voles voles (e.g., (e.g., Moilanen Moilanen et et aI., al., 1998). 1998). In In from each other, the distribution of dis disaddition, when patches were distant from persal distances by water vole was adjusted upward, upward, suggesting that that vagility does does not not necessarily limit limit dispersal dispersal in in metapopulations. metapopulations. Evidence Evidence of of aa rescue rescue effect effect often often involves involves no no more more than than aa positive positive relation relationship ship between between the the probability probability of of local local populations populations persisting persisting in in successive successive years years and and indices indices of of proximity proximity to to other other local local populations, populations, taken taken as as reflecting reflecting the the number of dispersers reaching a patch. By this measure, evidence of a rescue effect population ((but but effect in in small small mammals mammals was was restricted restricted to to the the Bodie Bodie pika pika meta metapopulation see see Clinchy Clinchy et et aI., al., 2002). 2002). This This may may be be because because the the rescue rescue effect effect is is unimportant unimportant ((Clinchy, Clinchy, 11997) 997) or, or, more more likely, likely, because the the assumption assumption that that connectivity connectivity is aa good good measure measure of of the the number number of of dispersers dispersers reaching reaching aa site site is is overly overly simplistic. simplistic. Indeed, or population Indeed, if the the per per capita capita emigration emigration rate rate changes changes with with density density ((or size), size), there there would would not not necessarily necessarily be be aa simple simple relationship relationship between between population population size size and and the the number number of of dispersers dispersers it it produces. produces. Considering Considering more more realistic realistic rela relationships tionships between between the the number number of of immigrants immigrants aa patch patch receives receives and and the the regional regional metapopulation metapopulation may may be be required required to to detect detect aa rescue rescue effect effect as as pervasive pervasive as as envis envisaged 1 997) . aged by by Stacey Stacey et et ai. al. ((1997).
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Using genetic data to directly estimate immigration gives compelling evi evidence of the importance of immigration within local populations. A genetic geneticbased based assignment assignment of of individuals individuals to to populations populations or or to to putative putative parents parents indicated a high fraction fraction of immigrants immigrants in colonies of both black-tailed prairie dogs (Roach et aI., 1 ) and water voles (Telfer al., 200 2001) (Telfer et aI., al., 2003b), including in the uplands where extinction rates were high, spatially correlated, and unrelated to levels of isolation. As the use of such genetic methods becomes more widespread to characterize metapopulations, a more quantitative evalu evaluation of the role of the rescue effect will become possible. We expect this will help resolve the apparent contradiction between what can presently be inferred from patterns of occupancy and what is known about the process of dispersal and colonization.
21.5 21 .5
THE THE GENETIC GENETIC STRUCTURE STRUCTURE OF OF SMALL SMALL MAMMAL MAMMAL METAPOPULATIONS METAPOPULATIONS Considerable emphasis has been placed on trying to model the effects of metapopulation processes on patterns of population divergence between patches, and levels levels of diversity within individual patches and across entire meta populations. Such efforts can provide information on the effects of metapopulations. extinction and recolonization on genetic diversity and concomitant viability, but also could potentially highlight metapopulation processes in the absence of ecological data on, for example, patch occupancy. Modeling approaches range from from necessarily simplistic models derived from island or stepping-stone that explicitly include ecologically models to coalescent and and simulation models that realistic processes and exclude some of the simplifying assumptions added for logistical and mathematical expediency (see ( see Chapter 8). 8). that more fragmented populations will lose Models commonly predict that genetic variability faster than than comparable comparable unfragmented unfragmented populations populations (Whitlock and Barton, 1997; 1 997; Nunney, 1999; 1 999; Chapter Chapter 7) and that population that population turnover can severely reduce turnover reduce neutral genetic diversity, diversity, particularly if combined with 1 999; Chapter 7). with propagule-pool colonization (Pannell and Charlesworth, 1999; However, Ray (200 1 ) showed that (2001) that the scale of correlated deaths or or local cer extinction affects how diversity is retained retained in structured populations. In cerpopulations with frequent tain cases, subdivided populations frequent local extinction may retain than populations populations of equivalent equivalent total size but but fewer and and larger larger more diversity than Furthermore, higher diversity can be expected in a metapopulation metapopulation with units. Furthermore, many subpopulation local populations populations than than in one with with populations populations of many small subpopulation populations frequently frequently provided immigrants variable size where larger source populations to the smaller ones. This conclusion conclusion is thus comparable comparable with with that that of of the the clasclas to more unrealistic the total total amount amount of variation variation is sic and more unrealistic island model, where where the high due to a lack of variance in reproductive success among local populations populations success among ( see Chapter different approach, but reaching similar conclusions, conclusions, (see Chapter 7). Using a different approach, but and Aliacar (200 1 ) highlighted the the contrasting contrasting effects of migration Wakeley and Aliacar (2001) and population turnover turnover on the site-frequency distribution distribution at polymorphic and They warned warned that, that, from from genetic genetic data, data, it will be impossible to to distinguish sites. They between changes in population population number number and and changes in the the rates rates and and patterns patterns between of migration migration and and turnover turnover as explanations explanations of of variable effective effective size over over time. of
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As As such, such, it it is is not not clear clear that that any any models models can can yet yet identify identify aa signature signature of of metapopu metapopuwith little underlying ecological data nor nor deter deterlation dynamics in a species with population processes to the mine the relative contributions contributions of of particular meta metapopulation the overall patterns patterns of diversity and divergence. The The problem of identifying a characteristic pattern pattern of population population genetic structure population is exacerbated structure from from a meta metapopulation exacerbated for for mammals. Those complex social structures structures and behaviors behaviors that that are common common if not not inherent inherent to to mammals may either confound confound or negate the effects of of metapopulation metapopulation processes processes on pat patterns terns of of genetic genetic diversity diversity or or divergence. divergence. Indeed, Indeed, parallel parallel to to the the effort effort invested invested in trying to understand understand the effects of metapopulation metapopulation dynamics on genetic parameters, considerable considerable emphasis has been been placed placed on on developing aa theoret theoretical framework framework to to model gene dynamics in socially structured structured populations populations ((Chesser, Chesser, 11991a,b; 99 1a,b; Sugg and 994; Sugg et aI., 996; Berg et aI., and Chesser, 11994; al., 11996; al., 1998). Rarely Rarely have have the the two two been been combined, combined, and and certainly certainly there there remains remains aa lack lack of of replicated replicated empirical empirical genetic genetic data data from from species species in in both both fragmented fragmented and and unfrag unfragmented landscapes landscapes where where realistic estimates of turnover turnover and and dispersal are available. considerable number number of of studies studies have attempted to describe describe the effects of A considerable meta population processes metapopulation processes on on population population genetic genetic structure structure from from empirical empirical stud studies ies of of allele allele frequencies frequencies at at markers markers such such as as microsatellite microsatellite DNA DNA polymorph polymorphisms. While the choice of markers and and subsequent subsequent analyses are generally sound, a majority of studies can still be criticized for for attempting attempting to to examine metapopulation metapopulation genetic structure structure when when there is sparse evidence that that a metapopulation 1; metapopulation framework framework is appropriate appropriate (e.g., Gerlach and and Hoeck, 200 2001; Burland et aI., 1 ; Schulte-Hostedde 1 ) . The al., 200 2001; Schulte-Hostedde eett aI., al., 200 2001). The patterns patterns observed could thus be attributable to other population other contemporary or historical population processes such as ancestral bottlenecks bottlenecks or limited dispersal in a semicontinuous distribution of individuals. Patterns population genetic structure Patterns of population structure have been examined to a certain certain degree in populations of pika that that are considered considered more consistent consistent with with the the classical population paradigm. classical meta metapopulation paradigm. Low Low allozyme allozyme diversity diversity has has been been detected, detected, which has been attributed attributed to repeated genetic bottlenecks and a concomitant concomitant reduction reduction in in Ne, Ne, which which suggests suggests widespread widespread extinction-recolonization extinction-recolonization dynam dynamics ((Hafner Hafner and Sullivan, 11995). 995). In more more recent recent studies, neutral neutral markers markers ((microsatellites) microsateIIites) have been employed to compare compare fragmented and unfrag unfragmented population founded mented populations. populations. The The fragmented fragmented Bodie meta metapopulation founded 40 40 gen generations that observed in a more erations ago had a level of heterozygosity as high as that more continuous habitat habitat (Peacock and and Ray, 200 2001). explained by lim limcontinuous 1 ) . This was best explained outside the meta metapopulation, with the the small ited immigration from outside population, combined with size limit on all local populations, populations, which which tends tends to equalize the the reproductive reproductive success of colonists. Peacock and Ray (200 1 ) suggest that (2001) that this system reproductive success due to local retains high variability because variance in reproductive extinction-recolonization extinction-recolonization dynamics is minimal when when local populations populations hold hold few individuals. In pikas, pikas, monogamy is the the modal modal mating mating system in patchy populations populations where patches patches are too too small to to host host more more than than one breeding female, and too too dispersed for a male to defend several, but but polygyny is more frequent 1 ). frequent in larger patches or continuous continuous populations populations (Peacock and and Ray 200 2001). This is thus also likely to be of importance importance for how how genetic variance may be retained in the more fragmented fragmented populations populations (e.g., Sugg et aI., al., 11996). retained 996).
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We We have have examined examined patterns patterns of of population population genetic genetic structure structure among among our our net networks Scotland with works of of water water vole vole populations populations in in Scotland with contrasting contrasting population population size size and turnover turnover (Table (Table 2 21.1). We have have also surveyed three three different different continuous continuous and 1 . 1 ). We also surveyed populations 1 ) River populations for for comparison: comparison: ((1) River Itchen Itchen in in southern southern England, England, where where water voles are much water voles are present present at at high high density density and and probably probably do do not not fluctuate fluctuate much between years; (2) linear population between years; (2) aa linear population from from Alajoki Alajoki in in southwestern southwestern Finland, Finland, where where animals animals were were sampled sampled along along aa stream stream at at trapping trapping stations stations 1 km km apart; apart; and 3 ) aa terrestrial and ((3) terrestrial (i.e., (i.e., two-dimensional) two-dimensional) population population from from Allkia, Allkia, western western Finland, sampled sampled within square km. Finnish populations Finland, within aa square km. Both Both the the Finnish populations were were sam sampled pled at at very very high high density, density, but but the the populations populations are are known known to to fluctuate fluctuate (Aars (Aars et et al. al. unpublished unpublished results) results).. Table 1 .2 summarizes Table 2 21.2 summarizes some some ooff the the genetic genetic patterns patterns observed observed iinn these these popu populations lations (see (see also also Aars Aars et et al. al. unpublished unpublished results) results).. Further Further genetic genetic characteriza characterization al. ((1999). 1 999). tion of of the the lowland lowland populations populations can can be be found found in in Stewart Stewart et et al. Comparing Comparing across across all all population population networks, networks, there there was was no no pattern pattern of of genetic genetic variability fragmentation. All variability that that correlated correlated to to the the degree degree of of habitat habitat fragmentation. All popula populations tions were were characterized characterized by by aa high high number number of of alleles alleles per per locus locus and and high high indi individual heterozygosity heterozygosity ((typically typically 0 . 7 to . 8 ) . Thus level of vidual 0.7 to 0 0.8). Thus the the high high level of fragmentation fragmentation of of water water vole vole habitats habitats was was not not associated associated with with any any significant significant loss of lowland metapopulation, loss of variability. variability. Except Except for for the the lowland metapopulation, which which had had the the low lowest est variability variability among among all all the the surveyed surveyed populations, populations, fragmented fragmented populations populations had 1 .2). This had variability variability comparable comparable to to the the continuous continuous populations populations (Table (Table 2 21.2). This pattern pattern was was consistent consistent over over all all six six upland upland low-density low-density survey survey blocks blocks and and over over 2 to to 55 yr yr of of sampling. sampling. Thus Thus even even the the extreme extreme rates rates of of local local turnover turnover observed observed 2 do do not not reduce reduce Ne Ne severely severely among among fragmented fragmented water water vole vole populations, populations, consis consistent 1 ) prediction. tent with with Ray's Ray's (200 (2001) prediction. The The upland upland metapopulations metapopulations are are bound bound together together by by dispersal dispersal across across large large scales, scales, with with numerous numerous small small colonies colonies thus thus experiencing experiencing frequent frequent turnover turnover due due to to stochastic stochastic demographic demographic events. events. In In all all areas, areas, environmental environmental stochasticity stochasticity also also appeared appeared influential influential with with occupancy occupancy rates 1), rates varying varying widely widely between between blocks blocks within within years. years. According According to to Ray Ray (200 (2001),
TABLE T A B L E 21 2 1 ..2 2
Summary Summary of of Genetic Genetic Properties Properties of of Water Water Vole Vole Population Population NetworksG Networks a
Number Number of of networks networks (sampling (sampling years) Lowland Lowland area Upland (Assynt (Assynt and Grampians) Islands Continuous
Ho
A A
Fsr FST
spatial spatial
Fsr FST
temporal temporal
(f) (f)
F F
11 (4)
0.65, 0.72 0.72 6.2, 6.2, 7.4
6 (2-5) 5 (2-3) 3 ((1) 1)
0.0 0.02, 0.73, 0.81 0.81 7.3, 7.3, 110.0 0.02, 0.21 0.00, 0.12 0.12 -0.16, -0.16, 0.00 0.00 -0.05, -0.05, 0.12 .9 0.52b 2.6, 2.6, 33.9 0.00, 0.38 0.00, 0.06 0.46, 0.52b 0.00, 0.02, 0.04 0.06 -0.02, 0.04 0.20, 0.20,0.32 na 0.74, 0.79 0.02 0.03, 0.06 0.79 6.0, 6.0, 9.6 0.03 0.03 [0.03,0.04] c [0.03,0.04]C [[-0.01,0.05] -O.Ol,O.OsF2
0.10, 0.10, 0.14 0.14
0.01, 0.01,0.03 0.03 -0.04, 0.01 0.01
0.06, 0.06,0.15
a Only Only colonies colonies with with more more than 10 individuals individuals are are included included in calculations calculations of A, Ho. Continuous Continuous networks a
include Finland populations and one sample sample from include two southwest southwest Finland from the River River Itchen Itchen in southern England. England. bb Island Island H H ex = 0.59 0 . 5 9 to t o 0.61 0.61 across across three three to five five islands. islands. expp = (9-km transect, c Continuous Continuous nn = = 11 for FST FsT and (f) ([) (9-km transect, sampled sampled each each kilometer). kilometer). C
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under under such such a pattern pattern of of extinction, extinction, highly fragmented fragmented populations populations will retain retain metapopulations with variability more more effectively than than less fragmented fragmented metapopulations with some persistent persistent local populations. populations. In metapopulations metapopulations with with very small local popula populations, founders founders of new new colonized colonized patches patches tend tend to be a mixture mixture from from different different local populations populations rather than than from from one larger source population, population, which which increases Ne. The high dispersal ability observed observed in water water voles implies that that local populations populations often often consist consist of animals from from different different patches, patches, considerably considerably increasing increasing Ne Ne compared compared to what what it would would be if local populations populations were were founded founded according according to a propagule-pool propagule-pool pattern pattern (Pannell and and Charlesworth, Charlesworth, 1999). Negative local inbreeding inbreeding coefficients coefficients (f) are a common common feature feature in fragmented fragmented water 1 .2) and correlated with the local water vole populations populations (Table 2 21.2) and the value is correlated population population size. This This also reflects the the fact that that parents parents tend tend to be of different different genotypes, genotypes, partly because they will be so by chance chance when when they are few (Wahlund, 928), but (Wahlund, 11928), but more more so the higher higher the dispersal rates. Within Within survey ust slightly positive, concon blocks, total total inbreeding inbreeding coefficients coefficients were were frequently frequently jjust sistent with with the high dispersal rates between between local populations populations revealed from from CMR CMR and and assignment assignment data. FST FsTvalues for among-local among-local populations populations differenti differentiation around 0 . 1 (Table 2 1 .2). This reflects high ation within within blocks were were typically around 0.1 21.2). kin similarity uveniles have dispersed. similarity within within local populations populations before before jjuveniles Considering populations are typically ust Considering only adults, adults, FST FsTvalues in the upland upland populations typically jjust above zero and frequently frequently non-significantly non-significantly positive (Aars et al. unpublished unpublished results). results). Temporally, Temporally, genetic genetic drift drift within within survey survey areas areas is is significant, significant, and and FST FsT correlated correlated positively with interval between between sampling within within survey blocks at least up to to 5 yr, both both on the islands islands and and in the mainland mainland metapopulations. metapopulations. Furthermore, Furthermore, it is correlated correlated negatively negatively to the the harmonic harmonic mean mean of animals animals trapped trapped within within the blocks. The The fact that that local genetic drift drift can be quite quite pro profound, found, yet populations populations are still characterized characterized by high genetic genetic variability, indi indicates that that these populations populations are part part of much much larger networks. networks. The high frequency frequency of solitary solitary breeding breeding pairs pairs or populations populations consisting consisting of very about an very few few adults adults with with about an even even sex sex ratio ratio may may contribute contribute to to low low variance variance in reproductive reproductive success between between individuals, individuals, particularly particularly among among males, and and thus thus contribute contribute to a high Ne Ne in very fragmented fragmented populations. populations. Variance of female and and male male reproductive reproductive success should should be studied studied in detail across differ different fragmentation fragmentation levels before conclusions conclusions are drawn drawn about about such effects, effects, par particularly as there there also is a possibility possibility that that polyandry polyandry (a single litter sired by more sys more than than one one father) father) could could be be more more prevalent prevalent in in denser denser less less fragmented fragmented systems and and have the opposite opposite effect (i.e., increase increase Ne Ne more more in continuous continuous or less fragmented fragmented colonies). colonies). Overall, these empirical empirical studies of water water vole population population genetics confirm confirm theoretical concerns that that characteristic characteristic patterns patterns of genetic divergence and diver diversity are difficult 1 ) little inference difficult to detect. As such, it is apparent apparent that that ((1) inference can be made about the relative contributions population dynamics, made about contributions of meta metapopulation dynamics, social structure, structure, and and contemporary contemporary and and historical historical microevolutionary microevolutionary processes to the the overall overall patterns patterns observed; observed; and and (2) metapopulation metapopulation processes processes may, in fact, retain retain comparable comparable levels of genetic diversity within within networks networks relative to populations populations with reduced reduced population population turnover turnover and and increased increased connectivity. Only the the isolated isolated populations show show any identifiable identifiable signature signature of reduction reduction in genetic vari variisland populations and these populations not experience experience extinction-recolonization extinction-recolonization ability, and populations do not dynamics. dynamics.
21.. SMALL SMALLMAMMAL METAPOPULATIONS 21
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CONCLUSION CONCLUSION Metapopulation persistence requires some some degree of balance between extinctions and recolonizations through dispersal as well as sufficient inde independence between between local populations so as to maintain asynchrony in extinc extinctions. In nearly all instances of mammalian metapopulations reviewed in this chapter, there was a relatively high degree of spatial synchrony in extinction, and dispersal was nonrandom with respect to the presence and abundance of conspecifics. These may be general features of mammalian metapopulations and, taken singly, singly, either of these features might call into question the adequacy of the metapopulation approach. Together they provide a strong rationale for using metapopulation theory to study these and other small mammal species. Empirical studies show that dispersal by small mammal is highly targeted and effective in linking local populations over larger scales than dictated by dispersal distance alone. Migration among small populations is also a function of of local demography demography with with aa higher higher immigration rate rate in in smaller smaller populations. populations. These aspects of dispersal behaviors typical of small mammals are not addressed within but see Chapter within most metapopulation metapopulation models ((but Chapter 4). In most instances, however, seasonally restricted natal dispersal by juveniles and subadults and not movements by adults is linking populations. The dynamics of local populations thus remain sufficiently independent and subjected to (e.g., water water voles in local stochastic processes, extinctions, and recolonizations (e.g., the lowland area in the absence of mink). Our review also highlights the fact that spatially correlated extinctions are widespread among small mammal. In some instances, they dominate the overall dynamics to such an extent that that the metapopulation meta population paradigm paradigm is not useful for predicting their dynamics (e.g., others, genuine genuine metapopulations prevail prevail where ( e.g., cyclic microtines). In others, biotic (predation and pathogens) and abiotic (local variation in climate) fac factors external to the focal species impose independent local dynamics at the scale of several local populations. populations. These agents of correlated extinction thus importance of metapopulation metapopulation dynamics in systems that that would would elevate the importance otherwise be more influenced by demographic interdependence among small local populations. Two scales are thus relevant relevant in small mammal metapopulametapopula tions: the scale of habitat patchiness or modal dispersal movements movements and the scale of correlated extinctions. If dispersal is sufficiently directed and effective to overcome external synchronizing influence and allowing recolonization recolonization of that have suffered correlated correlated extinctions (e.g., between blocks in upland upland areas that water despite spawater vole populations), populations), small mammal mammal metapopulations metapopulations prevail, prevail, despite spa tially correlated extinctions. If the spatially correlated the scale of of spatially correlated extinctions is too too large relative to the the tail of of the the distribution distribution of dispersal distances, the metapopulation metapopulation will be unable unable to to persist. In both water voles, comparative both the pika and and in water comparative studies on on population population fragmented populations populations failed to reveal genetics in continuous continuous and in highly fragmented failed to any loss of genetic variability in metapopulations. metapopulations. This This questions whether whether subsub division commonly population sizes sizes in most mammal mammal commonly will reduce effective population species, and and more important, important, the the potential for a genetic signature signature associated species, potential for with descriptions of with mammal mammal metapopulations. metapopulations. Frequency-based Frequency-based descriptions of population population structures are unlikely unlikely to to highlight highlight the the complex complex interplay interplay between between genetic structures metapopulation and and social processes that that define define the the genetic structure structure among among metapopulation
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patches. This This does does not not represent represent a major major paradigm paradigm shift shift in in thinking, thinking, but but patches. merely reiterates reiterates what what has has been been said said elsewhere, elsewhere, but but is frequently frequently overlooked overlooked merely Pannell and and Charlesworth, Charlesworth, 1999). 1 999). We advocate advocate that that genetic genetic data data can can (e.g., Pannell play an an important important role role in in defining defining a mammalian mammalian metapopulation, metapopulation, but but more more so play in delimiting pedigrees pedigrees within within natural natural populations populations to to identify identify relatives relatives and and in delimiting determine social structure, structure, and and more more directly estimating migration. migration. The The determine social directly in estimating now lies in accurately accurately characterizing characterizing the the complex complex dynamics dynamics of of disdis challenge now persal and and extinction and using using these these data data to to parameterize parameterize sufficiently sufficiently realisrealis persal extinction and models that that encompass encompass the the characteristics characteristics inherent yet unique unique to to mammals. mammals. tic models inherent yet
22
META PO P PUL ATION M ETAPO U LATI ON DYN AMICS AND AND DYNAMICS RESERVE RESERVE NETWORK NETWORK DESIGN DESIGN Mar Cabeza, Moilanen, and and Hugh Possingham Cabeza, Atte Atte Moilanen, Hugh P. Possingham
22.1 22.1
SYSTEMATIC ETHODS FOR SYSTEMATIC M METHODS FOR RESERVE RESERVE NETWORK NETWORK DESIGN DESIGN Choosing Choosing which which sites sites to to include include in in aa reserve reserve network network is is aa fundamental fundamental problem in places that problem in conservation conservation biology. biology. We We will will not not be be able able to to protect protect all all places that contribute because that would be place on contribute to to biodiversity biodiversity because that would be every every place on earth. earth. Given Given the conservation and the realities realities of of limited limited resources resources for for conservation and other other political political and and economical limitations, we economical limitations, we need need to to identify identify priority priority areas areas to to be be set set aside aside as as reserves. Over Over the past past two two decades three three philosophies philosophies have emerged in the the reserves. reserve reserve selection selection literature. literature. First, First, there there has has been been increasing increasing emphasis emphasis on on the the design design of of entire entire reserve reserve networks networks rather rather than than the the selection selection of of individual individual sites. sites. Second, Second, there there has has been been interest interest in in reserve reserve networks networks achieving achieving goals goals such such as as adequacy, adequacy, representativeness, representativeness, and and compactness. compactness. Third, Third, there there has has been been an an iinterest nterest in in these these goals goals being being met met efficiently. efficiently. We We now now realize realize that that ad ad hoc hoc reserve reserve selections, sites, competition selections, often often aa result result of of availability availability of of sites, competition with with alternative alternative land value, and land uses, uses, scenic scenic value, and other other factors, factors, can can lead lead to to very very inefficient inefficient steps steps toward toward constructing constructing reserve reserve networks networks that that conserve conserve the the majority majority of of species species ((Pressey, Pressey, 11994, 994, 11999; 999; Cowling 999; Margules Cowling et et aI., al., 11999; Margules et et aI., al., 2002; 2002; Stewart Stewart et et aI., al., 2003 2003).) .
Ecology, Ecology,Genetics, Genetics, and and Evolution Evolution of of Metapopulations
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Copyright Copyright 2004, Elsevier, Elsevier,Inc. Inc. 0-12-323448-4 0-12-323448-4
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The The ecological ecological theories theories of of island island biogeography biogeography and metapopulation metapopulation dynamics dynamics have have provided provided some some guidance guidance for for the the selection selection of of reserves, reserves, including including rules big reserves rules such such as as ""big reserves are are better better than than small reserves," reserves, . . . ." aggregated aggregated reserves are are better better than dispersed dispersed reserves," reserves, ...."minimize minimize patch loss/maximize loss/maximize the protect source the number number of of suitable suitable habitats," and and ""protect source populations populations and and ignore ignore sink see, e.g., 997). While sink populations" populations" ((see, e.g., Hanski and and Simberloff, Simberloff, 11997). While these these general general concepts concepts provide provide broad broad guidance, they they are are not not useful useful in in helping helping us us choose choose between between different different sites sites and and they they are are not not universally universally true true (Possingham (Possingham et 1 ; Cabeza 9 8 0s, attention et aI., al., 200 2001; Cabeza and and Moilanen, Moilanen, 2003 2003).). Since Since the the early early 11980s, attention quantitative methods that that use empirical has focused on developing systematic quantitative data and and economic considerations considerations to select a set of sites for biodiversity data conservation. biodiversity conservation. This This chapter chapter reviews reviews quantitative quantitative methods methods for for reserve reserve selection selection and and then then concentrates concentrates on on new new ideas ideas that that incorporate incorporate concepts concepts from from spatial spatial ecology, ecology, particularly particularly metapopulation metapopulation biology biology and and landscape landscape ecology. ecology. We We compare compare aa range which consider range of of methods, methods, which consider spatiotemporal spatiotemporal dynamics dynamics in in different different ways, ways, and and show show that that persistence persistence of of species species in in reserve reserve networks networks is is enhanced enhanced when when spatial spatial considerations considerations are are taken taken into into account account in in reserve reserve network network design. design. The The chapter chapter ends ends with with aa discussion discussion of of problems problems of of dynamic dynamic reserve reserve selection selection in in an an uncertain uncertain world world where where not not only only do do we we need need to to deal deal with with dynamic dynamic populations populations but also a dynamic landscape.
Key Key Concepts Concepts in in Reserve Reserve Network N e t w o r k Design Design the adequate adequate configur configurReserve network design methods seek to identify the ation ation of of conservation conservation areas areas for for biodiversity, biodiversity, including issues of of shape, shape, area, area, connectedness, 999). Reserve connectedness, management, management, and and scheduling scheduling (e.g., (e.g., Pressey, Pressey, 11999). Reserve selection selection algorithms algorithms are are computational computational tools that that are are often often used used as as part part of of the the reserve network design procedure. Reserve selection algorithms (also known as as area area selection, selection, or or site site selection algorithms) algorithms) aim aim at at meeting meeting conservation conservation goals efficiently in reserve networks. In In principle, principle, the the concept concept of of biodiversity biodiversity embraces embraces the the entire entire biological biological hierarchy, hierarchy, from from molecules molecules to to ecosystems, ecosystems, including interactions interactions and and processes, processes, although although most most often often we we consider consider our our biodiversity biodiversity elements elements to to be be simply simply species species and/or and/or habitat habitat types. types. In In marine marine conservation planning planning the the features features are are more more likely likely to to be be habitat types types and and biophysical biophysical domains due due to e.g., Leslie to aa lack lack of of comprehensive comprehensive information information on on species species distributions distributions ((e.g., Leslie et et aI., al., 2003 2003).) . Likewise Likewise in in terrestrial terrestrial environments, environments, planning planning is is hindered hindered by by aa lack of of knowledge, knowledge, for for instance, instance, about about the the existence existence and and location location of of species. protect aa sample species. Consequently, Consequently, selected areas will only protect sample of of biodiver biodiversity, representation" is goals for sity, hence hence the the term term ""representation" is used used to to describe describe the the goals for these these reserve Represent" implicitly reserve selection selection methods methods (Margules (Margules et et aI., al., 2002). 2002). ""Represent" implicitly means sample. " Therefore, means ""sample." Therefore, the the representativeness of of aa system system of of reserves reserves means the the extent extent to to which which it it adequately adequately samples all all the the targeted targeted natural natural fea features vegetation types), tures (e.g., (e.g., species, species, vegetation types), of of the the region. region. The The efficiency of of the the solution solution for for aa given given problem problem is is often often measured measured as as the the cost of of the the solution solution (where (where cost cost may may be be measured measured as as area, area, number number of of selection selection units, units, boundary boundary length, length, acquisition cost, cost, management management cost, cost, etc.; etc.; e.g., e.g., Pressey Pressey and and Nicholls, Nicholls, 11989; 989; Rodrigues 999). Rodrigues et et aI., al., 11999).
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Reserve selection algorithms algorithms have concentrated concentrated on the key concept concept of of
complementarity. 1 ) defined property complementarity. Williams (200 (2001) defined complementarity as a ""property of of sets sets of of objects that that exist exist when when at at least some some of of the the objects in in one one set set differ differ from from the the objects objects in in another another set." set." Complementarity-based Complementarity-based methods methods were were developed representation targets developed because they they can can achieve achieve representation targets more more efficiently efficiently (with less cost, area, or number of reserves than, for instance, reserves)) than, instance, scoring scoring approaches approaches or or approaches approaches based based on on hot hot spots spots of of richness richness or or rarity rarity (Williams, (Williams, 200 1 ; Possingham aI., 200 1 ) . Box 22. 2001; Possingham et al., 2001). 22.11 exemplifies exemplifies the the concept concept of of com complementarity by comparing comparing scoring scoring approaches approaches to to complementarity-based complementarity-based methods. methods. In practice, practice, the idea of complementarity refers to to the amount amount of new new features features that that aa site site would would add add to to the the already already represented represented features features in in the the set set of of selected selected sites sites in in iterative iterative heuristic heuristic algorithms algorithms (see (see section section Reserve Reserve Selection Algorithms) Algorithms).. Complementarity-based 1 983) Complementarity-based methods methods were were first first described described bbyy Kirkpatrick Kirkpatrick ((1983) and 980s and were developed simultaneously by several authors authors during the 11980s (Kirkpatrick, 11983; 983; Ackery and 984; Margules 988; and Vane-Wright, Vane-Wright, 11984; Margules et aI., al., 11988; Rebelo 990). The Rebelo and and Siegfried, Siegfried, 11990). The term term complementarity complementarity was was coined coined specifically specifically in 1 99 1 ) (for in the the context context of of reserve reserve selection selection by by Vane-Wright Vane-Wright et et al. ((1991) (for aa review, review, see 1). see Williams, Williams, 200 2001).
BOX 22.1
An Example of Advantages of Complementarity Methods
This sites x species matrix shows candidate sites and species occurring in them (indicated by a X) of a hypothetical system . Suppose that we could choose two sites. A scoring approach (Margules and Usher, 1 98 1 ) would select sites in decreasing order of richness (number of endemisms, site quality) starting from the richest site (a hot spot of richness), which in this example is site 2 with six species. The next rich est site would be site 3, with five species. Sites 2 and 3 would sample a total of seven species (c-i). However, another solution that represents more species with the same number of sites can be obtained when using the idea of complementarity. We can see that sites 3 and 4 would include all species and therefore complement each other better than sites 2 and 3. Scoring approaches ignore complementarity and tend to choose sets of sites that contain high levels of replication for some species while ignoring others and therefore are less efficient than complementarity methods (e.g., Williams, 200 1 ).
�
rJl
2
3
4
5
I
a
X
Ix
b
X
X
Species
c
d
e
X
X
X
X
X
X
X
X
9
h
X
X
X
X
X
X
X
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Reserve Selection Reserve S e l e c t i o n Algorithms Algorithms Two Two common common ways ways of of defining defining the the most most basic basic reserve reserve selection selection problem problem are ((1) 1 ) to to choose choose the the minimum minimum set of of reserves reserves that that contain contain all species species (or other other features) number of features) at at least a given number of times, known known as the the minimum minimum set covering problem (Underhill, 11994; 994; Csuti 997; Pressey ai., 11997), 997), covering problem Csuti et ai., al., 11997; Pressey et al., and (2) (2) to to maximize maximize the the number number of of features features represented represented when when there there is a and limit on that may be chosen, known as the on the the number number of of reserves reserves that chosen, known the maximal maximal coverage Camm et ai., 11996; 996; Church 996; Arthur coverage location location problem problem ((Camm et al., Church et et ai., al., 11996; Arthur et 997). et ai., al., 11997). These These approaches approaches share share several desirable desirable characteristics, characteristics, such such as being data data driven, driven, goal directed, directed, efficient, efficient, explicit, explicit, repeatable, repeatable, and and flexible flexible (Pressey, 11999). 999). These These features features make make these these methods methods valuable valuable for systematic reserve hoc decisions thumb derived from design in comparison comparison to to ad hoc decisions or rules of of thumb from ecological ecological theories. theories. Data Data Driven Driven
The features X The selection selection of of a set of of reserves is based based on on a ""features • sites" sites" matrix, matrix, usu usually indicating indicating the the occurrence occurrence (presence-absence (presence-absence or or probability probability of of occurrence) occurrence) of a feature feature in an area. The The features features can can be any any chosen chosen set of biodiversity biodiversity sur surof rogates rogates and/or and/or natural natural elements elements of of interest, interest, for for instance, instance, landscapes landscapes (Pressey and 989; Pressey Cocks and and Nicholls, 11989; Pressey et et ai., al., 1996), 1996), plant plant communities communities ((Cocks and Baird, 11989; 989; Nicholls 993; McDonnell Nicholls and and Margules, Margules, 11993; McDonnell et ai., al., 2002), 2002), habitat habitat types Olson and 998; Leslie types ((Olson and Dinerstein, Dinerstein, 11998; Leslie et et ai., al., 2003 2003),), environmental environmental vari variables (Arafijo (Araujo et ai., 1 ) , and Kirkpatrick, 11983; 983; Rebelo al., 200 2001), and species ((Kirkpatrick, Rebelo and and Siegfried, 992; Sa:tersdal 993; Kershaw 994; Church Siegfried, 11992; S~etersdal et ai., al., 11993; Kershaw et ai., al., 11994; Church et ai., al., 11996; 996; Csuti et ai., 997). There There are no al., 11997). no rules rules for for choosing choosing the the features features to to be represented, represented, and and often often the the choice choice is is determined determined by by the the available available information information for particular system of more discussion biodiver for the the particular of interest interest (see later later for for more discussion on biodiversity goals for for conservation). conservation). The The sites, or or selection selection units, are are any any discrete part part of landscape to contribution to of the landscape to be evaluated evaluated for for their their contribution to the the reserve system. They 998; They can can be regular regular (e.g., grid grid cells, Freitag Freitag and and Van Jaarsveld, Jaarsveld, 11998; Rodrigues 997) or Rodrigues et ai., al., 2000; 2000; hexagons, hexagons, Csuti Csuti et ai., al., 11997) or irregular irregular (e.g., woods, woods, Sa:tersdal ai., 11993; 993; pastoral ai., 11997) 997) and S~etersdal et al., pastoral holdings, holdings, Pressey et al., and can can be continuous continuous or or discrete discrete (e.g., forest forest fragments) fragments).. Goal Goal Directed Directed
Quantitative Quantitative goals are set for for all features. Most Most often the goals have been framed framed in in terms terms of of representation representation of of the the features features by by setting setting quantitative quantitative targets targets for for each each feature feature (e.g., at least one occurrence of of all species). Increasingly, other other goals goals that that incorporate incorporate considerations considerations of spatial spatial distribution distribution of of reserves, or or probabilities probabilities of of persistence persistence for for the the species, species, have been considered considered (Possingham ai., 2000; ai., 2002). (Possingham et et al., 2000; Noss Noss et et al., 2002). The The targets targets may may be be different different for for each of the the features. features. Efficient Efficient
Reserve Reserve selection selection methods methods acknowledge acknowledge the the limited limited resources resources for for conserva conservation tion and and therefore therefore aim aim at at minimum-cost solutions, solutions, given given the the goai. goal. The The cost cost is is usually usually the the number, number, or or total total area, area, of of selected selected sites. sites. Some Some authors authors consider consider
22. 22. METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICS AND AND RESERVE RESERVENETWORK NETWORK DESIGN DESIGN
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more more economically economically oriented oriented costs, costs, such such as as acquisition acquisition or or opportunity opportunity costs costs ((Ando Ando et al., aI., 11998; 998; Polasky et aI., 1). al., 200 2001).
Biodiversity Biodiversity Conservation Conservation Goals Goals and and Data Data Requirements Requirements Because Because all all biodiversity biodiversity cannot cannot be be measured, measured, there there is is aa need need to to base base reserve reserve bio selection on biodiversity surrogates. There There are three major types of biodiversity diversity surrogates: subsets subsets of of taxa, assemblages, assemblages, and and environmental environmental vari variables. ables. Different Different surrogates surrogates appear appear to to be be better better than than others others under under different different circumstances (Williams et aI., 1 ; Margules al., 2000; 2000; Araujo Aratijo et aI., al., 200 2001; Margules et aI., al., 2002). In practice, because available data for setting priorities priorities usually come from from different different sources, sources, some some combination of of these these surrogates surrogates should should be be used used (Nix et al., 2000). aI., 2000). Distributional Distributional data data for for species species can can be be constructed constructed from from records records held held by by museums and sets have and herbariums, herbariums, but but this is presence-only data. Most Most data datasets spatial or taxonomical taxonomical bias, as in the sampling process, process, certain kinds of sites and taxa taxa are favored favored (i.e., easily accessible sites and and vertebrates; vertebrates; Margules Margules and and Austin, 11994). Jaarsveld ((1998) problem of of using using Austin, 994). Freitag and van Jaarsveld 1 998) assessed the problem incomplete sets for incomplete data datasets for reserve reserve selection selection and and found found aa large large variation variation in in selected selected networks networks as as well well as as decreases decreases in in efficiency. efficiency. Andelman Andelman and and Willig Willig (2002) (2002) found found that the source that source of data affects the location of of reserves and and the efficiency of the solutions. Therefore, systematic and intensive biodiversity surveys are an important important element element of of efficient efficient conservation conservation planning. planning. Reserve Reserve selection selection algo algorithms rithms can can also also be be applied applied to to data data on on probabilities probabilities of of occurrence occurrence (e.g., (e.g., Araujo Araujo and and Williams, 2000; Williams and and Araujo, Aratijo, 2002 2002).). Analytical procedures procedures can be be used used to to address address the the problem problem of of bias bias in in data data and and provide provide probability probability values values for for all all points points in in aa map map based, based, for for instance, instance, on on environmental environmental variables variables (Elith, (Elith, 2000; aI., 2002). 2000; Williams Williams et et al., 2002).
22.2 22.2
M ETAPOPULATION DYNAMICS, METAPOPULATION DYNAMICS, HABITAT HABITAT LOSS, LOSS, AND AND RESERVE RESERVE SELECTION SELECTION ALGORITHMS ALGORITHMS Regional reserve networks should) have two networks ((should) two strategic goals: (i) to to effi efficiently ciently represent represent the the full full spectrum spectrum of biodiversity biodiversity within within aa system of protected protected areas and and (ii) (ii) to to ensure ensure the the long-term long-term persistence persistence of of that that biodiversity biodiversity (Frankel (Frankel and 9 8 1 ) . Until and Soule, Soul~, 11981). Until recently, reserve selection selection methods methods have generally focused focused on on the the goal goal of of representation representation and and the the goal goal of of persistence persistence has has been been neg negfew authors authors have paid paid attention attention to the role of spatiotem spatiotemlected. Moreover, few poral population dynamics dynamics and and the the effects effects of of landscape landscape changes. changes. After After formulating the basic reserve network network design problem, problem, most of this chapter chapter formulating focuses on these issues. Simple approaches approaches base the the selection of reserves on static patterns patterns of of species presence/absence ((often snapshot of incidence patterns). patterns). Several presence/absence often a single static snapshot studies studies have have analyzed analyzed the the persistence persistence and and turnover turnover of of species species in in reserves reserves designed 1 994) designed with with these these simple simple methods. For For instance, instance, Margules Margules et et aI., al., ((1994) evaluated evaluated aa procedure procedure for for identifying identifying aa minimum minimum set set of of sites sites that that would would capture capture all all rare rare or or uncommon uncommon plant plant species species in in the the region. region. They They compared compared the the occurrences occurrences in in the the minimum minimum set set solution solution with with the the occurrences occurrences after after 1111 yr in
546 541415
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the same same locations locations and and found found that that the the original site site selection selection was was inadequate inadequate to to the the initial initial species species (the loss loss of of species species in in the the selected selected network network was was 36% 36% preserve the even when when there there was was no no habitat habitat loss loss in in the the region). region). Other Other studies studies have have reached reached even the same same conclusion: conclusion: the the static static representation representation approach approach does does not not account account for for the spatial turnover, turnover, and and species may may be be naturally naturally lost lost from from the the selected selected species spatial short time (Nicholls, 1998; 1 998; Rodrigues Rodrigues et et al., 2000; 2000; sites after after a relatively short Araujo et et al., al., 2002). 2002). Arafijo selection approaches approaches have have considered considered criteria criteria for for persistence, persistence, Some reserve selection but mostly mostly in an an implicit way. These criteria criteria affect the selection selection units units but way. These affect (i) the including sites of of a minimum size), (ii) (ii) the the features features to to be considered considered (e.g., by including (ignoring representation representation of of common common species and and concentrating concentrating on on endangered (ignoring endangered or on on endemic endemic species), or or (iii) (iii) the the goal. Criteria Criteria that that affect affect the the species or goal include, include, for for example, approaches approaches based based on on abundance abundance data data (that (that aim aim at at goal where the the species is more more abundant abundant or or where where the the total total number number selecting sites where conserved is above above a viability threshold) threshold) or or "bet-hedging" " bet-hedging" approaches approaches (multiple conserved representations for four populations populations of each representations for all species, i.e., include at at least four species; and Moilanen, 1) . species; see see Cabeza Cabeza and Moilanen, 200 2001). Spatiotemporal landscape change change Spatiotemporal population population dynamics dynamics and and effects effects of landscape habitat loss and and degradation) degradation) have generally been ignored. Dynamics (including habitat concerning landscape changes changes have more commonly concerning have been more commonly considered, but but this this has been basically from the perspective perspective of of cost cost effectiveness, e.g., scheduling scheduling from the and acquisition acquisition of of sites (e.g., Pressey, 1999; and Taffs, and 1 999; Pressey and Taffs, 2001), 200 1 ), not not from from the perspective perspective of of linking landscape the landscape dynamics, spatial population dynamics, and reserve network network design. design. and reserve We important to to adopt population dynamics view We believe believe it is important adopt a spatial population view in combination with with a spatially structured structured landscape landscape to to determine the reserve combination network network that that will will conserve conserve biodiversity biodiversity in in the the long long term. term. For For instance, instance, Cabeza Cabeza and and Moilanen Moilanen (2003 (2003)) explored, explored, by by simulating simulating species-specific species-specific spatial spatial popula population dynamics, what what happens happens in a reserve reserve network network selected selected by reserve selection algorithms algorithms with with simple representation representation goals goals when when all all nonselected nonselected candidate candidate reserves are lost. Simple reserve selection algorithms tended tended to select small and overdispersed overdispersed reserve reserve networks. networks. Cabeza Cabeza and and Moilanen (2003 (2003)) showed showed that that when reserves are when selected selected reserves are far far apart apart and and only only one one or or aa few few representations representations of of each each species species are are considered, considered, numerous numerous extinctions extinctions can can be be expected. expected. The The chances of having species extinctions in such a reserve system are even larger if bitat surrounding if the the ha habitat surrounding the the reserve reserve is is lost. However, However, when when the the selected selected reserves reserves were were kept kept close close together, together, the the number number of of extinctions extinctions following following habitat habitat loss Cabeza and loss was was much much smaller smaller ((Cabeza and Moilanen, Moilanen, 2003). 2003). This This is is another another way way of of saying saying that that habitat habitat loss loss around around aa reserve reserve network network will will cause cause an an extinction extinction debt debt in 994; in the the region region and and that that extinctions extinctions in in the the region region will will follow follow (Tilman (Tilman et et al., al., 11994; Hanski Hanski and and Ovaskainen, Ovaskainen, 2002; 2002; Ovaskainen Ovaskainen and and Hanski, Hanski, 2002) 2002) In summary, the the ways in which persistence and and spatiotemporal population population dynamics dynamics have have been been considered considered in in reserve reserve selection selection algorithms algorithms can can be be classified classified into following Cabeza into four four broad broad problem problem categories categories ((following Cabeza and and Moilanen, Moilanen, 2003 2003).). CO. In this category, no criteria for persistence or spatiotemporal dynamics are used. This is the "representation" "representation" problem, which implicitly assumes assumes that that aa species species will will persist persist indefinitely indefinitely in in any any site site where where it it was was observed. observed. This This might might be be appropriate appropriate when when the the probability probability of of losing losing aa
22. 22. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS AND AND RESERVE RESERVENETWORK NETWORK DESIGN DESIGN
547 547
species from from a site is very low, which may be the the case when sites are rather rather large. large. C l . No although implicit or C1. No spatial considerations considerations are taken taken into into account, account, although or explicit explicit criteria criteria favoring favoring local local persistence persistence are are used, used, by by acknowledging acknowledging the population size the effects effects of of area area or or population size on on persistence. This This is is often often done by by using using thresholds thresholds to to define define the the minimum minimum size size of of selection units, units, by by basing basing the the selection selection on on abundance abundance data data and and species species abundance abundance targets, probability approaches targets, or or by by considering considering habitat habitat quality quality on on probability approaches (see (see Section Section 22.3 22.3 for for aa more more extended extended description description of of probability probability approaches). approaches). C2. C2. Spatiotemporal Spatiotemporal population population dynamics dynamics are are considered considered implicitly implicitly by by means means of compact reserves/ of keeping reserves reserves close together, together, which which results in in compact reserves/ reserve reserve networks networks that that are are less less susceptible to to negative negative external external effects effects than scattered reserve networks. networks. This class includes includes methods methods referred referred to spatial reserve to as as ""spatial reserve design" design" (Possingham (Possingham et et al., al., 2000). 2000). A A detailed detailed description description of of these these approaches approaches is is given given later. later. C3. These methods methods explicitly consider species-specific species-specific spatiotemporal spatiotemporal popu popuC3. These lation lation dynamics dynamics and and regional regional persistence, persistence, and and possibly possibly the the interaction interaction between population dynamics between population dynamics and and landscape landscape change. change.
22.3 22.3
COMPARISON COMPARISON OF OF DIFFERENT DIFFERENT ALGORITHMS ALGORITHMS This This section section describes describes in in increasing increasing order order of of complexity complexity the the site site selection selection algorithms algorithms used used in in the the following following comparison. We We start start with with the the minimum minimum set set covering covering formulation, formulation, with with aa simple simple (single) (single) representation representation problem: problem: "what "what is is the the smallest smallest set set of of sites sites that that contains contains at at least least one one representation representation of of each each species": species": N N
m in 2: min ~] Ii Ii
ii=1 =l
given given that that
(22. 1) (22.1) N N
all j. 2: IiPi IiPijj :::: >--: 11 for for all j.
i=l i=1
In In this this problem, problem, and and in in later later problem problem formulations, formulations, N N is is the the number number of of candi candidate date sites, sites, M is is the the number number of of species, species, Pi Psij is is an an element element of of an an N N X x M matrix matrix giving giving the the probability probability of of presence presence of of species species jj in in site site i,i, and and Ii Is is is an an indicator indicator variable variable indicating indicating whether whether site site ii is is selected selected or or not. not. The The minimum minimum set set coverage coverage formulation formulation may may be be extended extended easily easily to to cover cover aa variety variety of of situations. situations. To To do do so, so, we we introduce introduce some some new new symbols: symbols: let let n ~ be be the set of all sites, S be the index set of selected sites (for which Ii = 1 ), the set of all sites, be the index set of selected sites (for which I i = 1), T T ;i the cost of site i, Ai area the target target level level of of conservation conservation set set for for species species j, j, Ci ci the the cost of site A i the the area of of site site i, and and R;(S) Ri(S ) the the representation representation of of species species jj in in S. A A generalized generalized form form of of Eq. . 1 ) is Eq. (22 (22.1) is now now m m in in 2: ~] Cdii iES iES
548
MAR MAR CABEZA CABEZA ET ET AL. AL.
given given that that
(22.2)
Rj S ) == 2: Rj ((S) ~ Pijij 2: >- Tj Tj ii EESS P
for for all all j. j.
In this formulation 11 when minimizing the number formulation Ci Ci = ~ number of sites, Ci Ci = - - Aj A i if minimizing the area of the the solution, solution, or Ci ci may be equal equal to to a real site cost if such information information is available. The The commonly used site selection algorithms algorithms ijE{O,I}, but operate on presence-absence data, in which case operate on presence-absence data, in which case P pijE{0,1}, but P Piiij can can also also be be the the probability probability of of presence presence of of the the species species if if aa statistical statistical model model for for that that exists. exists. For For the the simple simple representation representation problem, problem, Tj T j -= 1, and and for for the the multiple multiple representation problem, Tj > 1 (often an integer) . The proportional representation problem, Tj > 1 (often an integer). The proportional cover coverIn PPsi, where a age problem is obtained by age problem is obtained by setting setting Tj Tj = - a cxE~ cx is is the the proportion proportion i" where of populations populations that that have to to be protected protected for for each species. Different Different weights weights can be given to different species by setting comparatively higher can be given to different species by setting comparatively higher targets targets for for species that that are considered important important in the the region. Note Note that that the problem definition definition applies equally to to any biodiversity elements (habitat (habitat types, ecosys ecosystem tem type, type, etc. etc.).) . Variants ooff the minimum set coverage problem problem are solved commonly Variants using a stepwise ai., 11997; 9 97; stepwise richness-based richness-based heuristic heuristic algorithm algorithm (e.g. (e.g.,, Csuti et al., Pressey ai., 11997). 997). Pressey et et al.,
Algorithm Algorithm 11:: Forward Forward Richness-Based Richness-Based Heuristic Heuristic 11.. set S S= = 0 O 2. for for all sites k not not in S, S, calculate calculate measure measure Ub Uk, which gives the proportion proportion of of underrepresentation underrepresentation covered covered by by the the addition addition of of site k; > kj / (Tj Rj(S)), Uk summing only over species j having Tj Uk = = I ~jPkj/(Tj Rj(S)), over Tj > Rj(S) Rj(S) jP 3. add site k with highest ratio Uk/Ck to S 3. add site with highest ratio Uffck to S 4. else quit 4. if if Rj(S) Ri(S ) < < Tj T i for for any any j, j, go go to to 22 or or else quit For the simple representation representation problem, problem, algorithm algorithm 11 adds adds the site iteratively, which adds adds the greatest number number of new species to the solution solution until all species are represented represented at least once. At each step you add the patch patch with the highest value in complementarity richness per patch patch cost. At least for for small and and moderate problems, Eqs. (22. (22.1) moderate 1 ) and (22.2) can be solved exactly using linear programming Cocks and 993; Rodrigues programming ((Cocks and Baird, Baird, 1989; 1989; Possingham Possingham et et ai., al., 11993; Rodrigues and Gaston, 2002), 2002), although the simple stepwise heuristic runs runs orders orders of magnitude quicker (Pressey et ai., that al., 1996; Possingham et ai., al., 2000). 2000). Note Note that in the final solution solution some species will have Rj(S) Rj(S) > > Tj, Tj, i.e., the species are over overrepresented. represented. Algorithm Algorithm 11 does not not give any consideration consideration to to the distribution distribution or value of overrepresentation overrepresentation among species. the minimum minimum set covering formulation formulation that that use presence-absence presence-absence Variants of the data belong to problem category CO because they do not not consider space and and spatial population population dynamics; effects of habitat habitat loss outside the reserve network network are not not considered, considered, and and it can be interpreted interpreted that that species are implicitly assumed to proba to occur occur forever forever in in the the sites sites that that they they were were observed. observed. If If P Psiij is is based based on on aa probability model for the presence of species, the algorithm partly belongs to cate catethat probabilities probabilities of presence in sites partially gory C1 as it is acknowledged that
22. 22. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS AND AND RESERVE RESERVENElWORK NETWORK DESIGN DESIGN
$49 549
reflect reflect their their capacity capacity to to support support species species (and (and presumably presumably good good sites sites are chosen), chosen), but but the the spatial spatial configuration configuration of of the the reserve reserve network network is is still still ignored. ignored. The simplest site selection method, method, which attempts attempts to to consider the the spatial configuration configuration of the habitat, habitat, is based based on an adjacency adjacency rule (Nicholls and and Margules, 993). This algorithm is similar to algorithm 11,, but Margules, 11993). but if there there is a tie in step 33 (several sites give the same contribution contribution to complementarity rich richness), then then the site closest to to any already selected site is chosen. In other other words, words, proximity proximity to to sites sites in in SS is used to break break ties. However, spatial configuration configuration is only of of secondary importance importance when when using using the the adjacency rule (richness (richness is the the primary primary objective of optimization), optimization), and and consequences consequences of using using the adjacency rule rule are are entirely entirely dependent dependent on on the the occurrence occurrence of of ties, ties, which which may may be be infrequent, infrequent, especially if if Ci ci is is based based on on site site area area and and Ai A i values values vary. vary. Consequently, Consequently, we we do do not not believe believe that that the the adjacency adjacency rule rule can can be be expected expected to to be be of of much much value value in in spatial spatial reserve reserve design design in in general. general. The The first first method method to to properly properly consider consider the the spatial spatial configuration configuration of of aa reserve reserve is is given given by by Possingham et et al. al. (2000), (2000), which which minimizes minimizes aa linear combination combination of of reserve reserve area area and and boundary boundary length. length. In In this this algorithm, algorithm, it it is is possible possible to to obtain obtain reserves reserves with with different different levels levels of of aggregation aggregation by by giving giving different different relative relative weights weights to to area area and and boundary length. length. Cabeza Cabeza et et al. (2004) (2004) described aa method that that combines the the probability of of occurrence occurrence approach approach (Araujo and and Williams, 2000; 2000; Williams and and Araujo, Araujo, 2000, 2000, 2002) 2002) with with the the method method of Possingham et al. (2000). (2000). In In this this method, method, the the probabilities probabilities of of presence presence of of species species jj on on patch patch i, P Pij, ij ' are obtained obtained by by fitting fitting aa habitat habitat model (e.g., (e.g., by by logistic logistic regression) regression) to to obser observations vations of of species species presence-absence. presence-absence. Optimization Optimization minimizes minimizes the the cost cost of of the the reserve network network penalized by the boundary boundary length length of the the reserve, reserve, and and targets targets are are set set as as expected expected numbers numbers of of populations. populations. By By setting setting aa high high penalty penalty for for boundary length, a relatively compact reserve is obtained, obtained, whereas whereas the the spatial configuration configuration of the reserve is of no no consequence consequence when when the boundary boundary length penalty approaches approaches zero. More More formally, the problem is m in L min ~ CCii + + bL bL
iES i~S
(22.3) (22.3)
subject subject to to
R j ( (S) S) R;
=L ~ , PPi ji; 2::: >- T Tjj
iES i~S
for all j for allj
in which L L is the boundary boundary length and bb is the penalty given to boundary boundary length length (relative (relative to to other other costs). costs). The The objectives objectives in in Possingham Possingham et et al. (2000) (2000) and and Cabeza et al. (2004) (2004) are very similar; both both essentially minimize a linear combin combination of cost (area) and boundary boundary length. The difference between between the methods methods is that in the latter the conservation conservation targets targets Tj T i are are framed framed in in expected expected numbers numbers of populations, as of populations, as P Piiij values values are are based on on aa probability model model for for the the presence presence of of the species. Cabeza Cabeza et al. (2004) (2004) used the ratio ratio of reserve boundary boundary length boundary length of the reserve (L) to reserve area (L') (L') instead of the boundary (L) directly. This is because L' L' is much much less dependent dependent on the absolute absolute size of of the the system than L' is more suitable to be used in the context than L, L, which which means that that L' context of aa stepwise stepwise heuristic heuristic optimization optimization algorithm algorithm where where the the number number of of sites sites (and (and L) L) varies during during the optimization optimization process.
MAR MAR CABEZA CABEZAET ETAL. AL.
550 $$0
Note Note that that the the methods methods of of Possingham Possingham et et ai. al. (2000) (2000) and and Cabeza Cabeza et et ai. al. (2004) (2004) both both belong belong to to problem problem category category C2. C2. This This means means that that reserve reserve aggregation aggregation is is achieved achieved in in aa qualitative qualitative manner, manner, without without any any estimate estimate of of the the species-specific species-specific effects effects of of aggregation aggregation on on spatial spatial processes processes and and persistence persistence of of populations populations per ust assumed per se. se. It It is is jjust assumed that that aggregation aggregation is is useful useful because because itit decreases decreases edge edge effects effects and and reserve reserve maintenance maintenance costs (Possingham (Possingham et et aI., al., 2000). It It follows follows that that an an important important question question is is how how much much aggregation aggregation in in the the reserve reserve network network can can you you get get with with little little or or no no increase increase in in reserve reserve cost? cost? In In the the example example of of Cabeza Cabeza et 0 % decrease et ai. al. (2004), (2004), it it was was typically typically possible possible to to achieve achieve aa 550% decrease in in L with with aa <2 <2 % % increase in in total total reserve reserve cost. cost. Similar reductions in boundary boundary length for for minimal minimal increases increases in in area area have have been been achieved achieved for for terrestrial terrestrial and and marine marine reserve reserve network network design design problems problems (McDonnell (McDonnell et et aI., al., 2002, 2002, Leslie Leslie et et aI., al., 2003). 2003). Cabeza Cabeza (2004) (2004) presented presented an an important important development development of of Cabeza Cabeza et et al. al. (2004). (2004). In In Cabeza Cabeza et et al. al. (2004) (2004),, Pij Psivalues values are are constant constant during during optimization; optimization; they they are are not not dependent dependent on on S. Connectivity Connectivity and and recolonization recolonization of of empty empty habitat habitat will, will, how however, ever, be be important important for for some some of of the the species. For For such such species species it it is is possible to to build build aa dependency dependency between between Pij Psi and and S into into the the optimization. optimization. This This means means that that habitat habitat loss loss will will decrease decrease Pij Psi values, values, especially especially near near the the location location where where habitat habitat has has been been lost, lost, and and Pij Psi will will decrease decrease possibly possibly substantially substantially when when site site i loses loses many many of of its its neighbors. neighbors. We We start start by by defining defining aa connectivity connectivity measure measure Gijii for for species species jj in in site site i:
Gij = ~
f (dik) Pkj,
(22.4) (22.4)
kENi(S)
in which Nj(S)ES Ni(S)ES is the the neighborhood neighborhood of of site j in S. NAS) Ni(S) may be equal to to S-{k buffer around around patch patch j, which is what what is used in the the S-{k},}, but but it can also be a buffer Moilanen and and Nieminen (2002) (2002) for for a on following example example [see [see Moilanen a discussion discussion on the distance distance connectivity measures used used in in metapopulation metapopulation models], models]. dik connectivity measures ask is is the ((dik) is is aa decreasing decreasing function function of of ask. dik• Then between between sites sites i and and k, and and f(dik) Then
psi(S) = f[hi, Gii(S)],
(22.5)
in which which hi vector of of habitat habitat variables, variables. pi(S) pAS) may may be fitted originally, originally, for for in h i is a vector be fitted example, using using logistic logistic regression, regression, the the original original presence-absence presence-absence information, information, example, and Gii(~) Gij(11) calculated calculated assuming assuming Pij in sites sites ii where where species species jj was was observed observed and Psi == 11 in and 0 otherwise. otherwise. If aa species species is not not affected affected by by connectivity, connectivity, then then simply simply put put and The optimization optimization problem problem is is what what Cabeza Cabeza (2003) (2003) called called the the Psi(S)--f(hi). Pi/ S) = ((hi). The dynamic probability probability problem. problem. "Dynamic" " Dynamic" comes comes from from the the fact fact that that the the probprob dynamic abilities are are reevaluated reevaluated during during optimization, optimization, meaning meaning that that the the method method takes takes abilities into account account that that habitat habitat outside outside the the reserve reserve will will be be lost: lost: into
bL' m Ill F = = ~, 2: CCii -I-+ bL' min i ES
i~S
(22.6 ) (22.6)
subject subject to to
Rj(S ) -= ~2: Pij Tj for for all all j.j. Pi; (hi, ( hi, S) S ) >� Tj Rj(S) i ES
iES
To solve solve Eq. Eq. (22.6), (22.6), all all probabilities probabilities Psi Pi; have have to to be be recomputed recomputed by by iterating iterating To Eqs. (22.4) (22.4) and and (22.5) (22.5 ) after after changing changing SS until until convergence. convergence. Any Any analogue analogue of of the the Eqs.
22. METAPOPULATION M ETAPOPULATION DYNAMICS DYNAMICS AND AND RESERVE RESERVE NETWORK NE1WORK DESIGN DESIGN 22.
551 551
forward stepwise stepwise heuristic heuristic (algorithm (algorithm 1) 1 ) will will perform perform poorly poorly with with Eq. Eq. (22.6). (22.6). forward Assume that that the the optimal optimal solution solution consists consists of of several several essentially essentially separate separate patch patch Assume aggregates. The The forward forward algorithm algorithm will will start start by by selecting selecting one one site site and and therethere aggregates. after itit will will overextend overextend this this first first cluster cluster because because starting starting aa new new cluster cluster from from one one after site (and (and thus thus very very low low connectivity connectivity and and low low Piis) will have have aa lower lower marginal marginal site Piis) will contribution than than extending extending the the existing existing cluster, cluster, which which has has high high connectivity. connectivity. contribution 2003; Better solutions solutions will will be be achieved achieved by by aa backward backward algorithm algorithm (Cabeza, ( Cabeza, 2003; Better Cabeza et et al., aI., 2004); 2004); this this algorithm algorithm starts starts by by selecting selecting all all sites sites and and then then Cabeza removes "bad" " bad" sites sites one one by by one one until until no no site site can can be be removed removed without without the the tartar removes get being being violated violated for for at at least least one one species. species. For For the the algorithm, algorithm, we we need need to to define define get the changes changes in in F F and and R R following following the the removal removal of of site site i:i: the
L + 6.L L ) - b(( L+AL - - L)
= -- ciCi ++ b\ A--(S A6.Fi Fi = {i})) A ( S ~_ -~i}
A(S)
(22.7) (22.7)
M APij A R i = E E Rj(S) - rj' i~sj=l
in which which Pij where A(S) A(S) is is the the area area of of solution solution S S and and Api 6.Pi;i (--<0) (:::::: 0 ) isis Pij'-Pii Pi!' - P'i in Pii where and Pi to the the converged converged values values of of Pi before and and after after the the removal removal and Pij' correspond to Psi; before ;' correspond 0) and :: O ) express express changes changes in cost cost and and of of site site i,i, respectively. 6.Fi AF i «(<0) and AR((:::: ARi'(<~O) boundary boundary length length and and changes in species species overrepresentation, overrepresentation, respectively. 6.R;l6.Fi During each step, the site with with the smallest smallest ratio ratio A During each optimization optimization step, R i / A F i is chosen removal. Note large chosen for for removal. Note that that aa large 6.R;f6.Fi A R i / A F i means means that that either either (i) A6.Rj R i is large and and therefore therefore removing removing site site i would would cause cause aa large large (negative) (negative) change change in in species species overrepresentation ii) 6.Fi overrepresentation (not desired) and/or and/or ((ii) AF i is is small, small, meaning meaning that that remov removing ing site site ii would would increase increase the the fragmentation fragmentation of of the the reserve reserve system system (not (not desired) or or the the cost cost of of site site i is is low, low, or or both. both. Westphal problem of Westphal and and Possingham Possingham (2003) (2003) considered considered aa similar similar problem of optimal optimal habitat habitat reconstruction reconstruction where where the the objective objective is is to to maximize maximize the the summed summed prob probabilities abilities of of occurrences occurrences and and these these probabilities probabilities are are modeled modeled as as aa function function of of the the entire entire landscape. landscape. They They use use simulated simulated annealing annealing to to find find good good solutions solutions to to their their problem. problem.
Algorithm Algorithm 2: 2: Backward Backward Stepwise Stepwise Heuristic Heuristic 11.. set set SS = = n calculate (:::::: 0 ) and calculate 6.Ri ARi(<~O) and 6.Fi AF i for for all all sites sites i E ~ SS let let K E SS be be the the set set of of sites sites k, the the removal removal of of which which does does not not violate violate the the constraint constraint T; T i for for any any species species jj and and for for which which 6.Fk ~lFk < < 00 4. 4. if if K = = 0, Q, optimal optimal solution solution = = S, S, quit quit 5. 5. remove remove site site k E K with with smallest smallest ratio ratio 6.Rk/6.Fk ARk/AFk from from S, S, go go to to 22
2. 2. 3.3.
Equations Equations (22.1 (22.1)) and and (22.2) (22.2) assume assume implicitly implicitly that that habitat habitat outside outside the the reserves reserves will will not not change change after after the the reserve reserve has has been been established established (or (or at at least least no no effects effects of of habitat habitat loss loss are are considered). considered). In In contrast, contrast, Eqs. Eqs. (22.3) (22.3) and and (22.6) (22.6) assume assume that that changes changes will will occur occur for for unselected unselected sites. sites. Equation Equation (22.6) (22.6) can can be be extended extended to to include Cabeza and include an an explicit explicit model model of of landscape landscape change change ((Cabeza and Moilanen, Moilanen, 2003). 2003).
MAR MARCABEZA CABEZAET ETAL. AL.
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Properties of different solutions that that are of interest include cost of solution, total total area, total boundary length, and the representation representation level calculated calculated either presence-absence observations or from the habitat model assuming from presence-absence probabilities with habitat loss or ((ii) probabilities (i) unchanged probabilities ii ) dynamic probabilities [requires iteration iteration of Eqs. (22.4) (22.4) and (22.5)]. (22.5)]. When comparing solutions obtained using two two different algorithms, you should set one aspect (e.g., cost) of obtained the solutions to be equal and compare compare the rest of the attributes attributes (e.g., boundary boundary species).. The problems and algorithms algorithms are compared compared length, persistence of species) using the same data set.
22.4 22.4
A CASE CASE STUDY STUDY A Data used here are from a survey of 26 26 butterfly and moth moth species, includ includtwo endemic races (Plebejus argus caernensis and Hiparchia semele thyone) ing two the Creuddyn Creuddyn Peninsula, Peninsula, north north Wales, United Kingdom ((Cowley living on the Cowley et al., ai., 2000; Gutierrez Gutierrez et ai., al., 2001; 2001; Menendez Menendez et ai., al., 2002; 2002; Thomas Thomas et ai., al., 2002; 2002; 2000; Wilson et al., ai., 2003; 2003; Cabeza Cabeza et ai., al., 2004). 2004). The The presence presence and and density of these butterflies have been surveyed in a 35-km 35-km 22 region divided into 500 500-m butterflies 500 Xx 500-m al., 2000). 2000). squares (Cowley et ai., For the purpose purpose ooff comparing comparing reserve selection algorithms, algorithms, two two statistical models were fitted fitted for each species ((Cabeza, 2003; Cabeza et ai., al., 2004). 2004). Both Cabeza, 2003; models models were logistic regressions explaining explaining the the observed presence-absence presence-absence of the species species by a density index index (a habitat habitat quality-based quality-based index, index, see see Cowley et ai., al., the 2000). The The other other model model also included a connectivity connectivity measure measure as an explanatory explanatory 2000). variable. We proportion of occupied cells in the neighbor variable. We used the proportion the one cell neighborhood connectivity, with on the hood as connectivity, with neighboring cells that that were on the ocean excluded from computation on the the assumption assumption that from the computation that the sea is a reflecting reflecting boundary boundary for the measure was was significant, for the butterflies. butterflies. Even this this simple connectivity connectivity measure and for 23 25 ((Cabeza, Cabeza, 2003). 2003). For the and often often highly significant, for 23 species out out of of 25 species of connectivity, connectivity, the the model was based based species not not having aa significant significant effect effect of on the the density density index. only on
Reserve Networks N e t w o r k s Selected Selected by by Different Different A lgorithms Reserve Algorithms compare here the site selection methods methods described described earlier butter We compare here the site selection earlier using using butterdata from from the the Creuddyn Creuddyn Peninsula. Peninsula. The The following following denotes denotes by BLM/PA BLMlPA and and fly data (22.3)] when when using BLM/prob the the boundary boundary length length minimization minimization method method [Eq. (22.3)] BLM/prob presence-absence data data or or probability probability data, data, respectively. respectively. The The probability probability model model presence-absence is based on (2000). By DP we we denote denote on the the grid density density index index of of Cowley Cowley et et al. (2000). the dynamic probability probability method method [Eqs. (22.4-22.6); (22.4-22.6); we we used used b -= 0]. 0]. Targets in the dynamic the following analyses are are proportional proportional coverage targets, with with the the same same propro the following analyses coverage targets, portion portion used used for for all species species (computed (computed from from full original original data). data). Note Note that that and BLM/prob BLM/prob methods methods with with the the penalty penalty set set to to zero zero (b -= 0) 0) reduce reduce BLM/PA and to (22.2)]. to variants variants of of the the ordinary ordinary multiple multiple representation representation problem problem [Eq. (22.2)]. Figure 22.1 22.1 shows shows sites with with the the highest highest species species richness richness in i n the the Creuddyn Creuddyn Figure region. region. As shown shown later, later, however, however, the the richest richest sites are are not not automatically automatically the the most most valuable valuable when when considering considering complementarity complementarity richness. richness. Some of of the the
22. METAPOPULATION ETWORK DESIGN METAPOPULATION DYNAMICS AND RESERVE RESERVEN NETWORK DESIGN
553 553
most most species-rich species-rich sites sites are are likely likely to to be be in in the the optimal optimal selection, selection, but but the the selection selection may may also be influenced influenced heavily by rare rare species, which do not not always occur occur on on species-rich species-rich sites. sites. For For this this problem the the minimum minimum set set coverage coverage solution solution [Eq. 1 )] consists [Eq. (22. (22.1)] consists of of only only three three 500 500 X x 500-m 500-m sites sites (not (not shown). shown). It It is is obvious obvious from from studies studies of of butterfly butterfly population population dynamics dynamics that that three three scattered scattered sites sites of of this this size size cannot cannot be be expected expected to to maintain maintain viable viable populations populations of of 26 26 butterfly butterfly and and moth 997; Hanski, 999). Following moth species (e.g., Thomas Thomas and and Hanski, Hanski, 11997; Hanski, 11999). Following habitat loss, such such aa "reserve "reserve network" network" could could be be expected expected to to lose lose species species quickly quickly (see (see Chapter Chapter 20). 20). Results Results in in Figs. Figs. 22.2-22.5 22.2-22.5 show show summary summary information information for for all all the the 26 26 species. species. Optimal Optimal solutions solutions for for different different site site selection selection methods methods and and increasing increasing target target levels are are shown shown in in Fig. Fig. 22.2. 22.2. Rows Rows one one and and four four correspond to to variants of of the the multiple representation representation problem. These These rows rows show show high high scatter scatter in in the the solution, which which is is expected expected because because no no spatial spatial component component is is included included in in optimization. optimization. It It is is easy easy to to imagine imagine that that the the reserves reserves systems systems selected selected in in rows rows one one and and four four would would be be difficult difficult to to start, expensive expensive to to maintain, maintain, and and susceptible susceptible to to habitat habitat loss/degradation loss/degradation between between reserve reserve sites. sites. Incidentally, Incidentally, use use of of the the simplest simplest method method of 993), of spatial spatial reserve reserve design, design, the the adjacency adjacency rule rule (Nicholls (Nicholls and and Margules, Margules, 11993), does does not not differ differ from from nonspatial nonspatial reserve reserve selection selection methods methods because because very very few few ties ties occur occur in the optimization and and thus thus the solutions solutions are almost entirely unaffected unaffected by the rule (not by the adjacency adjacency rule (not shown). shown). Having Having a penalty for for boundary boundary length changes optimal solutions solutions drastic drastically. ally. Reserves Reserves in in Fig. Fig. 22.2 22.2 obtained obtained using using BLM/PA BLM/PA or or BLMlprob BLM/prob with with b > > 00 are much more aggregated aggregated than the the b = 00 solutions. solutions. Of Of these, these, BMLlprob BML/prob produces more more compact compact reserves reserves with with less less bboundary o u n d a r y- it it will will be be difficult difficult to to find find compact compact reserves reserves using using BLM/PA BLM/PA if if the the species species is is often often missed missed in in the the field field because many gaps will be left in the the observed distribution distribution of the the species. Also, BLMlPA BLM/PA may be quite quite sensitive sensitive to errors errors and and sampling artifacts artifacts in empirical empirical data data collection, collection, which which makes makes it it more more sensitive sensitive to to data data quality quality than than B LM/prob, which BLM/prob, which relies on a habitat habitat model. model. Nonetheless, Nonetheless, BLMlprob BLM/prob is likely to to be be quite quite good good for for species species for for which which the the statistical statistical habitat habitat model model explains explains a major major proportion proportion of the variance in the occurrences occurrences of the species, but but
A
B
c
Fig. 22.1 22.1 Maps of the Creuddyn Peninsula showing sites sites having the highest species richness calculated from 0% (B), from presence-absence presence-absence data for the 25 butterflies: 5% (A), 110% (B), 20% 20% (C), and 40% 40% (D) of most species-rich sites.
554 554
.
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MAR MAR CABEZA CABEZA ET ET AL. AL.
T = 30%
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Fig. 22.2 22.2 A comparison of site site selection methods. The xx axis iiss the percentage level level for the proportional coverage target (T). The three uppermost rows correspond to presence-absence presence-absence data and the boundary length minimization method [Eq. [Eq. (22.3)] with with zero, intermediate, and high penalties for boundary length. Rows -3, but using probabilities calculated Rows 4-6 4-6 are as rows 11-3, from from the habitat model instead of PIA P/A observations. The bottom bottom row row is for for the dynamic prob probability method [Eqs. [Eqs. (22.4-22.6), b = = 0] with connectivity-dependent probabilities recalculated recalculated during each optimization step. As rows 11 and 4 have zero penalty for boundary length, they actually revert back to the multiple representation problem [Eq. [Eq. (22.2)] with a proportional coverage target.
BLM/PA BLM/PA might might be be best for for species species for for which which aa good good habitat habitat model model cannot cannot be be obtained. obtained. Interestingly, the actual areas selected by BLM/PA and and BLM/prob BLM/prob different. are quite different. The dynamic probability probability method method (Fig. 22.2, 22.2, bottom bottom row) consistently pro produces duces aggregated aggregated solutions, solutions, which which correspond correspond most most closely closely to to those those obtained obtained using BLM/prob BLM/prob ((bb > > 0). This is not not a coincidence coincidence as DP uses the the habitat habitat model model augmented augmented with with connectivity connectivity for for the the 23 23 species species for for which which connectivity connectivity No boundary length penalty was was used for the DP method method was significant. No [b [b = = 00 in in Eq. Eq. (22.6)] (22.6)] and and thus thus the the clustering clustering obtained obtained with with DP DP is is aa conse consequence of of the use of connectivity. It is encouraging encouraging that that including a component component of spatial population population dynamics into the site selection method method consistently results in clear reserve aggregation. The results of Fig. 22.2 22.2 are put put into perspective when when combined with with those those of Fig. 22.3, 22.3, which which shows shows the evolution of of reserve reserve network network area and boundary boundary length with with increasing b when when using BLMlPA BLM/PA and and BLMlprob. BLM/prob. The general trend boundary length can be obtained trend is that that a large reduction reduction in reserve boundary obtained with with
22. 22.
METAPOPULATION M E T A P O P U L A T I O N DYNAMICS D Y N A M I C S AND A N D RESERVE RESERVE NETWORK N E T W O R K DESIGN DESIGN
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Fig. 22.3 22.3 Behavior Behavior of of solution area and boundary boundary length length when using the boundary boundary length minimization minimization method method [Eq. [Eq. (22.3)]. Solid and dashed lines correspond to to presence-absence presence-absence and probability probability data, respectively. respectively. Lines with triangles and circles correspond correspond to to area and boundary boundary length, respectively. respectively. With With both both problem problem variants, a significant reduction reduction in reserve reserve boundary boundary length can be achieved with a minor area. Using PIA minor increase in reserve reserve network area. P/A observations observations produces more fragmented solutions than probability fragmented solutions probability data. Results Results shown are for the 5% tar target level.
aa minor minor (even (even zero) zero) increase increase in in reserve reserve area. area. If If reserve reserve aggregation aggregation can can be be obtained obtained for for free free in in terms terms of of cost cost (here (here cost cost = area), area), such such aggregation aggregation should should often often be be favored favored (but (but see see Section Section 22.6). 22.6). Very Very similar similar results results were were obtained obtained also also for targets other other than than 110% (Fig. 22.3). for 0 % (Fig. Figure Figure 22.4 22.4 compares compares different different site site selection selection methods methods in in terms terms of of the the expected expected number number of of populations populations (per species) calculated calculated using using the the effects effects of of connectivity connectivity and and assuming assuming nonselected nonselected sites sites are are lost. lost. In In this this comparison, comparison, DP DP does does best best and and averages averages about about 30% 30% higher higher in in terms terms of of populations populations than than the the simple simple multiple multiple representation representation variants. variants. Encouragingly, Encouragingly, BLMJPA BLM/PA and and BLMJprob BLM/prob with high b also do quite well, which indicates that that the qualitative clustering achieved achieved by by BLM BLM methods methods is is aa useful useful step step in in the the direction direction of of designing designing reserves reserves that that support support long-term long-term conservation conservation of of biodiversity. biodiversity. Another Another way way of of com comparing paring the the site site selection selection methods methods looks looks at at the the difference difference between between the the realized (Fig. 22.5). (using DP) and and target target representation representation (Fig. 22.5). BLM/PA BLM/PA systematically systematically fails fails to b. Best to achieve achieve the the set set target target regardless of of the the choice of of b. Best results are are achieved achieved with with BLMJprob BLM/prob with with high high b or or with with DP. DE BLM/prob BLM/prob can can produce produce an an overall overall overrepresentation overrepresentation of of the the target target even even when when evaluated evaluated using using DP. DE This This is is because because aa high high penalty penalty for for boundary boundary length length actually actually forces forces more more area area into into the the solution. When When comparing comparing solutions solutions of of the the same same size, size, DP DP still still achieves achieves highest highest expected expected numbers numbers of of populations populations (Fig. (Fig. 22.4). 22.4). Note Note that that some some overrepresentation overrepresentation in in the the solution solution does does not not mean mean that that any any site site can can be be removed removed from from the the solution with without the target target failing for at least one species. out The The effects effects of of reserve reserve aggregation aggregation are are not not equal equal for for all all the the species; species; those those species species that that show show strongest strongest effects effects of of connectivity connectivity are are likely likely to to be be affected affected most most
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Fig. 22.4 umber of 22.4 Different Different site site selection selection methods methods and and the the average average expected expected nnumber of popula populations dynamic probability tions as as aa function function of of solution solution area area calculated calculated over over all all species species using using the the dynamic probability method (evaluating the the effects of habitat loss). The dotted for the dynamic loss). The dotted line is for dynamic probability method [Eqs. are for the the boundary length minimization [Eqs. (22.4-22.6)]. (22.4-22.6)]. Solid lines are minimization problem problem [Eq. 3)] using using the and with high [Eq. (22. (22.3)] the habitat habitat model model for for probabilities probabilities and with zero zero (lower (lower line) line) and and high penalties (upper (upper line) for for boundary boundary length. Dashed lines are as the solid lines lines but for for presence-absence observations. The worst performers are problem variants with with zero 0). These penalty for boundary boundary length (b = = 0). These solutions are fragmented fragmented (see (see Fig. Fig. 22.2), which shows a comparatively populations when con nectivity effects are comparatively low expected expected number number of populations connectivity accounted accounted for. for.
adversely adversely by by fragmentation. fragmentation. In In these these particular particular data, Plebeijus argus is is both both an an important important endemic endemic race race and and also also aa species species showing showing strong strong effects effects of of con connectivity in statistical analysis. Figure 22.6 22.6 shows predicted effects of the site selection method for P. argus. When When accounting for the effects effects of connectivity (right bars), bars), the the species species is is expected expected to to be be practically practically extinct extinct from from any any solution solution ) . Thus with with significant significant scatter scatter (all (all solutions solutions with with PIA P/A data data or or with with b = = 00). Thus the the clustering clustering of of the the reserve reserve can can be be expected expected to to be be of of primary importance for for this this species. species. When When applying applying site site selection selection methods methods to to real real world world problems, problems, at at least least two two factors factors that that were were ignored ignored earlier earlier should should be be considered: considered: the the weighting weighting of of the the species species and and landscape landscape dynamics. dynamics. It It makes makes sense sense to to set set different different targets targets for for different different species species according according to to their their conservation conservation status. status. The The setting setting of of species species
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Fig. Fig. 22.5 22.5 Percentage Percentageof the optimization target realized realized for different site selection methods when calculated over all species species and evaluated using dynamic probabilities. A positive positive value indicates that the realized realized representation is higher than the target, which is possible possible because some common species species will have have more than the minimum required number of populations. The methods producing scattered solutions solutions perform worst.
weights is likely to be partially a political decision where where local and and global conservation conservation needs needs are balanced. balanced. In this particular particular case, giving high weight to to the the two two endemic endemic races races does does not not change change the the solution significantly significantly from from the the solutions solutions produced produced by by the the dynamic dynamic probability probability method method (not shown). shown). The The reason reason is that that the endemics have somewhat specialized habitat habitat requirements requirements that that influence influence the the solution solution disproportionately. disproportionately. The The BLM BLM and and DP DP methods applied applied to to the the case study study assume assume aa worst-case worst-case scenario scenario in in the the sense sense that that they they explicitly explicitly assume assume that that habitat habitat outside outside the the
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558 5 58
MAR CABEZA CABEZA ET ET AL. AL. MAR
selected reserve network network is lost. lost. In In some some cases, cases, there there may may be be more more knowledge knowledge selected reserve available about about what what can can be be expected expected to to happen happen for for nonselected nonselected habitat habitat and and available this knowledge knowledge could could be be integrated integrated into into the the reserve reserve selection selection process. process. this
222.5 2.5
USING A A STOCHASTIC STOCHASTIC METAPOPULATION METAPOPULATION MODEL MODEL USING IN SITE SITE SELECTION SELECTION IN Moilanen and Cabeza Cabeza (2002) (2002) described described how how a stochastic stochastic metapopulation metapopulation Moilanen and model can be be used used in the the site site selection selection process process in order order to to explicitly explicitly incorincor model can porate spatiotemporal spatiotemporal population population dynamics dynamics into into reserve reserve network network design design porate (category C3, C3, see Section Section 22.2). 22.2). They They ask ask the the question: question: which which subset subset of of sites S S (category do you you select select to to maximize maximize the the long-term long-term persistence persistence of of aa metapopulation metapopulation given given do that you you have have a parameterized parameterized metapopulation metapopulation model, model, unselected unselected habitat habitat is that lost, each each site site has has aa cost, cost, and and the the amount amount of of resource resource (e.g., (e.g., money) money) available available is is lost, here what what kind of results can can be expected when when applying limited? We We show here of results be expected applying this method. method. In In our example, we we use use the the incidence incidence function function model model (IFM; (IFM; see this our example, see Chapter and 5 references and of the the model). model). Chapter 4 and 5 for for references and a description description of The way of of integrating model in in simplest way integrating a stochastic stochastic metapopulation metapopulation model The simplest site selection selection is is to to use metapopulation model model to to find find the the set set of site use the the metapopulation of sites sites that that gives the lowest lowest metapopulation metapopulation extinction rate for simulations of of aa specified specified gives the extinction rate for simulations of sites that gives the metapopumetapopu length T. (Alternatively, length (Alternatively, one one could find find the set set of sites that gives the lation the longest average average lifetime.) lifetime. ) There two significant significant problems problems with with lation the longest There are are two this extinct, only rarely (or practically practically never), never), this appoach. appoach. First, if replicates replicates go extinct, only rarely a of simulation runs is needed to evaluate evaluate the extinction a very large large number number of simulation runs needed to the extinction probability population reliably, probability of of the the meta metapopulation reliably, which which will will slow slow down down optimization optimization considerably. considerably. Second, Second, the the simple simple measure measure is is unable unable to to distinguish distinguish between between solutions solutions that that are are always persistent and and between between solutions solutions that that always always lead to Moilanen and to extinction. extinction. Consequently, Consequently, Moilanen and Cabeza Cabeza (2002) (2002) used used aa measure measure of of the the persistence persistence of of the the simulation, F(S), F(S), which which can can distinguish distinguish the the quality quality of solutions solutions without without actually actually observing observing extinctions. extinctions. This This is is the the average average one-step one-step global probability of global extinction extinction probability of the the metapopulation metapopulation calculated calculated over over N N simula simulation runs: tion runs:
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in in which which Xn,t is is the the simulated simulated patch patch occupancy occupancy pattern pattern aatt time time tt in in replicate replicate simulation simulation nn and and ~(Xn,t) is is calculated calculated as as the the probability probability of of simultaneous simultaneous extinction extinction of of all all local local populations, populations,
I~(Xt) =
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22. 22.
METAPOPULATION DYNAMICS DYNAMICS AND AND RESERVE RESERVE NETWORK NETWORK DESIGN DESIGN METAPOPULATION
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rescue effect. effect. Functions Functions E() EO and and C() CO will will naturally naturally be be determined determined by by the the strucstruc rescue ture of of the the metapopulation metapopulation model. model. ture Equation (22.8) (22 . 8 ) is a function function of of the the metapopulation metapopulation model model and and its its paramparam Equation T. ItIt eter distribution, distribution, the the initial initial occupancy occupancy state state of of the the metapopulation, metapopulation, S S and and T. eter is related related to, to, but but not not identical identical to, to, the the extinction extinction risk risk of of the the metapopulation metapopulation and and is F(S)�l when when the the metapopulation metapopulation goes goes extinct extinct has the the following following properties: properties: F(S)--~I has almost immediately immediately and and F(S)--*O F(S)�O when when the the metapopulation metapopulation is highly persistent. persistent. almost When the the metapopulation metapopulation goes extinct extinct in a simulation simulation at at time time t, t, F(S) F(S) is When (T -- t)/(NT), and thus thus the value of of F(S) F(S) is always greater than than the the increased by (T increased t)/(NT), and the value always greater proportion of of time time the the metapopulation metapopulation is extinct the simulations. simulations. proportion extinct in the Importantly, Eq. Eq. (22.8) (22.8) is able able to to differentiate differentiate between between two two solutions solutions that that Importantly, F(S) is smaller smaller for the solution that persist of all simulation simulation runs; runs; F(S) persist until until the the end end of for the solution that is more more persistent. persistent. Moilanen Moilanen and and Cabeza Cabeza (2002) (2002) described described an an optimization optimization is technique that that is able able to to efficiently solve solve the the nontrivial nontrivial optimization optimization problem problem technique m of finding the optimal optimal set of of sites S* S" from the search search space space of of size 2 m,, where where of finding the from the m is is the number of patches in in the the metapopulation. metapopulation. Note Note that that the the difference difference m the number of patches between a population viability viability analysis analysis (PVA, (PVA, see, see, e.g., e.g., Murphy Murphy et aI., between a spatial spatial population et al., 1 990; Coulson aI., 200 1 ) and metapopulation site selection is that PYA 1990; Coulson et et al., 2001) and metapopulation site selection that a PVA alternatives, whereas whereas metapopulation metapopulation site selection only compares compares a few alternatives, selection actually searches for an an optimal optimal solution solution within within the the given constraints. constraints. actually searches for
Important the Selection Important Factors Factors Affecting Affecting the Selection of of the the Reserve Reserve Network Network Here we apply method to to a metapopu Here we apply the the metapopulation metapopulation site site selection selection method a metapopulation of heath fritillary Melitaea diamina. lation of the the false false heath fritillary butterfly, Melitaea diamina. M. M. diamina diamina lives on on moist moist meadows, meadows, which which are are nowadays nowadays being being overgrown overgrown rapidly. rapidly. This This poses poses persistence problems problems to to the the butterfly butterfly if if no no restoration restoration work work is is done done for for main mainpersistence taining taining the the quality quality of of the the meadows. meadows. A A system system of of 125 125 habitat habitat patches patches scattered scattered in in an an area area of of 20 20 X x 30 km km in in southern southern Finland (Fig. 22.7) 22.7) was was used used to to assess assess "which "which subset subset of of sites sites should should be be maintained maintained to to maximize maximize the the long-term long-term persistence persistence of of M. diamina, diamina, given given the the cost cost of of the the sites sites and and the the available amount amount of of resources? resources?"" A A brief brief overview overview of of the the effects effects of of different different factors factors on on optimal optimal selection selection is is given: given: the the value value of of the the dispersal dispersal parameter parameter a, cx, the the available amount amount of sites. of resources resources for for setting setting the the reserves, reserves, and and the the cost cost of of the the sites. The The dispersal dispersal ability ability of of the the species species (average (average dispersal dispersal distance distance is is given given by by 1/a) 1/0~) most important important factor in the meta metapopulation model affecting is possibly the most population model the reserve network. the configuration configuration of of the the reserve network. When When dispersal dispersal distances distances are are short short (large (large a), 0~),the the best best option option is is to to protect protect sites sites that that are are close close together together (Fig. (Fig. 22.8A). 22.8A). However, However, when when the the dispersal dispersal abilities abilities of of the the species species are are not not limiting limiting and and the the individuals individuals can can reach reach any any site site in in the the system, system, the the optimal optimal solution solution does does not not consist consist of of aa compact compact cluster, cluster, but but of of aa larger larger number number of of more more scattered scattered sites sites our example, example, to assess the effects of the dispersal dispersal parameter, parameter, (Fig. 22.8B). In our we we compared compared selections selections done done with with different different values values for for the the parameter: parameter: aa small small dispersal .5 ) and dispersal range range (a (cx = - 11.5) and aa large large dispersal dispersal range range (a (el = - 0.4). 0.4). The The configuration configuration of of the the final final reserve reserve network network might might not not be be so so intuitive intuitive as as shown shown here here when when the the real real costs costs of of the the sites sites vary vary greatly. greatly. The The real real value value of of this this algorithm algorithm comes comes to to play play when when the the costs costs of of the the sites sites are are considered. considered. An An expert expert knowing knowing the the system system and and the the dynamics dynamics of of the the species species might might be be able able to to choose choose aa good good set set of of sites sites for for species species persistence. persistence. However, However, when when the the resources resources
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are limiting and varies, it it is is very very difficult difficult to to identify identify from from aa map, map, are limiting and patch patch cost cost varies, without be the the without computational computational aid, aid, which which sites sites would would be the best best ones ones given given the amount for conservation. conservation. When When some some of of the the patches patches amount of of resources resources available available for are be proportionally proportionally more are considered considered to to be more expensive expensive than than others others (patches (patches with with commercial plantations were assumed to be 10 1 0 times times more more costly costly than than commercial forest forest plantations were assumed to be
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Distance (Km) (Km) Distance Fig. 222.8 Effects of of dispersal dispersal ability ability and and patch patch cost cost on on the the reserve reserve network network configuration configuration Fig. 2 . 8 Effects when using using the the metapopulation metapopulation approach. approach. The The area area shown shown is is aa subregion subregion of of the the complete complete when patch system system for for M. M. diamina diamina (see (see Fig. Fig. 22.7A). 22.7A). Sizes Sizes of of the the circles circles are are scaled scaled according according to to the the patch ()( = = 1.5, 1 .5, patch patch cost cost == patch patch area; area; area of of the the patch. patch. Dark Dark circles circles show show the the selected selected sites. sites. (A) (A) cx area (B) o~ ()( = = 0.4, 0.4, patch patch cost cost = patch patch area; area; and and (C) (C) ~()( == 0.4, 0.4, patch patch cost cost (white (white circles) circles) == patch patch area, area, (B) patch cost cost (dashed (dashed circles)= circles) = 10x l Ox patch patch area. area. For For the the remaining remaining IFM IFM parameters, parameters, standard standard patch M.diamina parameter parameter values values were were used used (see (see Moilanen Moilanen and and Cabeza, Cabeza, 2002). 2002). M.diemine =
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Fig. 22.9 22.9 Optimal selection at different resource levels. The shown is Optimal reserve reserve selection at different resource levels. The area area shown is aa sub subregion 7B). Panels 00 (see Fig. Fig. 22. 22.7B). Panels are are based based on on 1100 region of of the the complete complete patch patch system system for for M. diamina (see replicate optimizations replicate optimizations with with parameter parameter values values sampled sampled from from the the joint joint four-parameter four-parameter confidence confidence limits The color color of of the the patch patch shows shows how how often often the the patch patch was was selected: selected: White White = = limits for for M. diamina. The never selected; Black Black = selected. (A) (B) large never selected; = always always selected. (A) Small Small resource resource (50,000). (50,000). (B) Large resource resource (1 0 50,000). so, ooo).
those those natural natural meadows meadows in in early early successional successional stages; stages; Moilanen Moilanen and and Cabeza, Cabeza, 2002), 2002), keeping keeping all all other other factors factors equal, equal, the the optimal optimal solution solution changes changes greatly greatly (compare Figs. (compare Figs. 22.8B 22.8B and and 22.8C). 22.8C). Optimal Optimal solutions solutions with with aa large large amount amount of of resources resources do do not not always always build build on on solutions solutions found found with with aa smaller smaller amount amount of of resources. resources. Figure Figure 22.9 22.9 demonstrates demonstrates the the effect effect of of increasing increasing the the amount amount of of available resources. resources. When When the the amount amount of of resources resources is is limited, limited, it it is is optimal optimal to to select select only only aa small small cluster cluster of of sites sites (Fig. (Fig. 22.9A). 22.9A). However, However, when when the the amount amount of of resources resources is is tripled, tripled, the the solution solution consists consists of of not not only only aa larger larger amount amount of of sites, sites, but but also also aa rather rather different different set set of of sites sites (Fig. (Fig. 22.9B). 22.9B). This This result result provides provides an an important important message message for for planners planners that that often often have have thought thought that that site site selection selection algorithms algorithms only only provide provide the the core core of of the the reserve reserve network, network, which which will will be be extended extended later later on, on, by by adding adding more more sites sites to to the the core core network network when when more more resources available. The resources are are available. The optimal optimal solution solution may may strongly strongly depend depend on on the the amount amount of available. An of resource resource that that is is available. An ordering ordering in in which which patches patches should should be be conserved conserved can can only only be be given given if if the the total total amount amount of of available resources resources is is known.
22.6 22.6
DISCUSSION DISCUSSION In In order order to to optimize optimize in in situ situ conservation conservation of of biodiversity, biodiversity, and and given given limited limited resources, resources, major major effort effort has has been been placed placed on on the the development development of of reserve reserve network network design problems and design problems and algorithms to to solve solve those those problems efficiently. efficiently. Unfortu Unfortunately, problems have nately, most most of of the the existing existing problems have not not been been formulated formulated in in aa way that persistence and and hence solutions cannot cannot guarantee that is is focused focused on on persistence hence solutions guarantee the the long longterm problems have term persistence persistence of of biodiversity. biodiversity. Reserve Reserve selection selection problems have mostly mostly been formulated biodiversity, measured measured by formulated so so that that the the aim aim is is to to represent represent biodiversity, by aa snapshot snapshot of More of species species presence-absence presence-absence information, information, in the the most most efficient efficient way. way. More recent reasonable targets targets for for recent reserve reserve selection selection problem problem formulations formulations set set reasonable species for sensible species viability viability (e.g., (e.g., Noss Noss et et aI., al., 2002) 2002) and allow allow for sensible spatial spatial design design aI., 2002; aI., 2003 However, the dynamics dynamics of (McDonnell et et al., 2002; Leslie et et al., 2003).) . However,
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populations populations and and landscapes landscapes have been mostly overlooked, overlooked, and and much remains to to be achieved in formulating formulating and and solving reserve network network design problems in an an ever-changing ever-changing world. world. This chapter meta)population dynamics chapter addressed addressed the effects of spatial ((meta)population dynamics on the the persistence of biodiversity in reserves designed using using different different site selection manner with with which which they consider algorithms. These algorithms differed in the manner spatial dynamics during during the optimization process. We have shown shown how how the selection of sites for the conservation of 26 butterfly species in the Creuddyn Creuddyn Peninsula may be very different different depending depending on the reserve selection method. The The methods used here range from single representation from the simplest ""single representation heuristic" to to more more complex complex methods methods that that explicitly explicitly consider consider spatiotemporal spatiotemporal population population dynamics dynamics during during optimization. optimization. For the example concerning concerning the Creuddyn Creuddyn Peninsula, Peninsula, the single represen representation tation solution solution (i.e., (i.e., aa solution solution that that includes includes the the presence presence of of the the 26 butterfly butterfly species) population dynamics species) requires only only three three sites. sites. Studies of of butterfly butterfly meta metapopulation dynamics support the the view view that that we we cannot cannot expect expect all the the 26 species to to persist in in three three sites of 500 500 X 3< 500 500 m if all the remaining suitable habitat habitat would would be lost. We can appreciate from subsequent subsequent results that that the more more realism included in the methods (e.g., (e.g., larger number number of of representations representations per per species, species, spatial consid considerations), erations), the the larger larger the the amount amount of of sites sites in in the the solution solution and and the the better better the the prospects prospects for biodiversity persistence. Nonetheless, Nonetheless, the different different factors that that need to be considered in a reserve selection procedure procedure (including spatial popu population lation dynamics) dynamics) depend depend on on the the spatial spatial scale scale under under consideration. consideration. Reserve Reserve selection algorithms have been applied at worldwide worldwide or continental continental scales. At these these scales, scales, and and with with sufficiently sufficiently large large selection selection units, aa single representation representation for for each each species species might might be be enough enough to to ensure ensure viability, viability, especially if if the the aim aim is is to to demonstrate demonstrate the the most most efficient efficient way way of of concentrating concentrating conservation conservation efforts. efforts. However, at smaller spatial scales and with with smaller selection units, spatio spatiotemporal temporal dynamics dynamics should be be considered considered when when selecting selecting reserve reserve networks. networks. Note Note that that the the scale scale where where population population dynamics dynamics need need to to be be considered considered is is species species specific, specific, and and it it depends depends mostly mostly on on the the dispersal ability of of the the species species some bird species might show metapopulation metapopulation dynamics at a continental scale, whereas population dynamics whereas meta metapopulation dynamics would would be be quite quite localized localized for for snails. snails. The The simplest simplest site site selection selection methods methods (problem (problem categories categories CO CO and and Cl, C1, see see Section 22.2) 22.2) do not not include any notion notion of the spatial configuration configuration of the reserve, although although populations populations may may be be chosen in in aa way way that that aims at at local local persistence (problem category C l ). The C1). The simplest way of including including spatial considerations to reserve selection is to use some computational computational technique to to aggregate aggregate the the reserve reserve network network (problem (problem category category C2), C2), which which implicitly implicitly improves improves biodiversity biodiversity persistence by by minimizing negative negative external external effects. effects. In the example of the Creuddyn Creuddyn Peninsula reserve, aggregation could actually be be achieved achieved with with aa very very low low cost cost in in terms terms of of increased increased area. area. In In brief, brief, we we suggest suggest that that analysis of of the the cost cost of of reserve reserve aggregation aggregation should be be done done routinely routinely as as part of the the reserve selection process, process, and at least aggregation that that can be a part obtained obtained for for free should, should, in most most cases, be taken. taken. Given Given that that the maintenance cost cost of of aa compact compact reserve reserve is is likely to to be be smaller than than that that of of aa scattered scattered reserve reserve (Possingham prudent to (Possingham et et aI., al., 2000), 2000), it it is is economically economically prudent to pay pay aa little little extra extra for for aa compact compact reserve. reserve. Nevertheless, Nevertheless, from from the the perspective perspective of of species species persistence, persistence, there there might might also be be reasons reasons to to avoid avoid reserve reserve clustering. Where Where catastrophes catastrophes
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can impact large large areas causing causing extinctions, extinctions, it may may be less risky risky to conserve each species in at least two two or three separate places rather than than clustering those 996; Lei 997). However, those sites sites (Hess, (Hess, 11996; Lei and and Hanski, 11997). However, if if reserves reserves have have to to be be selected selected far far apart apart from from each each other, other, they they should be be large large enough enough to to allow allow species species persistence persistence independently. independently. Going one step further further from from qualitative reserve network clustering clustering to to the explicit species explicit consideration of of spatial dynamics dynamics requires information information on on speciesspecific parameters parameters of of spatial population dynamics. In the example for for the 26 butterflies, this was was done done by fitting fitting for every species a statistical model for for the function the probability probability of of occurrence occurrence of of the the species. species. This This model was was made made aa function of habitat quality and and connectivity (see also Westphal Westphal and and Possingham, Possingham, 2003 2003).). The The inclusion inclusion of of connectivity connectivity in in the the model model enables enables us us to to consider consider the the conse consequences probabil quences of of changes changes in in habitat habitat spatial spatial pattern pattern on on species' species' occurrence occurrence probabilities. ities. In In practical practical terms, terms, this this means means that that habitat habitat loss loss will will have have aa negative negative effect effect in the probabilities probabilities of occurrence occurrence in the regions close to to the site of of habitat habitat loss. This This is is aa natural natural consequence consequence of of decreased decreased immigration immigration and and increased increased edge effects. Estimates of population population numbers numbers in the selected selected reserve reserve network network differ differ quite quite significantly significantly when when effects effects of of connectivity connectivity are are excludedlincluded excluded/included in in optimization. optimization. Another Another way of considering considering spatial dynamics explicitly is integrating integrating sto stochastic metapopulation metapopulation models into the reserve selection procedure. procedure. We have presented presented an an approach approach for for selecting selecting the best reserve network network that that maximizes the the persistence persistence of of aa metapopulation metapopulation for for aa given given time time frame. frame. The The extension extension of of this this approach approach for for many many species species is is challenging, challenging, but but it it is is feasible feasible technically technically (Moilanen and and Cabeza, manuscript manuscript in preparation) preparation).. One One of the limitations limitations of of the the approach approach is is the the availability availability of of information information to to estimate estimate all all model model parameters parameters for for all all the the species. species. Reserve Reserve selection selection methods methods for for problems problems CO CO and and C1 assume assume aa best-case best-case scenario in the sense that that they they implicitly assume that that there there will be no no changes changes in the the landscape landscape outside the selected reserves. In contrast, contrast, methods methods for for classes C2 C2 and and C3 C3 assume assume aa worst-case worst-case scenario scenario in in that that all all nonselected nonselected habitat habitat is is assumed assumed to to be be lost, lost, which which of of course course will will not not always always be be the the case. case. It It is is possible possible to to improve improve the the dynamic dynamic probability probability method method by by including including information information on on threats threats and and vulnerability of sites into the optimization model (Pressey et a!., al., 11994; 994; Pressey and 1 , Cabeza and Taffs, 200 2001, Cabeza and and Moilanen, Moilanen, manuscript manuscript in prepara preparaSerneels tion). At a general level, this means means that that a model of of land-use change ((Serneels and 1 ; Veldkamp 1 ) would and Lambin, Lambin, 200 2001; Veldkamp and and Lambin, Lambin, 200 2001) would be be integrated integrated into into the the reserve reserve selection selection algorithm algorithm and and that that the the best-case/worst-case best-case/worst-case scenario would would be be relaxed relaxed and and modeled modeled more more realistically. realistically. The inclusion of landscape landscape dynamics into into reserve selection is in its infancy (Possingham et a!., 993; Costello and networks al., 11993; and Polasky, 2003). 2003). Reserve networks are are not not generally generally constructed constructed instantaneously instantaneously (except (except perhaps in in some some marine marine areas). In many many regions, sites can can only be selected selected if they they become become available for for acquisition. acquisition. While While sites are slowly being assembled assembled into into a network, network, some sites may be developed and and lost to the system. To take take this into into account, account, we we can can formulate formulate the the problem problem as a dynamic dynamic programming programming problem problem and and find find optimal optimal solutions solutions using using stochastic stochastic dynamic dynamic programming programming algorithms algorithms (Possingham (Possingham et 993; Costello and et a!., al., 11993; and Polasky, Polasky, 2003 2003).) . These These algorithms algorithms only only work work at at present that can can deliver present for for small small problems problems and and we we have have yet yet to to develop develop tools tools that deliver
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adequate adequate reserve networks networks when there are landscape landscape dynamics as well as spatial spatial population population dynamics in large systems. Preliminary results results suggest that that some measure of irreplaceability (Pressey et al., aI., 11994; 994; Pressey and Taffs, 200 1) 2001) may provide good solutions to the reserve network network design problem when there is landscape landscape change (E. Meir, personal communication) communication).. IIn n conclusion, the integration of spatial population population dynamics, landscape modeling, and scheduling of conservation action into the reserve selection problem should lead to reserve network network designs and acquisition strategies that that are better at achieving the goal of long-term biodiversity persistence. Many challenges remain in the proper proper formulation formulation of reserve network network design problems and in the development development of algorithms that that deliver robust solutions solutions uncertainty and change. in the face of several sources of uncertainty
23
ANA LYSIS VIABILITY ANALYSIS FOR ANGERED FO R END EN DAN G ERED META PO PU PUL ATIONS M ETAPO LATI O N S:: A DIFFUSION A DIFFUSION A PPROXIM ATION APPROXIMATION A PPRO A C H APPROACH E.E. E.E. Holmes Holmes and and B.X. Semmens
23.1 23.1
INTRODUCTION INTRODUCTION Population viability analysis (PVA) (PVA) assesses the rate of population decline and and the risks of extinction or quasiextinction over a defined time horizon for a population of concern ((Gilpin Gilpin and Soule, Soule, 11986; 986; Boyce, 1992; Morris and Doak, 2002). Although the techniques employed to conduct PYA PVA are varied, they typically involve building quantitative models that are parameterized by demographic and environmental data. PYA 980s PVA was first used in the early 11980s (Shaffer, 98 1 ), and in the (Shaffer, 11981), the past decade it has gained broad acceptance in the conservation community as a useful tool for assessing and managing ""at-risk" at-risk" species (Beissinger, (Beissinger, 2002; Morris and Doak, 2002; Reed et aI., al., 2002). This is particularly true for demogaphic PYAs, PVAs, due due in large part to the advancements (Beissinger, 2002). The in Monte Carlo techniques and desktop computers (Beissinger, International Union for the Conservation of Nature (IUCN)'s Red List Criteria,
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probably the most widely applied set set of decision rules for determining the sta stapartially defined defined by metrics metrics that require some form form of tus of at risk species, is partially PYA 994). For instance, under one of the Red List criteria, a taxon PVA (IUCN, 11994). may be classified as endangered endangered if a "reduction of at least 50%, projected projected or suspected suspected to be met within the next ten years or three three generations" generations" is predicted. Although many PYAs PVAs are focused on single populations populations in single sites, there are often needs for spatially explicit PYAs: PVAs: many populations populations of conservation concern concern are are distributed distributed across across multiple multiple sites sites and and additionally, additionally, the the primary primary anthropogenic anthropogenic threats threats facing facing at-risk at-risk species species are are habitat habitat destruction destruction and and alter alteretal., ation, which are fundamentally spatial processes (Wilcove et aI., 11998). 998). Several software packages have been written for spatially explicit PYA, PVA, includ includ997) and RAMAS 996), Metapop (Ak"akaya, (Ak~akaya, 11997) RAMAS GIS (Boyce, 11996), ing RAMAS Metapop ALEX (Possingham and 995), and 993). These and Davies, 11995), and VORTEX VORTEX (Lacy, 11993). models models incorporate incorporate aa diversity diversity of of demographic demographic and and spatial spatial attributes attributes such such as as distance-dependent distance-dependent migration, migration, allee allee effects, effects, social social population population structure, structure, hab habitat quality and spatial spatial arrangement, and and genetic variability. The The development itat sophisticated PYA PVA software software packages such as these has made the of flexible sophisticated construction construction and and simulation simulation of of spatially spatially explicit PYA PVA models models feasible for for those those who are not not highly skilled programmers programmers and and has greatly increased the number number who and scientists capable capable of using spatially realistic PYA PVA models. of managers and As As the use of PYA PVA has grown in conservation conservation science, so have concerns that that PYAs Reed et aI., PVAs are often overextended given limited data data sets ((Reed al., 2002). 2002). Beissinger and Westpahl ((1998) 1 998) suggested that that PYA PVA should should be limited to assess assessshort time frames flames using the simplest models that that can rea reaing relative risks over short sonably sonably be be justified. justified. For For single single species species with with spatially spatially simple simple structure, structure, data data needs needs can when Beissinger and and can often often be met when and Westpahl's call call for for model model moderation moderation and simplicity more complex spaspa simplicity are are heeded. heeded. When When one is faced faced with with species species with with more tial structure, structure, a much larger larger amount of data amount of data is needed to parameterize the populations, the levels and patterns of of dispersal, dynamics of of individual local populations, and patterns and the correlations among among local populations and the spatial pattern pattern of temporal correlations populations (e.g., Rails Ralls et aI., 2002). Unfortunately, Unfortunately, collection of data needed to parameter al., 2002). of data to parameterize a spatial model is rare of conservation concern, rare for species of concern, at least in the the disconnect between param United States (Morris (Morris et aI., al., 2002), 2002), and and there is a disconnect between the parameter PYA models and and the willingness eter requirements requirements for for spatially explicit explicit PVA willingness and/or and/or ability of management agencies to to collect the types of data data needed to to appropriappropri ately apply fulfill data apply such tools. tools. Because it is usually impossible impossible to to retroactively fulfill data requirements for spatial PVA PYA and and there there will always always be cases cases where collection requirements for a spatial where collection of spatial data data is infeasible, infeasible, managers managers require require PVA PYA tools tools that that can can help help guide concon of servation of of metapopulations meta populations in the the absence absence of of large amounts amounts of of spatial data. data. servation
D iffusion A pproximation ffor or M etapopulations Diffusion Approximation Metapopulations
One problem of limited population One approach approach to to the the problem of limited population data data is to to find find a diffudiffu sion approximation approximation that that correctly models models the the long-run statistical statistical properties properties of of complex population population process. process. This This approach approach has been used successfully for for a complex single population population models models (Karlin (Karlin and and Taylor, Taylor, 1981; 1 9 8 1 ; Lande Lande and and Orzack, Orzack, 1988; 1988; Lande, 1993; 1 993; Dennis Dennis et e t al., aI., 1991; 1 99 1 ; Hill Hill et e t al., aI., 2002; 2002; see also also Morris Morris and and Doak, Doak, Lande, 2002; 2002; Lande Lande et et al., aI., 2003) 2003) and and reduces reduces the the problem problem of of parameterizing parameterizing aa large large model model with with many many parameters parameters to to the the much much simpler simpler task task of of parameterizing parameterizing a
23. VIABILITY VIABILITY ANALYSIS ANALYSIS FOR FOR ENDANGERED ENDANGERED METAPOPULATIONS METAPOPULATIONS 23.
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two-parameter diffusion diffusion model. model. One One of of the the main main practical practical implications implications of of the the two-parameter diffusion approximation approximation approach approach is that that it it is not not necessary to to know know the the multimulti diffusion tude of of parameters parameters describing describing the the local dynamics, dynamics, dispersal dispersal levels, spatial spatial patpat tude terns of of dispersal, dispersal, and and spatial spatial synchrony between between local populations populations in in order order to to terns make basic basic predictions predictions about about the the statistical statistical distribution distribution of of the the long-term long-term make metapopulation or or local population population trajectories. trajectories. The The relevant relevant two two parameters parameters metapopulation for the the diffusion diffusion approximation approximation can be estimated estimated from from a simple time time series of of for counts from the the population population process. process. counts from This diffusion approximation approximation approach to model model the the longlong This chapter chapter uses the diffusion approach to run behavior of spatially structured populations. populations. Our Our focus is on on stochastic stochastic run behavior of spatially structured meta populations characterized characterized by structured structured population population size, density-indedensity-inde metapopulations pendent local dynamics, dynamics, and, and, in keeping with with the the assumption assumption of of density indeinde pendent pendence, a metapopulation metapopulation that that is declining as a whole. whole. Local populations populations are are pendence, assumed to to have patch-specific patch-specific structured and dispersal dispersal rates, rates, assumed structured local dynamics dynamics and with spatial spatial structure among local populations populations in terms terms of of both both their their local with structure among dynamics patterns. Description Description of the long-run statistical behavbehav dynamics and and dispersal patterns. population trajectories diffusion approximation ior of the meta metapopulation trajectories using a diffusion approximation allows PYA risk metrics rate of metapopula the estimation estimation of of PVA metrics such as the long-term rate metapopulation probability of reaching tion decline and and the probability reaching different threshold threshold declines over different horizons (i.e., probabilities probabilities of extinction or quasiextinction). different time horizons quasiextinction). These methods metapopulation PVA PYA metrics are illustrated illustrated using methods for for estimating metapopulation data metapopulations in the U.S. Pacific Northwest. data from from two two chinook chinook salmon salmon metapopulations Pacific Northwest.
23.2 23.2
A STOCHASTIC M ETAPOPULATION MODEL MODEL A STOCHASTIC METAPOPULATION Our populations, and Our focus is on declining meta metapopulations, and thus thus what what has been been termed termed nonequilibrium populations. We model a collection of local populations nonequilibrium meta metapopulations. populations connected connected by dispersal where where local populations populations have density-independent density-independent local dynamics, which sinks," but population as a which may be "sources" "sources" or ""sinks," but the meta metapopulation whole is declining. Dispersal levels could be very low, resulting in basically independent local populations, populations, or extremely high, resulting resulting in essentially one independent population. population. From a practical standpoint, standpoint, this approach approach is most appropriate appropriate when dispersal is not % per year localized dispersal not insignificant insignificant (e.g., above 22% or 0 . 1 % global dispersal), otherwise 0.1% otherwise parameterization parameterization of the model requires requires inordinately inordinately long time series. Data from this type of metapopulation metapopulation would would be characterized by fluctuating local population population trajectories, but actual extinc extinctions would be unusual until the meta population has very few individuals. Our metapopulation model assumes no density dependence nor carrying capacities within the indi individual local populations. populations. Such a model is only appropriate in cases where the population is declining and all local populations are well below their carrying capacities. Our example using data on chinook salmon illustrates a situation that that is is likely likely to to be be well well modeled modeled as as this this type type of of metapopulation. metapopulation. The following following section gives a rather parameter-intensive mathematical mathematical description of a stochastic, declining metapopulation. However, the reader should keep in mind that this model will not be parameterized. parameterized. Rather the asymptotic behavior of of this this model's trajectories will will be derived and that that informa information tion will will be be used used to to develop develop aa diffusion diffusion approximation approximation of of the the process. process. Time Time series series data data will will then then be be used used to to parameterize parameterize the the diffusion diffusion approximation. approximation.
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
568 568
The Model Model The Consider Consider an an individual individual local local population population i with with stochastic stochastic yearly yearly growth growth and and stochastic stochastic dispersal dispersal to to and and from from other other local local populations. populations. Such Such aa local local popula populaNi(t), could be described described as follows: follows" tion's numbers in year t, Ni(t),
Ni(t)t) = - growth g r o w t h -- dispersal out out + + dispersal dispersal in Ni(
== Ni(t Ni(t - l1)ezi( - l1)) )ezi( tt +
--
di(t - l1)Ni(t - l1 )ez )ezi(;(tt - l1)) di(t )Ni(t -
E o taj jii( (tt - -- l1 )d ) djj ((tt -�
jrl- i r
.1) (23.1) (23
Nj ((tt - - l1)ezi(t-l) )ezAt- l ) 11 )) N
where where Zi(t) zi(t) iiss the the stochastic stochastic growth growth rate rate ooff local local population population i iinn year year t and and is is random variable variable with with some unspecified unspecified statistical statistical distribution distribution with with mean a random J.1i ~i and and variance variance ITT. cr2. The The J.1i ~i term term will will be be referred referred to to as as the the local local population's population's growth rate; rate; it will not be observed, observed, as the local population population is sub subintrinsic growth ject to to immigration immigration and and emigration. emigration. Some fraction fraction of individuals, individuals, di(t), di(t), ject leaves local local population population i at at year year t and and disperses disperses to to other other local populations, populations, and dispersal dispersal into into local population population i occurs occurs from from other other local populations. populations. and The fraction fraction of dispersers dispersers from from local population population j that that go to to local local population population The i in in year year t is is a;i(t) otji(t ) and and can can vary vary depending depending on on the the destination, destination, i, thus thus allow allowfor spatially spatially structured structured dispersal. dispersal. The The dispersal dispersal parameters, parameters, di(t) di(t) and and ing for ;i(t) , are otji(t), are assumed assumed to to be be temporally temporally random random variables variables from from some some unspecified unspecified a statistical distribution. distribution. statistical
The Model Model in in Matrix M a t r i x Form Form The The model for the the entire entire metapopulation can be written using a random The model for metapopulation can written using random transition dispersal and transition matrix, matrix, A(t), which which encapsulates encapsulates both both dispersal and local growth: growth:
+ 1)1 )
N l ((tt + Nj
N2 N 2 ((tt ++ 1) 1)
+
N3 N 3 ((tt +
1 1))
N mlj
(t)
N2 (t) N2 A(t) X = A(t) 3< N3 N3 (t)
(23.2)
=
NNkk ((tt ++ 1) 1)
Nk (t) (t)
where where
( l - dj ) ezl F(i-dl)eZl
a 3 1 d3eZ3 c131d3ez3 ... ( 1 -- d2)ez2 d2 ) ez2 a112dleZ1 2d l ezl a32d3eZ3 ot (1 c132d3ez3 ... A ( t) = = p cx13dleZ1 ( 1. - d3 a 1 3dj A(t) . .ezl. . .ca23d2eZ2 ~23d2e.z2 . . . .(1-d3)ez3 . .)ez3. . ... a l kd l ezl L_~lkdleZl
a21d2eZ2 c121d2ez2
a2kd2eZ2 ~2kd2ez2
a3kd3eZ3 Ot3kd3ez3
...
akj dkeZk ~ CikldkeZk ~ak2dkeZk I ak3dkeZk ~ c~k3dkeZk
(23.3) (23 . 3 )
( 1 - dk ) ezk (1-dk)ezk]
The '(t)' ' (t)' on on the the d's, cx's, a's, and and z's have have been left off off to to remove remove clutter. clutter. There There may may The be be any any level level or or spatial spatial pattern p attern of of temporal temporal correlation correlation among among the the intrinsic intrinsic local local growth rates, zi's, z;'s, dispersal dispersal rates, rates, d;'s, and dispersal dispersal patterns, patterns, c~ii's. a/so growth rates, di's, and In the the matrix matrix model, model, each each row row represents represents I1 unit unit of of habitat. habitat. Local Local populations populations In with with multiple multiple units units of of habitat habitat appear appear as as multiple multiple rows rows with with very very high high dispersal dispersal
23. FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
569 569
between between the the units units of of habitat habitat in in that local local population. population. The The habitat habitat units units within within aa local population population could vary vary in in quality (i.e., (i.e., habitat within within aa local population population need need not not be be uniform) uniform) and and different different local local populations populations certainly certainly differ differ in in the the num num' S are ber ber of of habitat habitat units units they they contain. contain. The The d;'s di's and and elj; OLji'S a r e assumed assumed to to be be drawn drawn from from some distribution distribution that that can be different for each local population population or local popu population lation pair. pair. Although Although the the d;'s, di's, elji oLji and and z/s zi's are are temporally temporally random random variables, variables, they they are are assumed to to be be stationary, stationary, i.e., i.e., that that there there is is no no overall overall change change in in the the mean mean values over over time. time. For For the the purposes of of this this chapter, chapter, it it will will be be assumed that that the the d;'s, di's, ' elj; S, and aji's, and z;'s zi's are are all strictly strictly postitive, postitive, which means that that all local populations populations are are connected connected to to each each other other to to some (although (although possibly possibly very very low) degree degree and that that mean mean yearly yearly geometric geometric growth growth rates, rates, exp(J.L;)'s, exp(p~i)'s, while while possibly very very small small are are not not zero. zero. These These assumptions assumptions imply imply that that the the A(t) describe describe an an ergodic ergodic set set of of matrices matrices ((Caswell, Caswell, 200 1 ). The 2001). The assumption of of strict strict positivity positivity is is not not strictly strictly necessary. necessary. It It is is possible possible for for A(t) A(t) to to describe describe an an ergodic ergodic set set if if some some elements elements of of A A are are zero; zero; it it depends on on the the pattern pattern of of zeros zeros within within A A [ef. [cf. Caswell (2001 (2001)) for for aa discussion discussion of the the conditions under under which which matrices matrices are are ergodic]. ergodic]. The The model model is is very very general, general, allowing allowing some some sites sites to to be be dispersal dispersal sources sources and and others others to to be be dispersal dispersal targets, targets, allowing allowing any any spatial spatial pattern pattern of of dispersal dispersal or or spa spatially tially correlated correlated local local growth growth rates, rates, allowing allowing any any pattern pattern of of temporal temporal correl correlation ation amongst amongst local growth growth rates, and and allowing allowing any combination combination or or pattern pattern of habitat habitat sizes sizes of of local local sites. sites.
Using Using Random Random Theory Theory to to Understand Understand the the Model's Model's Statistical Statistical Behavior Behavior Together, Together, Eqs. Eqs. (2) (2) and and (3) (3) describe describe aa quite quite generic generic model model of of aa declining declining meta population with metapopulation with density-independent density-independent local local dynamics. dynamics. From From aa viability viability analysis Can one analysis perspective, perspective, one one might might ask ask the the question: question: ""Can one predict predict the the viability viability of what are of the the total metapopulation? metapopulation?"" In In more precise precise terms, this is is asking asking what are the the statistical properties of the meta population trajectories con metapopulation trajectories of this type of connected 3 ) ] ? Clearly, nected collection collection of of local populations populations [of [of the the form form in in Eqs. (2) (2) and and ((3)]? Clearly, the the matrix matrix A(t) A(t) has has aa large large number number of of parameters parameters that that would be be difficult, difficult, if if not not impossible, population of impossible, to to estimate estimate for for any any given given meta metapopulation of conservation conservation concern. concern. However, However, using using random random theory, theory, it it can can be be shown shown that that the the long-term long-term dynamics dynamics can can be be described described by by only only two two parameters parameters and and that that it it is is unnecessary unnecessary to to know know the the multitude multitude of of other other parameters parameters for for the the purpose purpose of of projecting projecting long-run long-run dynamics. dynamics. To To use use this this random random theory, theory, we we first first need need to to recognize recognize that that this this stochastic stochastic metapopulation model falls into into the class of random random processes that that involve meta population model products random matrices, products of of ergodic ergodic random matrices, in in this this case case products products of of A(t), A(t), which which can can be local population population sizes sizes forward: be seen seen by by using using Eq. Eq. (2) (2) to to project project the the vector vector of of local forward:
N(1) A(0)N(0) N( l ) = A(O)N(O) N(2) = A(O)A( A(0)A(1)N(0) N(2) l )N(O)
(23.4) (23.4)
o o o
N( )A(2) . . . A(t )N(O) N(t)t) = - A(0)A(1 A(0)A(1)A(2)... A(t - l1)N(0) where in Eq. where N(t) N(t) is is the the column column vector vector of of Ni Ni values values at at time time tt in Eq. (2). (2). Products Products of of random random ergodic ergodic matrices matrices have have aa well-established well-established theoretical theoretical foundation foundation and and certain well-studied well-studied asymptotic asymptotic statistical statistical properties. properties. A brief brief review review of of have certain two two of of the the key key results results from from this this theory theory is is provided provided in in Box Box 23.1 23.1 and and aa simulated simulated
570 $10
E.E. HOLMES AND B.X. E.E. HOLMES AND B.X. SEMMENS SEMMENS
BOX 23.1
Key Results from Random Theory
Two of the fundamental results from the theory of products of random matrices are reviewed and interpreted in the context of our metapopulation model. The reader is referred to chapter 1 4. 3 in Caswell (200 1 ) and Tuljapurkar (1 990) for other reviews interpreted in the context of demographic, single population models. The Metapopulatlon and Local Populations Decline at the Same Rate
One of the basic results from Furstenberg and Kesten's "Products of Random Matrices" (1 960) is that the product of ergodic random matrices asymptotically goes to an equilibrium. Say that Xt is an ergodic random "k x kIf matrix and that V (also a k x k matrix) denotes the product of n of the X matrices: X" XbX3, Xn. Then Furstenberg and Kersten's results say that V goes to an equilibrium state such that •
lim ! log k
t-'"
t
lEa
ki Vii
=
.
a constant which is the same for all a
•
(B1 )
We can use this result to show that the long-run exponential growth rate of the metapopulation and the local populations will be the same. N(t) = A( O) A( 1 ) A( 2). . . A( t - 1 ) N ( O )
Let V
=
A( 0) A( 1 ) A( 2 ) . . .A( t
Then log Nj( t)
=
log
a nd log M( t) = log Thus from Eq. (B1 ), lim ! log t
... "
t
1)
2: Vii + log N,{ O ) i
2: 2: Vii i i
2: Vii i
-
our metapopulation model
=
+
log M( O )
lim ! log /--4>'
t
2: 2: Vii i i
=
a constant
=
11m
The Distribution of Local Population and Metapopulatlon Sizes Is Distributed Lognormally
One of the most powerful results, for our purposes at least, concerns the statistical distribution of the metapopulation and local trajectories. This tells us what distribution of sizes we would see if we ran our model over and over again and allows us to make population viability analyses for metapopulations since we have a prediction about the likelihood of different metapopulation futures. Random theory (Furstenberg and Kersten, 1 960; Tuljapurkar and Orzack, 1 980) shows that any sum of the N,(t),s, such as the total metapopulation (all i's), a single local population (one i), or any other subset, goes to the same distribution: (82)
where the sum of local populations is denoted in matrix terms as cON(t) and c is a column vector with O's and 1 's to show which local populations to sum together.
23. 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR FOR ENDANGERED ENDANGERED METAPOPULATIONS METAPOPULATIONS
511 571
Example These results are simple to see with simu lations. An example of a linear chain of 1 0 local populations connected via 2% yearly dispersal to their nearest neighbors and 0.2% to nonnearest neighbors is shown. The local dynamics were eZ; where Zi is a nor mally distributed random variable, Normal(lJ.i, The local g rowth rates, IJ./s, for local populations 1 to 1 0 were, respectively, 0.97, 1 .00, 0.96, 0.83, 0.88, 1 .00, 1 .00, 0.89, 0.99, and 0.81 . Figure 2 3 . 1 A shows that the long-run g rowth rate of the local population a nd metapopulations is equal to the same constant. Fig u re 2 3 . 1 B shows that the distribution of metapopulation size after 1 00 yr is Normal(1 00IJ.m, 1 OO(J�). The expected distribution was specified using the maxi m u m likelihood (ML) estimates for 11m and (J� [Eq . (9)] from a single 1 000-yr time series of metapopulation counts. The M L estimate for (J� relies on a n assumption of normality for t 1 , although strictly speaking normality only holds for t large. However, it does quite well as can be seen in Fig. 2 3 . 1 B.
=
example . 1 , the example is is shown shown to to illustrate illustrate these these results. results. As As described described in in Box Box 23 23.1, the the theory ory demonstrates demonstrates that that this this stochastic, stochastic, density-independent density-independent metapopulation metapopulation will will have = '2.Ni(t), ENi(t), have an an asymptotic asymptotic growth growth rate rate and and that that the the metapopulation, metapopulation, M(t) = the Ni(t)'s, and and sets sets of of N/t)'s Ni(t)'s representing representing the the units units of of habitat habitat com comthe individual individual Ni(t)'s, prising prising aa semi-independent semi-independent local local population population will will be be distributed distributed lognormally lognormally parameters: with the same parameters: log t~2 Normal(tfLm, Normal(t~m, to';n) t(r2) log M(t)/M(O) � log log Ni(t)/Ni(O) � t~d Normal(tfLm, Normal(t~m, to';n) t~r2)
(23.5)
llog og ,L Ni(t)/ ,L Ni(O ~ Ni(t)/~a Ni(O)) � t--~ Normal(tfLm, Normal(tp~m, to';n) to"2) ll eEa a
E aa l~E
where a = {al, a2, . . . . , am} Figure 23 . 1 shows an example of this behavior. A meta population is simulated 23.1 metapopulation (described 1 ) and, (described in in Box 23. 23.1) and, over over time, time, the the metapopulation metapopulation declines declines at at aa con constant have the the same same long-term long-term fate. fate. When When viewed viewed over over short short stant rate rate and and all all Ni(t)'s have time . 1 , the time frames, frames, tt small small in in Fig. Fig. 23 23.1, the local local sites sites show show different different growth growth rates rates with with some some declining declining more more or or less less than than the the long-term long-term rate, rate, but but over over the the long longterm their their rate of decline is the same. The The model model studied studied here here approximates approximates the the local local dynamics dynamics by by aa simple simple expo exponential growth or decline) model. However, it has been shown that growth ((or that results results from random random theory theory (presented (presented in Box 23. 23.1) to a more more compli complifrom 1 ) also apply to cated cated metapopulation metapopulation model model where where local local dynamics dynamics are are described described by by stochas stochastic Sanz and 998). tic age-structured age-structured Leslie Leslie matrices matrices ((Sanz and Bravo Bravo de de la la Parra, Parra, 11998). Essentially, Essentially, this this occurs occurs because because even even when when the the local local dynamics dynamics are are described described by by aa local local matrix matrix model, model, the the system system can can still still be be described described by by products products of of random random matrices.
E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEM M ENS E.E.
5572 72
-
ro
OJ >-
ro
.� U OJ
OJ "0
'0
0.1 5
- Metapopulation - Local population
0. 1
A
0.05 0
OJ
� -0.05
"0 OJ
ro
.§ -0.1 in OJ
0
20
40
t (year)
60
80
1 00
800
B
700 600 C
:J 0 0
500 400 300 200 1 00 -6
0
-4
-5
-3
-2
-1
0
log metapopulation size at year 1 00
2
3
Fig. 23.1 Illustration Illustration of of two two of of the the main main results results from from random random theory. theory. (A) (A) All All local local popula populations go toward toward the same long-term long-term rate of population population growth growth (or decline) as t gets large. (B) The distribution distribution of log M(t) is a normal distribution distribution with with mean given by the long-term long-term rate of growth (or decline) multiplied multiplied by t and the variance given by t multiplied multiplied by the rate that that variance i.e., tt x /T)log M(t + T)/M(t) ~)/M(t) for for T9 not not overly overly variance increases increases in in an an individual individual trajectory, trajectory, i.e., • (1 (1/~)log small. Here the variance was estimated from time series 0 and and this is used to from one time series using T9 == 110 00. predict predict the the distribution distribution at at tt = 1100. =
23.3 23.3
DIFFUSION DIFFUSION APPROXIMATION APPROXIMATION The bution of log M(t) The asymptotic distri distribution M(t) in Eg. Eq. (5) has the same properties as the distribution of a diffusion process with drift; drift; it is normal and the mean and variance of the distribution of log M(t) increase linearly with time, t. This observation in the context of age-structured matrix population population models (Lande and Orzack, 1988; Dennis et aI., 9 9 1 ) led to the use of a diffusion approxi al., 11991) approxicalmation to enable parameterization using simple time series and to enable cal culation of extinction probabilities. Diffusion approximation methods for method for single population populations are an important and established method approximating stochastic trajectories (Lande and Orzack, 11988; 988; Dennis et aI., al.,
23. 23.
VIABILITY ANALYSIS ANALYSIS FOR FOR ENDANGERED ENDANGERED METAPOPULATIONS METAPOPULATIONS VIABILITY
5573 73
Chapter 3 in in Morris Morris and and Doak, Doak, 2002; 2002; Chapter Chapter 22 in in Lande Lande et et al., aI., 2003). 2003 ) . 1991; 1 99 1 ; Chapter Models for for single single populations populations are are mathematically mathematically analogous analogous to t o the the models models Models used here here for for metapopulations metapopulations with with aa stochastic stochastic process process involving involving products products of of used random matrices. matrices. However, However, in in single single population population models, models, the the matrix matrix represents represents random history matrix matrix rather rather than than a growth growth and and dispersal dispersal matrix, matrix, and and the the N(t) N(t) aa life history vector [in [in Eq. Eq. (2)] (2)] represents represents different different age or stage stage classes, classes, whereas whereas in in the the vector age or metapopulation matrix, matrix, it it represents represents different different local local sites sites and and populations. populations. metapopulation A diffusion diffusion approximation approximation with with drift drift is a stochastic stochastic process with the the A process with following properties properties (cf. (d. Karlin Karlin and and Taylor Taylor 1981): 1 98 1 ) : following
}
+ el, X( O X( tt )) -- X( = I~mt O )) = e for f.1mt + X( 1 "2 3 , for tt = 1,2,3 2 t) e -- normal(0 normaI ( 0 ,cr2t) ,(J"m
J
(23.6) (23.6)
. i. Q.
i
For any any nonoverlapping nonoverlapping pair pair of time periods, periods, t1 < t2 t2 and and t3 t3 < < t4, t4, X(t2) X(t2) -- X(ta), X(t1 ), For of time ta < and X(t4) independent random random variables. variables. X(t X(t + ~) T) is a random random and X ( t 4 )- X(t3) X(t3) are are independent variable Normal (X(t)+ (X(t) + ~m~, f.1mT, (r2~). (J";' T). Correspondingly, Correspondingly, the the variable with with distribution distribution Normal probability density function for for X(t X(t + ~) T) given X(t) is is probability density function given log log X(t) X(t)) (X(t + T) p(X(t ~ ) lI X ( t ) ) == p
[
11 [ --( X(X(t ( t + T+) -T)X (-t ) X( - ~ mt )~ ) 2 1f.1mT)2 . _ exp exp � X/2~rcr2.r 2(r2~ 2 2 (J"mT v 2TI(J"mT .
]
(23.7) (23.7)
Behavior Versus Diffusion Diffusion Behavior of of Metapopulation Metapopulation Trajectories Trajectories Versus Trajectories Trajectories approximation is based on the behavior of log M(t) as t goes to to infin infinDiffusion approximation ity; ity; however, however, in in PYA PVA settings settings the the time time frame flame of of interest is is substantially substantially less than than infinity infinity and and is is typically typically in in the the range range of of 25 25 to to 100 100 yr. yr. How How well well does does the the diffusion diffusion approximation approximation do do over over these these finite finite time time periods? periods? To To explore explore this, this, aa collection collection of of 50 50 local local populations populations were were simulated simulated that that were were connected connected by by global global dispersal dispersal ran ranging 5% per ging from from 0.1 0.1 to to 5% per year year and and that that had had correlated correlated local local dynamics, dynamics, zA!), zi(t), drawn drawn from 0.05, variance from aa Normal(mean Normal(mean = = -0.05, variance = = 0.09) 0.09) and and aa temporal temporal covariance of of 0.2 0.2 ':*- 0.09 0.09 between between the the zA!)'s zi(t)'s of of local local populations populations in in any any given given year. year. meta population trajectories If the log metapopulation trajectories behave behave like a diffusion process, process, and if we we repeatedly repeatedly generate generate aa large large sample sample of of replicate replicate metapopulation metapopulation trajectories, trajectories, the l/t)logM(t)/M( O) from the mean mean and and variance variance of of ((1/t)logM(t)/M(O) from those those trajectories trajectories should should be be aa constants l/t)logM(t)/M(O) constants over over the the time time period of of interest. interest. Additionally, Additionally, ((1/t)logM(t)/M(O) should should be be normally normally distributed. distributed. To To examine examine whether whether the the metapopulation metapopulation tra trajectories jectories had had these these properties, properties, the the simulations simulations were were started started from from aa distribution distribution of of local local population population sizes sizes selected selected from from the the equilibrium equilibrium set set of of local local population population distributions using the distributions and and then then run run forward forward for for 200 yr. yr. This This was was repeated repeated ((using the same 000 times same initial initial distribution distribution of of local local populations) populations) 11000 times to to estimate estimate the the distri distribution l/t)logM(t)/M(O). This bution c£ ef ((1/t)logM(t)/M(O). This process process was was repeated repeated for for four four randomly randomly cho chosen sen initial initial distributions distributions of of local local population population sizes. sizes. The The mean mean and and variance variance of of ((1/t)logM(t)/M(O) l/t)logM(t)/M(O) are t), respectively, are denoted denoted as as f.1m(t) I~m(t) and and (J";, (r2 ((t), respectively, in in Fig. Fig. 23.2 23.2 and and in in the the discussion discussion given given later. later. Figure % or Figure 23.2 23.2 illustrates illustrates the the results. results. For For dispersal dispersal levels levels 11% or higher, higher, the the trajec trajectories tories behaved behaved like like aa diffusion diffusion process process with with f.1m(t) l~m(t) and and (J";' (rZm(t) (t) roughly roughly constant constant and and the the distributions distributions approximately approximately normal normal according according to to aa Kolmogorov-Smirnov Kolmogorov-Smirnov test test
574 514
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
at %, the at P = 0.05. 0.05. For For low low dispersal, dispersal, 0.1 0.1%, the trajectories trajectories did did not not behave behave like like aa diffusion process for t less than 200 200 at least. The variance CT� (r2m(t) was not constant, constant, except 50, and except for for tt > > 1150, and the the normality normality assumption was was generally generally violated violated except except again again at at large large t.t. This This means that that when when dispersal is is very very low, low, diffusion diffusion population would be more approximate approximations for this meta metapopulation approximate than for meta populations with higher dispersal. metapopulations 23.2 illustrates results from from one particular model. Repeating Repeating this Figure 23.2 process for for aa number of of different different models models indicated indicated some some general general behaviors. behaviors. The higher the dispersal levels, the more trajectories behaved behaved like a diffusion diffusion process. process. Global Global dispersal levels of of at least 2 to 5 % were were generally generally high high enough enough to to result in in diffusion-like diffusion-like behavior behavior within within aa short short time time frame. Note Note that that local localized dispersal has the effect of of lowering lowering the effective effective dispersal rates. The The higher higher the populations in the amount amount of of temporal temporal covariance covariance between between local populations in terms terms of of their their yearly growth growth rates, the more the trajectories behaved behaved like a diffusion diffusion process. The simulations simulations were were done done with with the local population population sizes within within the equilib equilibrium set of population distributions of local population distributions ~ indeed the theory theory is predicated predicated on the local populations populations being near equilibrium. equilibrium. For metapopulations metapopulations with with 2 to 55% % dispersal, dispersal, the the local local populations populations equilibrated equilibrated fairly fairly quickly quickly starting starting from from all all populations with with equal numbers. numbers. However, However, at very very low low dispersal, dispersal, equili equililocal populations bration bration took took thousands thousands of of time time steps. steps. This This suggests suggests that that the assumption assumption of of equilibrium equilibrium should should be be viewed viewed cautiously cautiously for for metapopulations metapopulations that that have have very very low low dispersal dispersal rates rates between between local local populations. populations. =
0 . 1 % d i dispersal spersal 0.1%
0
11% % ddispersal ispersal
0
-0.01 -0.01
-0.01 -0.01
-0.02 -0.02
-0.02 -0.02
-0.02 -0.02
~ --0.03 0.03
-0.03 -0.03
-0.03 -0.03
-0.04 -0.04
-0.04 -0.04
-0.05 -0.05
-0.05 -0.05 00
2: "-
-~ if % 0
0 0
50 50
100 100
150 150
200 200
0.05 0.05 0.04 0.04
0.03 0.03
0.03 0.03
0.03 0.03
0.02 0.02
0.02 0.02
0.02 0.02
100 100
150 150
200 200
0.01 0
50 50
100 100
150 150
200 200
0.01 0.01
2
2
11.5 .5
1.5 1.5
1
1
0.5 0.5
0.5 05
0.5
0
0
1.5 1.5
'" "
�
100 100
0.05 0.05
2I
o c
50 50
-0.05 ' -0.05 200 00 200
0.04 0.04
50 50
0
50 50
100 100
150 150
200 200
' "~--
-0.04 -0.04 ' 150 150
0.05 0.05
0
�
-0.01 -0.01
_
0.04 0.04
0.01
...
_
5% 5 % ddispersal ispersal
0
0
50 50
100 100
150 150
200 200
00
'
'
50 50
100 100
150 150
200 200
50 50
100 100
150 150
200 200
- - pp=o.o5 1I- o.05 1I
0 0
50 50
100 100
150 150
200 200
t
Fig. Illustration the performance the Fig. 23.2 23.2 Illustration of of the performance of of aa diffusion diffusion approximation approximation for for modeling modeling the behavior with 50 50 local populations 1 , 11,, or 5% behavior of a metapopulation metapopulation with populations and and uniform uniform 0. 0.1, 5% yearly ddispersal. ispersal. The iffusion approximation frame when The ddiffusion approximation performs performs well well for for aa given given time time frame when JLm(t) /t)logM(t)/M(O) and I~m(t) = = (1 (1/t)logM(t)/M(0) and O"ii,(t) Cr2m(t)= = (l ( l /it) t ) var var [logM(t)/M(O)] [IogM(t)/M(O)] are are constants constants over over that that time time frame frame and when when log M(t)/M(O) is normal. normal.
23. ANALYSIS FOR FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ENDANGERED METAPOPULATIONS METAPOPULATIONS
23.4 23.4
515 575
ESTIMATING ESTIMATING THE THE PARAMETERS PARAMETERS Maximum Maximum likelihood likelihood estimates estimates of of fL i~m and a;' Cr2mcan can be be calculated calculated using using the the dif difm and Denote the series as fusion approximation fusion approximation for for log log M(t). M(t). Denote the observed observed time time series as M M = = M(O), M(0), M(1),), M(2), M ( 2 ) ,.. . .., , M(n). If If we we approximate approximate log log M(t) M(t) as as aa diffusion diffusion process, process, the the M(l 1 M ) is likelihood likelihood function function L(fL L(la,m m,, a;' o'2]M) is given by by the the product product of of the the probability probability M(t + function function distributions distributions for for the the transitions transitions from from log log M(t) M(t) to to log log M(t + 11),), which which is 1, over Eq. (7) with Eq. (7) with 'T~ = = 1, over tt = = 0, 0, 1, 1, 2, 2 , ... .. ,. , n. n. Thus Thus the the log log likelihood likelihood function function is is log
L(bl,m,, a� (r2 I[ M) M) = = - ((nn / 2) (2~rcr2m) L(fLm 2) log (21Ta�)
�
11 n - 2cr2�m ~ [10g(M(i) /M(i - 1 ) ) 2a ii=1 [log(M(i)/M(i - 1))
-
-
2
fLml ~m] 2
(23.8) (23.8)
Maximum likelihood estimates Maximum likelihood estimates are are obtained obtained by by solving solving for for fL tXm and a;', (r2m,which which m and maximize Eqn. (8), (8), maximize A = fLm ~m --
_
( ) 1 � [ 1 ( M( i ) ) ] fLm -;; i� M( i 1 ) 1 M( n ) log ;; -~ \ MM(O) (O) ]
A2 a6"2= og m ni=l log -
Mi;
_
1/ -
-
A
2
(23.9 (23.9))
Note n rather than than n. Note that that the the unbiased unbiased estimator estimator for for a;' Cr2mdivide by by ((n - 11)) rather n. The The estimates of variance from and a;' ~2 are are analogous analogous to to the the estimates of mean mean and and variance from n n sam samfl~mm and ples ples from from aa normal normal distribution, distribution, and and confidence confidence intervals intervals on on fL ~m and a;' cr2 are are m and analogous: analogous:
(~m -- G/2,n- 1X/' ~r2mIn, ~m q- G/2,n- 1V' ~2 In ) (nor m^21 X2,n -1 ,norm^2 i X 2 _ e~,n-1 )
(23.10) (23. 10)
where where tm G,qq iiss the the critical critical value value ooff aa tt distribution distribution aatt P = = ex ~ and and q degrees degrees of of freedom and and X�,q • the critical value value of of a x • 2 distribution distribution at at P = = ex cx and and q freedom is the 1991 ) for degrees of freedom. See al. ((1991) degrees of freedom. See Dennis Dennis et et al. for aa more more in-depth in-depth discussion discussion of of maximum maximum likelihood likelihood estimates estimates for for diffusion diffusion processes. processes. Following Following Dennis Dennis et et al.'s al.'s monograph, monograph, parameter parameter estimation estimation based based on on the the diffusion diffusion approxi approximation mation has has been been widely widely used used for for the the analysis analysis of of single single population population trajectories. trajectories. For For aa discussion discussion of of parameter parameter estimation estimation that that is is not not based based on on the the diffusion diffusion approximation, 1985). approximation, the the reader reader is is referred referred to to Heyde Heyde and and Cohen Cohen ((1985). Maximum Maximum likelihood likelihood estimates estimates assume assume that that the the metapopulation metapopulation has has reached reached aa stochastic stochastic equilibrium equilibrium and and thus thus that that the the diffusion diffusion approximation approximation is is reasonable. reasonable. When When exploring exploring these these methods methods using using simulations, simulations, it it is is important important to equilibrate, after simulation with to allow allow the the system system to to equilibrate, after starting starting the the simulation with something something peculiar like all local populations at the the same size. Equilibruim Equilibruim can can be moni monipeculiar populations at tored tored by by waiting waiting for for the the variance variance of of (log(NAt)) (log(Ni(t)) - 10g[mean(Nj(t))]) log[mean(Ni(t))]) to to stabilize. stabilize. In In simulations simulations done done for for this this chapter, chapter, the the distribution distribution stabilized stabilized relatively relatively quickly dispersal is quickly when when dispersal dispersal was was nonzero. nonzero. If If dispersal is zero, zero, however, however, the the distri distribution ) - 10g[mean(N;(t) )] ) bution never never stabilizes stabilizes and and the the variance variance of of (log(N;(t) (log(Ni(t))log[mean(Ni(t))]) increases wants to to increases continually. continually. For For an an actual actual metapopulation, metapopulation, for for which one wants
576 576
E.E. E.E. HOLMES HOLMES AND B.X. B.X. SEMMENS SEMMENS
conduct conduct aa PYA, PVA, it it is is also also critical critical to to test test the the appropriateness appropriateness of of the the diffusion diffusion approximation for for one's one's time time series series data. data. Dennis Dennis et et al. al. ((1991) and Morris Morris and and approximation 1 99 1 ) and Doak Doak (2002) (2002) reviewed reviewed how how to to do do this, this, which which is is based based on on diagnostic diagnostic proced procedures for for evaluating evaluating the the appropriateness appropriateness of of linear linear models. models. ures Parameter Bias Parameter Bias
The The estimators estimators are are unbiased unbiased maximum maximum likelihood likelihood estimators estimators for for the the diffu diffusion approximation, approximation, X(t). X(t). It It is is important important to to understand understand whether whether and and how how these these sion estimates population estimates are are biased biased when when working working with with short short time time series series of of meta metapopulation trajectories, trajectories, M(t), as as opposed opposed to to an an actual actual diffusion diffusion process. process. In In particular, particular, iT;' 82 is is certain certain to to be be biased biased to to some some degree, degree, as as it it reli.:s relics on on the the diffusion diffusion approximation approximation holding holding for for T~ = 1i in in log log M(t + 'T)/M(t), ~)/M(t), regardless regardless of of the the length length of of the the time time series series used used for for estimation. estimation. This This is is not not the the case case for for ,J.,m, Igm,which which is is also also an an unbiased unbiased predictor predictor for for M(t) given given aa long long time time series series (Heyde (Heyde and and Cohen, Cohen, 1985). 1985). To To numerically numerically explore explore parameter parameter bias bias from from short short time time series, series, simulations simulations were were used used to to look look at at the the difference difference between between ,J.,m lkm and and iT;' 82 from from aa 20-yr 20-yr time time series series versus versus their their true true values values JLm ~m and and a;'. Cr2m.An An example example metapopulation metapopulation of of 50 50 local local sites sites was was simulated simulated with with global global dispersal dispersal and and correlated correlated local local growth growth rates, rates, Zi(t), zi(t), drawn drawn yearly yearly from from aa normal normal distribution distribution with with mean mean = = JL ~i,i, variance variance = = <1>, &, and and covariance "'<1> between covariance of of 0.2 0.2*& between any any two two local local growth growth rates rates Two Two versions versions of of the the simulation simulation were were run: run: one one to to model model uniform uniform site site quality quality (spatially (spatially uniform uniform JLi 13,,i = -- -0.05) - - 0 . 0 5 ) and and one one to to model model highly highly variable variable site site quality quality (spatially (spatially variable variable JL;'s). IXi's). To To explore explore biases biases over over aa range range of of different different dispersal dispersal and and variability variability levels, levels, models models were . 1 and % per were run run with with dispersal dispersal between between 00.1 and 55% per year year and and local local variability, variability, <1>, &, between 0.1 These parameters parameters translated translated to population level level rates, rates, between 0.1 and and 0.5. 0.5. These to meta metapopulation in the to --00.05 metapopulation level level variability, variability, a;', in tXm, in the range range of of 0.01 0.01 to . 0 5 and and metapopulation (r2m,in JLm, the to 0.08. For each each dispersal dispersal and local variability variability pair, pair, 1000 1 000 the range range of of 0.001 0.001 to 0.08. For and local replicate metapopulation trajectories were simulated, each each with an initial replicate metapopulation trajectories were simulated, with an initial distridistri bution of local local population sizes selected selected randomly randomly from from the the equilibrium equilibrium set. bution of population sizes The ,J.,m and over the the dispersal and local vari The mean mean difference difference between between }~m and JLm ~m over dispersal and local variability parameter space space was was very very low, both uniform uniform and variable ability parameter low, <0.0015, <0.0015, for for both and variable JLi simulations. Overall bias in that rely rely pripri ~i simulations. Overall the the lack lack of of bias in ,J.,m ~m supports supports metrics metrics that marily such as as the metapopulation ~ A. (next (next section). section). For For marily on on this this parameter, parameter, such the metapopulation most of the parameter space explored, explored, 0 < < liT;' 6"2 - a;,1 Cr2ml< most of the parameter < 0.01, 0.01, representing representing a 0 to a;'. Larger Larger biases, biases, 16"2 liT;' - Cr2m a;,1 >> 0.01, 0.0 1 , reprep to 20% 2 0 % underunder- or or overestimation overestimation of of o.2. resenting aa > >20% under- or or overestimation, overestimation, were were seen seen for for some some parameter parameter resenting 2 0 % undercombinations. combinations. The The impact impact of of this this bias bias depended depended on on where where 6.2 iT;' was was used. used. For For instance, the the effect effect on on estimated confidence intervals on ~m JLm [Eq. (10)] ( 1 0)] was was instance, estimated confidence intervals on minimal with the width of the the interval changing by by aa median median 0.002. 0.002. The effect minimal with the width of interval changing The effect on estimated estimated passage passage probabilities probabilities was was higher, higher, although although not not dramatic. dramatic. For For on example, the the estimated estimated probability probability that that the the metapopulation will be be 10% 1 0 % of of example, metapopulation will current levels levels at at the the end end of of 50 50 yr yr was was decreased decreased by by 00 to to 0.04 0.04 (on (on aa scale scale from from current to 1) 1 ) for for the the uniform uniform Ixi JLi simulation simulation and and increased increased by by 00 to to 0.04 0.04 for for the the varivari 0o to able ~i JLi simulation. simulation. The The estimated estimated probability probability that that the the metapopulation metapopulation will will able pass below below 10% 1 0 % of of current current levels levels at at any any point point during during the the next next 50 50 yr yr was was pass iT;' bias bias was was low low in in these these simulasimula changed by by 00 to to 0.09. 0.09. Overall, Overall, the the effect effect of of 6"2 changed tions, but but this this will will depend depend on on the the particular particular metapopulation metapopulation and and will will need need to to tions, be be investigated investigated for for individual individual cases cases of of interest. interest. -
-
23. FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
577 577
In In practical practical applications, applications, one one must must contend contend with with other other factors factors that that can can lead lead to to parameter parameter bias, bias, but but which which are are outside outside the the scope scope of of this this chapter. chapter. In In particu particular, nonequilibrium local population distributions, distributions, and lar, observation observation error, error, nonequilibrium local population and tem temautocorrelation can problems are poral autocorrelation can lead lead to to parameter parameter bias. Such Such problems are being being studied population PYA. studied in in the the context context single single population PVA. Much Much of of this this work work is is likely likely to to be Morris and be relevant relevant for for metapopulation metapopulation PYA. PVA. See See Morris and Doak Doak (2003) (2003) and and Holmes Holmes (2004) (2004) for for aa review review and and discussion discussion of of current current work work in in this this area. area.
23.5 23.5
METAPOPULATION ETRICS METAPOPULATION VIABILITY VIABILITY M METRICS One One of of the the most most basic basic viability viability metrics metrics is is the the long-term long-term geometric geometric rate rate of of decline decline (or (or growth) growth) of of aa population, population, termed termed generally generally A k in in the the PYA PVA literature. literature. If .0, the population ultimately If A k is is less less than than 11.0, the population ultimately declines declines to to extinction extinction and and k) is is roughly roughly the the average average yearly yearly percent percent decline. decline. The The metapopula metapopula1100"(1 0 0 " ( 1 - A) tion tion A k is is exp(J.1m) exp(~m) and and its its estimate estimate is is then then
-
(23. 11) (23.11)
}~ = exp(gm).
definition of of A k follows follows Caswell's Caswell's use use of of the the symbol symbol As ks as as the the long-term long-term This definition average stochastic stochastic growth growth rate: rate: As ks = = [N(t)/N(O [N(t)/N(O)] lit as as tt � --) 00 oo (Caswell, (Caswell, 200 2001). )PIt 1). average This This iiss the the long-run long-run geometric geometric growth growth rate rate that that would would bbee observed observed iinn almost almost 1, the every every trajectory. trajectory. Defined Defined this this way, way, if if A k < < 1, the population population goes goes extinct extinct with with certainty, certainty, eventually. eventually. This This differs differs from from Dennis Dennis et et al.'s al.'s use use of of the the symbol symbol A k 2/2) and where used for long-term average average geometric geometric growth growth where A k is is used for exp(J.1 exp(Ix + ( cr2/2) and the the long-term maximum likelihood rate denoted by rate is is instead instead denoted by ex cx = - exp(J.1). exp(tx). The The maximum likelihood estimate estimate of of A k is is normally is aa biased biased estimator; estimator; because because ,Lm ~m is normally distributed, distributed, the the median median value value of of exp ( ,Lm) is mean value not. Dennis al. ((1991) 1 99 1 ) gave exp(O.m) is exp(J.1m) exp(tXm) but but the the mean value is is not. Dennis et et al. gave an an unbiased 1 9 8 1 ), unbiased estimator estimator [mean( [ m e a n (X.k) )= - A k]] based based on on Shimizu Shimizu and and Iwase Iwase ((1981), although differences between between biased although Dennis Dennis and and colleagues colleagues found found negligible negligible differences biased and and unbiased unbiased estimators estimators in in their their examples. examples. From the asymptotic asymptotic distribution distribution of log M(t), Eq. (5), the probability probability that that the the From population is metapopulation is below below aa threshold threshold b b at at the the end end of of y years years can can be be calculated calculated as as meta P [M( t ) P[M(t)
M( O )] = <<-bb [IM(0)] = ~
:s
(
)
Ilog(b/M(0)) Ixmt) Og( bIM( O ) ) - J.1mt -
wr;t
V~2~mt
(23 .12) (23.12)
Although Although this this uses uses the the asymptotic asymptotic distribution, distribution, this this iiss mitigated mitigated by by the the fact fact that that it it is is used used for for the the distribution distribution at at the the end end of of y years years but but not not at at any any time time before before replaces J.1m and a their estimates that. The that. The estimate estimate of of P[M(t) :S <_bb [I M(O)] M(0)] replaces ~m and cr;' 2 by by their estimates ,Lm ~m and and IT;'. ~2. Like Like the the estimate estimate of of A, k, the the median median estimate estimate of of P[M(t) :S <<-bb [JM(O)] M(0)] is is equal to to the the true true value, value, but but not not the the mean. mean. equal Some populations can Some meta metapopulations can have have aa low low long-term long-term risk risk of of being being below below aa .0, but risks of threshold due threshold due to to aa A k near near 11.0, but high high short-term short-term risks of hitting hitting that that threshold threshold due probabilities are due to to high high variability. variability. Such Such quasiextinction quasiextinction or or extinction extinction probabilities are commonly commonly used used and and very very important important PYA PVA metrics. metrics. The The diffusion diffusion approximation approximation for for log log M(t) can can be be used used to to estimate estimate these these probabilities probabilities for for the the metapopulation. metapopulation. The The probability probability of of that that the the diffusion diffusion process, process, X(t), experiences experiences aa decline decline below below
5578 78
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
threshold log log bb at at some some time time T T less less than than yy years years is is calculated calculated by by integrating integrating aa threshold over the the probability probability density function function for for first first passage passage times times for for aa diffusion diffusion over process with with drift (Karlin and and Taylor, Taylor, 1981). 1 9 8 1 ) . Lande Lande and and Orzack Orzack (1988) ( 1 988) go go process drift (Karlin through the the calculation, calculation, which which leads leads to to through
(-(X(O) -
) (- (X(O) -x/F2yb )
log b ) fLmY P(T P ( T <� Y) y) = = ~
fLm
)
Y + x p (- 22((X l o glog bb))fL b ~m/ m / C
\
+
V
/
X(O)
M(O).
Risk Metric Risk Metric Uncertainty Uncertainty The 00( 1 ) % confidence The 1100(1 - a c~)% confidence intervals intervals are are often often used used as as characterizations characterizations of of uncertainty. uncertainty. These These can can be be calculated calculated for for risk risk metrics metrics using using the the estimated estimated intervals for distributions of distributions of tlm ~m and and &?;, 6"2.. The The confidence confidence intervals for X. k are are ((exp exp ((~m -iLm -
tt~/2,n-1V'(r2mln ) ). exp ((~m + tt CJj2,n ~ / 2 , n-- 1t VV ' (& r 2�/ 1 n n )). CJj2,n- t V&�/ n ),), exp iLm +
(23.14) (23.14)
""q is where where tt~,q is the the critical critical value value of of aa t distribution distribution at at P = = a o~ and and q q degrees degrees of of freedom. freedom. The The corresponding corresponding significance significance level, level, a, cx, for for aa hypothesis hypothesis test, test, such such as as "Is "Is A k< < b" b" is is the the a cx such such that that tl~mm
-
-- log log b b
� v
/
�
= tt~,n-1. = et,n-t .
M
M(O)]
(23 .15) (23.15)
Confidence [ ( y) � Confidence intervals intervals oonn P(T P(T � <- y) and and P P[M(y) <- bb I] M(0)] can can bbee calculated calculated by by parametric parametric bootstrapping bootstrapping from the estimated distributions of tlm s and &?;,: 82: Normal(tlm, &?;'!n) )12, scale I(n - 11)). ) ). A Normal(s 82m/n) and and Gamma(shape Gamma(shape = = (n ( n -- 11)/2, scale = = 2&?;, 282m/(nA large large number of of ((stlh> &l) %) ^ 2 pairs are are generated generated randomly randomly by by sampling sampling from from these these distributions 1 3 ) or 1 2)] for distributions and and the the risk risk metric metric 'I' 9 isis calculated calculated [Eqs. [Eqs. ((13) or ((12)] for each each pair. pair. The The range range of of 'I' 9 over over the the ((itlkbb,' &l) 8~) bootstrapped bootstrapped pairs, pairs, for for which which both both parameters 00( I -a ) % confidence parameters are are within within their their respective respective 1100(1-o~)% confidence intervals, intervals, defines defines the 00( 1 - a ) % confidence the 1100(1u)% confidence interval interval for for '1'. ~. This This and and other other methods methods for for calculating calculating confidence confidence intervals intervals for for diffusion diffusion approximation approximation risk risk metrics metrics are are discussed 1 99 1 ) . discussed in in Dennis Dennis et et al. al. ((1991).
579 579
23. FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
An An alternate alternate way way to to present present the the level level of of uncertainty uncertainty is is to to estimate estimate the the data data support support for for different different values values of of aa risk risk metric. metric. There There are are both both frequentist frequentist and and Bayesian 1 ) for Bayesian approaches approaches for for this this [see [see Wade Wade (200 (2001) for aa review review geared geared toward toward conservation conservation applications] applications].. Holmes Holmes (2004) (2004) presented presented aa Bayesian Bayesian approach, approach, which probability distributions which uses uses posterior posterior probability distributions to to illustrate illustrate data data support. support. That That method support for method is is adapted adapted here here for for estimating estimating the the level level of of data data support for the the metapopulation probability that metapopulation risk risk metrics. metrics. Let Let 'I' 9 be be aa risk risk metric. metric. The The probability that 'I' xI, is is greater greater than than some some threshold threshold
P (xIt > q) l ~m'~'2) =
j
all (l&m,~) for which ~>q~
L(l& '
(r21~m)L((r21&2)Tr(l&m)Tr((r2)
(23 .16) (23.16)
Tl(~m, ~'2)
mlLm, u;;,ln), where where L(IL, L(Ix, u;;' 0"2 II P.m) JSbm) is is the the likelihood likelihood function function given given P. ~bm "-" Normal( Normal(b~m, r L(cr2 I1&;;,) 82) is the likelihood likelihood function function given &;;, 82 _ Gamma((n Gamma((n - 11)/2, 2cr2/(n - 11)), )/2, 2 u;;,/( n ) ), L(u;;, 1T( lLm) and ~r(tXm) and 1T(U;;') ~r(~r2) are are the the priors priors on on ILm ~ m and and u;;" ~r2, and and the the normalizing normalizing constant constant is is
J� at
iJ. m m ~= q -+00 0 0 0- 2 = ~ 0 0co
T I (( ", ~ mm' (, yrr2�I ) �
�
�OO
b~m= - ~
(23 . 1 71 L ( rr;, Iu;, I'IT ( �m ~ m I'M� Id�mdrr;, (23.~7)
m , C,rr;, r m l ~I ",m z m ) LI ( 0 L ( t z�m
L(
=0
The I P.m, &;;,) I P.m, The posterior posterior distribution distribution of of 'I' 9 isis [P('I' [P(~ < <
223.6 3.6
A A SIMULATED SIMULATED EXAMPLE EXAMPLE In In this this example, example, aa collection collection of of 49 49 local local populations populations in in aa 7 7 X x 7 7 grid grid was was simulated simulated with with neighborhood neighborhood dispersal. dispersal. Local Local populations populations were were specified specified with with variable variable mean mean local local growth growth rates; rates; thus, thus, some some lLi [s values values were were much much larger larger than than others. others. The The local local growth growth rates rates in in any any given given year year were were slightly slightly correlated correlated between sites. Thus between sites. Thus all all sites sites were were more more likely likely than than random random to to have have good good and and bad Dispersal was 0 % from bad years years together. together. Dispersal was variable variable between between 55 and and 110% from year year to to year year and and was was mainly mainly to to the the four four nearest nearest neighbors neighbors (or (or two two and and three three for for cor corner A(t) was ner or or edge edge sites). sites). In In specific specific terms, terms, A(t) was specified specified with with zM)'s zi(t)'s drawn drawn from from aa normal normal distribution mean = were distribution with with mean = lL ~ii and and aa variance variance of of 0.0625. 0.0625. The The lLi ~i were different local population between different for for each each local population and and were were chosen chosen randomly randomly between - 00.22 .22 and and - 00. 0. 011. . Each Each year, year, new new Zi(t)'S zi(t)'s were were selected selected from from the the normal normal distribution distribution for for that that local local population. population. The The zM)'s zi(t)'s were were correlated correlated among among the the local local popula populations 0 . 1 )(0.0625 ) . The tions such such that that the the covariance covariance of of zM) zi(t) and and Zj( zi(tt)) was was ((0.1)(0.0625). The di(t) di(t) var varied population, di( ied from from year year to to year. year. Each Each year year and and separately separately for for each each local local population, di(t)t) was selected from random distribution between 0.05 . 1 ; thus was selected from aa uniform uniform random distribution between 0.05 and and 00.1; thus the the dispersal dispersal varied varied from from year year to to year year and and between between local local populations populations in in any 0 % , was any given given year. year. Most Most of of this this dispersal, dispersal, 880%, was to to nearest nearest neighbors. neighbors. Thus Thus neighbors, (Xji for for nearest nearest neighbors, o~ji = 0.80 0.80 dj(t)/nn, dj(t)/nn, where where nn nn is is the the number number of of nearest nearest neighbors, neighbors, and and for for nonnearest nonnearest neighbors, neighbors, (Xji oLji = 0.2 0.2 dj(t)/nnn; dj(t)/nnn; where where nnn nnn is is the the number number of of nonnearest nonnearest neighbors. neighbors. =
- -
580 580
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
The The simulation simulation was was started started from from aa set set of of local local population population sizes sizes drawn drawn ran randomly domly from from the the stochastic stochastic equilibrium, equilibrium, and and starting starting sizes sizes were were drawn drawn anew anew from from this this distribution distribution for for each each replicate replicate of of the the simulation. For For each each replicate, replicate, aa 25-yr time ~m and and a;' (r2mwere were time series series was was generated, generated, and and from from this this time time series, fLm estimated estimated using using Eq. Eq. (9). (9). From From the the estimates, estimates, the the probability probability that that the the metapopu metapopulation % of lation would be be below below different different thresholds thresholds (50 (50 or or 75 75% of starting starting levels) at at the the end end of of 25-yr 25-yr was was predicted predicted and and compared compared to to the the actual actual probabilities probabilities obtained 1 000 times) obtained by by repeatedly repeatedly ((1000 times) running running the the simulation simulation for for 25-yr 25-yr starting starting from from the point point where the initial 25-yr 25-yr time series stopped. This simulation was was replicated replicated 500 times times to to generate generate the the distribution distribution of of estimated estimated probabil probabilities % decline ities of of 50 and and 75 75% decline in in 25-yr 25-yr versus versus the the true true probability. probability. Also, Also, from from 25-yr simulation, the metapopulation metapopulation A ~ was estimated and compared compared each 25-yr to 0000-yr simulation. to the the actual actual value value calculated by by running running aa 110000-yr simulation. For For each each estimated estimated risk risk metric, metric, confidence confidence intervals intervals were were estimated estimated via via the the methods methods in in Section Section 23.5. 23.5. Figure Figure 23.3 23.3 shows shows the the distribution distribution of of A ~ estimates estimates and and the the estimated estimated prob probabilities of 50 and 75 % decline versus true 75% true values. As expected, expected, the median median estimate (p.'m is estimate of of A ~ was was equal equal to to the the true true value value (l~m is an an unbiased unbiased estimator estimator of of fLm) ~m).' o actual relationship - correct relationship
- true value CJ distribution of estimates 1 50
1 00
� O.5 �
50
8.85 300
E 0 I&.L. Q)
0.9
0.95 A estimate
1 .05
_ _ _ _ _ _ _
£
0
Q)
'm c
c
8
iii £
!!i
0.5 confidence level
0.5
.!:
"E o prob. of 50% decline in 25 yrs (.)
'0
1 00
confidence level
c o
U
�
50
0.5
prob. of 75% decline in 25 yrs
confidence level
Estimated viability metrics and and their intervals versus the Fig. 23.3 23.3 Estimated viability metrics their estimated estimated confidence confidence intervals versus the true grid with 0% dispersal true values values for for aa 49 49 site site metapopulation metapopulation in in aa 77 x x 77 grid with 5-1 5-10% dispersal to to the the closest closest neighboring compared to distribution of neighboring sites. sites. (Left) (Left) True True metrics metrics compared to the the distribution of estimated estimated metrics metrics from from 500 simulations starting from initial conditions. (Right) Performance of the estimated from the same initial estimated confidence 00(1 -a)% confidence confidence intervals intervals by by looking looking at at the the fraction fraction of of estimated estimated 1100(1-c0% confidence intervals intervals that that contain contain the the true true values. values.
23. 23.
VIABILITY ANALYSIS ANALYSIS FOR FOR ENDANGERED ENDANGERED METAPOPULATIONS METAPOPULATIONS VIABILITY
581 581
The median median estimate estimate of of ~ 'A. was was 0.97 0.97 compared compared to to the the true true value value of of 0.97. 0.97. The The The median estimates estimates of of 50 50 and and 75% 75 % decline decline were were 0.63 0.63 and and 0.14 0 . 1 4 compared compared to to the the median true values values of of 0.62 0 .62 and and 0.12, 0 . 12, respectively. respectively. Although Although the the median median estimates estimates true were very very close close to to the the true true values, values, the the estimates estimates were were variable. variable. The The estimates estimates were of ~ 'A. ranged ranged between between 0.9 0 . 9 and and 1.0. 1 . 0. The The estimates estimates of of declines declines to to thresholds thresholds were were of also variable. variable. The The variability variability depended depended on on the the threshold threshold and and the the time time frame. frame. also In this this example, example, there there was was low low variability variability around around the the estimate estimate of of 50% 5 0 % decline decline In in 25-yr, 25-yr, but but high high variability variability in in the the estimate estimate of of 75% 75 % decline. decline. The The true true values values in for each each of of the the metrics metrics are are shown shown by the the solid lines in the the middle middle of of the the for distributions. distributions. The variability of the the estimates due to to the the stochastic stochastic nature nature of of the the process process The variability of estimates is due and is is not not aa fault fault of of the the estimation estimation methods methods per per se; se; by by chance, chance, short short trajectrajec and tories will will appear appear to to have have underlying parameters that that are are different different than than the the tories underlying parameters true underlying parameters, which which leads leads to to variability variability in in the the estimated estimated viability viability true underlying parameters, metrics. When When estimates estimates are are inherently inherently variable, variable, it it is is critical critical that that the the confi metrics. confidence be estimated correctly. Figure Figure 23.3 23.3 (right) dence intervals intervals for for the the estimates estimates be estimated correctly. (right) confirms intervals properly properly characterize characterize the confirms that that the the estimated estimated confidence confidence intervals the uncertainty 0 0 ( 1 - u ) % of of the time the uncertainty for for the the estimate estimate risk risk metrics: metrics: e.g., 1100(1-~)% the 1 00( 1 - u ) % confidence the true values. 100(1-~)% confidence intervals intervals contain contain the true values.
23.7 23.7
SALMON AS SALMON AS METAPOPULATIONS METAPOPULATIONS Salmonid Oncorhynchus spp.) spp.) show structur Salmonid populations populations ((Oncorhynchus show strong strong spatial spatial structuring Reiman and ing and and they they have have often often been been referred referred to to as as metapopulations metapopulations ((Reiman and McIntyre, 995; Policansky 998; Cooper McIntyre, 11995; Policansky and and Magnuson, Magnuson, 11998; Cooper and and Mangel, Mangel, 11999; 999; Hill ai., 2002) Hill et et al., 2002).. Spawning Spawning and and rearing rearing habitats habitats of of different different salmon salmon stocks stocks occur occur on on discrete discrete and and physically physically separated separated river river or or stream stream sections. sections. Salmon Salmon have have aa well-known well-known and and strong strong tendency tendency to to return return to to their their natal natal streams 0 % ) dispersal streams with with aa low low ((11 to to 2 20%) dispersal to to other other stocks stocks (Fulton (Fulton and and Pearson, Pearson, 11981; 9 8 1 ; Mathews 9 9 1 ; Quinn, 99 3 ) . Within u.S. Pacific Mathews and and Waples, Waples, 11991; Quinn, 11993). Within the the U.S. Pacific Northwest, anadromous salmon Northwest, collections collections of of anadromous salmon stocks stocks have have been been divided divided into into "evolutionary units" (ESUs) 9 9 1 ) , which sub "evolutionary significant significant units" (ESUs) (Waples, (Waples, 11991), which represent represent substantially stantially reproductively reproductively isolated conspecific conspecific groups groups that that can can be be distinguished distinguished based based on on their their coherence coherence on on aa genetic genetic level level and and known known dispersal dispersal between between the the stocks. stocks. Salmon Salmon within within aa stock stock spawn spawn on on individual individual streams streams or or river river sections sections and and the the majority majority of of offspring offspring return return to to spawn spawn in in their their natal natal stream stream or or river. river. Straying Straying of of returning returning adults adults to to nonnatal nonnatal streams streams is is spatially spatially structured structured and and occurs occurs more more frequently frequently within within subbasins. subbasins. Stocks Stocks within within an an ESU ESU have have some some of synchrony synchrony due due to to exposure exposure to to common common migratory migratory corridors corridors between between level of the the ocean ocean and and the the natal natal stream stream and and also also due due to to exposure exposure to to similar similar large-scale large-scale ocean 992; Ware, 995; Mantua 997). However, ocean dynamics dynamics (Pearcy, (Pearcy, 11992; Ware, 11995; Mantua et et ai., al., 11997). However, stocks stocks also also show show aa great great deal deal of of asynchrony asynchrony due due to to exposure exposure to to their their inde independent pendent spawning spawning and and juvenile juvenile rearing rearing habitats habitats and and variability variability in in migration migration timing between 00 1 ) . Throughout timing between stocks stocks (e.g., (e.g., PSTRT, PSTRT, 22001). Throughout the the Pacific Pacific Northwest, Northwest, most most salmonid salmonid populations populations show show regional regional decline decline with with the the major majority of of individual individual stocks stocks showing showing steady steady declines with densities densities well below his hisity torical Rieman and torical levels levels ((Rieman and Dunham, Dunham, 2000; 2000; McClure McClure et et ai., al., 2003 2003).) .
582 582
23.8 23.8
LE. AND B.X. E.E. HOLMES HOLMES AND B.X. SEMMENS SEMMENS
SNAKE SNAKE RIVER RIVER SPRING/SUMMER SPRING/SUMMER CHINOOK CHINOOK ESU ESU The The Snake Snake river river spring/summer spring/summer chinook chinook ESU ESU (Fig. (Fig. 23.4 23.4)) includes includes all all spring spring and and summer summer chinook spawning spawning within within the the subbasins subbasins of of the the Tucannon Tucannon river, river, Grande Ronde river, and and the south, south, middle, and and east fork Salmon rivers, which flow into the Snake river below the the Hells Canyon dam (Mathews (Mathews and and Waples, 11991). 99 1 ). Juvenile fish rear rear in the mountain mountain streams and and then then migrate down down the the Snake and and Columbia rivers to to the the ocean. ocean. After maturing maturing in the ocean, ocean, adult adult return to to spawn spawn at at variable ages ages between between 33 and and 55 yr yr (mean = = 4.5 yr). yr). fish return Tagging Tagging experiments experiments in in the the Columbia Columbia river river basin basin (which (which the the Snake Snake river river basin basin part of) have found found that that the proportion proportion of of individuals that that disperse and and is a part spawn indi spawn away away from from their their natal natal sites sites is is on on the the order order of of 1-3 1-3 % % for for wild-born individuals (Quinn, (Quinn, 11993). 993). The threatened The Snake river spring/summer spring/summer chinook chinook ESU was was listed as threatened under 992. Stocks within and under the the U.S. Endangered Endangered Species Act Act in 11992. within this large and complex basin, like salmon stocks stocks throughout throughout the the Pacific Northwest, Northwest, are impacted 1 994 ) and impacted negatively by a variety of of factors factors (Wissman (Wissman et ai., al., 1994) and many many have 1 998; McClure McClure et ai., have experienced experienced substantial declines (Myers (Myers et ai., al., 1998; al., 2003 habitat degradation 2003).) . There There is habitat degradation in many many areas areas related related to to forestry, forestry, graz grazand irrigation practices, practices, resulting of pools, high tem teming, mining, and resulting in lack of pools, high peratures, poor overwintering conditions, high sediment sediment peratures, low low flows, flows, poor conditions, and and high loads many areas. areas. At the the same time, a substantial substantial portion of the ESU is loads in many portion of protected as part part of Waples, protected of federally federally designated designated wilderness wilderness (Mathews (Mathews and and Waples,
Fig. Fig. 23.4 2 3 . 4 Map of the Snake Snake river spring/summer spring/summer chinook ESU. ESU. The ESU ESU includes stocks from the Snake Canyon dams. The Hells Snake river and its tributaries between Ice Harbor and Hells Hells Canyon Hells Canyon hydropower dam dam has no passage facilities and blocks the migration of salmon into their histor passage facilities historical habitat in the upper Snake ical Snake river basin.
23. VIABILITY FOR ENDANGERED VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
583 583
11991). 9 9 1 ) . The The official official ESU ESU designation designation does does not not include include salmon salmon in in the the Clearwater Clearwater basin, as as chinook chinook in in this this subbasin subbasin originate originate from from hatchery hatchery fish fish that that were were stocked stocked in in the the subbasin subbasin after after the the original original natural natural fish fish were were extir extirpated 940s. However, pated in in the the 11940s. However, from from aa metapopulation metapopulation dynamic dynamic perspective, perspective, current current stocks stocks in in the the Clearwater Clearwater river river basin basin interact interact with with stocks stocks within within other other subbasins. subbasins. Thus, Thus, in in this this analysis, analysis, all all stocks stocks in in the the entire entire Snake Snake river river basin basin were were analyzed together. together. A population level ESU from A total total meta metapopulation level time time series series was was available available for for this this ESU from counts counts of the the total number number of of wild-born wild-born spawners spawners returning returning through through the the Ice Harbor 3 . 4 ) . Returning Harbor dam at at the the downstream downstream end end of the the ESU ESU (Fig. (Fig. 2 23.4). Returning spawners spawners can can be be either either wild wild born born or or hatchery hatchery born born as as hatcheries hatcheries have have been been operating 970s. McClure operating in in the the basin basin since since the the early early 11970s. McClure et et aI. al. (2003) (2003) discussed discussed the the effects effects of of hatchery hatchery production production on on viability viability analyses. analyses. By By focusing focusing on on the the wild-born wild-born spawner spawner time time series series and and not not incorporating incorporating aa correction correction for for hatch hatchery ery production, production, the the in-stream in-stream viability viability metrics metrics assume assume that that hatchery-born hatchery-born fish fish all all return return to to the the hatchery hatchery and and do do not not spawn spawn in in stream stream (which (which would would produce produce wild-born wild-born offspring) offspring).. As As discussed discussed by by McClure McClure et et aI. al. (2003), (2003), this this means means our our viability viability metrics metrics are are optimistic optimistic upper upper bounds, bounds, as as some some unknown unknown fraction poten fraction of of hatchery hatchery fish fish do do stray stray to to the the wild wild spawning spawning grounds grounds and and potentially reproduce. reproduce. addition to the meta metapopulation dam count, count, time series of redds per In addition population level dam mile mile (rpm), (rpm), which which are are indices indices of of the the density density of of gravel gravel egg egg nests nests made made by by spawn spawning ing females, females, were were available for for the the majority majority of of stocks stocks within within the the Snake Snake river river basin. Redds Redds per per mile mile are are an an index index of of the the redds redds (and consequently consequently returning returning spawners) trend trend within within aa stock, stock, but but the the total total redds redds are are unknown, unknown, as as the the total spawning spawning habitat habitat is is not not surveyed. surveyed. The The majority of of rpm rpm and and dam dam data data are are avail available 3). able in in the the digital digital appendices appendices of of McClure McClure et et aI. al. (200 (2003).
Parameter Estimation Parameter Estimation Our 962 and 999. The Our Ice Ice Harbor Harbor dam dam time time series series starts starts in in 11962 and ends ends in in 11999. The wild-born 1 ), M(2), wild-born component component of of the the dam dam count count is is denoted denoted M(O), M(0), M( M(1), M ( 2 ) , .. ... . M(37), 962 count 999 count. M(37), where where M(O) M(0) is is the the 11962 count and and M(37) M(37) is is the the 11999 count. The The maximum 9 ) assume maximum likelihood estimates estimates presented presented in in Eq. Eq. ((9) assume that that data data do do not not contain contain sampling sampling error error or or other other nonprocess nonprocess error; error; however, however, salmon salmon data data typ typically have high levels of sampling error error and and boom-bust boom-bust cycles that that confound confound especially 0';' (Holmes, 200 1 ) . An approach estimation estimation of of i-Lm ~m and and especially Cr2m(Holmes, 2001). An alternate alternate approach uses uses data data filtering filtering and and examination examination of of the the rate rate at at which which variance variance increases within within the the time time series series to to improve improve parameter parameter estimation estimation and and separate separate out sampling error 1 ; Holmes error variance variance from from the time series (Holmes, 200 2001; Holmes and and Fagan, Fagan, 2002; 2002; Holmes, Holmes, 2004 2004 cf cf also also Morris Morris and and Doak, 2002). 2002). These These methods methods have have been been cross-validated extensively extensively with with salmon data data (Holmes and and Fagan, Fagan, 2002; 2002; Fagan et aI., al., 2003) 2003) and and are used here to estimate parameters. parameters. First, data data are are transformed transformed using using aa running running sum: sum:
(t) 4i?oM(t
t
1 J, M M (t) = M(t + jj)) for for t = - 00 to to 34 34 m
(23. 18) (23.18)
584 584
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
of f.Lm ~m and and u;, {r2 are then The estimates of
�
4 ) // /M ~ ((tt)))) - var var(log/~(t = 0.0353 var( logM ( t + 4) / M ( t )) = a;' ( logM ( t + 11 ))/_]~(t)) G == 51 ((var(logM(t
�
4 11 334 34 t=O
�
�
) / M( t ) = fL ~m m == 34 ~ ] llog o g /M( ~ ( tt + + l1)//~(t) = -0.0561 -0.0561
(23 .19) (23.19)
{r2muses the property property that that the variance ooff the underlying sto stoThe estimate ooff u;, chastic process process should should increase increase linearly linearly with with time: time: E[var(logM(t)/M(O))] E[var(logM(t)/M(O))] = u {r2m chastic ;, t. The ( t) are slightly The confidence confidence intervals intervals for for flm ~bm and and a;' {~2musing using M M(t)are slightly different different than than 2002):) : Eq. ((10) 1 0 ) (Holmes and Fagan, 2002 ( ~m
--
G/2,df V'~r2 /( t - L + 1 ), ~ m
+
G/2,dfX/8 2 /( t - L + 1 ))
(( df df a 42� //Xx�2, df' df , ddffa f 2� //X2-~, df)) x� -ex, df
(23.20) (23.20)
,
where L is is the the number number of of counts counts summed summed together together for for the the running running sum sum and and where d ff == 0.333 0.333 + + 0.212 0.212 (n + + 1) - 0.387 0.387 L = = 6.84 6.84 (L = = 4 4 and and n = = 38 here). here). The The d estimated 95% confidence m and estimated 95% confidence intervals intervals on on J..L ~rn and u;, {r2mare are ((- 00. . 11333, 3 , 0.020) and and 0.111), ((0.017, 0 . 0 1 7, 0. 1 1 1 ), respectively.
Metapopulation Viability Viability Metrics Metrics Metapopulation The The estimate estimate of of A ~ for for the the Snake Snake river river spring/summer spring/summer chinook chinook ESU ESU is is
�~ == exp(flm) exp(~m)== 0.94. To To the the extent extent that that long-term long-term trends trends continue, continue, the the expected population size 25 yr is 21% 21 % of = �25). point expected population size in 25 of current current levels ((= ~2s). The The point estimate drops below below 10% 1 0 % of current levels estimate of of the the probability probability of of that that the the ESU drops of current at any any time over the 25 yr yr is at time over the next next 25
(
)
log( 1 0/ 1 ) -- p,,m25) -log(10/1) fLm 25 0/1 ) fl m //~r &ht) + exp( e x p(--2Iog( 2 1 o g l( 1 0/1)~ _m 2) + V' u 6 2 25 � 25 m
(
)
- log( 1 0/ 1 ) + fLm 25 X {I)(-l~ + ~m25) X
(23.2 1) (23.21)
= .23 = 00.23 The corresponding corresponding estimate estimate of of 90% 90% decline decline over over the the next next 50 5 0 yr yr is 0.74. 0.74. The The The probability of of extinction was not not estimated, estimated, as this this requires requires an an estimate of the the probability extinction was estimate of total population returning spawners spawners is not total poputotal population size. The The number number of of returning not the the total popu lation size, as nonmature nonmature fish the ocean. ocean. However, However, if the the true true kA of of lation size, fish remain remain in the the metapopulation than 1.0, the metapopulation is less than 1 .0, the the population population will will eventually eventually go extinct. extinct. The posterior posterior probability probability density density functions functions [Eq. (16)] ( 1 6)] for for the the estimated estimated The metrics are are shown shown in Fig. 23.5. 23.5. The The posterior posterior probability probability distributions distributions give an an metrics indication of of the the degree degree to to which which data data support support different different risk risk levels. The The distridistri indication bution for for kA shows shows considerable considerable data data support support for for a kA < < 1, 1 , indicating indicating a declindeclin bution ing metapopulation. meta population. There There is also also strong strong data data support support for for aa high high risk risk of of 90% 90% ing 25 yr yr decline over over the the next next 50 50 yrs; yrs; however, however, the the estimate estimate of of 90% 90% decline decline over over 25 decline is very very uncertain. uncertain. The The mean mean value value is is 0.23, 0 .23, but but the the probability probability distribution distribution is is
23. VIABILITY FOR ENDANGERED METAPOPULATIONS VIABILITYANALYSIS ANALYSIS FOR
585 585 Puget Puget Sound Sound ESU ESU
Snake Snake River ESU
00
O
00.8 .8
11.0 .0
11.2 .2
.
.
0.8 0.8
.
.
.
11.0 .0
11.2 .2
Median lambda �
f :� :L >.,
c
I.D
"O ,m m
..0
.0 O Q.. a.. L O ;::
.o
I.(3
d
O
00 20 20
*oO
60 60
1100 00
0 0 20 20
60 60
1100 00
60
1100 00
Prob. of 90% 90% decline in 25 25 years
(D
a.. 13_
I.D
o0 20 20
60 60
1100 00
O
o0 20 20
Prob. of Prob. of 90% 90% decline decline in in 50 50 years years Fig. Fig. 23.5 2 3 . 5 Estimated posterior probability distributions for X. X and the probability of 90% 90% decline in 25 and 50 50 yr. Posterior Posterior probability distributions, which were calculated using using uniform and
very very broad broad over over the the 00 to to 1I range. range. This This illustrates illustrates that that uncertainty uncertainty in in estimates estimates of of probabilities of of quasiextinction quasiextinction can can vary vary widely widely depending depending on on the the time time frame frame over over which which one one is is interested. interested.
23.9 23.9
PUGET CHINOOK ESU P U G E T SOUND SOUND C HINOOK ESU The The Puget Puget Sound Sound ESU ESU is is aa subset subset of of the the major major chinook chinook salmon salmon group group in in Washington's Washington's northern northern coastal coastal basins basins and and Puget Puget Sound. Sound. The The ESU ESU (Fig. (Fig. 23.6) 23.6) includes includes all all spring, spring, summer, summer, and and fall fall runs runs in in the the Puget Puget Sound Sound region region from from the the north Nooksack river north fork fork Nooksack river to to the the Elwha Elwha river river on on the the Olympic Olympic peninsula peninsula (Myers (Myers et 998). The basins of et aI., al., 11998). The Elwha Elwha and and Dungeness Dungeness coastal coastal basins of the the Strait Strait of of Juan Juan de de Fuca, Fuca, Hood Hood Canal, Canal, and and the the Puget Puget Sound Sound area area north north to to the the northern northern Nooksack Nooksack river river basin basin and and the the U.S. U.S. Canadian Canadian border border are are all all aa part part of of the the Puget Puget Sound Sound ESU. ESU. Basin-to-basin . 1 and % based Basin-to-basin dispersal dispersal rates rates have have been been observed observed at at between between 00.1 and 66% based on 1 ). Fish in this this on recoveries recoveries of of tagged tagged juveniles juveniles returning returning as as adults adults (PSTRT, (PSTRT, 200 2001). Fish in typically mature mature at at ages ages 33 and and 4 4 and and are are coastally coastally oriented oriented during during the the ocean ocean ESU typically of their their life life history. history. The The Puget Puget Sound Sound ESU ESU does does not not include include Canadian or or phase of
E.E. HOLMES AND AND B.X. E.E. HOLMES B.X. SEMMENS SEMMENS
586 586
,. - Seattle Chinook Selmon ESUs
i;v,. I:' �o Scale:
--
Fig. 23.6 23.6
I
Map of the Puget Puget Sound Sound chinook ESU. ESU.
coastal coastal Washington Washington populations. populations. The The Puget Puget Sound Sound ESU was listed as as threatened threatened under 999. Trends under the the Endangered Endangered Species Act Act in in March March of of 11999. Trends in in abundance abundance throughout throughout the the ESU ESU are are predominantly predominantly downward, downward, with with several several populations exhibiting declines. Degraded exhibiting severe severe short-term short-term declines. Degraded spawning spawning and and rearing rearing habitats, habitats, as as well well as as access access restrictions restrictions to to spawning spawning grounds grounds and and migration migration routes, routes, have have all all likely likely contributed contributed to to population population declines. declines. Salmon Salmon in in this this ESU ESU do do not not migrate migrate through hydropower system through aa hydropower system as as the the Columbia Columbia river river ESUs ESUs do. do. Data Data for for this this ESU ESU consist consist of of yearly yearly estimates estimates of of the the total total returning returning spawn spawners ers (wild-plus (wild-plus hatchery-born) hatchery-born) to to the the 44 44 separate separate river river and and creek creek systems systems feed feeding ing into into the the Puget Puget Sound (Fig. (Fig. 23.6). 23.6). These These time time series series were were compiled compiled by by the the National Seattle, WA) National Marine Fisheries Fisheries Service Service ((Seattle, WA) based on on aa variety variety of of data: data: redd redd counts, counts, carcass carcass counts, counts, in-stream in-stream harvest harvest records, records, weir weir counts, counts, and and hatch hatchery counts. An ery return return counts. An independent independent metapopulation metapopulation level level count count was was not not avail available; unlike unlike spawners basin, spawners able; spawners returning returning to to the the Columbia Columbia river river basin, spawners here here do do not not pass pass through through aa hydropower hydropower system system where where they they can can be be enumerated. enumerated. 979-1997 index Instead, aa 11979-1997 index of of the the metapopulation was was constructed constructed by by adding adding together together the the 29 29 time time series series for for the the local populations populations with with data data over over the the
23. FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
587 587
11979-1997 979-1997 period. period. As As for for Snake Snake river river analyses, analyses, our our viability viability metrics metrics implicitly implicitly assumes that not been assumes that hatchery hatchery fish fish have have not been reproducing reproducing and and will will be be optimistic optimistic if if some instead spawn some hatchery hatchery fish fish do do not not return return to to the the hatchery hatchery and and instead spawn success successfully fully in in the the wild. wild.
Metapopulation Metapopulation Viability Viability Metrics Metrics Parameters Parameters were were estimated estimated as as for for the the Snake Snake river. river. The The parameter parameter estimates estimates are a r e tlm ~m = = 0.0036, 0.0036, and and fr;, ~2m = = 0.012. 0.012. The The estimate estimate of of A. k for for the the Puget Puget Sound Sound chi chinook ESU .003. The point estimate probability that nook ESU is is X. k = = exp(tlm) exp(Ikm) = = 11.003. The point estimate of of the the probability that the 0 % of the ESU ESU drops drops below below 110% of current current levels levels at at any any time time over over the the next next 25 25 yr yr is is 0.000 50 yr 1. 0.000 and and over over the the next next 50 yr is is 0.00 0.001. The The posterior posterior probability probability distributions distributions (Fig. (Fig. 23.5, 23.5, right) right) illustrate illustrate the the high high uncertainty, data, as declining, stable, uncertainty, given given the the data, as to to whether whether this this ESU ESU is is declining, stable, or or increasing. most that that can these data that there data increasing. The The most can be be said said from from these data is is that there is is low low data (A. < increasing (k (A. > support severely declining declining (k support for for aa severely < 0.9) 0.9) or or increasing > 1.1) 1.1) metapopula metapopulation. tion. Interestingly, Interestingly, the the low low support support for for small small A. X values values translates translates into into high high data data support yr). Over support for for aa low low risk risk of of 90% 90% decline decline in in the the short short term term (over (over 25 25 yr). Over the the longer population is longer term, term, however, however, the the uncertainty uncertainty as as to to whether whether the the meta metapopulation is declining declining or or increasing increasing gives gives rise rise to to aa U-shaped U-shaped distribution, distribution, meaning meaning that that data data give 1, reflecting give the the most most support support to to aa probability probability of of 00 or or 1, reflecting that that A. k could could be be either either less than .0. This less than or or greater greater than than 11.0. This example example illustrates illustrates that that while while data data may may be be ? ", data equivocal questions of conservation concern, "is A. equivocal on on some some questions of conservation concern, such such as as "is k< < 11?", data may questions, such "is the may still still give give information information on on other other questions, such as as "is the short-term short-term risk risk of of severe severe decline decline high? high?""
23.1 23.100
USING USING THE THE STOCHASTIC STOCHASTIC METAPOPULATION METAPOPULATION MODEL MODEL TO TO INVESTIGATE EFFECTS OF OF MANAGEMENT INVESTIGATE EFFECTS MANAGEMENT Determining Determining how how to to distribute distribute effort effort in in order order to to recover recover an an at-risk at-risk species species is is aa routine, routine, and and challenging, challenging, task task of of conservation conservation managers. managers. For For salmon, salmon, man management agement actions actions tend tend as as aa generality generality to to affect affect an an entire entire ESU ESU or or multiple multiple ESUs ESUs or reductions or to to affect affect individual individual stocks. stocks. Management Management actions actions such such as as harvest harvest reductions or increases to between spawning or increases to survival survival during during migration migration ((between spawning areas areas and and the the ocean) improvements to estuarine environments examples of ocean) or or improvements to estuarine environments are are examples of actions actions that that will will tend tend to to improve improve conditions conditions for for all all stocks stocks within within an an ESU ESU or or multiple multiple ESUs. ESUs. Habitat Habitat improvements improvements or or protections protections that that affect affect spawning spawning areas areas and and management of of in-stream in-stream water water levels levels are are examples examples of of actions actions that that tend tend to to management affect affect individual individual stocks. stocks. Without Without knowing knowing the the local local stock stock dynamics dynamics or or disper dispersal about how sal rates, rates, one one can can still still give give certain certain types types of of guidance guidance about how much much effort effort is is required population and required for for recovery recovery of of aa declining declining meta metapopulation and about about how how effort effort should all local should be be distributed distributed across across all local populations. populations.
Metapopulation Level Metapopulation Level Actions Actions When local populations When management management actions actions affect affect all all local populations roughly roughly equally, equally, it can be how change the metapopulation it can be estimated estimated how change would would change change the metapopulation A.. k. Mathematically, Mathematically, this this means means that that all all JLi ~i values values increase increase by by some some dJL. d~.
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
588 588
An An absolute absolute dfL d~ change change in in all all fL ~ij values values is is equivalent equivalent to to multiplying multiplying all all ele ele(d fL) . The ments ments in in A(t) A(t) by by aa constant constant= - exp exp(d~). The mean mean of of the the distribution distribution of of 10gM(t)/M(0) dfL)t == (logA. g(A.new/A.old) == df logM(t)/M(O) becomes becomes (fLm (~m + d- dl~)t (logknew)t. log(knew/kold) d~.L . new)t. Thus 10 The can be into currency is more The change, change, df dp,L , can be translated translated into currency that that is more meaningful meaningful from aa management standpoint by by using the the relationship relationship A. k = = RO Ro Il/x, IT, from management standpoint between reproductive rate, Ro, and and the between A., k, the the net net reproductive rate, R0, the mean mean generation generation time, time, T T (Caswell, 1 ) . This This is is illustrated here for and hydropower (Caswell, 200 2001). illustrated here for harvest harvest and hydropower effects effects on ESU (d. on salmon salmon in in the the Snake Snake river river spring/summer spring/summer chinook chinook ESU (cf. McClure McClure et et a!., al., 2003). 2003). Harvest
In rates for In the the Pacific Pacific Northwest, Northwest, harvest harvest rates for salmon salmon are are generally generally expressed expressed in in terms terms of of the the fraction fraction of of spawners spawners that that did did not not return return to to the the spawning spawning grounds grounds but but that that would would have have without without harvest, harvest, e.g., e.g., aa harvest harvest rate rate of of 0.8 0.8 indi indiwould cates number of cates that that the the actual actual number of returning returning spawners spawners is is 20% 20% of of what what it it would have Harvest rates expressed in have been been if if there there had had been been no no harvest. harvest. Harvest rates are are expressed in this this way way so so that that harvest harvest that that occurs occurs in in the the stream stream versus versus in in the the ocean ocean can can be be compared compared via via aa common common currency. currency. We We can can write write the the net net reproductive reproductive rate rate using using fecund fecund1 ) as as ity ity and and age-specific age-specific survival survival (d. (cf. Caswell Caswell 200 2001)
Ro 1 F1( 1 Ro = = ss1F1(1 - h)f h)f + ss1(1 - F1)S2F2( F1)s2F2(11 - h)f h)f l( l -
+ (1 )S2(1 (1 + SI s1(1 - F1 F1)s2(1 - F2)S3F3 F2)s3F3(1 - h)f h ) f .·. .
.
(23.22) (23.22)
where where h h is is the the harvest harvest rate, rate, S$ij is is the the survival survival from from age age i - I1 to to i, Fi F i is is the the frac fraction tion of of spawners spawners that that return return at at age age i, and and f f is is the the mean mean offspring offspring per per spawner. spawner. Using Eq. Eq. (22), change in in h alone is Using (22), the the change in A. k from from aa change change in h alone is
) (- ) -
R o,new lI1/TT --_ (11 - hhnew~ A.new )knew = ((RO,new) new lI1/TT = = RO,old A.~kold 1-1 hhold R old ] old O,old
(23.23) (23.23)
Hydropower Hydropower
Juvenile river basin migrate through Juvenile salmon salmon from from the the Snake Snake river basin must must migrate through the the mainstem of the the Snake Snake river, river, enter enter the the Columbia Columbia river, river, and and descend down down the the mainstem of Columbia ourney to ocean. This Columbia river river on on their their jjourney to the the ocean. This migration, migration, and and the the return return migration migration of of spawning spawning adults, adults, involves involves passage passage through through four four large large hydropower hydropower dams dams on on the the Columbia Columbia river river and and four four Snake Snake river river hydropower hydropower dams. Improving Improving the uvenile and adult fish dams. the survival survival of of both both jjuvenile and adult fish migrating migrating through river hydropower hydropower systems systems has has been through the the Columbia Columbia and and Snake Snake river been the the focus much effort one of impacts that has been been relatively focus of of much effort and and is is one of the the human human impacts that has relatively well well quantified. quantified. Following Following aa strategy strategy similar similar to to that that used used for for harvest, harvest, the the effect effect of of changes changes in in survival survival through through the the hydropower hydropower system system on on the the rate rate of of decline decline at at the the ESU ESU u the level level can can be be estimated. estimated. Denoting Denoting by by Ccdd and and CCu the proportional proportional increase increase in in down- and upstream passage improvement in hydropower downand upstream passage survival survival due due to to improvement in the the hydropower system, improved net is system, the the improved net reproductive reproductive rate rate is
Ro,new- CdCu(SlFlf + S1(1 - F1)s2F2f + Sl (1 - F1)s2(1 - F2)s3F3f ...).
(23.24)
23. 23.
VIABILITY ANALYSIS ANALYSIS FOR FOR ENDANGERED ENDANGERED METAPOPULATIONS METAPOPULATIONS VIABILITY
5589 89
Thus, for for assessing assessing the the impacts impacts of of increased increased survival survival through through the the hydropower hydropower Thus, system: system:
Ra,new )l1/T Anew (( RO,new~ ~knew /T ( C Cu ) l T / d 1/T Xold -\ RO,old fl = (CdCu) Aold R ,old =
=
a
(23.25) (23.25)
Estimates of of the the Impacts Impacts of of Harvest Harvest and and Hydropower Hydropower Changes Changes to to Estimates the Snake Snake River ESU the
The mean ocean and in-river in-river 1980-1999 1980-1999 harvest harvest rate the Snake Snake river river The mean ocean and rate for for the spring/summer chinook chinook ESU ESU was was hh = = 0.08 0.08 (McClure (McClure et et al., aI., 2003). 2003). By By setting setting spring/summer hneww == 0, 0, we we can can examine examine the the effect of successful successful selective selective harvest harvest managemanage hne effect of ment that that would would substantially substantially eliminate eliminate harvest harvest impacts impacts on on salmon salmon in in this this ment (23) and and a mean mean generation generation time time of of 4.5 4.5 yr, the estimated estimated Using Eq. Eq. (23) ESU. Using yr, the roughly 2%. 2 % . NMFS NMFS (2000) (2000) has has required required that that increase increase in in A X with with h hne neww =- 00 isis roughly agencies operating the the Federal Federal Columbia Columbia river river power power system implement aa varvar agencies operating system implement iety of activities, including including increased increased spill, spill, improved improved passage passage facilities, facilities, and and iety of activities, increased salmon around as a means of improving sursur increased barging barging of of salmon around the the dams dams as a means of improving vival through the system. improvement in in passage passage survival survival from vival through the system. The The estimated estimated improvement from the improvements proposed by are on the order i.e., CCsCu the improvements proposed by NMFS NMFS are on the order of of 55-6% - 6 % ((i.e., dCu = 1 .05-1.06) for (McClure et aI., 2003). 1.05-1.06) for the the Snake river spring/summer spring/summer chinook chinook (McClure al., 2003). This translates into % improvement improvement in A for for this this particular ESU using using This translates into aa 11% in ~ particular ESU Eq. if the the combined effects of of substantially substantially reduced reduced harvest harvest and and the Eq. (25). (25). Thus Thus if combined effects the % increase increase in in A proposed passage passage improvements improvements are additive, then then roughly roughly aa 33 % proposed are additive, is less than % increase is estimated estimated for for these these actions. actions. If If the the true true A ~ is is less than 0.97, 0.97, aa 33% increase would would not not be be sufficient sufficient to to achieve achieve A X> > 11.. Figure Figure 23.5 23.5 indicates indicates that that data data can cannot not rule rule out out that that the the A ~ in in this this is is ESU ESU is is greater greater than than 0.97, 0.97, but but data data certainly certainly give more more support support to to aa lower lower A. k. This This suggests suggests that that other other recovery recovery actions, actions, such such give as also be as improvements improvements at at the the stock stock level, level, will will also be necessary. necessary. =
Local Local Population Population Level Level Actions Actions The The effects effects of of changes changes to to individual individual units units of of habitat habitat are are harder harder to to quan quantify tify than than the the effects effects of of metapopulaion metapopulaion level level changes. changes. The The change change in in A k achieved achieved by by aa change change at at the the level level of of aa specific specific unit unit of of habitat habitat depends depends on on the the level of that habitat level of dispersal, dispersal, the the spatial spatial pattern pattern of of dispersal, dispersal, whether whether that habitat is is connected connected to to source source or or sink sink habitat, habitat, the the level level and and pattern pattern of of synchrony synchrony between between sites, sites, and and so so on. on. In In other other words, words, it it depends depends on on the the type type of of detailed detailed information information that that has has traditionally traditionally been been difficult difficult to to obtain obtain for for metapopula metapopulations tions of of conservation conservation concern. concern. Interestingly, Interestingly, although although it it is is difficult difficult to to deter determine mine how how much much change change in in A X can can be be achieved, achieved, it it appears appears possible possible to to estimate estimate where where the the largest largest dA dk from from aa given given dfL dp~ change change (per (per unit unit of of habitat) habitat) in in the the local local growth growth rate rate is is achieved, achieved, even even though though the the size size of of the the resultant resultant dA dX can cannot not be be determined. determined. Recall Recall that that each each row row of of A A represents represents aa unit unit of of habitat habitat and and that that aa local local popu population lation is is composed composed of of some some set set of of units units of of habitat habitat with with high high connectivity. connectivity. When When the the intrinsic intrinsic growth growth rate, rate, fL p~j, in aa unit unit of of habitat habitat jj is is changed changed by by dfL, d~, to to j' in
590 $90
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
exp(J.L exp(l~ij + + dJ.L), d~), all all the the aij aq elements elements of of column column jj in in matrix matrix A(t) are are multiplied multiplied by by exp(dJ.L). exp(d~). The The goal goal is is to to calculate calculate the the total total change change in in A k from from this this dJ.L d~ change change to to all elements in column column j by summing summing over over rows rows i:
d k = z~. a~~iidP = ~ i I a logk db~ 9
9 k 3log
aq
(23.26) (23.26)
where A/dlogaij is where aij aq is is the the value in in row row i column jj in in matrix matrix A. The The term term dlog 31ogk/31ogaii is the the elasticity elasticity of of A. X. Chapter Chapter 14 in in Caswell Caswell (2001) (2001) presented presented the the calculation calculation for for X for for products products of stochastic matrices: the elasticity of A a log
A
n-l
Olog k = hm lim 11 n-1 � t-+oon L.aOlog log aij =O aq t--~-~ tt~o .
=
aqvi(t + 1)wj(t) R(( tt)vT( R )vT( t + 11 )w( t + 1 )
(23.27) (23.27)
R(t) is the relationship relationship between between the right eigenvector eigenvector and and A(t), 1) == A(t)w(t). R(t)w(t R(t)w(t + 1) A(t)w(t). Thus, Thus, the the dA dX from from aa dJ.L dl~ change change in in aa unit unit of of habitat habitat jj can can be be solved solved for for by by summing Eq. Eq. (27) (27) over over i:
where where
�
ddb~ J.L lim � n- l aijVi(t )wj(t) a,vi(t + + l1)wj(t) lim 1 n~l ~'~ n A �OO tt=O � R(t) dkj = k n--~oo-~ R(t)vT( + 11)) vT(tt + l1)w(t )w(t +
(23.28) (23.28)
� lw d J.L )A(t) (t + [dkl dA dk2.. dXk] = dp. lim lim 1 nn~l t ) o vT vT(t + l1)A(t) i w lT( (t) [dAl k] = 2 . . . dA n A �OO vT(t-t ++ i)w(t " ;k n--~oo ~- tt ==0O R(t) R(t-~T( 1) l )w(t + 1)
(23.29) (23.29)
d Aj
=
n
This This can be translated translated into into matrix matrix form form for for all units of habitat habitat 11 to K: 0
n
where 0 " denotes where ",,o,, denotes the the scalar scalar product. product. Using Using the the relationship relationship between between the the left left eigenvector and and A(t), Q Q(t(t + l1)v + l1)A(t), eigenvector )vTT (t) - vTv (t(t + )A(t), =
[dkl
dk2.. dkk] d' 1\.2 • • • d' I\.k ] "
nL.(t) (23.30) )wT (t) 0o vT �l QQ(t(t ++ l1)wT(t) = dJ.L d__~_~l'lim vT(t) (23.30) 1m 11 n~l A)k nn--)oo~�oo n tt -=OO R(t) vT(tt + -l~w-(t R(t)vT( l )w(t +-1i + 1)
=
-
The The denominator denominator reduces reduces to to aa constant constant that that depends depends on on t but but not not j. j. Thus Thus dA dk from from aa change change in in aa unit of of habitat habitat jj is is aa weighted weighted temporal temporal average average of of the the reproductive reproductive value of local population population j times its density: d Aj dkj
1l nn- --l1
= ;; n ~2: c( t)Wj( t )wj( t)Vj( t )vj( t) t) t= t oO
=
(23.31)) (23.31
where where c(t) is is aa constant constant that that depends depends on on t but but not not j. j. A A local local population population a is is composed , a3, . . . , am), where lab a2, composed of of units units of of habitat habitat in in the the set set a = = lab {al, aa2, a 3 , . . . , am}, where {al, 2 a3, .99. .9, am) am} denotes denotes which which rows rows of of A corresponding corresponding to to the the units units of of habitat habitat in in local population population a. The The dA dX per dJ.L d~ per per unit unit habitat habitat for for a particular particular local popupopulation lation a is is
dAa 1 1m ) 2: dka = = ((l/m) ~ jj~a dkj, where where m m is is the the number number of of units units of of habitat habitat E a dAj,
in in local local population population a. In In words, words, this this means means the the change change in in A k is is proportional proportional to average" density to the the product product of of the the ""average" density of of individuals individuals in in aa particular particular local local population population times times the the "average "average"" reproductive reproductive value value of of its its units units of of habitat. habitat. unknown, there many cases where Although reproductive reproductive values are are unknown, there are are many where the product VjWj the product viw i is is aa positive positive function function of of Vj v i as as long long as as dispersal dispersal is is not not too too
591 591
23. FOR ENDANGERED ETAPOPULATIONS 23. VIABILl1Y VIABILITYANALYSIS ANALYSIS FOR ENDANGERED M METAPOPULATIONS
unidirectional unidirectional (meaning, dispersal dispersal from from A to to B but but not not B to to A). This This can can be shown analytically in three three extreme extreme cases: ((a) uniform and and equal equal shown a ) 1100% 00 % uniform dispersal, b ) all J.I~j .lj values equal, a) dispersal, ((b) equal, or or (c) dispersal dispersal extremely extremely low. In cases ((a) and b ) , the l l equal and ((b), the reproductive reproductive values are aall equal and and VjWj vjwj = ((aa constant) constant) X 3< Vj' vj. In case (c), Wj wj '" = Vj vj and and VjWj vjwj '" = (vy. (vj)2. However, However, this positive relationship relationship was was also also found found in in simulations simulations with with variable variable local local growth growth rates, rates, neighborhood neighborhood dispersal, dispersal, and and dispersal dispersal sources sources and and targets. targets. An obvious obvious exception exception to this positive relationship relationship is if dispersal dispersal is unidirectional, unidirectional, for for example, example, a linear linear chain chain of of local populations populations with with dispersal dispersal via a steady steady directional directional wind wind or or ocean ocean current. current. However, However, as the the following following simulations simulations illustrate, illustrate, the the general general relationship can not relationship can still still hold hold even even when when dispersal dispersal is is strongly, strongly, although although not strictly, directional. directional. =
Density Density and and A. X Sensitivity Sensitivity Three different population models were used to different types of meta metapopulation to look look at the relationship relationship between average densities in units of of habitat habitat versus the dx. dk from from a small increase in the local growth rate in each unit of of habitat. habitat. In each model, model, dispersal dispersal was was nonuniform nonuniform among among the the local local populations populations so so that that some some sites sites were were dispersal sources (more dispersal out than than in) and and others dispersal dispersal targets (more dispersal in in than than out). out). In In the the first first model, local growth growth rates rates were were equal equal among among all all sites sites and and dispersers dispersers were were distributed distributed globally globally among among all all sites. sites. In In the the second second model, model, local local growth growth rates rates were were variable so so that that some some sites sites had had much much higher higher local growth growth rates rates than than others others and and dispersal dispersal was was mainly mainly to to nearest nearest neighbors. neighbors. In In the the third third model, model, local local growth growth rates rates were were again again variable variable and and dis dispersal persal mainly mainly to to the the two two south south and and east east neighbors; neighbors; however, however, aa small small propor proportion tion of of dispersers dispersers were were distributed distributed globally. globally. Thus Thus the the three three examples examples illustrate illustrate global, local, and and directional directional dispersal. A A hundred hundred randomly randomly generated generated matrix matrix models models in in each each of of these these three three cat categories egories were were made made and and dX.j dki calculated calculated via via Eq. Eq. (30). (30). Figure Figure 23.7 23.7 shows shows the the rela relationship tionship between between the the average average density density of of aa local local population population and and the the dx. dk from from increasing the local growth growth rate rate in that that unit unit of habitat. habitat. The The x x axis ranks ranks the increasing dX.j, 1 " indicates the local population with dkj, thus thus ""1" with the the highest dX.j dkj in any simu simulation lation and and "49" "49" the the lowest. lowest. The The yy axis axis shows shows the the corresponding corresponding mean mean density density rank 1 " indicates rank of of that that local local population; population; ""1" indicates the the population had had the the highest highest density among among the 49 49 sites and and "49" "49" the lowest. Results from from the 100 100 ran randomly domly generated generated models models are are summarized summarized by by showing showing aa box box plot, plot, which which shows shows the ldJ..l the median median and and range range of of all all density density ranks ranks for for the the sites sites with with aa given given dX.j dkj/d~ rank. Thus, 1 " shows Thus, the box box plot plot at the x x axis position position ""1" shows the range of density ranks habitat with .l in ranks for for the the units units of of habitat with the the highest highest dX.ldJ. dk/dp~ in each each model. model. Model Model results show show a strong strong positive relationship relationship between the relative density rank rank within within a unit unit of habitat habitat and and which which unit unit of habitat habitat produced produced the largest increases in the meta population X. metapopulation k for for a given dJ..l d~.. The The two two to to three units units of habitat habitat with with the highest average densities were were consistently the units that that pro produced .l. This duced the the largest largest dx. dk for for aa given given dJ. d~. This suggests suggests that that plotting plotting the the distribution distribution of the relative densities population could densities within within local populations populations in a meta metapopulation give give aa rapid rapid indication indication of of the the sensitivity sensitivity of of the the metapopulation metapopulation to to changes changes to to individual local populations. populations.
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
592 592
Uniform Uniform Local Local Growth Growth Rates Rates Dispersal Dispersal Sources Sources and and Targets Targets with with Global Global Dispersal Dispersal
f? o . $ '
11 55 ~ · 9 110 0 115 5 20 20 25 25 30 30 35 35 40 40 45 45
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Rank Rank of of Influence Influence of of Habitat Habitat Unit Unit on on A
Variable Variable Local Local Growth Growth Rates Rates Dispersal Dispersal Sources Sources and and Targets Targets with with Neighborhood Neighborhood Dispersal Dispersal < � Cl:i
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20 25 20 25 30 35 30 35 Unit on on X, A Rank of of Influence Influence of of Habitat Habitat Unit Rank
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Variable Local Growth Rates Variable Local Growth Rates Dis 3ersal Sources Dispersal Targets Dispersal Sources and and Tarc ets with with Directional Directional Dispersal 1 1 5 5 1100 5 115 2O 20 25 25 3O 30
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Fig. 3.1 Fig. 223.7 Relationship between between the the influence influence of of aa given given habitat habitat unit unit on on the the metapopulation metapopulation k" Relationship and the the average average density density in in that that habitat habitat unit. unit. One One hundred hundred 77 •x 77 metapopulations meta populations with with spatially spatially and variable rates (some targets) were variable dispersal dispersal rates (some sites sites dispersal dispersal sinks sinks and and others others targets) were generated generated randomly randomly in in each of of three three classes: classes: (1) (1 ) spatially spatially uniform uniform growth growth rates and global global dispersal, dispersal, (2) (2) spatially spatially variable variable each rates and growth with neighborhood neighborhood dispersal, dispersal, and and (3) (3) spatially spatially variable variable growth with directional directional growth rates rates with growth rates rates with neighborhood dispersal dispersal to to the the SS and and EE two two neighbors neighbors only. only. The The xx axis axis shows shows the the rank rank in in terms terms of of neighborhood cfA/dj.L, and and the the yy axis axis shows shows aa box box plot plot of of the the distribution distribution of of density density ranks ranks for for sites sites with with aa given given clk/d~, cfA/dj.L rank across all 100 1 00 models models in in each each class. class. Thus Thus the the box box plot plot at at xx == 11 shows shows the the distribution distribution dk/dp, rank across all of ranks ranks for for the the sites sites with with the the highest highest dX/dl~ cfA/dj.L in in each each model. model. The The line line in in each each box box shows shows the the median median of density rank rank for for the the sites sites with with aa given given dX/d~ cfA/dj.L rank, rank, the the box box encloses encloses 50% 50% of of the the ranks, ranks, and and the the density whiskers whiskers show show the the range range from from all all 100 1 00 randomly randomly generated generated models. models.
23. FOR ENDANGERED 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS METAPOPULATIONS
593
One One application application of this would would be to estimate where where negative impacts would would lead lead to to the the greatest greatest decrease decrease in in 'A, k, thus thus suggesting suggesting where where protection protection in in most most critical. critical. It would would also suggest suggest where where local improvements would would be most effect effective for a given increase in the local growth rate. rate. However, in actual manage management ment situations situations where where improvements improvements are are being being sited, sited, one one is is generally generally trying trying to to maximize bang per maximize the the ""bang per buck," d'A/d$ dkJd$ = d'A/dlLj dkj/dl~j X • dlL/d$. dl~Jd$. The The cost, cost, d$, as, is is the the actual monetary monetary cost or some combination combination of monetary, logistical, logistical, and and polit political ical costs costs and and dlL/d$ dlxJd$ is is the the cost cost of of aa unit unit improvement improvement to to aa unit unit of of habitat habitat j. j. Thus Thus d'A/dlL dkj/dl~ is is one one part part of of the the equation, equation, and and the the other other part, part, the the cost cost of of aa unit unit improvement in different different habitats, would would have to come from from a specific analy analysis of the costs and estimated estimated effects of of management management actions on different different local populations. populations. =
Example Example Using the the Snake Snake River River ESU
The The overall level of salmon salmon dispersal dispersal between between and and among among stocks stocks within within this ESU is known known to be fairly low low and and spatially localized (Mathews (Mathews and and Waples, 11991; Quinn, 11993). addition, there there is high variability in the the Waples, 99 1 ; Quinn, 99 3 ) . In addition, habitat habitat quality between between stocks, stocks, with with some stocks relatively relatively pristine and and pro protected tected within within wilderness wilderness areas, areas, whereas whereas others others are are exposed exposed to to high high and and mul multiple tiple impacts impacts (such (such as as stream stream degradation degradation and and disturbance, disturbance, pollution, pollution, in-stream in-stream harvest, harvest, and and irrigation irrigation impacts). Figure 23.8 23.8 (top) shows shows the the distribution distribution of of average average normed normed redds redds per per mile mile for for 50 50 Snake Snake river river spring! spring/ summer 9 80 and 995, the redds summer chinook chinook stocks. stocks. For For each year between between 11980 and 11995, reddsper-mile count count for for each stock stock was was divided by the the maximum maximum count count among among the the 50 stocks stocks in in that that year. year. The The average average over over the the 1166 yr yr was was then then used used as as an an estimate estimate of of the average normed normed redds redds per mile. The The long-tailed distribution distribution is the expectation expectation from from theory theory given given low low dispersal dispersal and and high high variability variability in in stock stock habitat habitat quality. Estimation of the average normed normed redds redds per per mile was was repeated repeated using a var variety of different different time periods. Regardless of the time period period or or number number of of years used used for for averaging, averaging, six six stocks stocks consistently consistently appeared appeared among among the the top top five five stocks stocks with Johnson Cr., with the the highest highest density density of of redds: redds: Johnson Cr., Poverty Poverty Cr., Cr., and and Secesh Secesh R. R. in in the subbasin, the south south fork fork of of the the Salmon Salmon R., R., the the Lostine Lostine R. R. in in the the Grande Grande Ronde Ronde subbasin, Marsh Marsh Cr. Cr. in in the the middle middle fork fork of of the the Salmon Salmon R., and and the the Imnaha Imnaha R. R. Perhaps Perhaps not not surprisingly, surprisingly, all all of of these these are are in in relatively relatively low low impacted impacted regions regions of of the the ESU. ESU. At At aa subbasin level, south fork fork of level, the the overall overall highest highest redd redd density density was was in the the south of the the Salmon river where where summer-run summer-run chinook chinook primarily occur occur (Fig. 23.8, 23.8, bottom). bottom). The The other other regions regions are are primarily primarily spring-run spring-run chinook. chinook. The The south south fork fork of of the the Salmon river is relatively pristine and and few hatchery hatchery fish have been released into into this this subbasin; subbasin; the the stocks stocks presumably have have experienced experienced relatively relatively low low inter interbreeding breeding with with hatchery-reared hatchery-reared stocks. In In addition, addition, the the later later run run timing timing may may somehow somehow be be associated associated with with less less straying, straying, lower lower harvest, harvest, or or lower lower hydropower hydropower impacts. This This analysis analysis predicts predicts that that the the 'Ak of of the the Snake river river spring/summer spring/summer chinook chinook ESU would would be most most sensitive to to changes changes to to the the summer-run summer-run stocks stocks in the the south south fork fork of of the the Salmon Salmon river river and and to to the the spring-run spring-run stocks, stocks, the the Lostine Lostine R., Imnaha Imnaha River, River, and and Marsh Marsh Creek Creek and and should should be be protected protected preferentially preferentially from from impacts. imag impacts. This This can can be be counterintuitive counterintuitive in in some situations. situations. For example, example, imagine ine making choices about about where where to to allow allow aa limited limited catch-and-release catch-and-release sport sport
E.E. E.E. HOLMES HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
5594 94
Snake River sprlsum chinook
25 '" ::t:. U
.8
'"
'0 Q; .0 E � c
0.8 0.6 average normed rpm 1 980: 1 995 0.7 E e-
" Q)
0.6 0.5
E 0.4 o c � 0.3 o .� 0.2 0.1 o
GR
S FS
MFS
us
c
Fig. 23.8 23.8 Distribution Distribution of of densities densities of of redds redds in in the the Snake Snake river river spring/summer spring/summer chinook chinook ESU ESU at at aa stock stock and and subbasin subbasin level. level. The The average average normed normed redds redds densities densities (top) (top) are are shown shown for for the the 50 50 stocks with 980-1 995 data (the with 11980-1995 (the years were chosen chosen to maximum maximum the number number of of stocks with data). 6 yr data). For For each each stock stock the the normed normed redd redd density density was was averaged averaged over over the the 116 yr to to get get an an estimate estimate of the normed average density. In the lower plot, relative average densities over all stocks within within different different basins basins are are shown shown (with (with the the number number of of stocks stocks in in each each basin basin shown shown above above the the bars). bars). The The basin designations are GR, Grande Ronde; I, Imnaha; SFS, SFS,south fork salmon; MFS, MFS, middle fork salmon; salmon; US, US, upper upper salmon; salmon; C, C, Clearwater. Clearwater. Redds Redds due due to to hatchery hatchery fish fish released released into into stocks stocks were were removed removed before before doing doing these these analyses, analyses, as as the the density density will will be be artificially artificially high high simply simply due due to to hatch hatchery could not ery fish fish releases. releases. This This correction correction could not be be done done for for the the upper upper salmon salmon or or Clearwater Clearwater regions regions because the fraction of of spawners that are hatchery strays were were unknown; however, the hatch hatchery ery releases releases are are very very high high in in these these basins basins and and thus thus the the corrected corrected relative relative densities densities would would be be much much lower lower than than shown. shown.
fishery. Sites with the highest density would seem to be the prime prime candidates, whereas the analysis of d"AldJL dk/d~t indicates just the opposite. opposite. In terms of deter determining where to direct improvements, the d"A/dJL dk/d~ suggests that these pristine sites are where a given dJL d~ would produce the greatest metapopulation metapopulation "A; k; however, the regions dk/d~ is the highest highest are not not necessarily the regions where where d"A/dJL regions where f.J., i~ is improved improved most easily. Indeed a given given unit of improvement improvement regions may be more difficult in pristine sites. Choosing Choosing where to direct stock improvements improvements requires consideration consideration of the cost and difficulty of a given df.J., d~ for different stocks in combination combination with the estimate of the sensitivity of "A to to local local changes. changes.
VIABILITYANALYSIS ANALYSIS FOR ENDANGERED METAPOPULATIONS 23. VIABILITY FOR ENDANGERED
23. 11 23.11
595 595
POPULATION IN PRACTICE POPULATION VIABILITY VIABILITY ANALYSIS ANALYSIS IN PRACTICE The The purpose purpose of of this this chapter chapter is is to to present present aa theoretical theoretical framework framework for for metapopulation metapopulation PYA PVA using using time time series series data data and and diffusion diffusion approximations. approximations. These These methods populations. The methods are are then then illustrated illustrated using using data data from from two two salmon salmon meta metapopulations. The salmon salmon analyses analyses are are intended intended as as an an example example of of how how to to calculate calculate the the diffusion diffusion parameters parameters and and metrics. metrics. An An actual actual PYA PVA must must grapple grapple with with other other important important issues issues that that are are outside outside the the scope scope of of this this chapter, chapter, but but which which anyone anyone contemplat contemplating ing an an actual actual PYA PVA must must be be aware. aware. Morris Morris and and Doak (2002) (2002) gave gave aa review review of of the the criticisms criticisms and and caveats caveats surrounding surrounding the the use use of of PYA PVA and and outlined outlined general general recom recommendations mendations and and cautions when when conducting conducting aa PYA. PVA. In In the the context context of of diffusion diffusion approximation approximation methods methods in in particular, particular, Holmes (2004) (2004) outlined outlined an an approach approach using using matrix matrix models models to to conduct conduct sensitivity sensitivity analyses analyses in in order order to to choose choose among among different parameterization methods different parameterization methods and and metrics metrics for for aa specific specific PYA PVA application. application. One One of of the the issues issues that that is is especially especially pertinent pertinent for for our our chapter chapter is is the the issue issue of of variability variability in in estimated estimated risk risk metrics. metrics. A A number number of of recent recent PYA PVA cross-validations cross-validations using using actual actual data data on on aa large large number number of of different different populations populations have have shown shown that that careful careful PYA PVA analyses analyses give give unbiased unbiased risk risk estimates estimates (Brook (Brook et et aI., al., 2000; 2000; Holmes Holmes and and Fagan, Fagan, 2002; 2002; Fagan Fagan et et aI., al., 2003). 2003). Although Although this this is is very very encouraging, encouraging, aa dif difficult ficult issue issue is is the the high high inherent variability variability associated associated with with estimated estimated probabil probabilities ities (such (such as as probability probability of of extinction), extinction), even even though though they they may may be be unbiased unbiased (Ludwig, 996, 11999; 999; Fieberg (Ludwig, 11996, Fieberg and and Ellner, Ellner, 2000; 2000; Holmes, Holmes, 2001 2001;; Ellner Ellner et et aI., al., 2002). 2002). How How to to properly properly use use risk risk metrics metrics that that have have high high variability variability is is currently currently being don't use being debated debated within within the the field field with with arguments arguments ranging ranging from from ""don't use them" them" (Ludwig, 996, 11999; 999; Fieberg use to (Ludwig, 11996, Fieberg and and Ellner, Ellner, 2000), 2000), to to ""use to estimate estimate risks risks within within collections collections of of populations" populations" (Fagan (Fagan et et aI., al., 2001 2001;; Holmes Holmes and and Fagan, Fagan, 2002), 2002), to to ""use use where Coulson et aI., 2001 where data data are are extensive extensive and and high high quality" quality" ((Coulson et al., 2001),), to to "PYA "PVA metrics metrics based based on on data, data, even even if if variable, variable, are are better better than than the the alternatives" alternatives" (Brook (Brook et et aI., al., 2002). 2002). An An encouraging encouraging aspect aspect of of diffusion diffusion approximation approximation methods methods is is that that cross-validations cross-validations using using real real time time series series data data have have indicated indicated that that the the uncertainty uncertainty in the estimated metrics appears to be characterized properly (Holmes and Fagan, Fagan, 2002). 2002). Nonetheless, Nonetheless, how how to to use use and and present present metrics metrics with with high high variability, variability, albeit albeit well well characterized, characterized, is not not an easy easy question question to answer. answer. Presentation Presentation of of 1100(1-0~)% 00( 1 - a ) % is is an an oft-used oft-used approach, approach, but but experience experience in in the the forum forum of of salmon salmon recovery recovery planning planning in in the the Pacific Pacific Northwest Northwest has has shown shown that that it it is is easy easy to to misin misinterpret % confidence terpret confidence confidence intervals. intervals. For For example, example, it it is is easy easy to to interpret interpret 95 95% confidence .0 as intervals intervals for for A ~ that that overlap 11.0 as an an indication indication that that data data are are equivocal as as to to whether whether the the population population is is declining or or increasing, whereas there there may may be be consid considerable erable data data support support for for aa declining declining population. Graphic Graphic presentations presentations of of data data support support for for different different risk risk levels levels have have been been more more compelling compelling and and informative, informative, although although translating translating levels levels of of data data support support into into numbers that that policy policy makers makers can use use to to take take uncertainty uncertainty into into account account in in policy decisions has been been challenging. challenging.
23. 2 3 . 1122
DISCUSSION DISCUSSION This This chapter chapter focused focused on on the the calculation calculation of of metapopulation metapopulation PYA PVA metrics; metrics; how however, ever, there there are are other other more more general general PYA PVA insights insights from from an an examination examination of of stochastic stochastic meta populations and and of this specific metapopulations of this specific class class of of declining declining density-independent density-independent
596 .$96
E.E. HOLMES E.E. HOLMES AND AND B.X. B.X. SEMMENS SEMMENS
metapopulations. pop metapopulations. First, by definition the the trajectory trajectory of a stochastic stochastic meta metapoppopulation trajectory ulation is is subject subject to to random random processes processes and and thus thus the the meta metapopulation trajectory observed in any one snippet of time is unlikely to to capture capture the long-term dynamics. The shorter the time frame, the farther farther the observed trend trend is likely to be from the long-term trend. trend. Thus the trends trends in any two two adjacent adjacent time periods periods are unlikely to to be identical, and and the difference indicates not not necessarily a change in the underlying rate of decline, but but can be due simply to chance. The The variability of of observed observed rates rates of of decline decline can can be be estimated estimated from from the the level level of of the the variability variability driving the long-term dynamics, and and thus thus statistical tests performed performed to to deter determine the likelihood that apparent change in trend that an apparent trend occurred due to to the stochastic nature of the process rather rather than than an an underlying change in conditions. conditions. Second, the local populations populations within within a metapopulation metapopulation are linked and experi experience the same long-term growth growth rates, regardless of of the underlying difference difference in local population "). population conditions conditions (i.e., whether they are "sources" "sources" or "sinks "sinks"). However, However, measured measured over over aa short short time time period, period, there there will will bbee differences differences iinn the the observed local population population trends due to chance chance and local conditions. conditions. This means that that over a given time period, local populations populations will appear to be declining at different different rates, but this is not an indication the long-term trends trends and not not necessarily related to to local conditions conditions being better or or worse worse than than other other areas. areas. That That the the long-term long-term trends trends of of the the individual individual local local populations populations are are the the same as the metapopulation metapopulation has a direct impact on PYA PVA for for local populations populations within within a metapopulation. The rate of decline observed among among the different local populations populations will differ, as will the apparent apparent level of variability in the local time series. Thus Thus if an individual viability analysis is done done using parameters parameters estimated from from local population population time series alone, it will appear that that there is tremendous tremendous variability among among the local populations populations risk levels when in fact their long-term risks are similar. When When looking looking at the long-term risks, use of metapopulation metapopulation level parameters parameters leads to better estimates of the long-term local population population risks. Short-term risks, however, are still strongly influenced by local conditions. conditions. Clearly estimates of both both short-term and long-term risks are are needed needed to to capture capture the the whole whole viability viability picture picture for for aa metapopulation. metapopulation. Although populations modeled here Although local populations populations within within the type of meta metapopulations will be eventually repopulated repopulated by dispersal if they undergo undergo extreme declines, the resulting loss of genetic diversity leads to to a gradual gradual erosion of the genetic health of of the the metapopulation. metapopulation. Indeed this this has happened for for salmon salmon species species throughout throughout the the Pacific Pacific Northwest. Northwest. Recovery planning planning for endangered endangered and threatened threatened species typically requires determining where to put put the most most effort. effort. Rarely is it the case that that maximal effort effort can be applied everywhere. Using the stochastic stochastic metapopulation metapopulation model, a sensitivity analysis was used to look for for local characteristics characteristics that that predict where where local changes would produce produce the biggest change in the metapopulation metapopulation growth growth rate. Interestingly, local density (not absolute numbers) numbers) was a strong predictor of where a unit change in local growth growth rates led to to the largest metapopulation metapopulation growth growth rate. This relationship relationship was observed even in simula simulations tions with dispersal sources and and targets targets and and strongly strongly directional directional dispersal, although although it will break break down down when when dispersal is strictly unidirectional. unidirectional. Determining which local populations populations are best best suited for restoration restoration efforts also requires assessing assessing the the feasibility, cost, cost, and and acceptance of restoration restoration efforts. Indeed Indeed when when it it comes comes to to actually actually implementing implementing recovery recovery actions, actions, optimizing optimizing
23. FOR ENDANGERED ETAPOPULATIONS 23. VIABILITY VIABILITYANALYSIS ANALYSIS FOR ENDANGERED M METAPOPULATIONS
597 591
the efficiency of effort effort in terms affecting recovery requires solving a complex complex function of biological, economic, economic, and political information. information. However, under underfunction standing the population population dynamics of the species of concern concern and gaining insight regarding regarding how the demography demography of the species will respond respond to alternative management management actions are fundamental fundamental and primary components components of this conser conservation equation. equation.
sdfsdf
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INDEX INDEX
A A
effects, 200 variance, 1160 60 alternative stable equilibria, 112 2 strategies, 141 strategies, American mink (Mustela vison), 520, 522 population population size size and extinction risk, 527 Ochotona princeps), 1100, 00, American pika ((Ochotona 1132, 32, 503, 5 17 517 75 ancestral process, 1175 ancient lineages in a metapopulation, 88 metapopulation, 1188 androdioecy, 245 anther smut, 473-4 473-4 anthropogenic threats, 566 assignment test, 371 assortative mating coefficient, 82 coefficient, 377, 3382
abundance distribution, 36 distribution, 1136 adaptation, 8 , 223 adaptation, 118, in subdivided 65 subdivided populations, populations, 1165 to local conditions, conditions, 295 to marginal habitats, habitats, 409 to sink habitats, habitats, 406, 409-10 409-10 adaptive landscapes, 118, 8 , 280 plasticity, 263 additive additive genetic effects, 264 genetic variance, 203, 205-6, 13, 205-6, 208, 2213, 264, 348-9 348-9 age at maturity, 246 at reproduction, reproduction, 246 age-structure, 252, 399 age-structured population, 02 population, 1102 aggregation, 62, 64 A land Islands, 74, 492, 500-1, 506 Aland 500-1,506 Allee effect, 113, 3 , 87, 90, 346, 346, 458 alleles average effects of, 208 allelic diversity, 349
B B
badger, 28 Bateson-Dobzhansky-Muller model, 279, 296 Bay checkerspot butterfly, butterfly, see Euphydryas editha Bayesian approach, 1184 84 approach, 86 approach, hierarchical, 378, 383, 3386 approach, 84 approach, potential potential problems, 3384
683 683
684 684
Continued) Bayesian ((Continued) inference, 374 model choice, 380 BDM model, 279, 296 296 multilocus generalization of, 281 bet-hedging strategy, 243 Bicyclus anynana, 359 biodiversity, 541 black hole sink habitat, 158 adaptation adaptation to, 405 Cynomys black-tailed prairie prairie dog ((Cynomys 1 7, 5 19 ludovicianus), 5 517, 519 boundary length minimization problem, 556-7 556-7 breeder's equation, 203 breeding dispersal, 528 phenology, 4 11 411 value, 201 , 208 201,208 bubonic plague, 435, 438-40 438-40 Burramys parvus, 532
C C Carlina vulgaris, 246 carrying capacity, 389 Centaurea jacea, 459 chaotic fluctuations, 346 childhood diseases, 428 chinook salmon meta population, 567 metapopulation, clade disparity, 289 Clarkia concinna, concinna, 458 classic meta population metapopulation dynamics, 506 model, 9 theory, 12, 83 small mammal, 5 17 517 climate change, 507-9 507-9 coalescence approach, 7, 368 approach, 117, in metapopulations, 186 in panmictic populations, 1175 75 79 in two-sex diploid populations, 1179 times, 1176 76 coalescent inference inference analytical methods, 1182 82 computational method, 1183 83 iin n metapopulations, 1196 96 76 interval, 1176 77 standard model, 1177 coalescent predictions in meta populations, 1194 94 metapopulations, coalescent process, 1175, 75, 1178 78 at a multisite genetic locus, 1178 78
INDEX making inferences with the, 1182 82 simulations, 193 standard, 1175-6, 75-6, 1192 92 theory, 1174 74 coevolution in a metacommunity, 270 of dispersal and habitat specialization, 326 of phenotype and context, 259-60 259-60 coevolving species, 270 coexistence, 146-7, 146-7, 149 of multiple species, 1134 34 collecting phase, 1189-93 8 9-93 colonization, 85, 87, 247, 367, 370, 3385, 85, 505 dataset, 1116 16 density-dependent effects on, 376 distance effects effects on, 464 events, 3382 82 evolutionary effects of, 275 rate, 92, 1130 30 related ttoo connectivity, 1115 15 colonization and extinction processes, 82 rates, rates, 78 colonization-competition trade-off, 146, 149 colonizer syndrome, 238, 25 2511 colonizing groups, composition of, 378 common lizard, 3313, 1 3, 3316, 1 6 , 3318, 1 8 , 323-5 323-5 community heritability, 271 estimate of, 272 competition, 251 competitive ability, 148, 272 interactions as forms of IGEs, 269 complementarity, 543 connectivity, 25, 1100, 00, 1102, 02, 108, 1114, 14, 527, 550, 552, 563 asymmetrical, 28 in a metacommunity, 1138 38 measure of, 1111,498 1 1, 498 of networks, 501 conservation, 8, 5 11 511 biology, 20, 338, 541 conspecific attraction, 1111,315, 1 1 , 3 1 5, 5 1 9, 529, 519, 5 3 1 , 534 531,534 context evolving, 263 kinds of, 262 variance in, 262 continuum models, 50 core-satellite distributions, 90 correlated landscape, 424 random walk, 1133
INDEX INDEX corridors, 144 coupled-map lattice, 48, 100 critical community size, size, 423, 429 CSI method, 385 cyclic populations, populations, 530
D D Daphnia, 0 1 , 360 Daphnia, 1101,360 decision theory, 127 decline long-term geometric rate of, 577 rate of, 596 declining-population declining-population paradigm, 339, 365 deleterious 62 deleterious alleles, alleles, 1162 fixation of, 353 deleterious deleterious mutations, mutations, 327, 327, 350, 353-4, 353-4, 363 accumulation accumulation of, 356 demographic rescue, 358 stochasticity, stochasticity, 84, 339-40, 339-40, 344, 346 variance, 340 density dependence, 267, 338 density-dependent density-dependent dispersal, 5 1 , 3 14, 329 dispersal, 2251,314, immigration, immigration, 533 DIC, 3381 81 diffusion diffusion coefficient, 886 6 modeling, 461 approximation approximation approach, 572 approach, practical implications, 567 direct genetic effects, 272 disease colonization colonization event, 480 colonization colonization rate of, 482 decline, 486 dynamics, global coupling, 4 19 419 effect o onn population population growth, 475 eradication, 442 extinction, 482 impact on population population extinction, extinction, 475 incidence, 476 local abundance, 476 persistence, persistence, 423 prevalence, 476, 486-7 486-7 regional abundance, 476 resistance, resistance, 350, 473, 487 spread of, 488 transmission, transmission, 356, 473, 473, 483, 487 dispersal, dispersal, see see also also migration, 230, 250, 307, 389 balanced, 395, 402 breeding, 528
685 61~$ causes of evolution, 309 condition-dependent, condition-dependent, 307, 322 cost and benefits, benefits, 234, 322 curve, curve, 460 density-dependent, 25 1 , 3 1 4-5, 329 251,314-5, directed, 5 30 530 effects of, 3397, 97, 40 4011 ES rate, 2 3 1 , 233, 249-50 231,233,249-50 evolution of, 233, 255 exploratory exploratory movement, 534 forces selecting for, for, 230 functions, 3313, 1 3 , 462, 14 462, 5514 habitat 10 habitat heterogeneity, heterogeneity, 3310 iin n meta populations, 230, 528 metapopulations, inbreeding, 3 13 313 kernel, 52, 85-6 85-6 kernel, Laplace, 379 landscape structure, 3312 12 long-distance, 462 morphs, 3 19 319 mortality due to predation, predation, 529 natal and breeding, 309 natural natural selection selection on, 402 negative density dependence, 329 one-way, 394 parental control, 323 passive, 86, 402, 402, 405 pattern, 402, 402, 409 physiological physiological and behavioral control of, 320 see see plant plant predation, 12, 323 predation, 3312, presaturation, presaturation, 308 propensity, 234, 234, 235, 403 propensity, evolution of, 234 proximate control 19 control of, 3319 rate, 248, 390, 394, 409 rate, habitat-specific, 3 11 311 rate, optimal, 253 rate, plasticity of, 403 saturation, saturation, 308 social factors, 3 14 314 species interactions, interactions, 332 state-dependent, 1 3 , 3318-21 1 8-21 state-dependent, 3313, stepping-stone, stepping-stone, 534 strategy, conditional, conditional, 253 success, 28-30, 28-30, 32 swamping effect of, 406 symmetric, 397 threshold, 30 distribution distribution cauchy, 462 gaussian, 462 negative negative exponential, exponential, 462
686 686
disturbance, disturbance, 145 effects of, 250 effect on dispersal rate evolution, 249 249 rate, 144 diversification dynamics of, 283 patterns patterns of, 298 phase, 286 time homogeneous, 289 diversity, 140, 149 dominance, 1165 65 coefficients, 62 coefficients, 1162 effects, 362 drift load, 69 load, 1169 dynamic probability method, 554, 556 problem, 550 dynamic theory of island biogeography, 6-9, 83
E E
ecological niches, 408 evolutionary dynamics of, 408 ecotone, 396 edge effects, 38, 5 566 sensitivity, 38, 39 effective colonization colonization rate, 94-5 94-5 density, 328 dispersal, dispersal, 328 extinction rate, 93, 95 isolation, 27, 28 metapopulation metapopulation size, size, 94-7, 166, 328 migration rate, 1171,215 7 1 , 215 number of habitat patches, 95, 98 population population size, size, 154, 156-7, 391 size, 1155, 55, 1175, 75, 193, 326, 370 size with source-sink structure, 1159 59 elasticity 90 elasticity ooff long-term growth rate, 5590 emigration, 3361 61 main proximate proximate cause ooff extinction, 534 probability, 330 rate, 266-7 266-7 endangered endangered species, species, 566 recovery planning for, for, 596 endogenous heterogeneity, heterogeneity, 45 environment coarse-grained, 390 spatially heterogeneous, 392 environmental deterioration, deterioration, population population responses to, 356 heterogeneity, heterogeneity, 55
INDEX INDEX stochasticity, 3 9-40, 344 stochasticity, 84-5, 3339-40, variance, 203, 340-1 340-1 epidemic epidemic theory demographic stochasticity, stochasticity, 420 traveling waves, 426 epidemics, epidemics, 67 epidemiology, 1102, 02, 4 16 416 models, 103 theory, 12 epistasis, 262 epistasis, 200-1, 200-1,262 epistatic interaction, 1164 64 equilibrium equilibrium assumption, assumption, 495 eradication eradication threshold, 442 ESU, 8 1-2, 585-6 ESU, 5581-2, 585-6 Euphydryas 30, 494-5 aurinia, aurinia, 1130, 494-5 editha, 9, 338, 400-1, 400-1,491,503, editha, 491, 503, 505, 507, 5 14 514 European nuthatch, 32-3 evolution of dispersal, 1 7, 467 dispersal, 3317, 467 dispersal, kin selection selection model, 231 dispersal, dispersal, multiple factors, 317 dispersal, dispersal, theoretical studies, studies, 230 host-parasite interactions, 468 interactions interactions among individuals, 260 life span in a meta population, 237 metapopulation, migration rate, 118, 8 , 1102, 02, 3 1 7, 467 317, species' ranges, 8-9 ranges, 118-9 evolutionary equilibrium, 408 genetic theory, 259, 262 load, 349 significant 81 significant units, units, 5581 evolutionary dynamics in metapopulations, metapopulations, 275 of reproductive systems, systems, 467 exogenous heterogeneity, heterogeneity, 45 extinction, extinction, 20, 345 caused by disease, 5 19 519 correlates of, 526 dataset, 1116 16 debt, 41, 94, 452, 494, 496, 532 effect of habitat patch size, size, 357 genetic effects effects of, 157 mechanisms of, 355, 365 probability of, 252 related 15 related to patch patch area, 1115 rate, 344, 521 rate and and emigration, 521 risk, 34, 346, 357 risk, effect of gene flow, 358, 362 risk, effect of migration, 358, 362
INDEX INDEX risk in dynamic landscapes, 40 risk, scaling with carrying capacity, 343 threshold, 34-7, 34-7, 42, 89, 94, 98-9, 98-9, 423, 465, 490, 494, 496 time, 80 vortices, 355 extinction-colonization 07-8, 1111, 1 1 , 1114 14 probabilities, 1107-8, processes, 230 rates, 83 stochasticity, 96, 98, 122, 533
F F
fadeouts, 42 4211 fecundity, 248 fence effect, 308 Microtus agrestis), 30, 5 1 7, 520 field vole ((Microtus agrestis), 1130, 517, Fisherian quantitative genetics, 201 , 223 201,223 fitness, 350 measure of, 404 sensitivity of, 404 variation, 263 fixation 69 deleterious alleles, 1169 index, 180 180 rate, 285 flour beetles, 267 foot-and-mouth epidemic, 428 forest models, 48 founder effect, 241 fractal landscape, 30-1 30-1,, 35 fragmentation, 35 effects on extinction threshold, 36 effects on species richness, 145 fragmented communities, 144 frequency dependence, 1163 63 frequency-dependent transmission, 4 18 418 Fst, 60, 1162, 62, 1173, 73 , 1187, 87, 1180, 80, 368-9 Fst, 153-4, 1160, 368-9 as an estimator of dispersal, 1173 73 fugitive 35, 1139-40, 3 9-40, 145 fugitive species, 1135,
G G
gametic disequilibrium, 371 equilibrium, 371 gene by deme interaction, 209-10 209-10 flow, 359, 3 6 1 , 405 361,405 for gene ((GFG) GFG) resistance, 468 frequency change, 1154 54 genealogies, 1174-5 74-5 genealogies, effect of metapopulation meta population structure on, 1175, 75, 1189 89
687 687
interaction and speciation, 220 genealogical approaches, 368 branching patterns, 77, 1190 90 patterns, 1177, histories, simulations of, 1183 83 genealogies, branching patterns of, 1177, 77, 1190 90 genealogy of a sample, 1176 76 generation overlap, 349 genetic architecture, architecture, 297 297 architecture of sex, 241 background, 200 basis of dispersal, 3319 19 contexts, intraindividual, 262 data to directly estimate immigration, 535 differentiation between populations, 405 differentiation between source and sink, 405 diversification, 2 81 281 diversity, 252, 349 diversity, effects of extinction and recolonization, 535 drift, 1153-4, 5 3-4, 230, 348 effects of population structure, described, 1173 73 factors, relative importance, 3338 38 load, 3361 61 load iin n subdivided populations, 1168 68 mixture coefficients, coefficients, 376 neighborhoods, 474 rescue effect, 359, 365 resemblance, 242 stock identification method, see see GSI structure structure of metapopulations, metapopulations, 1166 threats, 347 variability, 348-9 348-9 variability, habitat fragmentation, 537 variability in metapopulations, 539 variability, loss loss of, 347 variance, 348 variance among demes, 208 variance components, 203 variance partitioning, 2 16 216 variance, fragmented populations, 536 variation, 207, 207, 347, 499 variation in a sample, effect of metapopulation structure, 93 structure, 1193 variation, effect of extinction, 187 187 variation, effect of recolonization, 1187 87 variation, loss of, 350 variation, maintenance of, 412 genotype fitness, 1160, 60, 1163 63 genotype-by-environment interactions, 263 genotypic value, 205, 2 16 216
~688 8 geographical geographical information systems, systems, 43 information distributions, 508 fritillary Glanville fritillary (Melitaea cinxia), cinxia), 36, 36, 74, 74, 83, 83, butterfly (Melitaea 89-90, 92-3, 92-3, 129, 129, 146-7, 146-7, 229, 229, 233,237, 233, 237, 89-90, 3 15, 358, 358, 491-3,500-1,503, 491-3, 500-1, 503, 505 315, model, 95-6 95-6 model, global eradication eradication (extinction) of disease, 423 graph theory, 32-3 32-3 gray seal, 375 great tit, tit, 340 growth growth rate long-term, 341 stochastic, 341 stochastic, GSI, 371,373, 371, 3 73, 376-7 376-7 gynodioecious species, 239
H I-I
habitat and meta population approaches, 497 metapopulation 497 area, 392 choice, active active dispersal, 403 choice, evolution of, 412 configuration, 463 connectivity, 32, 456, 456, 459 corridors, 3312, 12, 465 deterioration, 490 fragmentation, 99, 234, 492 fragmentation and extinction thresholds, 35 fragmentation, reduced reduced species species richness, 459 heterogeneity, 140 loss, 34, 99, 364, 493, 545, 563 loss and fragmentation, 42 loss, lagged response, 37 management, 508 networks, 500 patch, 75 quality, 129, 3315, 1 5, 392, 497, 508, 5512 12 quality threshold, 497, 5512 12 selection, 11 selection, 3311 specialization, 1137, 37, 140 value of patches, 99 Halicoerus 75 Halicoerus grypus, grypus, 3375 hard selection, 60-1, 1 6 6 selection, 1160-1,166 haystack model, 5 heritability, 203 of dispersal, 3319 19 heritable variation, 262 hermaphrodites, 245 245 Hesperia 498, 507-9, 5513 13 Hesperia comma, comma, 235, 491, 491,498,
INDEX INDEX
heterogeneous heterogeneous environments, 65 landscapes, 28 landscapes, metapopulation, 95 metapopulation, patch networks, 81 81 patch SPOM, 11, 1 1 , 76 heterosis, 171,323, 1 7 1 , 323, 327, 357 heterosis, heterozygosity, 348 loss of, of, 357 heterozygous advantage, 351 351 heterozygous landscapes, 15, 73-5, 73-5, 496 highly fragmented landscapes, metapopulation biology, biology, 6 history of metapopulation adaptive landscapes, landscapes, 18, 1 8, 280, 297 holey adaptive host 480 colonization, 480 extinction, 480 occurrence, occurrence, 476 population, population, 415 host-pathogen system, simulation of, 486
I 1 1 , 332 ideal free free distribution, 3311,332 ideal population, 1155, 55, 347 IGE, 259, 264, 268, 273 effects of, 269 immigration, 254 density-dependent, 17 density-dependent, 5517 negative density-dependent, 533-4 density-dependent, 533-4 inbreeding, 245 14 avoidance, 3314 coefficient, 206, 2 1 3 , 3351,538 5 1 , 538 213, decreasing resistance, 458 depression, 1162, 62, 3314, 14, 3317, 1 7, 350-2, 355 depression, effect effect of environmental conditions on, 352 depression, 51 depression, purging of, 3351 effect of, 457 incidence dataset, 1116 16 function model (IFM), (IFM), 33, 82, 87, 90, 1100-1, 00-1, 108, 12, 1116, 1 6 , 428, 453, 492, 108, 1112, 495, 503, 508-9, 5518, 1 8 , 521, 558 521,558 matrix, 141 indirect genetic 8 , 259, 272 genetic effects, effects, 118, in meta populations, 266 metapopulations, individual by deme interaction, interaction, 209 movement, 27, 505 individual-based individual-based model, 46-7, 98-9 infinite sites 79 sites model, 1179 integrodifference equations, 50 interacting particle systems, systems, 48 interdemic interdemic selection, selection, strength of, of, 265
689 689
INDEX INDEX
interspecific competition, competition, 65, 65, 273 273 interspecific intragenomic conflicts, conflicts, 241 241,243 intragenomic , 243 introductions of of introductions Melitaea cinxia, cinxia, 506 506 Melitaea species, 504 504 species, invasion capacity of of aa network, 89-90 invasions, 65 65 invasions, island biogeographic biogeographic theory, theory, 6-9, 6-9, 883 island 3 island model, model, 111, 74, 776, island 1 , 115-6, 5-6, 74, 6 , 1155, 55, 1168, 68, 1172, 72, 1187, 87, 2213, 1 3, 3368 68 of dispersal, dispersal, 3328 of 28 isolation bby y distance, 3370 70 iteroparity, 2238 38
K K
key patches, 1100 00 kin 231,235, competition, 2 3 1 , 235, 3316, 16, 3319 19 cooperation, 3316 16 231,255 selection, 23 1 , 255 kin-based interactions, 3317 17 Krebs effect, effect, 308
L L
Lactuca mura/is, muralis, 467 Lactuca lacunarity index, 3311 Lande's model, 79, 8811 landscape change, 5551 51 24-9, 32-3 connectivity, 24-9, dynamics, 44, 563 ecological perspective, 25 7-8, 111-2, ecology, 7-8, 1-2, 23, 102 1 02 fragmentation, evolutionary consequences of, 234 matrix, 24 mosaics, 24 structure, 34, 36, 76, 98, 101,491 1 0 1 , 491 lattice lattice model, 46, 48, 48, 68, 98 98 Leadbeater's Leadbeater's possum, 99 Levins model, 74, 74, 77, 77, 79, 81, 8 1 , 93, 101 101 rule, rule, 81, 8 1 , 100 100 Levins-type models, models, 90 90 life span, 235, 237 237 span, 235, table, table, 38 life history, history, 347 347 density-dependent, density-dependent, 251 25 1 evolution, evolution, effect of of genetic structure structure on, on, 255 255 evolution, evolution, in in metapopulations, metapopulations, 228, 228, 255-6 255-6 evolution, evolution, metapopulation metapopulation effect on, on, 228 228 strategies, strategies, 252 252
strategies, strategies, evolution evolution of, of, 246 246 strategies, strategies, evolutionarily evolutionarily stable, stable, 255 255 syndromes, syndromes, 237, 237, 250 250 theory, theory, 227, 227, 229, 229, 237 237 traits, traits, 227 227 traits, traits, evolution evolution of, of, 243, 254 life-time life-time reproductive reproductive contribution, 340 340 success, success, 396 396 likelihood 80 likelihood ratio ratio test, 373, 373, 3380 linkage 71 linkage disequilibrium, disequilibrium, 3371 Linum 68 Linum marginaie, marginale, 4468 load iin n subdivided populations, 71 populations, 1171 local adaptation, 326, 407, 457 and and regional patterns of of diversity, diversity, 142 average effects, 18 effects, 207, 209, 212, 2218 breeding value, 205-6 205-6 breeding populations, 209 breeding value in meta metapopulations, context, 261 diversity, 144-6, 148 drift load, 1171 71 dynamics, 248 extinction, 84 extinction, effect of inbreeding on, 352 extinction, evolutionary effects effects of, 275 275 extinction in the meta population metapopulation context, context, 358 mate competition, 3 17 317 population decline rate, 570 population size, 234 population population size size distribution, 570 population trends, differences in, 596 resource competition, 317 317 resource selection, 405 specialization, 501 specialization, regression, 552 552 logistic regression, long-term dynamics, 132 model, 52 Lotka-Volterra model,
M M
macroparasites, 416 416 macroparasites, mainland-island mainland-island epidemic metapopulation, 426 metapopulation model, model, 413, 4 1 3 , 531 531 metapopulation metapopulation structure, structure, 490 490 metapopulation major histocompatibility histocompatibility complex, complex, 350 350 major maladaptation, 411 411 maladaptation, habitats, 410 410 in sink habitats, actions, 587 587 management actions, harvest, 588 588 harvest, local population population level, 589 589 local river hydropower hydropower systems, systems, 588 588 river
INDEX INDEX
6690 90
Mantel test, test, partial, partial, 385 385 Mantel marginal habitats, habitats, adaptation adaptation to, to, 404 404 marginal Markov Chain, Chain, 80 80 Markov see MCMC MCMC Monte Carlo Carlo method, method, see Monte Markov process, process, 80 80 Markov mark-release-recapture, 505 505 mark-release-recapture, of seeds, seeds, 461 46 1 of techniques, 28 techniques, marsh fritillary fritillary butterfly, butterfly, see see Euphydryas Euphydryas marsh
aurinia aurinia mass effects, effects, 140 140 mass mate choice, choice, 310 310 mate maternal effects, 325 maternal and state-dependent state-dependent dispersal dispersal cues, cues, 321 321 and matrix, 313 313 matrix, population models, 12, 102 1 02 population MCMC, 117-8, 1 1 7-8, 183 183 MCMC, mean additive genetic variance, variance, within additive genetic demes, 206 genetic variance, variance, effect of drift additive genetic on, 217 217 on, allele frequency, 60 frequency, 1160 210 local average effect, 210 phenotypic values, 200 200 161 relatedness of individuals, 161 time ttoo extinction, 778, 8 , 353-4 353-4 measles, 420-1, 423, 426, 429-30, 429-30, 420-1,423,426, 432, 436 mega population, 500 megapopulation,
Melitaea Melitaea cinxia, cinxia, see see Glanville Glanville fritillary diamina, diamina, 233-4, 233-4, 559-61 metacommunity, 1133, 33, 270 diversity, diversity, 140 140 indirect effects effects in, 270 mass effects perspective, 1136 36 neutral perspective, 1136 36 patch-dynamic perspective, 1134 34 species-sorting 35 species-sorting perspective, 1135 meta population metapopulation approach, 3, 9, 111,514 1 , 514 approach, criticism of, 14-5 biology, biology, fundamental processes, 367 citations citations to, to, 66 classic, population classic, see see classic classic meta metapopulation capacity, 1 , 93, 98, 465, capacity, 35, 88-9 88-91, 465, 495, 495, 500-1 , 506, 512 500-1,506, 512 concept, concept, 243, 243, 413 413 concept, concept, plant-specific plant-specific problems, problems, 451 451 decline, decline, 492, 492, 567, 567, 570 570 diffusion diffusion approximation, approximation, 566 566 dynamic 7 dynamic connectivity, connectivity, 887 ecology, ecology, 99
effect, 467 467 effect, endangered, endangered, 565 565 evolution, 18 18 evolution, extinction, extinction, 40, 40, 78, 78, 98,~100, 98, 1 00, 311,362, 3 1 1 , 362, 364, 496 496 364, extinction time, time, 127, 127, 558 558 extinction genealogies of of samples samples from, from, 188 188 genetic structure, structure, 273 273 genetic genetics, 15 15 genetics, level management management actions, actions, 587 587 meltdown, meltdown, 363 363 stochastic, 567 567 model, stochastic, models, structured structured by the the sizes of of local populations, populations, 101 101 o birds, 8 off birds, of butterflies, butterflies, 8 of of fishes, 8 of of mammals, 8 of plants, 8 patterns, 6, 494, 499 persistence, 34, 42, 129, 498 498 processes, 5 processes, butterflies, 14 processes on population genetic structure, structure, 536 genetics, 200, 200, 205, 205, 223 quantitative genetics, site selection, 559 91, 328 size, 91,328 size distribution, 570 spatially realistic theory, 9, 111, 1 , 76, 83 stochastic theory, 94 structures, 5 theory 8, 228 trajectories, 573 viability metrics, 577, 584, 587 with very small local populations, 538 Metropolis-Hastings algorithm, 1184 84 MHC MHC loci, 350, 352 Microbotryum violaceum, violaceum, 473, 473,487 Microbotryum 487 microsatellite DNA polymorphism, 536 genotyping, genotyping, 528 528
Microtus Microtus oeconomus, oeconomus, 530 530 rossiomeridionalis, rossiomeridionalis, 532 532 migrant pool, pool, 158 158 migrant model, model, 369 369 migration, see see also also dispersal, dispersal, 153, 153, 360, 360, 505 505 migration, corridor, corridor, 360 360 cost cost of, of, 505 505 costs costs and and benefits benefits of, of, 323 323 density density dependence, dependence, 528 528 diversity, diversity, 144 144 in aa meta metapopulation context, 385 385 in population context,
691 691
IINDEX NDEX limitation, limitation, 140 load, 1170 70 process, 367 rate, 372 rate and diversity, 139 139 migrational 361,364 migrational meltdown, 3 6 1 , 364 formulation, 547 minimum set set coverage formulation, missing missing data, 1 8-20 data, 1118-20 117 years, 117 moment moment closure, closure, 58-9 58-9 moment moment equation equation approaches, approaches, 68 Moran Moran effect, 426 mosaic mosaic models, 456 movement parameters, parameters, habitat-specific, 27 multi locus multilocus properties of, 282 BDM model, properties genotype approaches, approaches, 368 genotype methods, methods, 20, 20, 370 370 multiple equilibria, equilibria, 90 multitrophic interactions, 459 multitrophic 459 mutation, 7, 1162, 62, 1179 79 mutation, 117, accumulation, accumulation, 353 and 81 and genealogical genealogical process, 1181 average fitness effects, 354 load, 68 load, 1168 number of, 1181 81 79 parameter, 1179 rate, genome-wide, 355 rate, per-genome, 354 mutational meltdown, 1170, 70, 327, mutational meltdown, 327, 354, 356, 363, 365 mutation-selection 62 mutation-selection balance, 1162 mutualistic mutualistic relationship, 459
N N
natural natural experiments, experiments, 401 selection, 62 selection, 1162 populations, 259 selection in meta metapopulations, Ne, 156, 1173, 73, 349, 354 effect of population population structure structure on, 157 network network connectivity, connectivity, 501 neutral landscape models, 32, 34 metacommunity perspective, 143 theories of community structure, 110 0 neutrality tests, 1180, 80, 1185 85 non-equilibrium non-equilibrium systems, 490 non-linear interaction, 260 non-random 55, 1163 63 non-random mating, 1155, number number of alleles, 80 alleles, 1180 of species, species, 276-7, 276-7, 292
o O Oncorhynchus species, species, 581 581 open and closed population population structures, 490 attraction, 534 opposite-sex attraction, Orkney Isles, 376 outbred vigor, vigor, 360 outbred outbreeding depression, 361-2 3 6 1-2 outbreeding overdominance, 163, 351 351 overdominance,
p P
pair approximation, approximation, 49 panmictic population, population, 175 1 75 parapatric speciation, 289 289 Pararge 10 Pararge aegeria, aegeria, 5510 parasitoid-prey system, 5 1 , 6611 51, parentage parentage assignment, 528 parental control of offspring parental control dispersal, 323 dispersal, passive propagule passive propagule dispersal, 86 patch areas, 76 connectivity, 26 connectivity, dynamics, 139, 456 homogeneous, 77 network, homogeneous, occupancies, 1114 14 occupancy models, 77 turnover, 40 values, 88, 91, 93, 1100 values, 00 pathogen pathogen colonization, colonization, 485 extinction, 485 occurrence, 476 percolation, percolation, 29 theory, 26, 331, 1 , 1100 00 threshold, threshold, 29-30, 29-30, 32 phase locking, 425 phenotypic effects of alleles, 200 of genes, 262 phenotypic plasticity, 457 457 plasticity, variance, 201 , 203, 263 201,203, variation, partitioning partitioning of, 263 philopatry, 3316 16 phylogenetic envelope, 91 envelope, 3391 147-8 pitcher plants, 145, 147-8 plant dispersal, 460-1 dispersal, molecular markers, 462 dispersal, dispersal, distance distance curve, 460 metapopulations, 449, 449, 475 475 metapopulations, interactions, 459 plant-animal interactions, plant-pathogen metapopulation, metapopulation, 471 plant-pathogen
6692 92
plant-specific problems problems with with metapopulation metapopulation plant-specific concept, 451 451 concept, plasticity, 251 251 plasticity, Plebejus argus, argus, 131, 1 3 1 , 234, 234, 491,501-4, 491, 501-4, 512, 5 12, Plebejus 556-7 552, 556-7 point speciation speciation model, 277 point pollinators, 458 458 pollinators, polymorphic polymorphic nucleotide sites, sites, number number of, 181 181 nucleotide sites, expected number number of, 177 1 77 sites, 181 sites, expected value, 181 total number number of, 181 181 sites, total polymorphism, protected, protected, 247 247 polymorphism, pool frog pool frog (Rana lessonae), 129 population biology, perspective in, 338 population population population age-structured, 395 bottleneck, 348-50 bottleneck, 348-50 connectivity, 498 density, 246, 387 246, 387 differentiation, 1 3 , 223 differentiation, 2213,223 dynamics, 250, 250, 345-6 345-6 dynamics, transient, 256 256 extinction, 337, 355 extinction, extinction, ceiling model, 343 extinction, extinction, ecological factors, factors, 339 extinction, 347 extinction, genetic factors, factors, 347 extinction extinction rate, 358 fluctuations, 340, 344 genetic methods, 3368 68 genetic processes, 1172 72 growth rate, 344, 346 history, inferences about, 82 about, 1182 mean fitness, 353 neighborhood, neighborhood, 63-4 63-4 persistence, effects of source-sink structure, structure, 398 regulation, 337-8, 390 size, effect of source-sink structure, structure, 397 stability, effects of source-sink structure, structure, 398 stage-structured, 395 structure, inferences about, 1182 82 synchrony, 329 turnover, 455, 502-3, 502-3, 524, 526 turnover, genetic effects of, of, 368 viability analysis, 113, 3, 20, 338, 565, 595 portfolio effect, 421 posterior distribution, distribution, 374 predator-prey interaction, 66 models, 54 54
INDEX
prior distribution, distribution, effect of, of, 384 3 84 prior probability of of probability coexistence, 332 332 coexistence, common origin, origin, 370 370 common extinction, 355 355 extinction, fixation, 165 fixation, fixation, effect of of population population structure structure fixation, 168 on, 168 fixation o alleles, 165-7 165-7 fixation off beneficial alleles, fixation of of additive additive alleles, alleles, 165 fixation fixation of neutral neutral mutations, mutations, 285 285 fixation fixation of of new mutations, mutations, 165-6 1 65-6 fixation Proclossiana eunomia, eunomia, 491 productivity, 142, 1 42, 248 248 productivity, index, 378 378 index, propagule-pool propagule-pool colonization, 535 model, 369-70 369-70 pattern, 38 pattern, 5538 propagules, 69 propagules, colonizing, 3369 polymorphism, 247 protected polymorphism, 247 PVA, 565-6 565-6
Q Q
quantitative genetic parameters, effect migration on, 213 213 of migration quantitative 99 quantitative genetics, 1199 quasiequilibrium, quasiequilibrium, 495 495 quasistationarity, 14, 122, 1131 31 quasistationarity, 1114, assumption, assumption, 122 quasistationary quasistationary distribution, distribution, 78-80, 78-80, 82, 97 equilibrium, 1116 16 state, 1107-8, 07-8, 124
II R
radiation adaptive, 298 duration duration of, 289, 297 297 dynamics of, 289 time to beginning of, 297 random fission model, 277 random walk, 886 6 model, 28 range boundaries, 507 expansion, 509 margins, 507 rate of expansion, 508, 5510, 1 0, 5512 12 habitat destruction, destruction, 4411 reaction norm, 252 252 reaction-diffusion, 50
IINDEX NDEX reciprocal transplant transplant experiments, 457 recolonization, 241,243, 241, 243, 245, 254, 254, 526 genetic effects of, 1157 57 recombination, 1153 53 genealogies and, 1178 78 recruitment from seed rain vs seed bank, 455 Red List List criteria, 566 regional coexistence, 146 diversity, 144, 146, 148 dynamics, 248 stochasticity, 89, 96, 1100, 00, 121, 1130, 30, 363, 5 14 514 stochasticity, migration and, 363 relatedness, 242 remnant populations, 452-3, 455-6 455-6 representativeness, 542 reproduction reproduction assurance, 241 reproductive effort, 246, 248, 250, 252 age-specific, age-specific, 238 ES, 235, 235, 248 evolution of, 235 in a meta population, 235 metapopulation, in a metapopulation, metapopulation, evolution of, 246 reproductive isolation, 2 8 1 , 284, 296 281,284, genetic architecture of, 279 genetics of, 278 threshold effect, 297 reproductive success in sink habitats, 4 10 410 reproductive value, 349, 3391,404-5 9 1 , 404-5 habitat 91 habitat specific, 3391 rescue effect, 113, 3 , 887, 7 , 90, 1101, 0 1 , 1107-8, 0 7-8, 1111-2, 1 1-2, 1115, 15, 1130, 30, 3310, 1 0, 328, 464, 5 1 8-9, 529, 518-9, 531, 534 531,534 reserve aggregation, 550, 554-5, 554-5, 562 reserve network, 541 design, 542 design, landscape dynamics, 546 design, spatial population population dynamics, 546 dispersal ability, 560 patch cost, 560 reserve selection, 541 algorithms, 542, 544 algorithms, persistence, 546 amount of resources, 61 resources, 5561 resistance, fitness costs of, 473 resource partitioning, 37 partitioning, 1137 restoration restoration efforts, 596 Reversible Jump Markov Chain Monte Carlo, 3381 81
693 693 risk metric confidence intervals for, 578 uncertainty, 578 variability of, 595 risk of extinction, extinction, 340, 344, 353, 356-7, 356-7, 359 long-term, 596 short-term, 596 RJMCMC, 81 RJMCMC, 3381
$ S
81 salmon, 5581 scattering phase, 1189, 89, 1191-3 9 1-3 scoring approaches, 543 seasonal forcing, 426 seed banks, 454, 456 456 dimorphism, 3 19 319 sowing, 451 segregating sites expected number of, 1177 77 expected value, 1181 81 segregation load, 1169 69 SEIR model, 422 selection, 153, 2 1 9, 230, 237 219, 237 among local populations, populations, 1188 balancing, 1163 63 differential, 203 directional, 221 diversifying, 200 hard, 1160-1, 60-1, 1166 66 heterogeneous, 1164 64 in meta populations, 264 metapopulations, in subdivided populations, 59 populations, 1159 individual, 266, 268, 270 interdemic, 266, 268-70 268-70 response to, 1160, 60, 200, 203, 203, 266, 404 soft, 1160-1, 6 0-1, 1166 66 spatially heterogeneous, 1159, 59, 1164, 64, 245 stabilizing, 165 strength of, 1160 60 temporal heterogeneity in, 245 uniform, 1159-60 59-60 selfing 245 semelparous organisms, 246 semi-independent patch patch network (SIN), 499-500 499-500 Senecio jacobaea, 146 settlement decision, 334 pattern, 529 success, 324-5 324-5
694 694 sex sex allocation, 238, 243, 245 inheritance, mode of, 242 ratio, 241-2 241-2 ratio, evolution, 239 ratio, evolution in a meta population, 239 metapopulation, ratio variation, 239 sex-biased dispersal, 1 3-4, 3317, 1 7, 521 dispersal, 3313-4, sexual selection, 13 selection, 3313 sexually transmitted diseases, 427 shifting 8 , 1186 86 shifting balance theory, 118, shrews, 1130, 30, 344, 347, 5 31 531 Silene alba, 360 latifolia, lati[olia, 457, 473, 487 Silene-Microbotryum population, Silene-Microbotryum meta metapopulation, 475, 485 sink, 40 absolute, 3 88 388 black-hole, 394 habitat, habitat, 388, 392, 398, 40 4011 populations, 58 populations, 136, 141, 1158 relative, 88 relative, 3388 SIR model, 4 1 6 , 427, 442 416, SIS model, 427 SIS site selection algorithms, 542 stochastic metapopulation model, 558 dispersal dispersal ability, 559 site-frequency distribution, 85 distribution, 1185 size-structured populations, 02 populations, 1102 15 small mammals, 5515 small-population 3 9 , 365 small-population paradigm, 3339, snapshot 12, 1114-5 14-5 snapshot data, 1112, social contact networks, networks, 67 context, average, 264 context, genetic variation in, 265 fence, 329 fence effect, 534 soft selection, , 1 66 selection, 160-1 160-1,166 Sorex, 344 araneus, 130, 5531 31 caecutiens, 1130, 30, 5 31 531 minutus, 1130, 30, 5 31 531 source source habitat, habitat, 388, 392 populations, 158 populations, 136, 141, 141,158 source-sink concept, applied aspects aspects of, 414 dynamics, 37, 143, 497 dynamics, ecological consequences of, 397, 399
INDEX INDEX dynamics, influence influence on population population size, size, 401 meta populations, 117, 7, 19, 3 11 metapopulations, 311 meta populations, evolutionary models of, metapopulations, 408 models, predictions of, of, 402 population 87-8, 408 population dynamics, 3387-8, populations, populations, 404, 453 potential, 337-8 7-8 relations, 39 relations, 1139 structure, 3387, 87, 398-400 398-400 structure, 13 structure, concept of, 4413 structure, ecological ecological consequences consequences of, 400 structure, effect of, 158 structure, effect on adaptive evolution, 405 structure, evolutionary consequences, consequences, 403 structure, evolutionary stability of, 402 structure, patch model of, 3389 89 structure, reverse, 394, 4 10 410 system, experimental, 401 system, polygenic model of evolution evolution in, 406 systems, 358 systems, laboratory, 412 spatial aggregation, 4 configuration, configuration, 497 correlation, correlation, 97 coupling coupling in epidemic metapopulations, metapopulations, 433 ecology, 4, 9 genetic structure, 327 locations, 76 logistic model, 64 moment equations, 46, 5 1, 9 51, 988 point process, 47 population population viability analysis, analysis, 559 reserve design, 547 synchrony, 50 spatially correlated correlated environmental environmental conditions, conditions, 96 correlated 1 8 , 527, 532-3 correlated extinction, extinction, 5518, 532-3 explicit explicit approach, 396 explicit explicit models, 47, 395 explicit models, consequences consequences for ecological processes, 471-2 471-2 explicit models, consequences consequences for genetic processes, processes, 471-2 471-2 realistic Levins model, 82, 84 realistic metapopulation theory, 9, 111, 1 , 76, 83 realistic realistic models, 234 speciation, 221, 223, 278, 296, 298 221,223, by random random drift and mutation, mutation, 300 dynamics, 296 in metapopulations, metapopulations, 275 mechanisms, 221
INDEX INDEX probability of, 278 rate rate of, 291 species extinction, 364 range, 364, 397 sorting, 1139-40, 39-40, 147 species-area species-area curves, 276-7 276-7 relationship, 6 SPOM, 111, 1, 74-75, 05-6, 1112, 12, 428 74-75, 1105-6, classification, classification, 89 deterministic approximation approximation of, 76, 82 homogeneous, 111, 1 , 76 homogeneous, homogeneous, deterministic deterministic approximation approximation of, l11l SRLM, 93-4 93-4 stable equilibria, 346 stationary state, 244 statistical genetics, 20 methods, limitations limitations of, 375 stepping-stone dispersal, dispersal, 534 model, 116 6 stochastic and seasonal forcing, 420 cellular automata, automata, 48 forcing of epidemics, epidemics, 420 logistic model, 78-9, 881, 1 , 98 logistic model, deterministic mean-field approximation approximation of, 77 meta population equilibrium, 82 metapopulation metapopulation metapopulation model, effects effects of management, 587 metapopulation, metapopulation, trajectory of, 596 patch patch occupancy models, see SPOM theory, 94 structured structured coalescent, 1187-8 87-8 meta population model, 12, 440 metapopulation populations, 86 populations, genetics of, 1186 successional 13 successional habitats, habitats, 5513 susceptible-infected-susceptible (SIS) (SIS) model, 103
T T
tagging experiments, 582 temporal variation, 97 temporally varying environmental conditions, conditions, 9966 territoriality, 3314 14 threatened species, species, recovery planning for, 596 threats to populations, populations, 365
695 695
threshold 1 , 889 9 condition, 79, 881, in connectivity, connectivity, 29 response, 3311 value, 880 0 (lag), 454-6, 454-6, 465, 494 time delay (lag), in meta population response, 494 metapopulation time to extinction, 9 1 , 344, 452, 495-6, 91, 495-6, 499, 503 most recent common ancestor, 92 ancestor, 1192 total coalescence 78 coalescence times, expected, 1178 77 total length of the genealogy, distribution, distribution, 1177 trade-off, 1137-8, 3 7-8, 146, 148-9 148-9 between competitive ability and predator predator tolerance, 148 between colonization colonization and competitive abilities, 135, 146 transient transient dynamics, 24, 93 dynamics, evolutionary importance importance of, 254 dynamics of speciation, 288 growth, 248 metapopulation metapopulation dynamics, 490 period in population population dynamics, 243-4 243-4 population population dynamics, 250 transmission transmission model density-dependent, 482 frequency-dependent, 482 transplantations, transplantations, 451 traveling waves of infection, 434 434 tree frog (Hyla 09-10, (Hyla arboreal, arborea), 106, 1109-10, 1122-4, 22-4, 128 Tribolium castaneum, 266, 271-2 271-2 trophic cascades, 43 cascades, 1143 turnover, 125, 127, 1132 32 14-5, 1117-8 1 7-8 data, 1114-5, events, events, 506 underestimate 32 underestimate of, of, 1132
U U
unstable equilibrium, 346 Urophora cardui, cardui, 129 Urtica Urtica dioica, dioica, 459 Uta stansburiana, 1 6 , 320 stansburiana, 3316,
v V
vaccination pulse, 442-4 442-4 random, random, 443 strategies, strategies, 442 Vancouver Island marmot marmot (Marmota vancouverensis), 532
INDEX INDEX
696 696
variance in average effects, effects, 2210 local average 10 on, 2217 local average effects, effect ooff drift on, 17 reproductive success, 155-7 reproductive viability analysis, 565 metric validation, simulations for, 579 of total metapopulation, 38, 569
W W
water populations, 522 water vole meta metapopulations, wavelet wavelet phase analysis, analysis, 434 white-backed woodpecker (Dendrocopos (Dendrocopos leucotos), leucotos), 8811 whooping cough, 421-2 421-2 Wright-Fisher model, 1176 76
