Contributors
Numbers parentheses indicate the pages on which which the authors' authors' contributions begin. Numbers in parentheses indicate the pages on contributions begin.
N. H. Barton ((183) 1 83 ) Institute of Cell, Animal, and Population Biology, University
of Edinburgh, Edinburgh EH9 3JT, Scotland Patrick 2 1 5 ) Department of Biological Sciences, California State Uni Patrick Foley ((215) Uni-
versity, Sacramento, Sacramento, California 958 19 95819 Steven A. Frank Uni Frank (325) Department of Ecology and Evolutionary Biology, Uni-
versity of California, Irvine, Irvine, California 927 17 92717 Barbara Barbara E E.. Giles (429) Department of Genetics, Umea Ume~i University, S-90 S-9011 87
Umea, UmeL Sweden Michael E. Gilpin 1 65, 407) Department of Biology, University of California, Gilpin ((165,407)
San Diego, La Jolla, California 92093 Jerome J~rbme Goudet Goudet (429) Institut de Zoologie et d'Ecologie Animale, Universite Universit6 de
Lausanne, CH 0 1 5 Lausanne, Switzerland CH-- 11015 Pierre-Henri Pierre-Henri Gouyon (293) Evolution et Systematique Syst6matique des Vegetaux, V6g6taux,,, Univ Univ-
ersite 1405 Orsay Cedex, France ersit6 Paris-Sud, 991405 Mats Gyllenberg (93) Department of Mathematics, University of Turku, FIN-
200 1 4 Turku, Finland 20014
xi
Contributors Contributors
xii xii
I1kka 359) Department Ilkka Hanski Hanski (5, (5, 69, 69, 93, 93,359) Department of of Ecology Ecology and and Systematics, Systematics, Division Division
of 14 Helsinki, of Population Population Biology, Biology, University University of of Helsinki, Helsinki, FIN-000 FIN-00014 Helsinki, Finland Finland Susan Susan Harrison (27) (27) Division Division of of Environmental Environmental Studies and and Center Center for for Popu Popu-
lation 16 lation Biology, Biology, University University of of California, California, Davis, Davis, Davis, Davis, California California 956 95616 Michael P. 1 23) Department of P. Hassell Hasseli ((123) of Biology, Imperial College at Silwood
Park, Park, Ascot, Berkshire SL5 7PY, United United Kingdom Alan Alan Hastings (93) Division Division of of Environmental Studies and and Institute of Theoret Theoret-
ical Dynamics, University of California, Davis, Davis, California 956 16 95616 Philip W. Hedrick 1 65) Department of Zoology, Arizona State University, Hedriek ((165)
Tempe, Arizona 85287 Robert D. Holt 1 49) Department of Systematics and Ecology, Natural History Holt ((149)
Museum, The University of Kansas, Lawrence, Kansas 66045 Rolf A. Ims Ires (247) Department of Biology, Division of Zoology, University of
Oslo, N-03 N-0316 1 6 Oslo 3, Norway Veronica A. Johnson Johnson (267) Program in Ecology, Evolution and Conservation
89512 Biology, University of Nevada, Reno, Nevada 895 12 Robert 1 23) Department ooff Zoology, Oxford OXI Robert M. May ((123) OX1 3PS, United King King-
dom Sean Nee ((123) 1 23) Department of Zoology, Oxford OXI OX1 3PS, United Kingdom
' Evolution, Universit6 Universite Montpel Isabelle Olivieri (293) Institut des Sciences de ll'Evolution, Montpellier 2, 34095 Montpellier cedex 05, France Daniel Simberloff Univer Simberloff (5) Department of Biological Sciences, Florida State Univer-
sity, Tallahassee, Florida Florida 32306 Andrew T. Smith Andrew Smith (407) (407) Department of Zoology, Arizona State University, Tempe, Arizona 85287 Peter B. Staeey Stacey (267) Program in Ecology, Evolution and Conservation Biology, Peter University of Nevada, Reno, Nevada Nevada 89512 895 12 Mark L. L . Taper Taper (267) Department of Boze Mark of Biology, Montana State University, Bozeman, Montana 59717 597 1 7 Andrew D. Taylor Taylor (27) Department o Andrew offZoology, University ooffHawaii, Honolulu, Hawaii 96822 Chris D. D. Thomas Thomas (359) (359) Department of of Biology, University University of of Leeds, Leeds, Leeds Leeds LS2 Chris 9JT, United United Kingdom Kingdom Ed van van der der Meijden Meijden (387) (387) Institute Institute of of Evolutionary Evolutionary and and Ecological Ecological Sciences, Sciences, Ed Leiden University, University, 2300 2300 RA RA Leiden, Leiden, The The Netherlands Netherlands Leiden Catharina A. A. M. M. van van der der Veen-van Veen-van Wijk Wijk (387) (387) Institute Institute of of Evolutionary Evolutionary and and Catharina Ecological Sciences, Sciences, Leiden Leiden University, University, 2300 2300 RA RA Leiden, Leiden, The The Netherlands Netherlands Ecological Michael C. C. Whitloek Whitlock (183) ( 1 83) Department Department of of Zoology, Zoology, University University of of British British CoCo Michael lumbia, Vancouver, Vancouver, British British Columbia, Columbia, Canada Canada V6T V6T 1Z4 l Z4 lumbia,
Contributors
xiii xiii
John A. Wiens Wiens (43) (43) Department of of Biology Biology and Graduate Degree Program in John Ecology, Colorado Colorado State State University, University, Fort Collins, Collins, Colorado Colorado 80521 805 2 1 Ecology, G . Yoccoz Yoccoz (247) (247) Laboratoire de d e Biom6trie, Biometrie, G6n6tique Genetique et e t Biologie Biologie des des PopPop Nigel G. ulations, URA URA CNRS 2055, 2055, Universit6 Universite Claude Bernard, Bernard, F-69622 F-69622 Villeurbanne Villeurbanne ulations, of Zoology, University Cedex, France; and Department of Biology, Division of of Oslo, N-0316 N-03 1 6 Oslo 3, Norway of
sdfsdf
Preface
the past past few few years, the metapopulation concept has has become become widely and and In the firmly established population biology and in conservation. conservation. The The number number firmly established both in population of papers papers on metapopulations metapopulations is growing growing exponentially, with with a doubling doubling time time of of of less than than two years. The The metapopulation metapopulation concept concept is beginning beginning to appear appear in text textbooks, and the metapopulation replaced the dynamic metapopulation theory has has to a large large extent replaced dynamic theory theory of of island biogeography biogeography in conservation conservation biology. As observed observed by Science magazine, metapopulation approaches approaches are are now "all the rage." rage." Our Our previous previous book, Metapopulation Dynamics: Empirical Empirical and Theoretical 99 1 ), brought together a Investigations (Gilpin and Hanski, Hanski, Academic Academic Press, 11991), range of of viewpoints and ecological models bearing bearing on spatially fragmented fragmented pop populations. The unmet demand. The book book sold out rapidly, leaving leaving an unmet demand. In considering considering the the need updated need for for aa new new book book on on the the same same subject, subject, we we had had aa choice choice between between an an updated second edition and an entirely new new volume. We We chose the second second alternative for two reasons. First, the fi eld of metapopulation biology has advanced field advanced considerably, with a vigorous interplay among theory, models, and fi eld studies, and we wanted field to ect the to refl reflect the depth depth and and the the breadth breadth of of this this growth growth in in the the new new volume. volume. Second, Second, we wanted wanted to shift some emphases emphases and expand expand along new lines of of inquiry. The The first volume was biased toward toward a conceptual conceptual analysis of metapopulation ecology. In this volume, we cover more more thoroughly both empirical empirical studies and more ad adgevanced theories, and we have now included more information pertaining to ge netics and evolution. The rapid progress that has occurred in field field studies is ev-
Xu xv
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Preface Preface
ident on the covers of the two volumes. Whereas the cover of the previous book butterfly madepicted a metapopulation of a hypothetical butterfl y species, Euphydryas ma cintoshus G., the cover of this volume illustrates the fragmented landscape of a butterfly Melitaea cinxia). real butterfl y metapopulation (the Glanville fritillary, Melitaea This volume consists of solicited chapters from selected authors working in This the general area of metapopulation biology. We are pleased that everyone whom we asked to contribute did contribute. Several chapters chapters in this volume are pri prichapters practically marily empirical, while others are highly theoretical. Some chapters chapignore ecology; many more ignore genetics and evolution. However, a few chap ters describe both theory and empirical results, and others cover both ecology and genetics, a trend that we hope will become more prominent in the near future. readers will equally appreciate every chapter, chapter, but we would be very dis disNot all readers appointed (and truly surprised) if most readers would not be better informed and indeed stimulated by most of the chapters. The scope of the chapters in this represents our attempt to sketch the general limits of metapopulation volume represents biology. We We hope that this volume will disseminate disseminate ideas, results, and and conclusions conclusions across the customary academic academic confi confines. across nes. One recent nite broadening broadening of recent trend trend that that we have noted is a defi definite of the meaning meaning of the term term "metapopulation." In the first volume, population turnover, local ex exof tinctions and and colonizations, was considered to be the key and and practically indis indistinctions pensable feature of of metapopulation dynamics. In this interpretation we followed pensable conceptual guidance guidance of of the root of of all metapopulation metapopulation models, the Levins Levins the conceptual continue to think think that metapopulation dynamics dynamics in this nar narmodel. Although we continue conceptual core of of this area of of population biology, it is row sense forms the hard conceptual accept that that a broader broader perspective perspective is needed. needed. This volume promotes promotes now time to accept such of the such a view. Inevitably, an expansion expansion of the metapopulation concept concept will attract attract applications to an even wider of situations can foresee, and some wider range range of situations than than we can foresee, and some of these applications will not turn of these applications tum out to be productive. productive. During During a period period of of rapid rapid growth, excesses excesses may occur and crossed. This is the time-honored and limits limits may be crossed. time-honored process of any worthprocess by which which the the scientific scientific community tests the the applicability of worth while idea or model. idea or We thank Chuck Crumly of Academic Press for for encouraging this We thank Chuck Crumly of Academic Press encouraging us to edit this second and for for all his assistance assistance during during the the process. following col colsecond volume volume and process. The The following leagues greatly helped chapters: Milo Adkinson, Richleagues greatly helped us in reviewing individual individual chapters: Milo Adkinson, Rich ard Barnes, Barton, Jan Jan Bengtsson, Berkson, Ian Ian B illick, Ted Ted Case, ard Barnes, Nick Nick Barton, Bengtsson, Jim Berkson, Billick, Case, Diane Gordon Fox, Andy Hansen, Diane Debinski, Debinski, Torbj6rn Torbjorn Ebenhard, Ebenhard, Gordon Fox, Andy Hansen, Alan Alan Hastings, Hastings, Phil Tad Kawecki, Phil Hedrick, Hedrick, Anthony Anthony Ives, Tad Kawecki, Joshua Joshua Kohn, Kohn, Russ Russ Lande, Lande, Simon Simon Levin, Trevor Price, Levin, Sean Sean Nee, Nee, Isabelle Isabelle Olivieri, Olivieri, Trevor Price, Jonathan Jonathan Silvertown, Dan Dan SimSim berloff, berloff, Monte Monte Slatkin, Slatkin, Peter Peter Stacey, Stacey, Mark Mark Taper, Taper, Chris Chris Thomas, Thomas, Rick Rick Walker, Walker, Christian Wissel, Wissel, and and Greg Greg Witteman. Witteman. We We also also thank Pia Vikman Vikman for her secresecre Christian thank Pia for her tarial contribution to Deborah Moses tarial contribution to the the project. project. Chuck Chuck Crumly Crumly and and Deborah Moses of of Academic Academic Press Press have have been been a pleasure pleasure to work work with.
Ilkka Hanski Hanski Ilkka Michael E. Gilpin Gilpin Michael
P
A
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CONCEPTUAL CONCEPTUALFOUNDATIONS FOUNDATIONS
The three chapters in this section explore the scope of of the applications. Hanski and SimSim metapopulation concept and its applications. berloff the history of studies and and the berloff sketch the of metapopulation metapopulation studies of approaches, both both theoretical and empirical, empirical, that have range of been used in single-species single-species studies. Harrison and Taylor assess assess critically the pertinence the metapopulation critically pertinence of of the metapopulation approach approach to field studies situations. studies and and expand expand their review review to multispecies multispecies situations. Wiens more more directly connects connects the metapopulation metapopulation concept concept to the complexities of of real landscapes. landscapes. Hanski and Simberloff Simberloff outout line in some detail the use (and misuse) misuse) of of the metapopulation metapopulation concept conservation, where an apparent paradigm shift concept in conservation, apparent paradigm shift has occurred from the dynamic theory of island biogeography to occurred from the dynamic theory of island biogeography to the metapopulation metapopulation theories. theories. The gradual gradual unfolding unfolding and and evolution evolution of of the the metapopulation metapopulation The concept from from the the pioneering pioneering studies studies of of Sewall Sewall Wright, Wright, AndreAndre concept wartha and Birch, Huffaker, Huffaker, Den Den Boer, Ehrlich, Gadgil, and and wartha have been been narrated narrated previously previously and and are are summarized summarized here here Levins have Hanski and and Simberloff Simberloff and and by by Harrison Harrison and and Taylor. Harrison Harrison by Hanski and and Taylor Taylor make make the the interesting interesting point point that that the the origin origin of of the the metapopulation metapopulation idea idea is different different in single-species single-species and and in mulmul-
tispecies tispecies studies. studies. Single-species Single-species studies have tended tended to empha emphats of size the benefi benefits of migration in leading to the establishment establishment of of new populations populations and thereby compensating compensating for for extinctions extinctions in small habitat species metapopulation habitat patches. patches. In the multi multispecies metapopulation sce scenarios, narios, the key issue has been been the locally locally unstable unstable interaction interaction among competitors competitors and between between a prey and its predator. predator. Habitat fragmentation beneficial in creating the possibility fragmentation can be beneficial possibility for asynchronuous uctuations, which asynchronuous fl fluctuations, which can enhance enhance regional sta stability. A high rate of of migration may eliminate eliminate such asynchrony, asynchrony, and is hence potentially harmful for for regional persistence persistence in mul multispecies tispecies metapopulations. metapopulations. Multispecies Multispecies metapopulation metapopulation theory is further further discussed discussed by Nee, May, and Hassell and by Holt in Part Part II. etapopulation Dynam In the predecessor predecessor of of this volume, M Metapopulation Dynamics: Empirical and Theoretical Investigations (Gilpin and Han Han99 1 ), metapopulation ski, 11991), metapopulation dynamics dynamics was seen to imply signif significant turnover turnover of of local populations, populations, local extinctions, extinctions, and ' s orig colonizations. colonizations. This notion notion follows follows directly from Levins Levins's original concept concept of of a metapopulation metapopulation as a population population of of populations, populations, analogous analogous to a population population of of individuals individuals with finite lifetimes. lifetimes. This narrow classical view of of metapopulations metapopulations has now now become become superceded superceded by a broader broader view, where where any assemblage assemblage of of dis discrete local local populations populations with with migration migration among among them them is consid considered ered to be a metapopulation, metapopulation, regardless regardless of of the rate of of population population turnover. turnover. (In a nonequilibrium nonequilibrium metapopulation metapopulation declining declining toward extinction extinction even among-population among-population migration is not a necessary criterion, but a system with no turnover mi turnover and no no migration would not classify as a metapopulation.) im metapopulation.) There are important portant questions questions to be asked about the role of of migration migration in (local) population dynamics, and these questions questions are most most nat naturally asked in a metapopulation (regional) (regional) context. Meta Metapopulation dynamics in the narrow narrow sense, with significant significant pop population turnover, turnover, is of of course course included included in metapopulation metapopulation dynamics in the broad broad sense. The realization that natural natural populations populations exemplify many many kinds of of spatial population population structures structures has stimulated stimulated a termi terminology, originally due to Susan Susan Harrison, Harrison, and including including entries such as patchy populations populations (not really metapopulations), metapopulations), clas classical (Levins) metapopulations, island metapopula metapopulations, mainlandmainland-island metapopulations, source - sink metapopulations, source-sink metapopulations, and nonequilibrium nonequilibrium metapopulations. metapopulations. These These concepts concepts and types of of metapopulation metapopulation structures structures are discussed discussed by Hanski and Simberloff Simberloff and and by Har Harrison and Taylor. The danger danger here is that too much much emphasis is cation, defi nition of given to classifi classification, definition of ideal types, which which in itself itself
does not guarantee any better understanding understanding of of the ecology, genetics, and evolution evolution of of metapopulations. What What matters is what what works. Does the "metapopulation approach" approach" help answer answer important questions questions about about spatially structured populations? populations? Does Does it provide provide us with scientific insight to the problems problems in which we are are interested? All this being said, there still is a need need to be concerned concerned with the type of of spatial spatial structure of of populations populations in any empirical study and in an application of of the metapopu metapopulation concept and models to real populations. populations. One should avoid the temptation of of pigeonholing every population with some form of of patchiness as a "metapopulation," as Harrison Harrison and Tay Taylor warn. In the worst case, this may obscure what what is important and draw attention to elements that are less critical. Unfortu Unfortunately, there are no easy answers answers here; one simply has to know the species and one has to understand the interactions interactions between the populations and their environment. Metapopulation biology may be a multifaceted subject, but there is one common element that characterizes characterizes this approach approach to population biology. The The metapopulation approach approach is based on the notion notion that space is not only discrete but that there is a binary distinction between suitable and unsuitable unsuitable habitat types. If this does not fi fitt one's one's idea of of a particular environment, one is probably better off off in using some approach approach other other than than the metapopulation metapopulation approach. An An important reason reason for the appeal of con of the metapopulation metapopulation concept concept comes from our our subjective conviction that natural lansdscapes truly are, for for many species, patchworks patchworks of of one or several several habitat types. Though Though the metapopulation view of of nature nature is complex enough, enough, it appears to be hopelessly hopelessly simplified in comparison of of how landscape ecologists view reality. Wiens in his chapter chapter lists four four components components that characterize landscape landscape ecology: variation in patch quality, variation in the quality of of the surrounding surrounding en environment, boundary boundary effects, and how how the landscape affects patch patch connectivity. Wiens Wiens is correct in suggesting that that most of of these elements are by and and large missing missing from from metapopulation models, which which are typically focused focused on idealized habitat patches in a featureless featureless landscape. Recent studies of of Andren Andr6n and Green (cited by Wiens) appear appear to suggest that where the suitable habitat fragments for for some species species cover only a rela relatively small fraction fraction of of total area (let us call these LC land landscapes, for for low coverage), patch area and isolation effects tend to be signifi cant; but where significant; where much of of the area is covered by more or less suitable habitat (HC landscapes, landscapes, for for high coverage), coverage), other a comother factor", factors, such as exactly how how individuals move in acom-
plex landscape, landscape, begin to dominate. Now, it so happens that the classical metapopulation concept implicitly assumes a LC land landscape, hence the tradition of of representing representing habitat patches as dots on maps, rather rather than drawing them as realistic habitat frag fragments. There appears to be a real difference difference between between the two traditions here, as they have been largely concerned concerned with either LC landscapes landscapes (metapopulation ecology) or HC landscapes (landscape ecology). As Wiens stresses, it is imperative for for the practical application land application of of both metapopulation biology and landscape ecology in conservation and planning that more common ground is established by developing appropriate appropriate theory and de designing appropriate field studies. Some necessary constituents of a more more unified approach seem relatively easy to achieve. For instance, it should not be too diffi cult to correct among-patch distances by taking into difficult account how the features of landscape affect of the intervening landscape ovement behavior. On the individual individual m movement the other hand, hand, when con considering HC landscapes, landscapes, patch inade patch models models are likely to be inadequate anyway. Metapopulation theory may well remain a useful practical tool for LC landscapes, landscapes, with with relatively small and iso isolated lated fragments of of suitable suitable habitat, habitat, but the "reserve "reserve mentality" that that this approach approach implies should give away, as Wiens Wiens argues, to "mosaic management" management" of of the the environment in HC landscapes. landscapes. Today, we do not yet have a conceptual and practical practical synthesis of of metapopulation biology and landscape landscape ecology, but no doubt the time will come when we will.
The Metapopulation Approach, Approach, Its History, History, Conceptual Conceptual Domain, and Application Application to Conservation Conservation Ilkka llanski Hanski
Daniel Simberloff
I. INTRODUGION INTRODUCTION At no period in the history of of ecology has the spatial structure of of populations populations and communities been entirely ignored, but the role that space plays in forming ecological patterns patterns and in molding processes has been viewed very differently in different times ((Mclntosh, McIntosh, 11991). 99 1 ). In the 11960s 960s and 1 970s, theoretical ecology and 1970s, was largely focused on issues other May, 11976a), 976a), with other than than spatial spatial dynamics dynamics ((May, notable MacArthur and Wilson, 11967), 967), and notable exceptions exceptions ((MacArthur and field ecologists tended tended to and space is introduced introduced in various follow suit. Today, space is in the forefront and ways into all fields of ecology and population biology more generally. Whether Whether one is interested in processes occurring at the level of genes, individuals, popu populations, or communities, spatial structure is widely seen as a vital ingredient ingredient of of better and more powerful theories, and good empirical work involving space is seen as a great challenge ((Kareiva, Kareiva, 11990). 990). Five years ago, before the publication of the predecessor of this volume (Metapopulation (Metapopulation Dynamics: Dynamics: Empirical and Theoretical TheoreticalInvestigations, Investigations, Gilpin and Hanski, 11991), 99 1 ), the metapopulation concept concept was new to most biologists. Since then, literature on metapopulations has grown exponentially, with a doubling time of less than Fig. 11). ). The metapopulation concept than 2 years ((Fig. concept has by now been firmly established established in population population biology and beyond; we review and analyze in this
Metapopulation Metapopulation Biology Biology
Copyright Academic Press, Inc. Inc. All onn reserved. Copyright © 9 1997 1997 by Academic All rights rights of of reproduction reproduction in in any any fform reserved.
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chapter the spread spread of of the metapopulation metapopulation concept concept to conservation conservation biology and and chapter applications. applications. What is the metapopulation metapopulation approach? approach? A more more complete complete explication is given What nutshell the two key premises premises in this approach approach to popUlation population below, but in a nutshell biology are that populations populations are spatially structured structured into assemblages assemblages of of local biology breeding breeding populations populations and that that migration among among the local populations populations has some some effect on local dynamics, including including the possibility of population population reestablishment reestablishment effect contrast with those of of standard models of following extinction. These premises contrast demography, population population growth, genetics, and community interaction interaction that assume demography, a panmictic popUlation population structure, with all individuals equally likely to interact
1 The The Metapopuiation MetapopulationApproach Approach
7 7
with any others. Population Population biology has has made productive use of of such models models for for at least 1100 00 years, but need to account for position of of but today today there is a distinct need for the position individuals populations in space. This need individuals and and populations need has arisen from from the intrinsic intrinsic development development of of population population biology as a science, science, but the trend trend has has clearly been been strengthened strengthened by the demand demand for for professional professional advice on environmental environmental issues typ typically involving space. space. In the past few few years, metapopulation metapopulation studies studies have have shed shed new new light on such phenomena phenomena as patterns patterns of of distribution distribution and and population population turnover turnover dynamics dynamics in frag fragmented Hanski, this volume; mented landscapes ((Hanski, volume; Harrison Harrison and Taylor, this volume; volume; Thomas volume; van der Thomas and and Hanski, Hanski, this volume; volume; Smith Smith and and Gilpin, Gilpin, this volume; der Meijden Meijden and vol and van der der Veen-van Veen-van Wijk, Wijk, this volume), landscape landscape ecology (Wiens, (Wiens, this volume) Holt, this volume), population population viability and ume) and and community community structure structure ((Holt, and time to extinction volume), coexistence extinction (Gyllenberg (Gyllenberg et et ai., al., this volume; Foley, this this volume), coexistence of of competing Nee et volume), competing species, species, and and of of prey and and their their natural natural enemies enemies ((Nee et ai., al., this volume), evolution evolution of of migration migration rate and and other other life-history life-history traits (Olivieri and and Gouyon, Gouyon, this volume), Ims and volume), ecological ecological consequences consequences of of migration migration ((Ims and Yoccoz, Yoccoz, this this volume; Stacey Stacey and and Taper, this volume), volume), unexpectedly unexpectedly high high levels of of inbreeding inbreeding and and low heterozygosity in natural Hedrick and volume), patterns patterns natural populations populations ((Hedrick and Gilpin, Gilpin, this this volume), of (Barton and of genetic differentiation differentiation (Giles and Goudet, this volume), volume), adaptation adaptation (Barton and Whitlock, Frank, this volume). As Whitlock, this volume), volume), and coevolutionary coevolutionary processes processes ((Frank, is apparent apparent from from the citations, citations, these these developments developments are are well represented represented in the the chapters literature at chapters in this volume, volume, which which provide provide an excellent entree entree to the the literature large. There are many advantages metapopulation approach, advantages of of a metapopulation approach, but success success may also breed breed problems. problems. As As in any scientifi scientificc field experiencing experiencing rapid rapid growth, growth, there there is the danger danger of of blurring blurring of of concepts. There There is the temptation temptation to view any system with any kind of of patchiness patchiness at any spatial spatial or or even temporal temporal scale as a "metapop "metapopulation." 1 99 1 , 11994b; 994b; Harrison ulation." Harrison Harrison ((1991, Harrison and Taylor, this volume) volume) cautions us about this tendency. Anticipating the kind kind of of verbal entropy that has has enveloped enveloped many terms 1 99 1 volvol terms in population population biology, Hanski Hanski and and Gilpin Gilpin sketched sketched in the 1991 ume of scale, hihi ume the meaning meaning of of the term term "metapopulation," "metapopulation," highlighting highlighting issues of erarchy, and and a requirement requirement for some population population turnover. turnover. We We feel a need need to dwell on the same same issues in this chapter, chapter, and and we provide provide a revised revised succinct succinct glossary glossary of of the commonly commonly used used terms in the literature. literature. First, however, however, let us examine examine briefl brieflyy the history of of the metapopulation metapopulation concept. concept.
II. BRIEF OF METAPOPULATION BRIEFHISTORY HISTORYOF METAPOPULATIONSTUDIES STUDIES The metapopulation metapopulation concept concept has has a pedigree dating dating back back to the early part of of this century, but but until recently this tradition tradition played only a minor minor and and episodic episodic role in the intellectual population biology. For For a long time, the pre intellectual advance advance of of population prevailing view was one one emphasizing emphasizing persistence persistence and and stability of of local populations, populations, or as McIntosh 1 99 1 ) put it, "the Mclntosh ((1991) "the great great tradition tradition of of balance balance of of nature, nature, going going back back
88
IIkko Honski and Daniel Simberloff Ilkko Honski and Daniel Simberloff
to antiquity, imputed imputed to nature nature homogeneity, homogeneity, constancy, or equilibrium equilibrium and ab abhored hored thoughts thoughts of of extinction and randomness." randomness." In evolutionary 1 93 1 , 11940) 940) had the insight that evolutionary biology, Sewall Sewall Wright ((1931, evolution evolution might proceed proceed rapidly in spatially structured structured popUlations, populations, especially if there are are local local extinctions extinctions and and recolonizations. recolonizations. Wright's Wright's shifting balance balance theory has remained an intriguing, understood, and little tested model intriguing, imperfectly understood, model ever since since (Barton (Barton and and Whitlock, Whitlock, this volume). Wright's Wright's work work may may have stimulated stimulated popUlations in the fi rst half interest in spatially structured structured populations first half of of this century, repre represented for instance by studies of Boycott ( 1 930), Diver ( 1 938), and Lamotte studies of Boycott (1930), Diver (1938), ((1951) 1 95 1 ) on ecology and genetics of of snail populations populations (for a more thorough thorough dis discussion, 996a). Pioneering cussion, see Hanski, Hanski, 11996a). Pioneering quantitative quantitative studies studies in epidemiology epidemiology ((Ross, Ross, 11909, 909, Kermack and 927; see Anderson 1 99 1 ; and McKendrick, McKendrick, 11927; Anderson and and May, 1991; theo Nee et et al., al., this volume) volume) are now seen as closely linked linked conceptually and theometapopulation studies, retically to metapopulation studies, but that connection connection remained remained without com comment May, 11991; 99 1 ; Lawton Lawton et 994; Nee, 1994). 1 994). ment until recently recently ((May, et al., al., 11994; The The ecological ecological implications implications of of the metapopulation metapopulation concept concept were were not consid considered 954, when ered before before 11954, when Andrewartha Andrewartha and and Birch Birch published published their their distinguished distinguished text text on animal animal ecology. Drawing Drawing on their wide experience experience from insect population population ecology, Andrewartha factors" Andrewartha and Birch found found the "dogma "dogma of of density-dependent density-dependent factors" unacceptable. They emphasized popUlations, documented unacceptable. emphasized wild oscillations oscillations of of populations, documented fre frequent quent local extinctions, extinctions, but also recognized recognized the possibility possibility of of reestablishment reestablishment of populations 1 954) advo populations at vacated vacated localities. localities. In brief, Andrewartha Andrewartha and Birch ((1954) advocated the view that phenomenon: that local population population extinction extinction was a common common phenomenon: "spots that that are occupied today may become become vacant vacant tomorrow tomorrow and and reoccupied reoccupied next week 954, p.87). week or next year" year" (Andrewartha (Andrewartha and and Birch, 11954, p.87). However, However, why did their their ideas fail to gain wider acceptance? acceptance? We We believe the reason reason is their nearly nearly cate categorical rejection rejection of of the concept concept of of density-dependent density-dependent population population regulation. regulation. The The Andrewartha Andrewartha and Birch notion about population population dynamics in space was largely ignored ignored and eventually forgotton. forgotton. The incipient metapopulation metapopulation concept concept nonethe nonethe950s and 960s, in works works of Huffaker ((1958), 1 95 8), less had a quiet existence existence in the 11950s and 11960s, of Huffaker den 1 968), Ehrlich and Raven 1 969), Gadgil ((1971), 1 97 1 ), and undoubtedly den Boer Boer ((1968), Raven ((1969), undoubtedly a few others. The 1 963, 11967) 967) dynamic theory of The MacArthur MacArthur and Wilson ((1963, of island biogeography metapopulation concept, even if biogeography has much much in common common with the metapopulation MacArthur multi species communities, MacArthur and Wilson were primarily concerned with multispecies communities, as we discuss discuss below. below. The The term term "metapopulation" "metapopulation" was introduced introduced in the works works of of Richard Richard Levins Levins in 11969 969 ((1969a) 1 969a) and 970. The and 11970. The word word itself itself suggests suggests a population population of of populations, populations, with colonization metapopulation likened colonization and and extinction extinction of of local populations populations in a metapopulation likened to births hence the to births and deaths deaths of of individuals individuals in a local population population ((hence the emphasis emphasis on population population turnover turnover in "classical" "classical" metapopulation metapopulation studies). Levins' Levins'ss work work marks the beginning though, beginning of of contemporary contemporary metapopulation metapopulation biology. It is puzzling, puzzling, though, that the early lead that that Levins provided provided was followed followed by a period period of of nearly 20 years of Fig. 11). ). We return to the possible below, of recess ((Fig. possible reasons reasons for for this delay below, in the the section section on metapopulations metapopulations and and conservation conservation biology.
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CONCEPTUAL DOMAIN DOMAIN AND AND METAPOPULATION METAPOPULATION APPROACHES APPROACHES III. CONCEPTUAL A fundamental fundamental assumption assumption of of the the original original metapopulation metapopulation concept concept (Levins, ( Levins, A 1 969a) is that that space space is discrete discrete and and that that it it is is possible possible and and useful useful to to distinguish distinguish 1969a) between habitat patches that suitable for for the the focal focal species species and and the the rest rest of of the the between habitat patches that are are suitable environment, often called called the the matrix. matrix. In this this respect respect the the metapopulation approach environment, often metapopulation approach akin to to the the dynamic theory of of island island biogeography (MacArthur ( MacArthur and and is closely akin Wilson, 1967) 1 967) but but differs differs from from landscape landscape ecology (Wiens, ( Wiens, this volume). The The Wilson, metapopulation concept concept also presumes that that the the habitat habitat patches patches are are large large enough enough metapopulation to accommodate accommodate panmictic panmictic local populations, popUlations, but but not not larger. larger. Other Other fields fields of of ecolecol with spatial spatial patchiness, patchiness, but but either either at a smaller smaller (foraging (foraging thethe ogy are are concerned concerned with Krebs and and Davies, Davies, 1984) 1 984) or or at a larger larger scale (e.g., (e.g., much much of of landscape landscape ories; Krebs Forman and and Godron, Godron, 1986; 1 986; GAP GAP analyses: Scott Scott et et al., ai., 1991; 1 99 1 ; ecology: Forman geographical ecology: Ricklefs Ricklefs and and Schluter, Schluter, 1993) 1 993) than than the the scale scale of of (panmictic) ( panmictic) geographical local populations. The The concept concept of of an ideal ideal metapopulation metapopulation a la Levins includes local three habitat patches patches have equal areas isola three other simplifying assumptions: habitat areas and isolametapopulation have entirely independent independent (uncor(uncor tion, local populations in the metapopulation related) dynamics, and the exchange exchange rate of individuals among populations related) rate of among local populations migration has no real effect effect on local dynamics in the the existing existing is so low that migration populations: local dynamics occur occur on a fast fast time scale scale in comparison comparison with metameta population dynamics. dynamics. No real metapopulation metapopulation completely satisfies all these these requirements. requirements. However, However, No assumptions, such as equal patch isolation, can can be the more more specific specific assumptions, patch areas areas and and isolation, relaxed without need for a major major conceptual conceptual amendment. amendment. This is not unlike how relaxed without need unlike how com the popUlation population concept concept is used in population population biology: no real population population completely satisfies all the criteria panmictic, population. What criteria of an ideal, closed and panmictic, really matters is the notion of of discrete local breeding populations connected by migration. We We suggest that if this assumption cannot be defended, defended, some other approach should be used instead of the metapopulation approach; and conversely, the more more distinct distinct and and smaller the local breeding breeding populations populations are, the more more useful the metapopulation approach 1 99 1 ) used poppop approach is likely to be. Hanski and Gilpin ((1991) ulation turnover, local extinctions and colonizations, as the hallmark hallmark of of true meta metapopulations. By this definition, the mainland-island mainland-island systems studied in the dy dynamic theory of island biogeography and in recent metapopulation models 99 1 ; Hanski and 993) would not count as metapopula (Gotelli, 11991; and Gyllenberg, 11993) metapopulations. Following the current usage of the term, we now include mainlandisland mainland-island structures among other metapopulation structures. It has been suggested that "much" migration among local populations popUlations makes the metapopulation approach less useful ((Harrison, Harrison, 11994b). 994b). While it is true that the classical concept (Levins, 11969a) 969a) implicitly assumes a low migration rate, so low that migration plays no role in the dynamics of existing local populations, more recent theoretical Hassell et 99 1 a, 11994, 994, Gyllenberg and Hanski, 11992; 992; theoretical ((Hassell et ai. al.,, 11991 Nee Hanski et 995a,b) has made Nee et et ai. al.,, this volume) and empirical work ((Hanski et ai., al., 11995a,b) good use of the metapopulation concept even when some tens of percents of
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individuals per per generation generation leave leave their their natal natal patch. patch. An An important important issue here here is is the the individuals spatial scale scale of of migration. migration. Theoretical Theoretical studies studies suggest suggest that that aa low low rate rate of of longlong spatial distance migration migration has has often often about about the the same same consequences consequences as as aa high high rate rate of of shortshort distance distance migration migration (Nachman, (Nachman, 1991). 1 99 1 ). Clearly, Clearly, if if migration migration rate rate is is very very high, high, say say distance > 50%, 50%, and and if if migration migration distances distances are are not not limited, limited, a metapopulation metapopulation approach approach is is > unlikely to be helpful. The fundamental criterion, however, is whether or not the unlikely to be helpful. The fundamental criterion, however, whether or not the metapopulation approach approach is useful useful in in elucidating elucidating the the questions questions in in which which we we haphap metapopulation pen to to be be interested, interested, not not whether whether migration migration rate rate is high high or or low. From From the the perper pen spective of of traditional traditional population population biology, the the question question is whether whether the the implicit implicit spective assumption that that migration migration makes makes no no difference difference to to the the dynamics dynamics of of the the focal focal poppop assumption ulation useful approximation approximation or or not. not. ulation is a useful Our remarks have Our remarks have been been directed directed at at the the population population ecological ecological properties properties of of metapopulations. Genetic Genetic and and evolutionary consequences consequences of of these these metapopulation metapopulation metapopulations. structures enlarge enlarge the the biological domain domain of of the the metapopulation metapopulation concept concept as dede structures scribed and Whitlock Whitlock (this scribed by Olivieri and Gouyon (this volume) and Barton Barton and volume). volume). metapopulation book, 199 1 ) defined defined a set In the the previous metapopulation book, Hanski Hanski and and Gilpin Gilpin ((1991) set of key metapopulation metapopulation terms terms in the terminol of the hope hope of of promoting a more uniform terminolWe repeat exercise here, here, with a of terms terms (Table (Table ogy. We repeat this exercise a revised and expanded expanded list of but a few comments are warranted. The The I). This list is largely self-explanatory, self-explanatory, but few comments are warranted. source-sink concept literature. Pulliam (1988) ( 1 988) source-sink concept continues continues to cause cause confusion in the literature. efined sources sources and sinks on whether emigration emigration exceeds exceeds immigraimmigra d defined on the basis of of whether or vice versa, at equilibrium. equilibrium. This definition definition is useful for population genetic genetic tion, or for population emphasizing asymmetry in gene flow, which which may have important purposes, in emphasizing consequences for consequences for genetic structure structure and adaptation adaptation (Barton (Barton and Whitlock, this volume; Giles and Goudet, this volume). The definition definition given in Table Table I, which is based on the expected expected population growth growth rate at low density, in the absence absence of of intraspecifi intraspecificc density dependence, dependence, may often be preferable preferable for ecological purposes. In the latter latter case, sinks are populations that that would go extinct in the absence absence of of immigration (by Pulliam's Pulliam's definition, a sink population may decline to a low but positive equilibrium Suth equilibrium value in the absence of immigration; Watkinson and Suth995). A third and potentially misleading sense in which the sourcesink erland, 11995). source-sink concept is often used is for a mixture of of small and large large habitat habitat patches. Popu Populations in small patches patches typically have a high risk of of extinction, extinction, but they are not necessarily "sinks" in the sense of 1 988) or Table I; small populations of Pulliam ((1988) have a high risk of stochastic extinction, even if the expected growth rate at low density and the expected equilibrium population size are positive ((Foley, Foley, this volume). volume).
A. Modeling Modeling Approaches Approaches The traditional approach to population biology assumes spatially unstruc unstructured populations. Modeling approaches to spatially structured populations can be divided conveniently into two classes, based on whether the model deals with
1 The The Metapopulation Metapopulation Approach Approach TABLE I
1II1
Metopopulotion Metapopulation Terminology Terminologya a
Term Term
Synonyms and and definition Synonyms
Patch Patch
Synonyms: Habitat Habitat patch, (habitat) (habitat) island, (population) (population) site, locality Definition: A continuous continuous area area of of space with all necessary resources resources for unsuit for the persistence of of a local population and and separated by unsuitable habitat from from other other patches (at any given time, a patch patch may be occupied occupied or empty) empty) Synonyms: Population, Population, subpopulation, subpopulation, deme deme Definition: Set of of individuals individuals that that live in the same habitat habitat patch patch and therefore interact with each other; most most naturally applied to "pop "populations" living in such small patches that all individuals individuals practi practically share a common common environment environment Synonyms: Composite Composite population, population, assemblage (of (of populations) populations) [pop [population (when "local populations" are called "subpopulations")] "subpopulations")] Definition: Set of of local populations populations within some some larger larger area, area, where where typically migration migration from from one local population population to at least some some other patches is possible (but see nonequilibrium nonequilibrium metapopulation) metapopulation) Synonyms: Metapopulation Metapopulation type Definition: Network Network of of habitat habitat patches patches which is occupied by a meta metapopulation population and which which has a certain distribution distribution of of patch patch areas areas and interpatch migration migration rates Synonyms: Classical metapopulation metapopulation Definition: Metapopulation Metapopulation structure structure assumed assumed in the Levins model: a large network network of of similar small patches, with local dynamics oc occurring curring at a much faster time scale than than metapopulation metapopulation dynamics; dynamics; in a broader sense used for systems systems in which all local populations, populations, even even if they may differ in size, have a significant risk of of extinction Synonyms: Boorman-Levitt Boorman- Levitt metapopulation metapopulation Definition: System of of habitat habitat patches (islands) located within dis dispersal distance distance from a very large habitat habitat patch (mainland) where where the local population never goes extinct extinct (hence (hence mainland-island mainland-island metapopulations metapopulations do not go extinct) extinct) Definition: Metapopulation Metapopulation in which there there are are patches patches in which the population population growth rate rate at low density and in the absence absence of of im immigration migration is negative (sinks) and patches patches in which the growth growth rate at low density is positive (sources) (sources) Definition: Metapopulation Metapopulation in which which (long-term) extinction rate rate ex exceeds ceeds colonization colonization rate or or vice versa; versa; an extreme extreme case is where local populations populations are located so far far from from each other other that there is no migration migration between between them and hence no no possibility for for recoloni recolonization Synonyms: Colonization-extinction Colonization-extinction events events (or dynamics) Definition: Extinction Extinction of of local populations populations and establishment establishment of of new local populations ex populations in empty habitat patches patches by migrants migrants from from existing local populations populations Synonyms: Expected Expected life-time of of a metapopulation metapopulation Definition: The Definition: The length of of time until all local populations populations in a meta metapopulation population have gone gone extinct
Local population population
Metapopulation Metapopulation
Metapopulation Metapopulation structure
Levins metapopulation metapopulation
Mainland- island Mainland-island metapopulation metapopulation
Source-sink Source-sink metapopulation metapopulation
Nonequilibrium Nonequilibrium meta population metapopulation
Turnover Turnover
Metapopulation Metapopulation persistence persistence time
(continues)
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TABLE TABLE II (continued) Term
Synonyms and definition
Patch Patch model
Synonyms: Occupancy Occupancy model, presence/absence presence/absence model Definition: A metapopulation metapopulation model in which local population population size is ignored patches ignored and the number number (or fraction) of of occupied occupied habitat habitat patches is modeled modeled Definition: 1 969a; see Hanski, this Definition: The The model presented presented by Levins ((1969a; volume) volume) Definition: A model population sizes model in which the distribution distribution oflocal of local population sizes modeled is modeled Definition: A model model of of the stationary probabilities probabilities (incidences) (incidences) of of patches functions of patches being occupied, occupied, generally generally assumed assumed to be functions of the sizes and isolations isolations of of the patches Synonyms: Island model Definition: Model in which all local populations populations are equally con connected; patch models are patch models and structured metapopulation metapopulation models spatially implicit implicit models Synonyms: Lattice (grid) model, model, cellular automata automata model, stepping steppingmodel stone model migration is distance-dependent, Definition: Model in which which migration distance-dependent, often restricted to the nearest habitat habitat patches; the patches patches are typically identical identical cells on a regular regular grid, and only presence presence or absence absence of of species in considered (the model is called a coupled the species in a cell is considered map continuous map lattice model if population population size in a patch is a continuous variable) Synonyms: Spatially explicit model (note that we make make a distinction distinction between spatially explicit and spatially spatially realistic realistic models) between Definition: Model that assigns assigns particular areas, spatial spatial locations, locations, and possibly other other attributes attributes to habitat patches, patches, in agreement with real real patch networks; spatially realistic models include simulation mod models and the incidence function model model
Levins Levins model model Structured Structured metapopulation model Incidence Incidence function function model Spatially implicit implicit metapopulation metapopulation model model
Spatially Spatially explicit explicit metapopulation metapopulation model
Spatially realistic metapopulation model metapopulation
a
Modified from Hanski and Gilpin, 11991, 99 1 , and Hanski, 11996a. 996a. "Modified
interactions populations connected interactions among among two conspecific populations connected by migration, migration, or with interactions interactions among among many many local populations. populations. The former former approach approach is useful useful when when the focus focus of of the study is specifically on the effect effect of of migration migration on local dynamics dynamics and one is willing to assume assume that that populations populations are so effectively regulated regulated that that extinctions Levin, 11974; 974; Holt, 11985; 985; Gyllenberg 993). In extinctions do not occur occur ((Levin, Gyllenberg et et a!. al.,, 11993). metapopulation metapopulation studies in the narrow narrow sense, when there there is population population turnover, turnover, it is necessary to resort resort to modeling modeling approaches approaches assuming assuming many many habitat habitat patches patches and and local populations. populations. Among Among these these approaches, approaches, we distinguish distinguish between between spatially im imHanski, 1994c). 1 994c). plicit, spatially explicit, and spatially realistic realistic approaches approaches ((Hanski, 1. Spatially Implicit Approaches Approaches
Truly signifi cant insights cation of significant insights are are often often based based on a critical critical simplifi simplification of what what at fi rst appears first appears a hopelessly hopelessly complex complex problem. problem. The model model that that Levins Levins (l969a, (1969a,
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11970) 970) constructed constructed to caricature caricature metapopulation dynamics is an excellent excellent example. example. Instead of attempting to extend a model of a single population to many popula Instead of attempting of population populations connected by migration, Levins modeled the changes in the number connected changes number of of such populations, effectively ignoring what what happens happens in each one of them and where where in space they happen to be located ((Hanski, Hanski, this volume). For the latter latter reason, reason, the Levins model and other related patch models (Table (Table I) are spatially implicit; the habitat patches and local populations are discrete (and are generally assumed to have independent independent dynamics), but they are are assumed to be all equally connected connected to each other. In spite of of this simplifying assumption, which can be generally de defended only for metapopulations close to steady state and with no strong spatial for strong aggregation, the patch models allow us to analyze many interesting questions about metapopulation dynamics, starting with the conditions of of metapopulation persistence persistence in a balance balance between between local extinctions and colonizations. The advan advantage of of the spatially implicit approach approach is that it greatly facilitates facilitates the mathematical mathematical and conceptual analysis; the disadvantage is that it can be used to study only a subset of of all interesting questions. Thinking about the restrictive assumptions assumptions of the Levins model and other patch models, ecologists have asked what happens when local dynamics are inin cluded in the metapopulation metapopulation model. What happens when the habitat patches are of different sizes and when the local populations have different extinction prob probabilities? abilities? What if migration rate rate is high enough to "rescue" local populations before extinction? What are are the consequences consequences of of real real spatial locations of of local populations? metapopu populations? What if extinction events are are correlated over the entire entire metapopulation? What What if there there is spatial asymmetry and source and sink populations? populations? Some of of these questions have been explored in the context of of spatially implicit models 988; Harrison and Quinn, 1989; 1 989; Hanski and 1 993; Gyl (Pulliam, 11988; and Gyllenberg, 1993; Gyllenberg et et at. al.,, this volume), but it comes as no surprise that that at some point we have to tum turn to models that that incorporate specific specific information on the spatial loca locahave tions of of populations. populations. Incidentally, most analyses of metapopulation genetics genetics (Bar (Barton and Whitlock, this volume; Hedrick and Gilpin, this volume) have been based on the Levins model, which is essentially equivalent to what population population geneti geneticists call the "island model." As in ecology, there there is an increasing increasing need to add space in a more explicit manner manner to metapopulation metapopulation genetic models. 2. Spatially Explicit Approaches
Under Under the rubric rubric of spatially explicit approaches approaches are are several related modeling modeling frameworks, such as cellular 1 993), interinter cellular automata automata models (Caswell and and Etter, Etter, 1993), acting particle systems ((Durrett, Durrett, 11989), 989), and coupled map lattice models ((Hassell Hassell et at., 11991a). 99 1 a). These et al., These modeling modeling approaches approaches assume that that "local populations" are arranged as cells on a regular lattice), with popUlation regular grid ((lattice), population sizes modeled as either discrete discrete or continuous variables. The key feature that distinguishes spatially localized interactions: explicit approaches from spatially implicit approaches is localized populations are assumed to interact only with populations in the nearby "cells." "cells." Localized interactions can have have profound dynamic consequences, consequences, such as very
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long times before the metapopulation settles to a steady state Hastings and Hig state ((Hastings Hig99 1 a; Nee et aI., this gins, 11994) 994) and spatially chaotic Hassell et chaotic dynamics ((Hassell et ai., al., 11991a; et al., volume). The disadvantage is that the the state state of of the metapopulation cannot cannot be de described presences and scribed simply by the fraction of of cells occupied; an entire vector vector of presences absences absences is needed. Such models require considerable considerable computation. An advantage is that, since each cell on the grid has a constant area area and constant spacing, the mathematical mathematical rules that govern local behavior are the same from cell to cell, and it is easy to write a computer program to model the dynamics. Lattice-based models and raster-based raster-based GIS descriptions descriptions in landscape landscape ecology share the the same format of representing space. Thus it is possible to develop com complex models that blur the distinction between between spatially explicit and spatially re realistic models ((below). below). From raster-based raster-based description of habitat suitability, one can aggregate "cells" "cells" into patches on which local populations may exist, thus reverting to a patch-based patch-based metapopulation model for for a dynamic analysis (Burg (Burgman et et al., 993; Ak�akaya, 994). al., 11993; Akqakaya, 11994). 3. 3. Spatially Spatially Realistic Realistic Models Models
Spatially realistic realistic models allow one to include in the model the specifi specificc ge geometry of particular particular patch networks: how many patches are there in the network, how large are they, and where infor where exactly are are they located? located? Including all this information in the model is necessary if one is interested interested in making specific quanti quantitative predictions about the dynamics of of real metapopulations. For instance, if we want to assess the likely consequences of destroying some particular particular patches patches in patch network, we need a spatially spatially realistic model. For For obvious reasons, the a patch spatially realistic field studies. realistic approach is closely linked with empirical field The incidence IF ) model ((Hanski, Hanski, 11994a,b, 994a,b, this volume) is perhaps incidence function ((IF) the simplest spatially realistic metapopulation model. The IF model is concep conceptually related related to the Levins model, but with the following critical differences: differences: there is a fi nite number finite number of of habitat habitat patches, and hence hence the model is stochastic stochastic in contrast to the deterministic Levins model; the patches are allowed to differ in area, which is assumed assumed to affect local extinction probabilities; probabilities; and and the the patches have specific spatial locations, which affect their probabilities probabilities of recolonization. Alternative Alternative spatially realistic approaches are based on extensive simulation of of many local populations connected by migration ((Hanski Hanski and Thomas, 11994; 994; Ak Ak�akaya, 994). Several generic models of this type are already available (Ak�ak qakaya, 11994). (Akqakaya, 11994; 994; Sjogren 996). Not surprisingly, meaningful appli Sjtigren Gulve and Ray, 11996). meaningful application of these models assumes much data. The extreme extreme approach approach is to simulate the birth, movements, reproduction, and death of individuals ((DeAngelis DeAngelis and Gross, 11992), 992), but this approach, which can be used for for any population structure, does not really take advantage of of the metapopulation concept. An individually based based modeling approach may nonetheless provide provide valuable insight into key pro processes affecting metapopulation dynamics, such as migration among populations ((Kindvall, Kindvall , 11995). 995).
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B. Empirical Empirical Approaches Approaches In a standard metapopulation study, a key initial task is to make standard ecological ecological metapopulation make a practical distinction between between habitat habitat and nonhabitat nonhabitat and and to delimit delimit the suitable habitat habitat patches patches in the study area. Suitable Suitable habitat habitat may be defined subjectively subjectively or or with the help Lawton and 1 99 1 ). An help of of statistical statistical methods methods ((Lawton and Woodroffe, Woodroffe, 1991). An experi experimental approach approach may be used used to test the accuracy accuracy of of an existing existing habitat habitat classifi classification: experimental ( Harrison, experimental introductions introductions to empty habitat habitat should should succeed succeed (Harrison, 11989; 989; Oates 990; Thomas, 992; Massot 994); introductions introductions Oates and and Warren, Warren, 11990; Thomas, 11992; Massot et et al., al., 11994); to nonhabitat Metapopulation studies focused nonhabitat should fail. Metapopulation focused on assemblages of of ex extinction-prone tinction-prone local local populations populations typically proceed proceed to record the the presence presence or ab absence of habitat patches of the focal species in the habitat patches and then then to analyze the effects effects of Verboom et al., 1991b; 1 99 1 b; of various various environmental environmental factors factors on patch patch occupancy occupancy ((Verboom et al., al., 11995a,b) 995a,b) and on the rates of Thomas 992; Hanski Thomas and Harrison, 11992; Hanski et et al., of extinction extinction and 99 1 ; Eber 994; Hanski 995b). and colonization colonization (Sjogren, (Sj/3gren, 11991; Eber and and Brandl, Brandl, 11994; Hanski et et al., al., 11995b). Other Other field studies have have been been concerned concerned with with more more permanent permanent local populations, populations, but ones migration (Stacey and ones whose whose dynamics dynamics are are significantly affected by migration Taper, this volume). volume). Experimental Experimental studies studies have attempted attempted to demonstrate demonstrate the pre predicted populations in a metapopulation dicted temporal stability of of local populations metapopulation as opposed opposed to that Murdoch et 1 996; Harrison that in isolated isolated local populations populations ((Murdoch et ai., al., 1996; Harrison and and Taylor, this volume). Landscape ecology ((Forman and Godron, Godron, 11986; Turner, 11989; Landscape Forman and 986; Turner, 989; Wiens, Wiens, this volume) volume) and and metapopulation metapopulation ecology share share a common common focus focus on space space and patchiness. The difference difference is primarily in the complex complex mosaic structure of of patchiness. mosaic structure object of volume). real landscapes landscapes that is the object of landscape landscape ecology ecology (Wiens, (Wiens, this volume). In contrast, contrast, metapopulation metapopulation studies studies typically typically assume assume that that the the patches patches which which are though this assumption used by the focal species species are of of the same type, though assumption is made made primarily for the sake of of keeping the models reasonably simple simple (see Holt, Holt, this re volume, for for metapopulation metapopulation models with two patch patch types). types). Empirical research in landscape landscape ecology has been been reluctant to use the the population population dynamic dynamic theory even if if in a rudimentrudiment theory that that metapopulation metapopulation ecology ecology purports purports to provide, provide, even ary form, two fields have form, and and consequently the two have developed largely independ independently. One One trend trend that is beginning beginning to change change this situation situation is the use of of GIS GISbased models based landscape landscape descriptions descriptions in generic metapopulation metapopulation simulation simulation models (Ak�akaya, 994). Today, ecologists have bases of digi (Ak~akaya, 11994). have access to huge data data bases of digitized information imminent arrival of low-cost information about about landscape landscape structure, and and the imminent arrival of low-cost global research in this global positioning positioning systems systems will greatly greatly facilitate further further empirical empirical research area. It should should come come as no no surprise surprise that the bulk bulk of of current current empirical empirical research research that is conceptually conceptually related related to the metapopulation metapopulation notion is conducted conducted in conservation conservation biology. We We therefore therefore devote devote the rest of of this chapter chapter to a more more thorough thorough scrutiny of conservation of the past past and present present links between between metapopulation metapopulation biology and and conservation biology.
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IV. METAPOPULATIONS METAPOPULATIONSAND AND CONSERVATION CONSERVATIONBIOLOGY BIOLOGY Conservation biology changed dramatically, beginning ca. 11975, 975, from a heavy emphasis on habitat relationships of individual species to a focus on refuge refuge design, guided by the dynamic theory of of island biogeography and the genetic deterioration 988). The two halves deterioration owing to drift drift and inbreeding (Simberloff, 11988). of this "new "new conservation biology" did not fi fitt together well, as the former dealt with species richness richness of communities, while the latter latter aimed at the population level. Currently, a replacement replacement of the island biogeographic component of of con conservation biology by metapopulation thinking is providing a more comfortable metapopulation fi t. Although fit. Although the incorporation of metapopulation models models into conservation bi biology has spurred spurred important insights, it has also led to some misfocused proposals.
A. The and Fall Fall of the Theory Island Biogeography The Rise Rise and Theory of Island Biogeography The theory of island biogeography ((MacArthur MacArthur and Wilson, 11963, 963, 11967) 967) quickly attracted Fig. 11)) by using attracted much attention from ecologists ((Fig. using simple math mathematics to focus on an easily obtained statistic (species richness) and depicting a dynamic nature that is nonetheless readily understood because because it is divided into small units, namely real or habitat 974, 11978a). 978a). The theory habitat islands (Simberloff, 11974, posits species richness on each island as a dynamic equilibrium maintained by continuing immigration of of all species, balanced by ongoing local extinctions on the island, primarily owing to demographic and genetic stochasticity. Clearly the island biogeographic theory shares key underpinnings underpinnings with meta metapopulation modelsmodelsmthe of nature nature into discrete entities, with movement the division of of of individuals among relatively unstable local populations. There is also an an ap apparent island biogeographic individ parent differencedifferencemisland biogeographic theory treats communities, not individual species. Its key statistic biogeo statistic is species richness. However, some island biogeographic graphic models are formally composites of of models for for individual species, with the community-wide immigration and extinction rates being simply sums of of the respective species-specific 969, 11983; 983; Gilpin and Diamond, species-specific rates (Simberloff, 11969, Diamond, 11981). 98 1 ). In these latter models, the underlying underlying conception conception of of what what is happening in nature - island version of Levins's Levins 's metapopulation nature is just a mainland mainland-island metapopulation concept (Hanski, (Hanski, this volume). However, even a species-based model of of this type, and even one with migration from from several sources, is still focused on a single island and on questions about the number number of of species and immigration and extinction rates on that island. A metapopulation model, even one in which the different sites and local populations are not modeled explicitly, focuses on the entire meta metapopulation population of of one or two species, using statistics such as the number of of sites occupied. In both types of models, an element element of arbitrariness arbitrariness is just how much move movement there there is between between sites for the models to remain remain useful. For For metapopulations, the bone of of contention is whether whether the movement is so frequent that one is dealing with a single population rather than a metapopulation, even if that that population
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may may be be so so large large that that individuals individuals are are likely likely to to interact interact with with their their neighbors neighbors only only ((Harrison, Harrison, 11991). 99 1 ). For For island biogeographic biogeographic models, models, the the argument argument is is whether whether groups of of conspecific conspecific individuals individuals on on sets sets of of islands islands are are separate separate populations populations or or groups just transient 975; Simberloff, transient parts parts of of one one widely widely ranging ranging population population (Smith, 11975; Simberloff, The latter argument has recrudesced recently with important conservation 11976). 976). The 1 988), extrapolating implications. Pimm et et al. al. ((1988), extrapolating from records of of breeding birds on small British islands, suggested suggested guidelines for how many individuals should should con constitute stitute propagules propagules for for reintroduction reintroduction efforts efforts based based on on the the body body size size of of the the species. species. Aside Aside from problems problems with the statistics statistics of extrapolation (Tracy (Tracy and and George, George, 11992), 992), Haila and 1 993) saw a more fundamental fl aw: the assumption and Hanski ((1993) flaw: that the birds on each island constituted a population, and their disappearance an are parts of wide-ranging, large large popula populaextinction. In their view, all these birds are tions and their disappearances from specifi specificc islands within the range are not pop population phenomena. Diamond and Pimm ((1993) 1 993) retorted that the birds of each island could be viewed as a population within a metapopulation. Within about a decade ((Fig. Fig. 11), ), the theory of island biogeography came to dominate much of conservation biology, with a series of nearly simultaneous papers (Terborgh, 11974, 974, 11975; 975; Diamond, 11975a; 975a; Wilson and Willis, 11975) 975) all advocating a set of "rules" of refuge design ostensibly based on the theory ((Fig. Fig. guration that would maximize 2). The rules each suggest a refuge confi configuration maximize species richness, and the papers describing the rules rules apparently stemmed stemmed from lectures given by E. O. Willis beginning in 11971 97 1 (Willis, 11984). 984). Although some of of the rules in fact were not based on the theory (references 1 988); they (references in Simberloff, Simberloff, 1988); publication in 1980 1 980 became popular in conservation circles, circles, particularly after their publication rst synthetic plan for dealing first dealing with a perceived disastrous wave of of extinc in the fi Union tions, World World Conservation Strategy, jointly authored by the International International Union for of Nature Nature and Natural Resources, the United United Nations, and for the Conservation of Natural Resources, and World Wildlife Wildlife Fund. With this imprimatur, imprimatur, it is unsurprising un surprising that that these these rules, the World and the the theory theory that that supposedly supposedly supported became the the governing paradigm and supported them, them, became paradigm conservation biology, reproduced in textbooks and published published in newspapers. newspapers. in conservation The The dominance dominance of of the the island island biogeographic biogeographic paradigm paradigm was was so strong strong that that even even studies that that today would would be be seen seen as metapopulation metapopulation research research were were published published as studies island island biogeographic studies, studies, with no no mention mention of of the the term term "metapopulation" "metapopulation" (e.g., (e.g., Fritz, Fritz, 1979). 1 979). was noted noted early that that most most ecological ecological publications publications citing citing island island biogeobiogeo It was graphic theory theory simply simply interpreted interpreted aa species-area species-area relationship relationship in in terms terms of of the the thethe graphic ory, ory, when when alternative alternative explanations explanations were were also also possible possible (Simberloff, (Simberloff, 1974), 1 974), and and that that there there was was little little empirical empirical evidence evidence for for continuing continuing local local extinctions extinctions of of the the sort sort envisioned by by the the theory theory (Lynch (Lynch and and Johnson, Johnson, 1974; 1 974; Simberloff, Simberloff, 1974). 1 974). Further, Further, envisioned as noted noted by by Smith (1975) ( 1 975) and and Simberloff Simberloff (1976), ( 1 976), by by defining defining the the comings comings to to and and as goings goings from from local local sites sites of of individuals individuals within within continuous continuous populations populations as as "immi"immi gration" and and "extinction," "extinction," one one could could almost almost always always claim claim that that extinctions extinctions and and gration" colonizations were were occurring, occurring, even even if if the the theory theory really really envisioned envisioned most most recruitrecruit colonizations ment to to local local populations populations as as being being by by in in situ situ reproduction reproduction rather rather than than immigration. immigration. ment
1188
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The 1 975; The "island "island biogeographic" biogeographic" rules rules for for refuge refuge design design (after Wilson Wilson and Willis, 1975; International Union Union for for the the Conservation Conservation of of Nature Nature and and Natural Resources, each rule, the International Resources, 11980). 980). For each design on alternative on design on the the left left is is seen seen as as superior superior to to the the alternative on the the right. right.
Nevertheless, Nevertheless, ecologists ecologists on the whole tended tended to view the theory theory favorably until around 11980, doubt about about the existence existence of widespread local extinction extinction be bearound 980, when doubt of widespread came pervasive (Gilbert, 1980; 1 980; Schoener 1 987b; Williamson, 989). Schoener and Spiller, 1987b; Williamson, 11989). The involves a subset The prevailing prevailing view view now now is that, in most most systems, "turnover "turnover involves subset of of fugitive fugitive populations, populations, with with many others, others, mostly much much larger, being permanent" permanent" (Schoener 987b). The decline (Schoener and Spiller, Spiller, 11987b). decline in citations of of "island "island biogeography" biogeography" ((Fig. Fig. 1) 1 ) reflects the declining longer seen as declining faith faith in the theory. Though Though it is no longer a model for biogeographic theory provided theoretical for much much of of nature, nature, island biogeographic provided a theoretical perspective from which to view a number number of - area perspective from of patterns, patterns, such as the species species-area relationship Haila and Jarvinen, 1982). 1 982). relationship ((Haila The biogeography The key key conservation conservation legacies of of the the dynamic theory of of island biogeography were ((1) 1 ) the interest in the the metaphor metaphor of of a refuge refuge as an island island or spaceship, (2) interest fragility of refuges and causes of of the biota of of individual refuges of this fragility (Soule (Soul6 and Simberloff, 986; Simberloff, 994a), and Fig. 2). Simberloff, 11986; Simberloff, 11994a), and (3) the rules of of refuge refuge design design ((Fig. The The recognition recognition that some of of the the rules, including including the the most most widely debated debated one (SLOSS, related to the theory (Soule (SLOSS, single large or several small; Fig. 2b) are not not related theory (Soul6 and 986, and and Simberloff, Simberloff, 11986, and references references therein), lessened lessened conservation conservation interest in
1 The The Metapopulation MetapopulationApproach Approach
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the theory, while documented exceptions to some of the rules, including SLOSS, led to their fall from status of conventional wisdom. For example, the the third edition Ecology ((Krebs, of one of the most widely used introductory ecology textbooks, Ecology Krebs, popularized by the IUCN and 11985, 985, p. 559), reprinted the figure of the rules as popularized described them as flowing from island biogeographic theory. The fourth edition figure, of the rules, and cites the ((Krebs, Krebs, 11994) 994) omits the fi gure, makes no mention of criticism of the theory as "true but trivial" by Williamson ((1989). 1989).
B. Paradigm Shift waning of the theory theory of island biogeography as a dominant conservation The waning paradigm in the late 11980s 980s coincided with the burgeoning burgeoning interest among biolo biologists in the metapopulation concept ((Fig. Fig. 11). ). As does Hanski ((1989), 1 989), Merriam "Metapopulation models have ((1991, 1 99 1 , p. 1134) 34) explicitly claims a paradigm shift: "Metapopu1ation of thinking thinking about largely replaced equilibrium island biogeography as a way of heterogeneous terrestrial en enterrestrial habitat islands, fragmented habitats and heterogeneous conservation biologists, the shift is tacit vironments in general . . . " For other conservation of an assertion that nature nature is structured structured as metapopulations metapopulations and consists simply of discussion of of what actions are required required to preserve followed by discussion preserve metapopulations Perhaps most telling is The Diversity of of Life, (e.g., Noss, 11993). 993). Perhaps Life by Wilson ((1992), 1992), founder of of the the theory of of island biogeography, biogeography, in which species species are typically a founder seen as structured as metapopulations consequences of metapopulations and and the consequences of this structure for conservation are explored. The causes of lit of the shift shift are are many. One One must be the growing growing ecological litdescribed above, doubting doubting the verisimilitude of of island biogeographic biogeographic the theerature, described ory. However, data and However, scientific data and the weakness of of a prevailing prevailing paradigm paradigm alone are paradigm shift (Kuhn, ( Kuhn, 1970; 1 970; Haila, 1988), 1 988), and and we are unlikely to precipitate a paradigm must seek other other prevailing that, fundamentally, prevailing currents. currents. It is worth worth recalling recalling that, fundamentally, the the theory can be construed multispecies version version of theory of of island biogeography biogeography can construed as just just a mUltispecies of an analogous to imagine objective objective scientific scientific analogous metapopulation metapopulation theory, so it is hard hard to reasons for accepting one while rejecting other. reasons for accepting one rejecting the other. One among conservation One possible possible explanation explanation is a shift among conservation biologists biologists and and ecolecol ogists from conception of of nature that of ogists from the the conception nature as an equilibrium equilibrium world world to to that of a nonnon equilibrium equilibrium one one ((Wiens, Wiens, 1977, 1 977, 1984; 1 984; Chesson Chesson and and Case, 1986). 1 986). Island Island biogeobiogeo graphic of course, course, but but the emphasis emphasis is on on equilibrium equilibrium species graphic theory theory is dynamic, of richness, richness, hence hence the the nickname, nickname, "equilibrium "equilibrium theory," theory," and and even even the the underlying underlying imim migration and are seen migration and extinction extinction rates rates are seen as constant. constant. Though Though metapopulation metapopulation thethe ories are are not not any more, more, or or less, "equilibrium" "equilibrium" theories theories than than the the theory of island island ories theory of biogeography, biogeography, the the emphasis emphasis in in the the latter latter on on equilibrium equilibrium species species richness richness and and in in the created the the former former on on population population turnover turnover may may have have created the sense sense of of aa conflict conflict bebe tween tween an an equilibrium equilibrium and and aa nonequilibrium nonequilibrium theory. theory. The The key key point, point, of of course, course, is that that in both both theories theories there there is no no equilibrium equilibrium at the the population population level. However, However, the the modus operandi operandi of of the the island island biogeographic biogeographic theory theory is to to ignore ignore the the changes changes in in modus the the presences presences and and absences absences of of individual individual species species and and to to focus focus on on the the equilibrium equilibrium ,
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IIkka Hanski Hanski and and Daniel Daniel Simbedoff Simberloff Ilkka
pattern of of species species richnesses; richnesses; this this theory theory is is spatially spatially implicit implicit in in our our taxonomy. taxonomy. pattern Yet two two growing growing interests interests in in conservation conservation are are spatially spatially explicit explicit models, models, to to aa large large Yet extent fostered fostered by by an an increase increase in in spatial data data and and the the use use of of GIS, GIS, and and maintenance maintenance extent of species species that that are are destined destined to to be be locally locally ephemeral, ephemeral, such such as as fugitive fugitive species species and and of successional ones. The The metapopulation metapopulation theory fits fits well with these interests. interests. early successional Indeed, a critical critical difference difference between between the the models models of of MacArthur MacArthur and and Wilson Wilson (1967) ( 1 967) Indeed, and Levins Levins (1969a) ( l 969a) is the presence of of a permanent permanent mainland mainland population population in the the and the presence former but not in the latter. former but not the latter. addition to to island island biogeographic theory, the the other other main main component of of the the In addition "new conservation conservation biology" is population population genetics, genetics, particularly particularly the study of of drift drift "new and and inbreeding inbreeding in small populations populations (Simberloff, (Simberloff, 1988). 1 988). This research research tended tended to the focus of of conservation conservation biologists biologists from communities to species species and and poppop shift the from communities ulations. Ecological Ecological aspects of of conservation conservation began also to be be seen in terms terms of of ulations. populations rather than than species--the populations rather species -the role role of of demographic demographic and and environmental environmental stosto chasticity in setting minimum viable population population sizes is the the prime prime example example (Sim(Sim berloff, 1988). 1 988). Again, a focus on populations populations rather rather than on communities communities is bound bound berloff, island biogeographic theory relevant. to make island theory seem seem less relevant. rescued small sites from their devaluation Finally, metapopulation metapopulation models models rescued devaluation by island biogeographic theory. The The main main ecological data interpreted island interpreted in ter.TIS tev,~as of of island biogeographic theory were species- area relationships, relationships, showing island were simply species-area other things things being equal, equal, large tend to have more species than that, all other large sites tend more species than small of refuge relationship as a ones. The The first rule of refuge design (Fig. 2a) expresses expresses this relationship mandate for planners. The The rules, and the theory, were widely used used mandate for conservation planners. to argue that that large refuges are needed needed and and the the elevated elevated extinction rates in small small refuges are extinction rates render them depauperate (e.g., Diamond, 1972; 1 972; Soul6 Soule et et al., al. , ones will inevitably render them depauperate 1 979). Indeed, to the extent that environmental stochasticity and catastrophies 1979). extinguish small populations, mathematical mathematical modeling suggested that even popu popuparks in the United lations in enormous refuges, the size of the largest national parks States, would would be States, be subject subject to to collapse. collapse. Conservationists eventually recognized recognized that that astute opponents could tum turn this emphasis on inviable small populations populations against against conservation. conservation. For example, the refuge system of of the small nation of Israel consists of some 200 reserves, many managed to various degrees by of which are very small. These are protected and managed the Nature Conservation Authority, and and the Authority was under under great pressure during the 11980s 980s to abandon abandon some small refuges, not because specifi specificc research research showed declining populations within them but because island biogeographic the theory, ory, codified in the refuge design rules, shows that they will inevitably lose species ((R. R. Ortal, 984). Ortal, personal personal communication, communication, 11984). This threat from island biogeographic theory to the maintenance of small reserves area relationship was reserves was forestalled in several ways. The speciesspecies-area shown to have such wide confi dence limits that an assertion of imminent faunal confidence collapse could not be sustained (Boecklen and Simberloff, 11986). 986). Some popu populations that had persisted as very small populations for millennia were adduced as cautions against taking the theory too literally (e.g., Walter, 11990). 990). However,
1 The Metapopulation Metopopulotion Approach Approach
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the the main main salvation salvation of o f small small sites sites was was the the shift shift by b y conservationists conservationists to to the the metameta population paradigm. paradigm. In In the the Levins Levins model, model, at at least, least, small small sites sites containing containing small small population populations were were the the only only homes homes of of aa species species and and thus thus the the proper proper locus locus of of concon populations servation concern. concern. The The model model even even suggests suggests that that aa certain certain number number of of unoccupied unoccupied servation required for metapopulation persistence persistence (Lande, (Lande, 1988a; 1 9 88a; Hanski, this volvol sites is required for metapopulation thus relieving relieving beleaguered beleaguered conservation biologists from having to to justify justify aa ume), thus conservation biologists from having refuge for for a given species by by confirmed confirmed residence. A A famous famous example example in which which refuge local extinction extinction rates high and and aa supply supply of of suitable suitable empty empty sites sites is necessary necessary local rates are high is Pedicularisfurbishae, Pedicularis furhishae, the Furbish Furbish lousewort lousewort (Menges, ( Menges, 1990). 1 990). In sum, sum, from from a conservation conservation standpoint, standpoint, it is not not surprising surprising that that citations citations of of metapopulation studies increase exactly when those of island biogeography de metapopulation studies increase exactly when those of island biogeography decline (Fig. (Fig. 1). 1 ) . These trends represent represent aa paradigm paradigm shift. shift. cline These trends
Misuse of the Metapopulation Concept Concept in Conservation Conservation Biology C. Use and Misuse Hanski and 1 99 1 ) observed have recently Hanski and Gilpin ((1991) observed that that "metapopulation "metapopulation ideas have become become the vogue in conservation biology," and and numerous numerous conservation strategies strategies are explicitly explicitly based based on metapopulation models (references metapopulation models (references in Harrison, Harrison, 1994b). 1 994b). The general effect been to draw draw attention to landscapes and networks, The effect has has been landscapes and networks, as opposed individual reserves reserves in isolation, isolation, for metaphor of of island opposed to individual for which the island metaphor biogeographic theory is appropriate. This is a salutary development. Even Even if if there biogeographic appropriate. This were no different refuges, no significant significant interactions interactions among populations populations in different refuges, it would would be good good to have have multiple refuges refuges simply as insurance insurance against against local catastrophes catastrophes (Soule and 1 986). Doak Doak and and Mills (1994) ( 1 994) and and Harrison Harrison (1994b) ( 1 994b) argue (Soul6 and Simberloff, Simberloff, 1986). argue not structured that, even if most species species are are not structured as Levins-type metapopulations in nature, the rise of of the metapopulation paradigm has served and continues continues to serve function by forcing conservation conservation biologists to gather gather data that are imim a useful function portant to effective conservation strategies of of individual speciesspecies--movement portant movement rates at different different sites, rates from site to site, relative reproduction reproduction and mortality rates and 1 982) for and the like. This is precisely the the view of of Haila and and Jarvinen J~irvinen ((1982) for island biogeographic biogeographic theory. theory. The problems arise, for for metapopulation models as for for island biogeographic theory, when it is assumed without empirical evidence that all species, or all species in a large class, conform to some particular model ((Doak Doak and Mills, 11994; 994; Harrison, 11994b). 994b). If a conservation conservation strategy is based based on metapopulation dynamics that do not exist, it can misfire. Thus, for example, Murphy et al. ((1990) 1 990) suggested that small-bodied, short-lived species with high reproductive rates and high hab habitat specificity typically constitute Levins metapopulations, but there there are are simply insuffi cient data to evaluate this claim ((Harrison, Harrison, 11991, 99 1 , 11994b). 994b). To focus auto insufficient automatically on metapopulation dynamics for such species would not constitute ef effective science. No wide-ranging generalizations are yet possible, because few data really demonstrate the existence of classical metapopulations. Harrison ((1991) 1 99 1 ) could cite only pool frogs (Rana lessoniae) in Sweden and waterfties waterflies (Daphnia) in rock pools as unequivocal cases (for some new examples, see Har-
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Ilkka Hanski HanskJand and Daniel DanielSimberloff Simberloff IIkka
rison and Taylor, this volume). The endangered endangered Glanville fritillary butterfly (Mel (Melitaea Hanski et al., 11995a,b) 995a,b) and itaea cinxia) cinxia) in Finland is another another good example ((Hanski may represent many other butterfly species ((Hanski Hanski and Thomas, 11994; 994; Hanski and Kuussaari, 11995; Thomas and Hanski, this volume). In such instances, an 995; understanding of metapopulation dynamics is crucial to effective conservation. understanding metapopulation Research is required required on local extinction and migration rates rates and how these are Research affected by patch size and isolation (c. (C. D. Thomas et al., al., 11993; et al., al., 993; Hanski et affected D. Thomas 11995b). 995b). Harrison ((1994b) l 994b) suggests that the the species most most convincingly conforming conforming to the Levins model occupy habitats that inevitably change because of of succession succession the (see also Thomas Thomas and and Hanski, Hanski, this volume). In many such species, the extinction populations is deterministic deterministic rather rather than stochastic, stochastic, but this fact fact does not of local populations conception of metapopulation metapopulation dynamics. The fundamentally undercut the Levins conception lousewort, for example, requires a riverside habitat that is endangered Furbish lousewort, neither too little nor too heavily disturbed disturbed (Menges, (Menges, 11990). However, any local 990). However, population is ultimately destroyed destroyed by ice scour and and bank bank slumping, slumping, so the metapopulation meta population requires a supply of of temporarily suitable suitable sites that are not not too isolated isolated population colonized. A metapopulation to be colonized. metapopulation analysis ((Menges, Menges, 11990) 990) including including observaobserva and recolonization suggests that the species is in decline decline tions on local extinction and rather than than at equilibrium equilibrium and and that that tempering of the disturbance disturbance (flow) regime rather tempering of exacerbate the situation. situation. Further, Further, in this species species as in the Levins scewill likely exacerbate Levins sce nario (Lande, 1988a; 1 988a; Hanski, this volume; Nee et al., this volume), volume), nario in general general (Lande, et al., restriction of of conservation conservation measures measures to occupied occupied sites only would would be fatal. A restriction of nature nature would not have have led to the recognition recognition static, nonmetapopulation nonmetapopulation view of of the importance importance of of currently currently unoccupied habitat. Much Much of of the history of of refuge refuge of unoccupied habitat. establishment simply of apparently healthy populations and preestablishment consists consists simply of locating locating apparently populations and pre serving their sites sites (Simberloff, (Simberloff, 1988). 1 988). Metapopulation Metapopulation models models have have been been used used to to deduce deduce the the minimum minimum viable viable metameta population population (MVM) ( MVM) size under under certain certain assumptions assumptions (Hanski ( Hanski et et al., al., 1996b). 1 996b). This This concept analogous to the concept is analogous the minimum minimum viable viable population population (MVP) ( MVP) size (Shaffer, (Shaffer, 1981), 1 98 1 ), but with the the critical difference difference that MVM MVM involves involves both the the minimum minimum viable number number of of populations of suitable suitable habitat habitat patches ( Han viable populations and and the availability availability of patches (Hanski et et al., al., 1996b). 1 996b). In practice, practice, use use of of these these concepts concepts may degenerate degenerate into specious specious "magic more constructive constructive approach approach is to use "magic numbers." numbers." A more use metapopulation metapopulation models models to rank scenarios of of landscape landscape change persistence of of a focal focal to rank alternative alternative scenarios change in terms terms of of persistence species. entire removal species. One One may may ask, ask, for for instance, instance, whether whether the the entire removal of of one one large large habitat habitat patch is more more detrimental detrimental to to aa metapopulation metapopulation than than reducing reducing the the areas areas of of several several patch patches patches (Hanski, (Hanski, 1994a,b; 1 994a,b; Hanski Hanski et et al., al., 1996c; 1 996c; Wahlberg Wahlberg et et al., al., 1996; 1 996; note note the the connection to to the the SLOSS SLOSS rule, Fig. 2b). The The theory theory of of island island biogeography biogeography inin connection spired spired the the rules rules of of refuge refuge design design discussed discussed above above (Fig. ( Fig. 2). The The analogous analogous contricontri bution from from metapopulation metapopulation theory theory is predictions predictions about about the the relative relative performance performance bution of particular particular species species in particular particular fragmented fragmented landscapes landscapes based based on on relatively simsim of ple ple but but spatially realistic realistic models models (Fig. ( Fig. 3, 3 , Hanski, Hanski, 1996b). 1 996b). There There are are two two reasons reasons to to expect expect the the latter latter sorts sorts of of predictions predictions to to be be more more helpful helpful than than the the island island biogebioge-
1 The The Metapopulation Metapopulation Approach Approach a
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FIGURE examples of the same FIGURE 3 Four Fourexamples same landscape landscape fragmented fragmented in different different ways ways (scenarios (scenarios a to d). Spatially model (Hanski, 1 994a), can Spatially realistic realistic metapopulation metapopulation models, models,such such as the incidence incidence function function model (Hanski, 1994a), be used time of the focal used to rank rank the alternative alternative scenarios scenarios in terms terms of the persistence persistence time focal species. species. Note Note the conceptual link to the SLOSS 2b). conceptual link SLOSS rule rule (Fig. (Fig. 2b).
ographic Fig. 2) ographic rules rules of of refuge refuge design. First, First, the the rules rules of of refuge refuge design ((Fig. 2) are are static, static, even even those those actually actually flowing flowing from from the the theory. theory. For For example, example, the the fundamental fundamental con concept Fig. 2a) which in applications is cept in in rule rule aa ((Fig. 2a) is is the the species-area species-area relationship, relationship, which in applications is seen xed area. seen as as meaning meaning aa fixed fixed number number of of species in in aa fi fixed area. In In contrast, contrast, the the meta metapopulation predictions explicitly address the dynamics of of species survival. Sec Second, the rules of refuge design contrast fixed general alternatives (such as in Fig. 2b), 2b), whereas whereas the the spatially spatially realistic realistic metapopulation metapopulation models models practically practically force force one one to Fig. 3). to compare compare specific specific fragmented fragmented landscapes landscapes ((Fig. 3). We We now now tum turn to to potential misuses of of the metapopulation concept in conser conservation. island (Levitt vation. To To start start with, with, if if aa species species is is structured structured as as aa mainlandmainland-island (LevittBoorman) Boorman) metapopulation, metapopulation, population population turnover turnover in in the the peripheral peripheral "island" "island" popu populations lations may may be be irrelevant irrelevant to to the the persistence persistence of of the the metapopulation metapopulation as as aa whole, whole, though though the the dynamics dynamics are are crucial crucial to to the the persistence persistence of of the the peripheral peripheral populations populations ((Doak Doak and 994; Harrison, 994b, Simberloff, 994b). More and Mills, Mills, 11994; Harrison, 11994b, Simberloff, 11994b). More generally, generally, emphasis emphasis on on metapopulation metapopulation models models can can potentially potentially harm harm conservation conservation by by draw drawing ing attention attention away away from from single single populations populations on on the the grounds grounds that that no no one one of of these these is Harrison, is crucial crucial to to aa species species'' persistence persistence and and it it is is the the ensemble ensemble that that matters matters ((Harrison, 11994b). 994b). Another Another example example of of the the metapopulation metapopulation concept concept used used in in misguided misguided attempts attempts
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Ilkka Hanski Hanskiand and Daniel DanielSimberloff Simberloff IIkka
to to deemphasize deemphasize single single populations is is the the hype hype surrounding surrounding movement corridors corridors we observe observe that, that, strictly strictly speaking, speaking, metapopulation models tend tend to to em em(though we phasize phasize connectance connectance among among habitat habitat patches, patches, not not corridors). corridors). One One rule rule of of refuge refuge design associated associated with island island biogeographic biogeographic theory theory is is that that aa set set of of refuges refuges con condesign nected nected by corridors corridors will contain contain more more species than than an an otherwise otherwise identical set set without corridors corridors (Fig. 2f). 2f). In the original original formulation of of this rule, the focus was on on aa community-level community-level statistic, species species richness, and corridors were were assumed to to increase increase this statistic statistic by increasing increasing immigration rate. rate. However, for most proposed corridor systems, systems, there there is scant evidence that that the corridors corridors would be used for movement or that they would actually forestall extinction Hobbs, 11992; 992; Sim movement or that they would actually forestall extinction ((Hobbs, Sim992). Even more berloff et et at., al., 11992). more troubling, investment in corridors can be expen expensive and can detract from efforts to protect particular populations that require a 992). specific refuge that is not part of a network (Simberloff et at., al., 11992). Some examples of classical metapopulations in human-fragmented land landscapes may represent transient, nonequilibrium situations, in which a previously more continuous population becomes divided into smaller units, with consequent consequent assemlocal extinctions, but no functional metapopulation was created, merely an assem blage blage of of populations populations all all slowly slowly declining declining to to extinction. extinction. It It seems seems likely likely that that almost almost any any gradual gradual extinction extinction would would appear, appear, during during some some parts parts of of the the decline, decline, as as aa non non994b ). Even in this case, unun equilibrium metapopulation situation (Simberloff, I1994b). derstanding derstanding its current current dynamics can aid in the maintenance maintenance of of the species species in a fragmented Harrison, 11991). 99 1 ). In SLOSS fragmented or otherwise changed changed landscape landscape ((Harrison, SLOSS termi terminology ((Fig. Fig. 2b), a single large large population might might have been better, better, but if all we small ones, their their interactions may be crucial to their their survival. have left is several small For example, the the metapopulation analysis by Beier of cougars (Felis con1 993) of (Felis con For Beier ((1993) color) color) in the the Santa Santa Ana Ana Mountains Mountains of of California California showed showed that that the the species species currently exists as a collection of populations loosely linked linked by riparian riparian corridors, corridors, exists of small populations and radiotelemetry data and his radiotelemetry data on movement combined combined with a simulation model model of particular particular populations populations and could affect the entire entire suggested how loss of and corridors corridors could affect the Data on sources sinks in source-sink source- sink metapopulation metapopulation can metapopulation. Data sources and and sinks can also also be be key key to to maintaining maintaining a species. species. A A particular particular worry about about nonequilibrium nonequilibrium metapopulations in increasingly increasingly fragmented fragmented landscapes landscapes is that that we we might might not not recrec metapopulations ognize them them as as such such (Hanski, ( Hanski, this this volume), volume), which which would would give give us us a misleadingly misleadingly ognize rosy picture picture of of the the ability of of species species to to persist persist in present present landscapes. landscapes. rosy Attempts to to model model the the minimum minimum number number of of populations populations necessary necessary to to mainmain Attempts tain tain aa viable viable metapopulation metapopulation are are hampered hampered by by assumptions assumptions that that are are hard hard to to verify verify and and data data that that are are difficult difficult to to gather. gather. These These are, are, of of course, course, problems problems with with all all poppop ulation models models that that aim aim at at quantitative quantitative predictions, predictions, and and the the problems problems become become even even ulation more severe severe with with spatially spatially realistic realistic models models that that might might guide guide specific specific management management more 1 994). The The history history of of conservation conservation biology biology is is marked marked by by plans (Doak ( Doak and and Mills, Mills, 1994). plans many examples examples of of misused misused minima minima (Simberloff, (Simberloff, 1988; 1 988; Crome, Crome, 1993): 1 993): as as soon soon as as many minimum is is set set for for any any variable, variable, forces forces opposed opposed to to conservation conservation use use itit to to see see aa minimum how much much of of nature nature they they can can get get rid rid of. of. Thus, Thus, as as tentative tentative and and general general as as the the how MYM model model is, is, someone someone may may attempt attempt to to manage manage for for aa specific specific minimum minimum based based MVM
Metapopulation Approach 1 The The AAetapopulation Approach
25
discussed above, a less controversial controversial use of of spatially realistic on this model. As discussed metapopulation Fig. metapopulation models models is simply to rank rank alternative alternative management management scenarios scenarios ((Fig. 3) and recognize that making long-term predictions predictions about and to recognize making long-term about (meta)population (meta)population persistence persistence time time in our our rapidly changing changing world is practically practically hopeless. hopeless. The The shift from from island island biogeographic biogeographic to metapopulation metapopulation thinking thinking united united ecol ecologists and and geneticists geneticists in focusing focusing on populations. populations. Genetic Genetic concerns concerns have have been prominent 1 963) fi rst prominent in the interest in metapopulations. metapopulations. Kimura Kimura and Crow Crow ((1963) first pointed populations can maintain pointed out that occasional occasional migration migration between between local populations maintain genetic variation population, essentially because variation better better than would would a single large population, drift is likely to fi populations. However, the situ fixx different different alleles in different different populations. situation becomes becomes more complicated complicated when we allow for for local extinctions extinctions and recol recolonizations, modeling the way that such onizations, and recently much much effort effort has gone into modeling population turnover affects the maintenance of genetic diversity in metapopula population turnover affects maintenance of metapopulations (Wade and McCauley, 1 988; Hastings and Harrison, 1 994; Barton Barton and Whit (Wade McCauley, 1988; Harrison, 1994; and Whitlock, this volume). volume). On theoretical grounds, one might expect species species that naturally metapopulations not to be prone inbreeding depression because they exist in metapopulations prone to inbreeding lack genetic Harrison, 1994b), 1 994b), while local populations populations in a recently frag genetic load ((Harrison, fragmented mented large population population might might be particularly susceptible susceptible to inbreeding inbreeding depres depression because heterozygosity would 988; Hedrick would quickly decline (Simberloff, (Simberloff, 11988; Hedrick and Gilpin, this volume). movement volume). Under Under the latter latter circumstances, maintaining maintaining movement among among populations populations might seem particularly particularly important, important, and indeed indeed many man management plans for for declining declining populations call for for measures to enhance enhance population population agement interaction, cally to avoid inbreeding inbreeding interaction, such as translocation translocation and corridors, corridors, specifi specifically depression (e.g. 995). However, eld evidence (e.g.,, U.S. Department Department of of Agriculture, 11995). However, fi field evidence inbreeding depression depression or other other problems problems in recently fragmented fragmented populations populations for inbreeding Harrison, 1994). 1 994). Lande l 988b) argues more generally that the imporimpor is scarce scarce ((Harrison, Lande ((1988b) tance of of genetic threats in conservation conservation has been overblown. overblown. His view is that, in naturally small populations, populations, the genes genes causing causing threatening inbreeding inbreeding depression depression would populations, ecological would have have been been selected out, while in recently reduced reduced populations, threats are immediate. Despite are more more immediate. Despite this widely cited statement, statement, genetic principles principles still underpin Harrison, 1994b). 1 994b). underpin many viability analyses and management management plans plans ((Harrison, Perhaps Perhaps the the very fact that genetic genetic modeling modeling is feasible feasible ensures ensures that it will be done. done, particularly if ecological prob ecological modeling, modeling, even if potentially more more useful, useful, is more prob1 995) appears lematic. The latest round round of of papers papers (e.g., Lynch et et al. al.,, 1995) appears to strengthen strengthen the genetic argument, but eld studies. but the most urgent urgent need need is for for relevant fi field studies. Thompson 1 996) contends Thompson ((1996) contends that that metapopulations metapopulations may be crucial crucial to the con conservation servation of of various various coevolutionary coevolutionary interactions, interactions, such as those those between between pathogens pathogens or parasites parasites and their hosts. In models, models, locally unstable unstable population population dynamics dynamics can 99 1 a; be stabilized by the addition Hassell et addition of of metapopulation metapopulation structure ((Hassell et al., al., 11991a; Nee metapopulation structure Nee et et al. al.,, this this volume). volume). In other other cases, the metapopulation structure stabilizes stabilizes evolutionary dynamics under certain dynamics of of the interaction. interaction. For For example, example, under certain circum circumstances, between a pathogen stances, the coevolutionary coevolutionary dialog dialog between pathogen and its host host can lead to the extinction extinction of of the host, if if a new virulence virulence gene gene in the pathogen pathogen spreads spreads rapidly rapidly enough. enough. A metapopulation metapopulation structure structure can then prevent the gene from eliminating eliminating
26 26
IIkka Hanski Hanski and and Daniel Ilkka Daniel Simberloff SJrnberloff
the entire ax (Unum ax rust (Melampsora entire species. Wild fl flax (Linum marginale) and flflax (Melampsora lini) may be a natural natural example example in which the host metapopulation metapopulation structure serves this function B urdon and Thompson, 1995). 1 995). Frank (this volume) presents function ((Burdon presents a thorough discussion of these issues. The focus on metapopulations, combined combined with that that on genetics, has led to the population and the species becoming the dominant con dominant levels of of concern in conservation. It is striking that the recent explosion of man of interest interest in ecosystem management agement is quite antithetic antithetic to a primary interest in populations and to single singlespecies management (Simberloff, 11996). 996). In fact, a key motivation of of ecosystem management management is that that research on species after species will be hopelessly expensive and inefficient, inefficient, and so will management management based on such research. Of Of course, course, both ecosystem management management and metapopulation metapopulation models share a concern with land landscapes and regions, rather than than highly local settings, and and one could imagine a landscape with a distribution of of habitat habitat patches that would maintain many meta metapopulations simultaneously. Also, the emphasis in ecosystem management management on maintaining processes rather 996) can accommodate rather than species (Simberloff, 11996)can concerns about the coevolutionary processes. Nevertheless, Nevertheless, the research research programs programs and primary goals of of these two approaches differ differ fundamentally and they will surely compete compete for both research funding and and influence in specific management management plans in the future.
V. CONCLUSIONS CONCLUSIONS The The changing pattern of of citations citations of of the the key words "island biogeography" and "metapopulation" represents a fascinating example of of a paradigm paradigm shift in population biology. This example example is the more more striking striking because the respective respective theories are are so closely related related that whatever evidence can be mustered for, or against, against, one theory is likely to serve serve the same function with respect to the other other theory. We We have discussed in this chapter chapter how it is largely the wider context that has made the difference. One apparently important important issue is the spatial scale. The dynamic theory of island biogeography was originally developed to explain explain pat patterns at large spatial scales, whereas the metapopulation concept is associated with fragmentation of of our ordinary landscapes. landscapes. Though the difference is in per perception ception only, it it matters. matters. Metapopulation Metapopulation models have contributed important insights to conservation, and they have inspired eld studies focused on collecting key data inspired fi field data on demography and movement. Nonetheless, the temptation to apply the metapopulation approach approach blindly to systems for for which there is no supporting evidence evidence can be counterpro counterproductive. Metapopulation Metapopulation maintenance maintenance may be crucial to a limited range of of species, probably dominated by those those characteristic characteristic of successional successional habitats. The The role of of metapopulation metapopulation dynamics in forestalling genetic deterioration is particularly particularly un unverified. verified.
II
Empirical Evidence Evidence for Empiricol Metopopulotion Metapopulation Dynomics Dynamics Susan Susan Harrison Harrison
Andrew Andrew D. Taylor Taylor
I. INTRODUCTION INTRODUCTION Underlying nements and elaborations metapopulation theory Underlying the many refi refinements elaborations of of metapopulation is the fundamental depends on their fundamental idea that the persistence of of species species depends their existence existence as sets of migration. of local populations, populations, largely independent independent yet interconnected interconnected by migration. Population Population structure structure at this large spatial scale is thought thought to alleviate the risks of of widespread widespread extinction extinction that that arise arise from from unpredictable unpredictable physical environments environments and from strong interactions interactions among among species. This concept concept has long attracted attracted many ecologists, ecologists, but more more for its plausibility plausibility than because because of of any compelling compelling empirical empirical evidence. evidence. Support Support for for metapopulation metapopulation theory theory has mostly consisted of of anecdotal anecdotal accounts uctuations, combined accounts of of local extinctions extinctions or asynchronous asynchronous population population fl fluctuations, combined with much much theoretical theoretical evidence evidence that that metapopulation metapopulation effects effects could occur. occur. Recently, however, however, as interest interest in metapopulation metapopulation dynamics dynamics and its conservation conservation applications applications has grown, grown, the number number of of more more substantial substantial studies studies has steadily increased. increased. Here we review review the current current body of of empirical empirical evidence evidence and ask whether whether and and how it supports brief review supports metapopulation metapopulation theory. We We begin with a brief review of of the theory and its origins, origins, to lay the groundwork for for specifi specificc criteria by which which to evaluate evaluate the evidence. In this review, we highlight differences differences between between metapopulation metapopulation theory theory for for single and multiple multiple species, species, which which will lead to somewhat somewhat different different criteria in the two cases. We We leave aside the genetic genetic and evolutionary evolutionary aspects aspects of Metapopu/alion g\! Metapopulation Bi% Biology
Copyright reproduction in any fonn Copyright © 9 1997 1997 by Academic Press. Press, Inc. All rights rights of of reproduction form reserved. reserved.
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Susan and Andrew Susan Harrison Harrison and Andrew D. D. Taylor Taylor
metapopulation dynamics, which have received comparatively little empirical work ((but but see Olivieri et a!., 1 990; Olivieri and Gouyon, this volume; McCauley, al., 1990; 11991; 99 1 ; Harrison and Hastings, 11996; 996; Barton Barton and Whitlock, this volume, for for re reviews). Single-species metapopulation theory arose largely from early observations of species in patchy and unpredictable unpredictable environments. The archetypal "shifting mosaic" or "blinking lights" species were insects whose populations populations were small and were prone prone to extinction extinction either either because of cli clior insular, fluctuated widely, and matic events or because Andrewartha and because their habitat habitat was transient (e.g., Andrewartha and Birch, 11954; 954; Ehrlich 1 972). Recent Ehrlich and Birch, 1967; Ehrlich Ehrlich et et al. at.,, 1972). Recent examples in the same Menges, 11990), 990), amphibians in small ponds vein include herbs on riverbanks ((Menges, (Gill, 11978a; 978a; Sjogren, 99 1 ; Sjogren 1 994), snails on rocky outcrops Sj6gren, 11991; Sj6gren Gulve, 1994), (Spight, 1974), 1 974), insects on weedy plants (van der Meijden, 1 979a; van der Meijden,1979a; der Meijden Meijden and van der Veen-van vulner Veen-van Wijk, this volume), and many cases of butterflies vulnerable to bad weather 979; Harrison et et al. 1 988; weather or habitat change (Shapiro, 11979; al.,, 1988; 994; Hanski Thomas and Harrison, 11992; 992; Thomas and Jones, 1993; 1 993; Hanski et et al. al.,, 11994; and Kuussaari, 995; Hanski, this volume; Thomas Kuussaari, 11995; Thomas and Hanski, this volume). Single-species metapopulation models, beginning 1 970), dem beginning with Levins ((1970), demonstrate that that such sets of transient demes may persist through a balance between 1 994; local extinction and recolonization (reviewed by Hastings and Harrison, 1994; Hanski, this volume). metapopula volume). Here we denote as "classical" "classical" single-species metapopulations sets of local populations that are all subject to extinction and persist at the metapopulation metapopulation level through through recolonization (Fig. 1l a). A very basic property of
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Different types of cir of metapopulation. Filled circles, occupied habitat patches; empty circles, vacant habitat patches; dotted lines, boundaries boundaries of of local populations; arrows, dispersal. (a) Classic (Levins); (b) mainlandisland; (c) patchy inter mainland-island; patchy population; population; (d) nonequilibrium nonequilibrium metapopulation; (e) intermediate case combining features of of (a), (b), (c), and (d).
22
Empirical Evidence Evidence for Metapopulation Empirical MetapopulationDynamics Dynamics
29
classical metapopulations metapopulations is that persistence requires requires an adequate rate of of migration among patches. Probabilities of of metapopulation metapopulation persistence also increase with the number of of habitat patches and local populations. populations. Besides resonating resonating with ecolo ecologists gists'' interpretations interpretations of of many natural systems, this verbal verbal and mathematical model is increasingly seen as relevant to how species persist, or fail to do so, in landland scapes recently fragmented by humans (Opdam, 11990; 990; Fahrig 994; Fahrig and Merriam, 11994; Harrison, 994b). Harrison, I1994b). Theory on the the metapopulation metapopulation dynamics of of multiple interacting species arose not so much from from natural history as from from mathematical and laboratory studies showing the instability of predatory predatory or competitive interactions interactions in simple envi environments Nicholson and Bailey, 1935; Gause, 11935; 935; Huffaker, 1 958). What ronments ((Nicholson Huffaker, 1958). What we may now all the "classical" multispecies model was fi rst demonstrated Huf first demonstrated in Huf' s ((1958) faker 1 958) mite experiments, showing undergo faker's showing that a predator and its prey undergo oscillations and crashes within patches patches of of habitat, yet coexist stably in a universe of of interconnected interconnected patches. Mathematical models building building on the same extinction extinctionand-colonization and-colonization format developed developed for for single species, as well as models of of other other types, have explored force in explored many aspects aspects of of spatial spatial subdivision as a stabilizing force both predatory and competitive interactions Harrison, interactions (reviewed by Hastings Hastings and Harrison, 11994; 994; Nee Nee et et at. al.,, this this volume). volume). Metapopulation Metapopulation theories for single single and and multiple multiple species thus address address related issues and are expressed in similar models, yet arose from different different concerns and arrive at partly confl icting answers. In single-species theory, the problem is the conflicting patchiness of habitats and the harshness harshness of of the abiotic environment, and the so solution is migration migration and recolonization. recolonization. Multispecies theory theory addresses addresses the the problem problem of intrinsically unstable interactions and identifi identifies subdivision as a soluof es spatial subdivision solu tion. Therefore, Therefore, migration among among patches patches always promotes promotes persistence in models models of metapopulations (though models that of classical single-species metapopulations (though less so in models that include the effects of 1 993; Olivieri of emigration on local populations; Hanski and Zhang, Zhang, 1993; and and Gouyon, Gouyon, this volume), but too too much much migration leads to instability in multispe multispecies meta populations. metapopulations. We note note that that the above above difference difference is not really due to the the number number of of species involved, but rather rather to the assumed causes of of local instability or extinction. If local local dynamics dynamics are are intrinsically unstable, high rates of of migration may destabilize metapopulation (Allen et et at., al., 1993). if multispecies a single-species metapopulation 1 993). Conversely, if systems are are subject subject to frequent frequent local extinctions from external causes, subdivision may possibly lose its stabilizing effect, an issue that that deserves deserves more more investigation. We may now now identify criteria for for jUdging judging the empirical evidence on meta metapopulations. rst whether whether all local populations populations populations. For single-species studies, we ask fi first are prone prone to local extinction and, second, whether whether persistence at the metapopu metapopurequires recolonization. first, lation level requires recolonization. For multispecies mUltispecies systems, we ask, fi rst, whether whether strong interactions interactions between between either competitors competitors or predators predators and prey cause local extinctions or popUlation population oscillations oscillations and, second, second, whether subdivi subdivipersistence or the temporal stability of of the system as a whole. sion increases the persistence Metapopulations ned more have just done, Metapopulations may be defi defined more broadly than than we have done, to include include any systems in which popUlations populations are are subdivided subdivided but exchange exchange some
30
Susan Susan Harrison Harrisonand and Andrew Andrew D. D. Taylor Taylor
migrants. In fact, since we find find few few systems that that conform well to the classical classical broader models, we we return return under under Discussion Discussion to to consider consider the the implications implications of of aa broader view. However, However, like Hanski (this volume), we take the classical classical models as a point of of departure, departure, since their their assumptions assumptions are are implicit in most cases cases in which the term term is used, including many of the more elaborate metapopulation models. Moreover, is used, including many of the more elaborate metapopulation models. Moreover, classical models yield the classical models the strongest strongest predictions predictions about about metapopulation metapopulation dynamics; in other other types of of metapopulation, metapopulation, as we will show, the causes causes of of persistence persistence or coexistence coexistence lie lie more more at at the the local level. level. Finally, Finally, we note note some some important practical practical reasons reasons for giving giving careful scrutiny to to metapopulation metapopulation ideas. ideas. In In conservation conservation biology, the the metapopulation metapopulation model model is sometimes used to support support the need for for numerous numerous reserves, corridors, or a land landscape-level How scape-level approach, approach, goals goals with with which which few few conservationists would would argue. argue. However, in in other other cases cases this this model model is is used more controversially, controversially, to to justify justify strategies strategies that would preserve only a handful handful of of well-spaced fragments fragments of of a habitat that is presently 993; Harrison, 994b; Doak and Mills, presently continuous continuous (see (see Harrison Harrison et et al al., 11993; Harrison, 11994b; Doak and 11994; 994; Noon 996; Gutierrez 1 996). Noon and and McKelvey, McKelvey, 11996; Gutierrez and and Harrison, Harrison, 1996). ..
II. SINGLE-SPECIES SINGLE-SPECIESMETAPOPULATIONS METAPOPULATIONS In reviewing empirical studies, we examine each each of of the critical assumptions assumptions of rst identifying of classical classical metapopulation models in tum, turn, fi first identifying studies that that appear appear not to meet them and then proceeding toward studies that appear appear to exemplify most or all of of them. Our purpose is not to criticize individual studies, studies, nor nor to impose categories categories for their own sake, but to ask whether available evidence sug suggests any systematic systematic pattern.
A. Are All Local Local Populations Subject to Frequent Populations Subject Frequent Extinction? Extinction? 1. Definition Definition and and Causes Causes of of Local Local Extinction Extinction
Since populations show spatial spatial structure at a hierarchy of of scales scales (e.g., Amar Amarasekare, 1 994), the definitions definitions of asekare, 1994), of local populations and hence hence local extinction are usually usually partly partly arbitrary. arbitrary. However, However, as as aa minimal minimal criterion, criterion, local local extinction may be be defi ned as the extirpation of ciently closed im defined of any population segment segment suffi sufficiently closed to immigration migration that, once extinct, typically remains remains so for for several generations generations or more. This This serves to exclude sUbpopulations subpopulations so tightly connected connected to others others that that their their "extinction" is caused caused more by the movement of of organisms organisms than than by their their mortality (to see why this is an important distinction, distinction, imagine birds in an orchard, orchard, under undergoing "local extinction" every time they fl y out of fly of a tree). In the much-used scheme 1 98 1 ), local extinction has scheme devised by Shaffer Shaffer ((1981), has four four general general causes. One of of these, demographic demographic stochasticity, is expected expected to affect affect only popUlations populations below a threshold size, and another, loss of of genetic variation, acts
22
Empirical Evidence for Metapopulotion EmpiricalEvidence Metapopulation Dynamics Dynamics
31
comparatively finish off population comparatively slowly. These These factors factors may help help to to finish off a declining declining population or or impede impede the establishment establishment of of new new ones, ones, but but are are unlikely to be ultimate ultimate causes causes of (including "catastro of local extinctions. extinctions. In contrast, contrast, environmental environmental stochasticity stochasticity (including "catastrophes") and and deterministic deterministic threats threats (e.g., loss of of habitat) may extirpate extirpate popUlations populations phes") habitat) may of a wide wide range range of of sizes and and thus are the most causes of of natural natural of thus are most likely ultimate ultimate causes rms ((Leigh, Leigh, 11981; 98 1 ; Karr, local extinctions, extinctions, as a variety of of empirical empirical evidence evidence confi confirms Karr, 11982; 982; Pimm et 988; Schoener, 983; C. D. Thomas 1 992; Thomas et al. al.,, 11988; Schoener, 11983; Thomas et et al. al.,, 1992; Thomas and and Hanski, this volume). volume). Environmental Environmental stochasticity stochasticity often often operates operates at a regional regional scale; for for example, example, weather synchrony over weather causes causes insect insect populations populations to fluctuate fluctuate in synchrony over broad broad geographic geographic areas Hanski and Woiwod, 1 993), and freeze may eliminate areas ((Hanski Woiwod, 1993), and a single drought drought or or freeze eliminate 972, 1980). 1 980). Such "re multiple conspecifi conspecificc butterfly populations populations (Ehrlich (Ehrlich et et al. al.,, 11972, "remetapopu gional stochasticity" stochasticity" reduces the likelihood likelihood of of persistence persistence for for classical classical metapopulations ((Hanski, Hanski, 11991). 99 1 ). In tum, importance of turn, it increases increases the potential potential importance of other other resistant life stages. mechanisms that enable enable persistence, persistence, such such as refuge refuge habitats habitats or or resistant Price 1 989) concluded undergoes Price and and Endo Endo ((1989) concluded that that because because Stephens Stephens'' kangaroo kangaroo rat rat undergoes extreme regionwide population fluctuations in response regionwide population response to weather, weather, a single large reserve is much preferable preferable to a proposed proposed design design of of multiple multiple small reserves reserves con connected nected by corridors. corridors. suc Deterministic causes of of local extinction include natural natural disturbance disturbance and succession cession and human human destruction destruction of of natural natural habitats. These These may may lead to classical metapopulation metapopulation dynamics, dynamics, but do do not always do do so, for for several reasons. reasons. Species Species adapted popula adapted to successional successional habitats habitats may be such such good good dispersers dispersers that that their their populations tions are are not not very subdivided. Species subjected subjected to longer-term longer-term habitat habitat change change shift their their spatial distributions distributions over over time without without ever approaching a dynamic dynamic may shift ever approaching balance Habitat fragmentation fragmentation may balance between between extinction extinction and and recolonization. recolonization. Habitat may pro produce populations or duce patches patches that that are too small to support support populations or too too isolated isolated to interact other patches. patches. with other 2. Some Some Populations Populations Are Are Highly Persistent Persistent
Most Most studies studies of of natural natural local extinctions extinctions have have taken taken place place on on small islands islands near 1 98 1 ; Peltonen Peltonen and 1 99 1 ; near the shore shore of of a lake or ocean ocean (e.g., Pokki, 1981; and Hanski, 1991; reviews 983; Diamond, 984). Local reviews in Schoener, Schoener, 11983; Diamond, 11984). Local extinctions extinctions affect affect the small (island) (island) populations, populations, but but the the system persists persists for for essentially the the same same length length of of time as do its larger Fig. 1l b). Many larger and and more more persistent persistent (mainland) (mainland) populations populations ((Fig. Many metapopulations metapopulations have have an essentially similar, similar, mainland mainland and and island island structure, structure, owing owing to high high variation variation in the the sizes of of habitat habitat patches patches or or populations. populations. Examples Examples include include metapopulations metapopulations of of spiders spiders on Bahamanian Bahamanian islands islands (Schoener (Schoener and and Spiller, Spiller, 11987a,b; 987a,b; Spiller 990), checkerspot patches of Spiller and and Schoener, Schoener, 11990), checkerspot butterflies butterflies on patches of ser serHarrison et 988), and pentine soil ((Harrison et al. al.,, 11988), and many others others (reviewed (reviewed by Schoener, Schoener, Harrison, 11991). For a system to have have mainlandmainland-island there 11991; 99 1 ; Harrison, 99 1 ). For island dynamics, there need not be a single single mainland of of extreme size. Substantial Substantial variance in patch patch or population pop population size means means that local extinctions extinctions will tend to strike the smallest populations, which are the ones with the least impact on metapopulation persistence. which are ones with impact on metapopulation persistence.
32 32
Susan Susan Harrison Harrisonand and Andrew Andrew O. D. Taylor Taylor
Heterogeneity in habitat quality, rather rather than patch or population size, may produce a similar effect. A recently popular idea is that dispersal from "source" "source" populations in high-quality high-quality habitat habitat may permit permit "sink" "sink" populations to exist in inferior habitat 988; Pulliam and Danielson, 1991). 1 99 1 ). Unlike island pop habitat (Pulliam, (Pulliam, 11988; populations, which are are merely small, sinks cannot cannot support support positive population population growth because of their poor quality. This idea remains largely untested, but several insect studies provide suggestive examples, with the sinks ranging from marginal marginal hab habitats that are occupied only during rare 1 979; Mur rare favorable years (e.g., Shapiro, 1979; Murphy and White, 11984), 984), to areas areas in which populations flourish most of of the time but cannot survive catastrophes (e.g., Strong et al. , 11990; 990; Singer et al., 11994). 994). et al., et al., 3. 3. Populations Populations Are Are Not Not Subdivided Subdivided Enough Enough to Permit Permit True True Local Extinction Extinction Local
extinction to occur, popUlations populations on separate patches must be rea reaFor local extinction sonably isolated from one another, with most recruitment recruitment coming from within the patch patch rather rather than than from immigration. At the opposite extreme extreme are are systems in which progeny from all patches patches are completely mixed and and reassorted among patches patches in each generation. Here the term term "patchy "patchy popUlation" population" is used for for systems toward the latter end of the continuum (Fig. l c). Sharp Sharp distinctions distinctions are are difficult in practice, practice, but if the average average individual inhabits more than a single patch in its lifetime, the patches clearly do not support separate populations. Local "extinc"extinc tions," presences presences followed by absences, may simply be the result of of individuals individuals'' foraging behavior or responses to conspecifics (e.g., the birds in an orchard). Importantly, unlike a metapopulation, the persistence of a patchy population is not not highly sensitive to the distances or rates of of movement among patches. patches. Invertebrates Invertebrates that specialize on fallen fruit, rotting logs, dung, carrion, or water-filled tree holes are sometimes regarded as forming classical metapopula metapopulations, colonizing and becoming extinct on their transient resource patches. How However, such species are typically highly mobile; each patch patch supports only one generation of the insect, and adults adults oviposit on numerous numerous patches in their their lifetimes (e.g., Kitching, 11971; 97 1 ; Kaitala, 11987; 987; Hanski, 1987). 1 987). Although weedy host plants are a slightly more permanent permanent habitat habitat than dung or carrion, the specialist insects feeding on milkweed (Solbreck, 1991; 1 99 1 ; Solbreck and Sillen-Tullberg, 990) and Sillen-Tullberg, 11990) 1 995) appeared ragwort ((Harrison Harrison et et al. al.,, 1995) appeared to disperse so well that their populations were effectively unsubdivided across large arrays of of patches (but see below and van der Meijden Meijden and van der Veen-van Wijk, this volume). Of Of course, at some larger scale (e.g., among different fields or forests) they may possibly show clas classical or nonclassical metapopulation structure. structure. Migration has long been been considered an important adaptation to environments that vary in space and time (e.g., den Boer, 11968; 968; Southwood, 1977). 1 977). Sessile marine invertebrates invertebrates with planktonic larvae show perhaps highest highest migration of of any organisms, relative relative to the scale of of the patches patches on which recruitment and growth growth occur, and they appear to persist longer in evolutionary time than than com comparable 99 1 ). Conversely, the parable taxa with nonplanktonic larvae (Jablonski, 11991). the evo-
22
Empirical Evidence Empirical Evidencefor Metapopulation MetapopulationDynamics Dynamics
33 33
lution of of flightlessness flightlessness in insects insects is strongly linked linked to stable, stable, continuous continuous habitats (Wagner 992). Thus, it is perhaps to be expected that in many (Wagner and Liebherr, 11992). cases, species in patchy and risky environments will disperse too well to form classical classical metapopulations on patches of of their habitat.
B. Does Does Recolonization Recolonization Balance Balance local Local Extinction? Extinction? Migration Migration and colonization colonization in metapopulations metapopulations have been reviewed by Eben Ebenhard 1 99 1 ), Hansson ((1991), 1 99 1 ), and Ims and Yoccoz (this volume). In classical hard ((1991), models, there there is a threshold rate of migration for the metapopulation metapopulation to persist. Above this level, patch patch occupancy achieves a stable stable equilibrium, equilibrium, arising arising from the fact that that every local extinction extinction makes an empty habitat available for colonization, in strict analogy to a density-dependent birth rate. There There are several reasons why this this assumption assumption may may not not always always hold. hold. 1. Nonequilibrium Declining) Metapopulations Nonequilibrium ((Declining) Metapopulations
Rather Rather than than being part part of a steady-state process, local extinctions may occur occur in the course of a species species'' decline decline to regional extinction, extinction, with recolonization occuring infrequently or not at all ((Fig. Fig. Il d), typically as the species species'' habitat habitat is undergoing reduction and fragmentation. ex fragmentation. A natural natural example is the series of of extinctions of mammal populations caused by the isolation of of mountaintop habitats habitats during post-Pleistocene 97 1 ). Many more examples can post-Pleistocene climate change (Brown, 11971). be found among species in habitats fragmented by humans, such as salamanders (Welsh, 11990) 990) and woodpeckers (Stangel et 1 992) on remnant et at., al., 1992) remnant patches of old oldgrowth forest. Hanski (this volume) discusses nonequilibrium dynamics in a butbut terfly metapopulation. The conservation of species in fragmented habitats habitats is an important area for the application of metapopulation concepts. In some cases, however, remnant populations are so isolated that there is little potential to manage manage them as an interconnected Harrison, 11994b), 994b), while in others, creating corridors or interconnected network ((Harrison, a dispersal-friendly matrix may be feasible (e.g., Noon and McKelvey, 11996). 996). 2. Nonequilibrium Habitat-Tracking) Metapopulations Nonequilibrium ((Habitat-Tracking) Metapopulations
Local extinctions are not always stochastic stochastic as most metapopulation theory assumes, but rather may occur when disturbance, disturbance, succession, or long-term habitat habitat change cause the loss of tum, colonization may occur only of suitable habitats. In turn, when new patches of of habitat are created near existing populations. populations. For example, example, the spatial spatial distribution distribution of many butterflies butterflies appears to be sensitive to vegetation age and and height, which are governed by grazing and other transient disturbances disturbances (Thomas and Harrison, 1992; 1 992; Thomas and Jones, 11993; 993; Thomas Thomas and Hanski, this volume). Thomas ((1994c) 1 994c) argues that deterministic extinction may be the rule and stochastic extinction the exception exception in real metapopulations. The spatial dynamics created created by disturbance disturbance and succession are interesting in their own right and are a subject of importance for of key importance for the conservation of of
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SusanHarrison Harrisonand ond Andrew AndrewD.D. Taylor Taylor Susan
many species. However, local extinctions caused by habitat loss violate an im important premise of the classical model, namely that extinctions make habitats metapopulation-level equilibrium pre preavailable for recolonization. The stable metapopulation-level from the inverse relationship between patch dicted by the classical model arises from density-dependent regulation of occupancy and patch availability, which creates density-dependent patch occupancy. In contrast, when patches are created created and destroyed destroyed by extrinsic species'' abundance abundance and distribution distribution will forces, no such regulation occurs. The species simply track the availability of habitat and and will remain roughly constant constant only if the rates of habitat 994c). habitat loss and renewal happen happen to be roughly equal (Thomas, (Thomas, 11994c).
C. C. Classical ClassicalMetapopulations, Intermediate Intermediate Cases, Cases, and Other Possibilities Possibilities Rana lessonae lessonae in ponds ponds along the Baltic coast of of Sweden The pool frog Rana (Sj6gren, 11991; Sj6gren Gulve, 11994) and the butterfly Melitaea Melitaea cinxia ((Hanski Hanski (Sjogren, 99 1 ; Sjogren 994) and al.,, 11994, and this volume) volume) on granite outcrops outcrops in southwest southwest Finland, Finland, form form et al. 994, and metapopulations metapopulations in which which all populations populations are susceptible susceptible to relatively frequent frequent extinction, migration migration among among popUlations populations is limited limited (i.e., most recruitment recruitment is lolo well cal), and extinctions extinctions create create vacant vacant habitats habitats which which are recolonized. recolonized. These These two two wellstudied systems appear appear to conform conform reasonably closely to the classical concept concept of of studied metapopulations in an extinction extinction-colonization balance. metapopulations -colonization balance. other systems resemble resemble classical metapopulations metapopulations in certain certain ways, for Many other example having having patchy distributions distributions with no obvious obvious "mainland" "mainland" patches patches ((Hanski example Hanski populations that that do not appear appear to be self selfand Kuussaari, 11995), 995), or having local populations sustaining (Stacey and Taper, 11992, volume). However, However, based based on the fore foresustaining 992, this volume). going evidence, we would argue that only only after after much detailed study can any going evidence, natural system be classical metapopulation. be described described as a classical metapopulation. 1. Mixed Structures Mixed Structures
Many of species patchy habitats habitats reveal Many studies studies of species distributions distributions in patchy reveal that that patches patciles are to other are more more likely to be occupied occupied the nearer nearer they are other occupied occupied patches patches (Brown and Kodric-Brown, Laan and and Verboom, (Brown and Kodric-Brown, 1977; 1 977; Fritz, 1979; 1 979; Opdam, Opdam, 1990; 1 990; Laan Verboom, 1990; 1 990; Lawton Lawton and and Woodroffe, Woodroffe, 1991; 1 99 1 ; C. D. D. Thomas Thomas et et al., at., 1992). 1 992). This This suggests suggests the of all different the possibility possibility that that many many real metapopulations metapopulations combine combine features features of different kinds of from clustered of metapopulation metapopulation structure, along gradients gradients from clustered central central patches patches to isolated ones (Fig. lI e). Central Central patches are united isolated peripheral peripheral ones patches are united by dispersal dispersal into into a single population, population, slightly more more isolated isolated ones ones undergo undergo extinction extinction and and recolonrecolon ization, ization, and and still more more isolated isolated patches patches are are usually vacant. vacant. other cases, cases, the the metapopulation metapopulation structure structure of of aa species species may may vary among among In other regions, regions, because because of of differences differences in the the configuration configuration of of habitat. habitat. Nine Nine metapopumetapopu lations lations of of the the silver-studded silver-studded blue blue butterfly butterfly (Plebejus (Plebejus argus) argus) in Wales Wales show show a continuum from nearly continuum from nearly equal-sized equal-sized patches patches to to aa mainland-island mainland -island configuration configuration (Thomas edithaa forms (Thomas and and Harrison, Harrison, 1992). 1 992). The The butterfly butterfly Euphydryas Euphydryas edith forms a mainmain land - island metapopulation metapopulation in coastal coastal California, California, but but shows shows a mixture mixture of of classical classical land-island and patchy patchy population population features et al., al. , 1988; 1 988; and features in montane montane California California (Harrison ( Harrison et Singer on patches Singer and and Thomas, Thomas, 1996). 1 996). Insects Insects on patches of of ragwort ragwort show show little little population popUlation
Empirical Evidence Evidence for Metapopulation Dynamics Dynamics 22 Empirical
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subdivision in in British British grasslands grasslands (Harrison ( Harrison et et al., al., 1995), 1 995), but but the the effects effects of of subsub subdivision division are are significant significant in in Dutch Dutch dunes dunes (van (van der der Meijden, Meijden, 1979; 1 979; van van der der Meijden Meijden division and van van der der Veen-van Veen-van Wijk, Wijk, this this volume). volume). and The northern northern spotted spotted owl owl (Strix (Strix occidentalis occidentalis caurina) caurina) occupies occupies aa once-cononce-con The tinuous tinuous but but now now coarsely coarsely fragmented fragmented forest forest habitat, habitat, while while the the Californian Californian subsub (S. o. o. occidentalis) occidentalis) lives (in part) part) in still-continuous still-continuous but but selectively logged logged species (S. species forests, and and the the Mexican Mexican subspecies subspecies (S. (S. o. o. lucida) lucida) inhabits inhabits insular insular mountaintops. mountaintops. forests, This natural natural and and unnatural unnatural variation variation in in habitat habitat structure, and presumed presumed metapopumetapopu This structure, and lation structure, feature of of conservation conservation strategies strategies for for the the spotted spotted owl owl lation structure, is a central central feature ( Noon and and McKelvey, McKelvey, 1996; 1 996; Gutierrez and Harrison, Harrison, 1996). 1 996). (Noon Gutierrez and 2. Metapopulations Metapopulations with with Little Little Turnover Turnover 2.
If local local populations populations fluctuate fluctuate fairly fairly independently independently of of one one another, another, but but exex If change low to moderate numbers of immigrants, metapopulation structure may change low to moderate numbers of immigrants, metapopulation structure may have an important important stabilizing stabilizing effect effect at the the regional level even even without without population population have regional level turnover. We We know know of of no no good good examples of this possibility, possibility, but but it could could be be tested tested turnover. examples of comparing the the magnitude magnitude of fluctuations in conspecific conspecific populations popUlations varying by comparing of fluctuations their degree degree of of isolation. isolation. in their Dynamics 3. "Local" 3. "Local" Spatial Spatial Dynamics
A growing number of studies suggest even in relatively concon A growing number of empirical empirical studies suggest that that even tinuous habitat, may be strongly tinuous habitat, the dynamics dynamics and and persistence persistence of of popUlations populations may affected small-scale habitat habitat heterogeneity, heterogeneity, localized interactions, interactions, and limited affected by small-scale and limited et al. 1 988; Harrison, Harrison, 1994a; 1 994a; Amarasekare, Amarasekare, 1994). 1 994). This This migration (e.g., Weiss Weiss et migration al.,, 1988; important class class of one that conceptual is an important of phenomena, phenomena, but one that lies outside outside the conceptual domain domain of of metapopulation metapopulation dynamics.
III. MULTISPECIES MULTISPECIESMETAPOPULATIONS METAPOPULATIONS We now now review empirical studies studies in which which it has been been proposed We proposed that predators predators competitors coexist coexist through through multispecies multispecies metapopulation metapopulation dynamics and prey or competitors earlier reviews by Bengtsson, 11991, "clas(see earlier 99 1 , and Taylor, 11991). 99 1 ). Here, the two "clas sical" conditions conditions we examine examine are that that the interspecific interspecific interaction interaction leads to local extinction extinction or instability and that that both both (or all) species have have something something like a clas classical metapopulation metapopulation structure at similar spatial scales, leading to greater stability or persistence at the regional regional level than than within each local patch. Once again, we begin by identifying ways in which these conditions conditions may not be met in some some natural systems.
A. Is the Interaction Locally Locally Unstable? Unstable? 1. All Local Populations Populations Are Stable or Persistent Persistent
Several recent studies have tested the metapopulation metapopulation explanation explanation for for coex coexistence by asking whether whether local populations populations of of predators and prey are are destabilized destabilized
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Susan and Andrew Susan Harrison Harrison and Andrew D. D. Taylor Taylor
by being isolated. Murdoch 1 996) found Murdoch et al. ((1996) found that populations populations of of the red scale Aonidella uctuate more more on Aonidella aurantii aurantii and and its parasitoid parasitoid Aphytis Aphytis melinus melinus did not fl fluctuate caged than on uncaged uncaged citrus trees. Similarly, C. J. Briggs (unpublished (unpublished manu manuscript) did not fi nd a signifi cant increase temporal variability populations find significant increase in the temporal variability of of populations of of the midge midge Rhopalomyia Rhopalomyia californica californica or its parasitoids parasitoids on caged versus uncaged uncaged coyote bushes (Baccharis (Baccharis pilularis). pilularis). However, the latter latter experiments experiments lasted only 3 --1100 insect generations, become generations, possibly too short for for the anticipated anticipated effects effects to become statistically signifi cant. significant. The interaction interaction between between prickly-pear cactus cactus (Opuntia (Opuntia spp.) and the moth moth Cactoblastis Cactoblastis cactorum, cactorum, which which was successfully successfully introduced introduced to control control the cactus cactus in Australia, Australia, has been described described as a classical case of of coexistence coexistence through through ex extinction Dodd, 11959; 959; A. J. Nicholson, tinction and and colonization colonization dynamics dynamics ((Dodd, Nicholson, as quoted quoted in Monro, 967). However, Monro, 11967). However, more more recent recent observations observations suggest that both both species species per perplants always survive sist locally and that the interaction interaction is stable because some plants attack Monro, 11967, 967, 1975; 1 975; Caughley, 11976; 976; Osmond Monro, attack by the moth moth ((Monro, Osmond and and Monro, 98 1 ). 1 98 1 ; Myers 1981; Myers et al., 11981). Local populations populations may survive survive strong predation predation or competition competition because of of a cryptic life stage. For example, early-successional early-successional plants may coexist coexist with su superior competitors by "recolonizing" perior competitors "recolonizing" newly disturbed disturbed sites from seed banks, rather than by dispersal. Resting stages appeared recolonization by appeared to explain the recolonization waterfl eas (Daphnia waterfleas (Daphnia spp.) of of cattle tanks from from which which they had been eliminated eliminated Murdoch et al. 1 984). Alternatively, prey by predatory bugs (Notonecta (Notonecta spp.) ((Murdoch al.,, 1984). populations may survive simply because predators predators leave a patch before before eliminat eliminating all prey; examples examples include include olive scales and their parasitoids parasitoids ((Huffaker ai., Huffaker et al., 1 986; Taylor, 11991) 99 1 ) and cottony cushion parasitoids 1986; cushion scale, scale, vedalia vedalia beetles, beetles, and and parasitoids 969). (Quezada, 11969). extinctions may may occur occur primarily for for reasons reasons other other than the inter interLocal extinctions specific interaction. Hanski and Ranta 1 983) and Bengtsson 1 989, 11993) 993) hyhy Ranta ((1983) Bengtsson ((1989, pothesized pothesized that three competing Daphnia Daphnia species in rock pools in the Baltic Sea coexisted cial pools, coexisted through through extinction extinction and and recolonization. recolonization. In studies studies with with artifi artificial pools, Bengtsson 1 989, 11993) 993) showed Bengtsson ((1989, showed that that extinction extinction rates rates were were higher higher in three-species three-species pools pools than than in two- or one-species one-species pools. However, However, species pairs and and even triplets could coexist coexist for for 44- 77 years even in very small pools. Bengtsson concluded concluded that that some apparent apparent extinctions were really pseudoextinctions pseudoextinctions caused by a cryptic rest resting stage, and that most natural natural extinctions were probably probably caused caused by predation, predation, low resource We note that if extinctions resource levels, salinity, or desiccation. We extinctions are caused caused by both both the the competitive competitive interaction interaction and and the extrinsic forces, forces, it creates creates the inter interesting possibility possibility that subdivision subdivision has both both positive positive and negative effects effects on sta stability. The parasitoids parasitoids Hyposoter Hyposoter horticola horticola and and Cotesia Cotesia melitaearum melitaearum parasitize The parasitize up to 90% of of the larvae of of the butterfl butterflyy Melitaea Melitaea cinxia, cinxia, and may contribute contribute to the observed Hanski et al., 11994; 994; Lei and Hanski, Hanski, observed local extinctions extinctions of of M. cinxia ((Hanski 11997), 997), but their importance importance relative relative to drought drought and and other other factors factors is not yet clear. Other parasitoid interaction include include spatial Other factors factors which which may stabilize stabilize this hosthost-parasitoid
22 Empirical Empirical Evidence Evidence for MetGpopulation Metapopulation Dynamics Dynamics
37 37
density dependence dependence in in the the mortality mortality caused caused by by aa generalist generalist hyperparasitoid hyperparasitoid (Lei ( Lei density and Hanski, 1997). 1 997). and Hanski, et al. al. (1994) ( 1 994) has has shown shown that that the the extinction extinction and and Recent work work by by Antonovics Antonovics et Recent alba) affects affects both both the the inin colonization of of populations populations of of white white campion campion (Silene alba) colonization (Ustilago violacea) and and the the distribution distribution of of the the cidence of of its anther anther smut smut disease disease (Ustilago cidence plant ' s disease-resistance disease-resistance genotypes genotypes among among populations. populations. However, However, there there is no no plant's evidence yet yet that that the the disease disease affects affects rates rates of of local local extinction extinction in the the plant. plant. evidence 2. Some Some Prey Prey Populations Populations Are Are Stable Stable or or Persistent Persistent ("Refuges") ("Refuges") 2.
analogy to to source-sink source - sink dynamics, dynamics, a prey prey or or inferior inferior competitor competitor In a close analogy may may persist persist because because it has has a particular particular type type of of habitat habitat in which which it escapes escapes its Balanus balanoides balanoides suffers suffers heavy heavy predator or or superior superior competitor. competitor. The The barnacle barnacle Balanus predator Urosalpinix cinerea cinerea in the the subtidal subtidal zone, zone, but but persists persists and and predation the snail snail Urosalpinix predation by the recruits in the the intertidal intertidal zone zone where where the the snail snail is absent absent (Katz, ( Katz, 1985). 1 985). European European recruits have a refuge refuge from predation in sprayed orchards orchards where where their their main main red mites have from predation predators are scarce Walde, 1991, 1 99 1 , 1994). 1 994). However, However, experiments experiments showed showed that that predators scarce ((Walde, refuges do do not explain the stability of interaction between between red red scale and and the the refuges of the interaction Aphytis melinus ( Murdoch et al. , 1 996). parasitoid parasitoid Aphytis (Murdoch al., 1996). In a slight modification modification of refuge pattern, pattern, interactions interactions may be stable stable in in of the the refuge may be Opheroptera some habitat types but not in others. For example, the winter moth some habitat types but not others. For example, the winter moth Opheroptera appears to coexist coexist stably with its parasitoids parasitoids in apple orchards, orchards, and and to brumata appears disperse from orchards orchards into into forests forests where where local extinctions extinctions are frequent (Murdoch (Murdoch disperse from are frequent 1 985; MacPhee MacPhee et al., 1988). 1 988). et al. al.,, 1985; Finally, patch effective refuges. patch size may may create create effective refuges. Lizards Lizards (Anolis spp.) concon tribute tribute to local extinctions extinctions of of spider spider popUlations, populations, but sufficiently sufficiently large large islands support (Schoener and Spiller, support stable popUlations populations of of both spiders spiders and lizards (Schoener Spiller, 11987a,b; 987a,b; Schoener, 99 1 ). Conversely, pool frogs frogs (Rana lessonae) persist Schoener, 11991). Conversely, pool persist better better ponds support pike (Esox lucius), which in small than in large ponds, since large ponds support pike major cause of of local extinctions of of the frog (Sjogren, (Sj6gren, 11991; Sj6gren Gulve, Gulve, are a major 99 1 ; Sjogren 11994). 994). 3. 3. Predator Predator Populations Populations Are Are Stable Stable (Generalists) (Generalists) Even Even if if a prey species species exists exists as a metapopulation, metapopulation, and its predator predator causes local extinctions extinctions or instability, the predator predator may persist persist stably if if it has alternate alternate prey. The predatory mite Typhlodromus frequently eliminates Typhlodromus pyri pyri frequently eliminates the European European red mite (Panonychus ulmi) from from individual apple trees, and migration among trees enhances enhances the persistence persistence of P. ulmi. Nonetheless, Nonetheless, T. pyri pyri is consistently abundant, abundant, since it can feed on pollen and the apple apple rust mite (Aculus schlectendali) as well as P. ulmi ((Walde, Walde, 11991, 99 1 , 11994; 994; Walde et al. 992). Similar examples al.,, 11992). examples include Daphnia and Notonecta ((Murdoch Murdoch et al., 11984), 984), the oak gall wasp wasp Xan Xantho teras politum and its parasitoids 98 1 ), and spiders thoteras parasitoids (Washburn (Washburn and Cornell, Cornell, 11981), and lizards (Schoener 987a,b; Spiller and Schoener, 990). It is (Schoener and Spiller, 11987a,b; Schoener, 11990). possible, possible, though though by no means means proven, that these systems function function as single-spe single-species metapopulations metapopulations for for the prey.
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Susan Harrison Harrisonand and Andrew Andrew D.D. Taylor Taylor Susan
B. B. Do Do Both Both (All) Species SpeciesShow Show Metapopulation Metapopulation Structures Structuresat Similar Similar Spatial Spatial Scales? Scales? Prey Is Subdivided Subdivided but Predator Predator Is Not Not 1. Prey Interacting species often differ in their mobility, with predators or parasitoids usually being better dispersers than their hosts or prey. This is clearly the case for goldenrod goldenrod aphids (Uroleucon nigrotuberculatum) nigrotuberculatum) and their ladybird beetle predator (Coccinella septempunctata) septempunctata) ((Kareiva, Kareiva, 11984, 984, 11987) 987) and may also be true predator for the red scale, olive scale, cottony cushion scale, gall midges, and their re respective parasitoids, parasitoids, discussed discussed above. In subdivided subdivided experimental populations populations of the intertidal snail Nucella Nucella ((= Thais) emarginata, emarginata, local extinctions extinctions were caused by predators predators (gulls, geese, and crabs) that are both mobile and generalists (Quinn et al., 11989). 989). If a predator predator is either very mobile or a generalist, generalist, it will be present present appear to on most prey patches most of the time, and subdivision would not appear for such a system to act as a single singleexplain coexistence. Again, it is possible for metapopulation for for the prey. species metapopulation =
None of of the the Species Species Is Subdivided Subdivided 2. None Mosquitos described as coexisting coexisting with their aquatic aquatic predators predators Mosquitos have been described through extinction extinction and recolonization, recolonization, but all individuals individuals disperse from from their natal through patch Murdoch et al. 985). A metapopulation patch and and oviposit oviposit on many patches patches ((Murdoch al.,, 11985). metapopulation explanation explanation for for coexistence coexistence has been been proposed proposed for for kangaroo kangaroo rats rats and smaller seed seedeating rodents 1 995), but these authors note that "the rodents by Valone and Brown Brown ((1995), "the responses observed observed largely represent represent habitat habitat selection selection by individual individual rodents" rodents" and and responses that typical persistence persistence times are on the order order of of a few few months, indicating that that months, indicating these dynamics dynamics take place within rather rather than among local populations. populations. Some suggests that cinnabar cinnabar moths (Tyria jacobaeae) may coSome evidence evidence suggests moths (Tyria jacobaeae) may co exist with distributed host (ragwort, Senecio and Senecio jacobaea), jacobaea), and with their their patchily distributed host plant plant (ragwort, with certain of their parasitoids parasitoids and competitors, through with certain of their and competitors, through classical classical multimulti species metapopulation metapopulation dynamics dynamics (van der Meijden species der Meijden, Meijden, 1979a; 1 979a; van van der der Meijden and and van van der der Veen-van Veen-van Wijk, Wijk, this this volume; Crawley Crawley and and Pattrasudhi, Pattrasudhi, 1988; 1 988; McEvoy al.,, 11993). in British McEvoy et ai. 993). However, However, one one test test of of this hypothesis hypothesis in British grasslands grasslands found found that that the moth moth disperses disperses so well, relative relative to the the distances distances between between ragwort ragwort patches, patches, as to preclude a metapopulation metapopulation explanation explanation for for coexistence coexistence (Harrison ( Harrison al.,, 1995). et al. 1 995).
C. Classical Classical Multispecies Multispecies Metapopulation Metapopulation Dynamics Dynamics Interactions between herbivorous Interactions between herbivorous and and predatory predatory mites mites in greenhouses greenhouses present present an an interesting interesting mixture mixture of of patchy-population patchy-population and and metapopulation metapopulation features features (e.g. Nachman, and Laane, et al., al., 1991; 1 99 1 ; van van de de Nachman, 1988, 1 988, 1991; 1 99 1 ; Sabelis Sabelis and Laane, 1986; 1 986; Sabelis Sabelis et Klashorst Klashorst et et al., al., 1992). 1 992). Metapopulation Metapopulation dynamics dynamics are are suggested suggested by by the the facts facts that mites between adjacent plants, and that mites have have very very limited limited mobility mobility between adjacent plants, and suitable suitable plants plants are are frequently frequently unoccupied. unoccupied. However, However, movement movement is more more frequent frequent and and more more directed directed than than most metapopulation metapopulation models models assume: assume: individuals individuals may may ococ-
22 Empirical Empirical Evidence Evidence for Metapopulation Metapopulation Dynamics Dynamics
39 39
cupy many many plants plants in in their their lifetimes, lifetimes, emigration emigration by by both both predators predators and and prey prey is is cupy density-dependent, and and predator predator dispersal dispersal may may respond respond to to chemical chemical signals signals by by density-dependent, the prey. prey. To To date date there there has has been been no no direct, direct, conclusive conclusive test test of of the the importance importance the of metapopulation metapopulation structure structure in in stabilizing stabilizing mite mite and and plant plant systems, systems, e.g., e.g., by by comcom of paring the the persistence persistence of of predators predators and and prey prey on on closely closely versus versus widely widely spaced spaced paring arrays of of plants. plants. arrays Conclusive experimental tests of of classical classical multispecies multispecies metapopulation metapopulation dydy Conclusive experimental tests namics are are exceptionally exceptionally difficult, difficult, and and few few have have been been done. done. (However, ( However, see see HolHol namics yoak and and Lawler, Lawler, 1996, 1 996, for for an an excellent excellent recent recent example.) example.) In In certain certain studies studies menmen yoak tioned above, above, such such as as that that of of Baccharis-feeding insects, insects, there there is is aa tantalizing tantalizing tioned suggestion of of metapopulation metapopulation effects, effects, but but strong strong tests tests are are precluded precluded by by too too few few suggestion patches and/or and/or generations. generations. In In others, others, such such as as that that of of Melitaea cinxia and and its its patches parasitoids, suggestive suggestive patterns patterns are beginning to to emerge. emerge. Finally, Finally, in in such such studies studies parasitoids, are beginning as that of rock rock pool pool Daphnia, some some of of the the conditions for metapopulation coex as that of conditions for metapopulation coexistence are met, but but it is difficult difficult to assess their their importance importance relative relative to to other factors, istence are met, to assess other factors, such as as dormant life stages stages and and abiotic causes of of extinction. such dormant life abiotic causes extinction.
IV. DISCUSSION IV. DISCUSSION Large-scale -metapopulation structure Large-scale spatial popUlation population structure structuremmetapopulation structure in the broad natural sys broad sense s e n s e-is m i s clearly important in a large number and and variety of of natural systems. Real populations populations often often behave behave very differently differently than than they they would would if if they they were were unsubdivided, habitats unsubdivided, and and both both the natural natural and and the human-caused human-caused discontinuity discontinuity of of habitats have large effects metapopulation approach effects on on how how populations populations function. function. Thus, a metapopulation approach is understanding and is essential essential to to understanding and managing managing many many natural natural phenomena. Our Our aim aim here is not to discount such an approach, approach, but rather rather to use empirical evidence to refine and clarify our our notions notions of of how how real metapopulations metapopulations work. The results cast some doubt species metapopu doubt on the classical models models of of both both single- and multi multispecies metapopulations. lations. We find that natural metapopulations Fig. 11), ), metapopulations have a variety variety of structures ((Fig. with implications for for persistence and coexistence coexistence that are correspondingly correspondingly varied. These different structures are are of of course not discrete entities, but rather lie along continuua in terms of patch structure and migration rates ((Fig. Fig. 2). Our review illustrates that when natural systems deviate substantially from the classical, ex extinction-and-colonization structure, their essential behavior behavior changes considerably as as well; well; in in all all cases, cases, persistence persistence and/or and/or coexistence coexistence become become more more dependent dependent on on local (within-population) (within-population) processes and and less so on on metapopulation ones. It is there therefore fore crucial crucial to to avoid avoid labeling labeling aa system system as as aa metapopulation metapopulation under under aa broad broad defi deftnition - e.g., because habitat is patchy, nitionme.g., patchy, some some local extinctions occur, occur, or or popu populations and then lations in in different different areas areas fluctuate fluctuate out out of of synchronysynchronymand then applying applying to to itit conclusions that nition. that follow follow from from aa narrower narrower (classical) (classical) defi definition. We We also also conclude conclude that that classical metapopulations metapopulations form form aa minority, minority, even even among among the the modest modest number number of of systems systems that that have have been been well well studied studied in in metapopumetapopu-
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SusanHarrison Harrisonand and Andrew AndrewD.D. Taylor Taylor Susan
Variance Variance patchsize size inin patch (or other other (or determinant determinant of population population of persistence) persistence)
MAINLAND-ISLAND MAINLAND-ISLAND ",_ .. - .. .. ..
NON NONEQUILIBRIUM EQUILIBRIUM
s , , , s
*, , , .
:'[ "CLASSIC" "CLASSIC" \", (LEVINS) :: �I (LEVINS) \ I
\�
9
.. ..
" ... '
s
"
PATCHY PATCHY POPULATIONS POPULATIONS
Dispersal Dispersal distance distance (relative (relative to to interpatch interpatch distances) distances)
FIGURE22 Relationships among among different different types of metapopulation. FIGURE
lation terms. tenns. We fi nd only a few find few examples of single species existing in a balance balance between the extinction and colonization colonization of populations populations and almost none none of of sys systems in which multiple species coexist through through tightly coupled coupled metapopulation metapopulation dynamics at comparable comparable spatial scales. Is this apparent apparent scarcity real? Clearly, the great difficulty of among-population processes of studying among-population processes in nature nature is aa major major ob obstacle to reaching reaching finn firm conclusions. conclusions. However, However, it may also be that systems in which migration too low, or which some populations migration among among patches patches is too high high or or too or in which some populations are much more more common approxi are highly persistent persistent (Fig. 2), are are truly much common than than ones ones approxistructure. mating the classical structure. If metapopulations sufficiently to include include the nonnon If we we broaden broaden our our view of of metapopulations classical how does perception of prevalence and imporimpor classical kinds, kinds, how does this affect affect our our perception of the prevalence tance of metapopulations? This This is clearly an an area area of of active work, work, as this volume volume tance of metapopulations? illustrates, but but we we will advance some of of our our own own speculations. speculations. Regional illustrates, advance some Regional persist persistence may may be be an overemphasized concern, concern, arising from models models that were based based ence an overemphasized arising from that were on a simple analogy and and that that overlooked overlooked variation patch size, on simple birth-and-death birth-and-death analogy variation in patch detailed local local population dynamics, explicit explicit spatial patches, and and detailed population dynamics, spatial relations relations among among patches, spatio-temporal correlation correlation in in the the environment. environment. Adding Adding such such real-world real-world refinerefine spatio-temporal ments to to models models may may have have the the general general effect effect of of reducing reducing the the relative relative importance importance ments of migration migration and and recolonization, recolonization, and and increasing increasing that that of of local population population processes, processes, of for regional regional persistence. persistence. If If this this is true, true, then then what what other other types of "metapopulation "metapopulation for types of effect" effect" may may we we seek seek in in nature? nature? When levels levels of of migration migration among among patches patches are are moderate, moderate, and and patches patches vary vary in in When their degree degree of of spatial spatial isolation, isolation, aa system system may may be be demographically demographically unified unified in in their central patches patches and and exhibit exhibit rescue rescue effects effects or or extinction extinction and and recolonization recolonization on on central increasingly marginal marginal ones. ones. The The defining defining feature feature of of such such aa metapopulation metapopulation is is not not increasingly the dependence dependence of of regional regional persistence persistence upon upon local local extinction extinction and and recolonization, recolonization, the but the the strong strong effect effect of of patch patch structure structure and and dispersal dispersal on on local local population population persistpersist but ence and/or and/or regional regional distribution. distribution. This This type type of of metapopulation metapopulation structure structure seems seems to to ence us us aa highly highly plausible plausible one. one. Consideration of of variation variation in in patch patch size size or or quality quality leads leads into into the the realm realm of of Consideration mainland - island and and source-sink source- sink dynamics, dynamics, where where again again the the appropriate appropriate quesquesmainland-island
22
Empirical EmpiricalEvidence Evidencefor Metapopulation MetapopulationDynamics Dynamics
41
tions are not about regional persistence, but about regional distribution, and about local persistence in habitats too small (islands) or poor in quality (sinks) to support Mainland-island are well documented, but long-lived populations. Mainland -island dynamics are source-sink source - sink dynamics are virtually untested; how frequently species are found in habitats where they are unable to replace themselves without immigration, and how much this affects their overall demography, remains a very open area for empirical research. The related refuge models of predation and competition, in which coexistence is made possible by the net dispersal of the victim species from habitats of low to those of high predation or competition (e.g., Hochberg and Holt, 11995), 995), also deserve more empirical study. another important area for for exploration is the For multispecies systems, another effect of trophic complexity. Both theory and empirical work have emphasized two or three tightly coupled species, but real food webs are nearly always more M. cinxia cinxia system comprises four important species on complex than this. The M. three three trophic trophic levels, plus some eight minor species of which some are gener generalists (G. Lei and Hanski, unpublished manuscript). manuscript). Four spider and three lizard Bahamanian islands (Schoener and species and their shared prey interact on Bahamanian sevSpiller, 11987a,b). 987a,b). Even relatively simple biocontrol systems typically include sev enemies (e.g., Quezada, Quezada, 1969; Murdoch et et at. al.,, 1996). eral important natural enemies 1 969; Murdoch 1 996). have yet to address systems of many species at multiple Metapopulation models have charactertrophic levels, each with different population dynamics and dispersal character istics, each coupled to other species to varying degrees (but see Holt, this volume). sitWhether spatial subdivision retains its potentially stabilizing effect in these sit other important important consequences, is an open area area for theoretical and uations, or has other empirical work. empirical In all of of these these extensions of of metapopulation metapopulation dynamics, an approach that comcom bines empirical general empirical work and modeling will be very helpful. helpful. For For example, example, no general guidelines are are available available for for empiricists empiricists to decide decide how much much migration migration is enough, too little or too much for metapopulation occur. Merely observing some metapopulation effects effects to occur. asynchrony in local population adequate, since this will be population fluctuations fluctuations is not adequate, shaped not only by migration, but by patterns patterns of of environmental variability and by the sampling regime. Moreover, the critical critical level of of migration will depend depend greatly on the exact hypothesis, or type of of metapopulation behavior, behavior, that is of of interest. Thus, combining combining field moderately detailed detailed system-specific interest. field work with moderately system-specific models models will be valuable valuable in many cases. In conclusion, this review review of of empirical empirical studies studies makes makes it clear clear that that a great great of spatial population structures exists exists in nature, nature, and and recognition of of this this variety of diversity suggests and theoretical suggests changes changes in how how both both empirical empirical and theoretical metapopumetapopu lation lation research research are are approached. approached. Whether Whether experimental experimental or or observational, singlesingle or or multispecies, empirical empirical studies studies need need to to take take fully into account account the the different different types of of metapopulation metapopulation structure structure that that are are possible, possible, perhaps perhaps treating them them as alternative alternative hypotheses to test. While empiricists empiricists attempt attempt to better better characterize characterize metapopulations for theorists metapopulations in nature, nature, an important important task for theorists is to to continue exploring exploring the ways ways in which which metapopulation metapopulation behavior behavior changes changes as as patch patch configuration, configuration, disdis persal, and local population dynamics are altered persal, and population dynamics altered in realistic realistic ways. Through Through this this
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SusanHarrison Harrisonand and Andrew AndrewD.D. Taylor Taylor Susan
combined c o m b i n e d effort, effort, we we will will be be much m u c h better better able able to to identify identify the the ways ways that that within withinand and between-population b e t w e e n - p o p u l a t i o n processes p r o c e s s e s interact interact to to determine d e t e r m i n e the the behavior b e h a v i o r of of natural natural systems. systems.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS We We thank thank Ilkka Ilkka Hanski. Hanski, Daniel Daniel Simberloff, Simberloff, Chris Chris Thomas, and and an an anonymous reviewer reviewer for helpful comments comments on on an an earlier earlier draft. draft. helpful
II
Metapopulation Dynamics and Landscape LandscapeEcology Ecology John A. A. Wiens Wiens John
INTRODUCTION II.. INTRODUGION The fusion of metapopulation metapopulation studies studies and and landscape ecology should should make make for for an exciting exciting fusion of landscape ecology scientific and Gilpin, Gilpin, 1991) 1 99 1 ) scientific synthesis synthesis (( HHanski a n s k i and
The synthesis o metapopulation studies and landscape landscape ecology anticipated anticipated The off metapopulation Hanski and and Gilpin Gilpin has has barely barely yet yet begun. There There are are at least two reasons reasons for for this by by Hanski ( Wiens, 1995a). 1 995a). First, First, as many many of of the the chapters chapters in this this volume volume illustrate, illustrate, metameta (Wiens, population theory theory continues continues to to be be tied tied to to a view of of spatial spatial patterning patterning of of environenviron population ments in which which patches patches are are embedded embedded in in a featureless featureless background background matrix. matrix. Second, Second, ments landscape ecology seems seems still to to be be in the the process process of of defining defining what what it is about and landscape ecology about and describing complex complex spatial spatial patterns, patterns, but but it it has has not not developed developed much much theory theory to to deal deal describing with spatial spatial patterning. patterning. By By focusing focusing on on some some shared shared areas areas of of interest, interest, perhaps perhaps the the with synthesis synthesis of of these these disciplines disciplines can can be be accelerated. accelerated. In this this chapter, chapter, I consider consider the the relationship relationship between between the the emerging emerging (but (but yet yet In immature) discipline discipline of of landscape landscape ecology ecology and and the the emerged emerged (but (but perhaps perhaps adolesadoles immature) cent) discipline discipline of of metapopulation metapopulation dynamics. dynamics. I will will argue argue that that considerations considerations of of cent) metapopulation structure structure may may often often be be incomplete incomplete unless unless they they are are framed framed in in the the metapopulation context context of of the the underlying underlying landscape landscape mosaic. mosaic. Metapopulation Metapopulation Biology Biology
Copyright Copyright 9 © 1997 1997 by by Academic Academic Press, Press, Inc. Inc. All All rights rights of ofreproduction reproduction in in any any form form reserved. reserved.
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John A. John A. Wiens Wiens
II. II. APPROACHES APPROACHESTO TO PATCHINESS PATCHINESS Ecologists have always known known that nature nature is patchy and heterogeneous, heterogeneous, even Ecologists if much much of of their their theory has not not treated it so. Habitats Habitats in areas used by humans if humans fragments, and the patchwork occur as sharply defined defined blocks blocks or fragments, patchwork nature nature of of the landscape landscape mosaic mosaic is especially evident in such environments. environments. Even in more more natural natural settings, habitats are heterogeneous heterogeneous at virtually any scale of settings, however, however, habitats of resolution. resolution. Although indistinct gragra Although patch patch boundaries boundaries in such situations situations may sometimes sometimes be indistinct dients 992b), the spatially variable dients rather than sharp discontinuities discontinuities (Wiens, (Wiens, 11992b), variable character follow the convenconven character of of environments environments still remains. remains. In this chapter chapter I will follow tion that has become become widespread under widespread in ecology of of considering considering such variation variation under the rubric though "patches" "patches" are not always evident rubric of of "patchiness," "patchiness," even though evident in nature. nature. Dealing Dealing with such spatial heterogeneity heterogeneity has been been a major major challenge challenge in both empirical and theoretical ecology. Faced with the daunting daunting complexity of of spatial patterns patterns in the real world, world, field ecologists ecologists historically tended tended to focus focus on patterns and dynamics dynamics of of ecological systems within relatively homogeneous homogeneous habitat habitat types (e.g., watersheds, watersheds, woodlots) woodlots) or aggregated spatial spatial variation into into dimensionless dimensionless indices indices of of heterogeneity heterogeneity or dispersion. dispersion. More More recently, recently, it has become become fashionable fashionable to map map spatial patterns patterns at broad broad scales using geographic geographic information information systems systems and impor spatial statistics, statistics, but the link between between such technologies technologies and ecologically important questions questions is not always apparent. apparent. Spatial variance also strains the capacities of of analytical models models and theory if it is viewed viewed explicitly (i.e., by location) location) rather than averaged as "noise." "noise." As a consequence, consequence, many theoreticians theoreticians concerned concerned with heterogeneity have contented themselves themselves with simple models in which which spatial patterning is collapsed into patches Kareiva, 11990b; 990b; Wiens, 1995a). 1 995a). patches and an ecologically neutral neutral "matrix" "matrix" ((Kareiva, Such patchmatrix theory is usually spatially implicit ( Hanski, 1994c), 1 994c), in that patch-matrix implicit (Hanski, the locations 996a). The inin locations of of patches patches in the matrix are not specified (Wiens, (Wiens, 11996a). teresting in teresting dynamics occur in the patches, patches, which are usually considered considered to be internally homogeneous; homogeneous; the matrix is viewed viewed as inhibiting inhibiting interactions interactions among among predators). patches (e.g., migration, migration, colonization, colonization, gene flow, prey discovery by predators). Traditional -matrix Traditional metapopulation metapopulation theory is an elaboration elaboration on this patch patch-matrix theme. 1 970; Hanski, theme. Levins' Levins' metapopulation model ((1970; Hanski, this volume) volume) considered the patches habitat of of a population population to be subdivided subdivided into an infinite number of of similar similar patches undefined locations mod occupying undefined locations in a background background matrix. As metapopulation metapopulation modeling has progressed, progressed, however, details about patch sizes, patch clumping, indi individual movement movement capacities, capacities, local patch dynamics, and explicit patch locations locations have been Hanski, 11994a,c; 994a,c; see Hanski, been incorporated incorporated ((Hanski, Hanski, this volume; Gyllenberg Gyllenberg et et al. al.,, this volume). volume). Most Most patch theory deals deals with with the the dynamics dynamics of of populations populations occupying occupying a patchy environment (Wiens, 976; Levin, 11976; 976; Kareiva, 11990b; 990b; Shorrocks Swing environment (Wiens, 11976; Shorrocks and Swingland, 990). Another heterogeneity has focused land, 11990). Another approach approach to heterogeneity focused on the the dynamics dynamics of of the patches patches themselves. themselves. Although Although the spatial pattern pattern of of some patches, patches, such as the islands islands considered considered in island biogeography biogeography theory, may be relatively static in ec-
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Metopopulation Dynamics and and Landscape MetapopulationDynamics LandscapeEcology Ecology
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ological time, the patch structure of most environments is not. Patches are de destroyed and generated by disturbances at multiple scales. They undergo change through successional development. These "patch dynamics" ((Pickett Pickett and White, 11985) 985) produce changes in the spatial patterns and relationships of patches in a matrix. Attempts to model these dynamics have generally followed analytical approaches 974; Hastings, 11991)or 99 1 ) or have approaches (patch demography; Levin and Paine, 11974; simulated the spatial and temporal dynamics of patchy environments (e.g., Fahrig, 11990). 990). Most of this work has followed the patch-matrix patch-matrix conceptualization of spatial patterns. The recent Risser et recent emergence of landscape ecology as a discipline ((Risser et al. al.,, 11984; 984; Forman and Godron, 11986; 986; Merriam, 11988; 988; Turner, 11989; 989; Wiens, 1992a; 1 992a; 993; Hobbs, 11995) 995) offers the prospect Wiens et et al. al.,, 11993; prospect for for going beyond a simple patch -matrix approach to adopt a more realistic, spatially textured patch-matrix textured view of het heterogeneity. In landscape ecology, the "matrix" is itself spatially structured, structured, and spatial relationships play an active role in determining determining the dynamics within the "patches" of interest. Patches are viewed as components in a landscape mosaic, and what happens happens within and among the patches patches in a landscape may be contingent on the composition and dynamics of other elements of of the landscape mosaic 993; Andren, 1 994; Wiens, 1995a, 1 995a, 11996a). 996a). (Wiens et et al. al.,, 11993; Andr6n, 1994;
III. WHAT WHATIS IS LANDSCAPE LANDSCAPEECOLOGY? ECOLOGY? One of the first first tasks of an emerging emerging discipline is to define its topic and itself. "Landscape" has been defi ned as "a heterogeneous land area composed of a clus defined cluster of interacting ecosystems" Forman and Godron, 11986), 986), "a mosaic of hetero ecosystems" ((Forman heteroUrban et 987), or "a geneous land forms, vegetation types, and land uses" ((Urban et al., al., 11987), (Turner, 11989). spatially heterogeneous area" (Turner, 989). Accordingly, "landscape ecology" is "a study of of the structure, function, and change change in a heterogeneous land area composed of interacting ecosystems" ((Forman Forman and Godron, 1986) 1 986) or "the inves investigation of ecosystem structure and function at the landscape scale" (Pojar (Pojar et et al. al.,, 1 994). It emphasizes "broad spatial scales and the ecological effects of the spatial 1994). patterning of ecosystems" (Turner, 11989) 989) and "offers a way to consider environ environmental heterogeneity or patchiness in spatially explicit terms" (Wiens et et al. al.,, 11993). 993). If these definitions are a bit nebulous, it may reflect the multifarious historical development of landscape ecology and continuing uncertainty or disagreement over what it is really about. Landscape ecology began in northern Europe during the 11960s 960s as a merging of holistic ecology with human geography, with infusions from land-use planning, landscape architecture, architecture, sociology, and other other disciplines 993) ((Fig. Fig. 11,, top). From (Turner, 11989; 989; Wiens et et al. al.,, 11993) From the outset, the emphasis was practical and applied: the focus was on the interaction of humans with their environment at a broad ((landscape) landscape) spatial scale. In the early 11980s, 980s, the discipline colonized North America (and other continents, most notably Australia). The
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JohnA.A.Wiens Wiens John Human Geography~
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Functional (holistic) Ecology
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Land-use e Policy
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FIGURF development of landscape ecology in Europe FIGURE |1 Contributors Contributors to the historical historical development landscape ecology Europe (top) (top) and North America America (bottom).
beachheads in in North North America America were were small small and somewhat isolated. isolated. Perhaps Perhaps beachheads and initially somewhat through founder founder effects effects or or mutations, mutations, the the development development of of landscape landscape ecology ecology there there through followed aa somewhat somewhat different different trajectory trajectory (Fig. ( Fig. 1, 1 , bottom). bottom). The The linkage linkage with with tratra followed ditional ecology ecology was was much much stronger stronger than than in in Europe, Europe, and and as as aa consequence consequence the the ditional questions asked asked and and approaches approaches used used differed differed considerably. considerably. There There was was aa more more questions self-conscious emphasis emphasis on on concepts concepts (Wiens, (Wiens, 1995a), 1 995a), aa greater greater reliance reliance on on quanquan self-conscious titative procedures procedures (Turner (Turner and and Gardner, Gardner, 1991), 1 99 1 ), and and an an application application of of the the landland titative scape scape perspective perspective to to aa broad broad range range of of basic basic as as well well as as applied applied problems. problems. These pathways pathways of of historical historical development development have have led led to to three three rather rather different different These views of of the the primary primary focus focus of of landscape landscape ecology. ecology. Continuing Continuing in in the the European European views
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Metapopulation Metopopulation Dynamics Dynamics and and landscape LandscapeEcology Ecology
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tradition, one view portrays landscape ecology as "a new holistic, holistic, problem-solv problem-solving approach to resource management" ( Barrett and Bohlen, 99 1 ). It is a syn management" (Barrett and Bohlen, 11991). synthetic, thetic, holistic, human human ecology. The The second second view, which has become become most prev prev' Neill alent among ecologists, treats treats "landscape" "landscape" as a level of of organization organization (e.g., O O'Neill et 986; Gosz, 11993) 993) or as a scale et al. al.,, 11986; scale of investigation (i.e., tens to thousands thousands of of 993; Hobbs, 1 994; Pojar et ha; Forman and 986; Hansen et and Godron, 11986; et al. al.,, 11993; Hobbs, 1994; et al., al., 11994). 994). In the the latter latter case, case, the the questions are often no different different from those that ecologists have always asked; they are just asked at a much broader broader scale. The third view more explicitly emphasizes the structure and dynamics of of landscape 989; Wiens et mosaics and their their effects on ecological phenomena (Turner, (Turner, 11989; et al., al., 11993; 993; Wiens, 11995a). 995a). Rather than restricting the the focus to broad scales, the the scale of investigation investigation is dictated dictated by the organisms studied and the questions asked ((Wiens, Wiens, 11989a; 989a; Haila, 11991; 99 1 ; Pearson et al., 1996). 1 996). In this view, landscape et al., landscape ecology is more than than just spatially explicit ecology, because because the patterns patterns and interactions interactions of entire mosaics are the focus of of investigations. This diversity of of views suggests that that landscape landscape ecology is "a science science in search of itself" Hobbs, 11994). 994). In addition to being a young discipline, intel itself" ((Hobbs, discipline, it is also intellectually immature, in that 987; Hagen, that it lacks conceptual unity (cf. Loehle, 11987; Hagen, 11989). 989). It has no well-defined theoretical framework (Turner, 989; Wiens, 11995a) 995a) (Turner, 11989; and 992a). Despite and tends to be more qualitative than quantitative quantitative (Wiens, (Wiens, 11992a). Despite all of of this, several prevailing prevailing themes of of landscape landscape ecology have emerged: •
patches) vary in quality in both space and 9 Elements in a landscape landscape mosaic ((patches) and time. In a landscape, landscape, patch quality is a continuous rather rather than than a categorical categorical (i.e., suitable vs unsuitable, or or patch-matrix) patch-matrix) variable. Patch quality can be viewed as a spatially dependent benefit function (Wiens et al., 993; Wiens, 11996a). 996a). dependent costcost-benefit (Wiens et al., 11993; 9 Patch edges edges or boundaries may play critical roles roles in controlling or filtering et al. 985; Holland flows of of organisms, nutrients, or materials over space (Wiens et al.,, 11985; et 99 1 ; Hansen and di Castri, 11992). 992). What happens at boundaries may have et ai., al., 11991; important effects on both within-patch within-patch and between-patch between-patch dynamics. landscape mosaic has 9 The The degree degree of of connectivity connectivity among among elements in a landscape major consequences Lefkovitch consequences on patch interactions and landscape landscape dynamics dynamics ((Lefkovitch 1 993). How disturbances and Fahrig, 11985; 985; Taylor et et al., al., 1993). disturbances propagate over a landscape, for for example, may be dictated by landscape landscape connectivity as well as boundary effects (Turner et 1 989). Connectivity involves much more than et al. al.,, 1989). corridors. 9 Patch context matters. What happens happens within a patch is contingent on its location, relative to the structure of of the surrounding mosaic. A patch of of the same habitat habitat may be of of quite different different quality, depending on the features features of of adjacent or nearby 993). Contrary to island nearby elements of the landscape landscape (Wiens et et al., al., 11993). island biogeography theory (or, implicitly, patch-matrix patch-matrix theory), no patch patch is an island (cf. Janzen, 11983). 983). It is this contextual dependency that requires requires landscape ecol ecology to be spatially explicit. •
•
•
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John A. Wiens John A. Wiens
IV. HOW IS IV. HOW IS LANDSCAPE LANDSCAPEECOLOGY ECOLOGYRElEVANT RELEVANTTO TO METAPOPULATION METAPOPULATIONDYNAMICS? DYNAMICS? To metapopulation dynamics, we must must To see see how how these these themes themes may may relate relate to to metapopulation dynamics, we review metapopulation theory theory (see Hanski and review briefl brieflyy the the essential essential features features of of metapopulation (see Hanski and Simberloff, ned in various Simberloff, this volume). volume). "Metapopulations" "Metapopulations" have been been defi defined various ways, but metapopulation is local (patch) but generally generally aa metapopulation is spatially spatially subdivided subdivided into into aa series series of of local (patch) populations. The view emphasizes populations. The classical classical view emphasizes aa balance balance between between extinctions extinctions and and recolonizations populations that long-term persistence recolonizations of of local local populations that facilitates facilitates long-term persistence of of the the 970; Hanski and Thomas, 1 994; Hanski, Hanski, this volume). volume). metapopulation Levins, 11970; metapopulation ((Levins, Hanski and Thomas, 1994; The dynamics dynamics of of local local popUlations populations are density-dependent density-dependent within within patches asynThe patches but but asyn chronous among among patches, patches, and and migration migration (dispersal (dispersal I1)) among among patches patches links them them chronous together. inter together. Interpatch Interpatch movement movement is the key. If If migration migration is large large relative to interpatch uncorrelated sources sources of popUlation variability variability patch distances distances (and (and other other spatially spatially uncorrelated of population are populations will be mixed are not not important), important), the dynamics dynamics of of local populations mixed together together and and they will act population. On the other hand, if if movement they will act as as aa single single large large population. On the other hand, movement among among patches recolonization of patches is infrequent infrequent it may not be adequate adequate to ensure ensure recolonization of habitat habitat patches have suffered extinction, dooming dooming the the entire patches in in which which local local populations populations have suffered extinction, entire metapopulation global extinction. metapopulation to to global extinction. The classical view view of metapopulations and The contrast contrast between between this this classical of metapopulations and aa landscape landscapebased view is perhaps most apparent graphically. In a traditional based perhaps apparent graphically. traditional (theoretical) (theoretical) metapopulation, metapopulation, local local populations populations occur occur in in habitat habitat patches patches in in aa featureless featureless matrix matrix ((Fig. Fig. 2A). Not all patches are occupied at a given time, and populations Not patches are given and local populations wink wink into into and and out out of of existence existence as as extinction extinction and and recolonization recolonization occur. occur. Patches Patches may vary in size or shape, patch-colonization shape, but but the primary determinants determinants of of patch-colonization probability Making a metapopula probability are are movement movement rates rates and and interpatch interpatch distances. Making metapopulation model not sufficient, to cast it in model spatially explicit is therefore therefore necessary, but not a landscape populations of metapopulation occur landscape context. context. In reality, the local populations of a metapopulation occur in in aa complex of other habitat patches, in habitat habitat patches patches that that are are immersed immersed in complex mosaic mosaic of other habitat patches, corridors, corridors, boundaries, boundaries, and the like (Fig. (Fig. 2B). The The most most obvious obvious effects of of this landscape landscape structure structure are are on on individual individual movement movement patterns patterns among among patches patches and, and, consequently, consequently, on patch-recolonization patch-recolonization probabilities. probabilities. In a landscape landscape mosaic, inter interpatch Fig. 2A), complex function function of patch distances distances are are not not Euclidean Euclidean (e.g., (e.g., Fig. 2A), but but are are aa complex of boundary boundary permeabilities permeabilities and and relative relative patch patch viscosities viscosities to to moving moving organisms organisms (e.g., (e.g., 993). Other Fig. Fig. 2B; Wiens Wiens et et al., al., 11993). Other aspects of of metapopulation metapopulation structure, structure, such as the dynamics of the patches themselves (and, consequently, patch-extinction dynamics of the patches consequently, patch-extinction probabilities), uenced by landscape probabilities), may also be infl influenced landscape structure. structure. Because very little empirical work that that directly directly links landscape landscape ecology to Because I
with usage this volume, "migration" rather rather than ITo be consistent consistent with usage elsewhere elsewhere in this volume, I use "migration" than "dispersal" "dispersal" to refer one-way movements Although "migration" refer to one-way movements of individuals individuals beyond beyond their their home home ranges. ranges. Although "migration" is customarily myself the customarily used used in this this sense sense by geneticists geneticists and entomologists, entomologists, to an ornithologist ornithologist like like myself term with birds, I will will term has a specific specificmeaning meaning that that is different different from from "dispersal." "dispersal." In examples examplesdealing dealing with therefore ( 1 992b) discuss therefore use "dispersal" "dispersal" rather rather than than "migration." "migration." Stenseth Stenseth and Lidicker Lidicker (1992b) discuss these these ter terminological minological issues. issues.
A
B
FIGURE inter FIGURE22 (A) Metapopulations Metapopulations in theory. The The solid patches patches are occupied occupied and are linked linked by intermittent migration, migration, whereas whereas the hatched hatched patch is suitable habitat habitat that is presently unoccupied. The The mittent background matrix has no effect on interpatch movements, movements, although the distance distance between patches patches background and their arrangement arrangement may. (B) Metapopulations in reality. The The patches patches are the same, but the "matrix" is a landscape various patches Movement pathways pathways among among suitable patches, landscape mosaic mosaic of of various patches and corridors. corridors. Movement patches, and the probability that migrating patches, are affected by the explicit migrating individuals individuals will reach reach the patches, spatial landscape. spatial configuration configuration of of the landscape.
50 SO
John John A. A. Wiens Wiens
metapopulation metapopulation dynamics dynamics has been been done, done, a discussion discussion of of how how the major major themes themes of of landscape landscape ecology m spatial and temporal temporal variations variations in patch patch quality, boundary boundary effects, landscape connectivity, patch context-affect connectivity, and and patch context--affect the three components components of local extinction, of metapopulation metapopulation dynamics dynamics ((local extinction, interpatch interpatch movement, movement, and recolo recolonization) nization) must must necessarily be somewhat somewhat abstract abstract and and conceptual. conceptual. It may be useful, therefore, therefore, to preface preface this discussion discussion with a few few examples of of the effects effects of of land landscape structure provided by Angelstam Angelstam structure in the real world. Additional Additional examples examples are are provided ((1992), 1 992), Fahrig and Freemark 1 993), and Hobbs ((1995). 1 995). Freemark ((1993), and Hobbs
A. Some Some Examples Examples of landscape LandscapeEffects Effects Some Some of of the effects effects of of landscape landscape structure are related related to patch characteristics characteristics such as patch patch size or spacing. spacing. For For example, the size of of habitat habitat patches patches has been been related Verboom et related to the persistence persistence of of local populations populations of of forest forest birds birds ((Verboom et al. al.,, 11991a; 99 1 a; Villard 992), and the degree habitat patches patches has been Villard et et al., al., 11992), degree of of spacing of of habitat has been shown shown to affect affect the likelihood likelihood of of recolonization recolonization of of vacant vacant patches patches by the Glanville Glanville fritillary (Melitaea Hanski et 995a). Both Both patch (Melitaea cinxia) cinxia) in Finland ((Hanski et al. al.,, 11995a). patch size and remnant forest spacing spacing influenced influenced the use by brown brown kiwis (Apteryx (Apteryx australis) australis) of of remnant forest fragments New Zealand Zealand (Potter, 990). Kiwis fragments in an agricultural matrix in New (Potter, 11990). Kiwis are flightless, isolated remnants. remnants. All fragments flightless, so they must must walk between between isolated fragments less than regardless of 80 m from from other other forest forest remnants remnants were used by the birds, regardless of their size. Movements Movements of of more more than than a kilometer kilometer from the reserve, reserve, however, however, were were accom accom"stepping stones." situation, the spatial plished by using small fragments fragments as "stepping stones." In this situation, interspersion of of habitat habitat patches patches was a critical factor factor in determining determining the effects effects of of patch isolation and, consequently, consequently, the potential for for metapopulation metapopulation dynamics. Patch guration may also be important. important. The emigration Patch edges edges and and their their confi configuration emigration of from patches of GIanviIIe Glanville fritillaries from patches of of suitable suitable habitat, habitat, for for example, increases increases with Kuus with the proportion proportion of of the patch patch boundary boundary that is bordered bordered by open open fields ((Kuussaari et 996). Gates 1 978) found passerine et al. al.,, 11996). Gates and and Gysel ((1978) found that that the abundance abundance of of passerine birds elds and forests, birds increased increased at the boundary boundary between between fi fields forests, and they suggested suggested that that individuals individuals might be drawn drawn to the edge edge as nesting nesting habitat habitat because because of of greater greater food 992; Andren, 992, food availability there. Numerous Numerous studies (e.g., Angelstam, Angelstam, 11992; Andr6n, 11992, 11995), 995), however, however, have have documented documented that predation predation rates may may be greater at such ecotones, ecotones, presumably presumably due to predators predators living in adjacent adjacent areas. For For some some species, species, edges individuals to areas in edges may function function as an "ecological "ecological trap" trap" by attracting attracting individuals which 978). Predation hab which predation predation losses are great (Gates and and Gysel, 11978). Predation risks at habitat edges 985; Angelstam, edges vary as a function function of of the surroundings (Wi1cove, (Wilcove, 11985; Angelstam, 11992; 992; Wiens, 11995b), 995b), so the landscape Pear landscape context context of of patches patches is also important. important. Pear' s ((1993) son 1 993) work on habitat occupancy also son's occupancy by birds in the Georgia Piedmont also illustrates illustrates the effects effects of of landscape landscape context. context. There, There, the composition composition of of the sur surrounding 74% of of the variance rounding matrix matrix explained explained as much much as 74% variance in habitat habitat occupancy occupancy by some but was unimportant unimportant for some species but for other other species. The The demographic demographic con consequences of sequences of such edge- and context-related context-related effects have have received received very little at attention, tention, but but they may have have important important effects effects on metapopulation metapopulation dynamics, es-
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pecially where where populations populations are subdivided subdivided among among many small habitat habitat patches patches and cant. predation risk is signifi significant. The effects landscape mosaic have effects of of corridors corridors linking linking elements in a landscape have also been eld studies been documented documented by by fi field studies (although (although not not to the degree degree that that the the widespread widespread adoption management option adoption of of corridors corridors as a management option would lead one to believe; Bennett, Bennett, ' s cock 11990, 990, 11991; 99 1 ; Hobbs, 992). In Western Hobbs, 11992). Western Australia, Australia, for for example, example, Carnaby Carnaby's cockfor atoos (Calyptorhynchus (Calyptorhynchus funereus) funereus) use roadside roadside vegetation vegetation as a pathway pathway for for foraging movements movements among among woodland woodland patches patches in their their large home home ranges ranges (Saunders, (Saunders, 11990). 990). Where linked or are Where woodland woodland patches patches are not linked are not visually apparent apparent to the cockatoos, they are not used, even though though food food may be available available there. On the other other hand, hand, singing honeyeaters honeyeaters (Lichenostomus (Lichenostomus virescens), virescens), which are habitat habitat generalists, y across farmland Merriam and Saun generalists, readily fl fly farmland with little vegetation ((Merriam Saunders, 11993) 993) and 1 984) found and apparently make make little use of of corridors. Osborne ((1984) found that richness in an area that hedgerow hedgerow area area was the best predictor predictor of of bird bird species species richness area of of Great Great Britain, and the presence presence of of red squirrels squirrels (Sciuris (Sciuris vulgaris) vulgaris) in wooded wooded fragments fragments in The Netherlands Netherlands was positively related related to the the amount amount of of hedgerow hedgerow surrounding Verboom and van Apeldoorn, 1990). 1 990). In Australia, surrounding the fragments fragments ((Verboom van Apeldoorn, the occupancy occupancy of of corridors corridors by arboreal arboreal marsupials could not be predicted predicted by habitat habitat features features within the corridor corridor but required required additional information information on the composition Lindenmayer and 1 993). composition of of the surrounding surrounding landscape landscape ((Lindenmayer and Nix, 1993).
B. B. Movement Movement and and Migration Migration Individual meta Individual movement movement is the most most important important unifying unifying element element in both both metapopUlation 1 99 1 ; Wiens, 1 992b, population dynamics dynamics and landscape ecology (Saunders (Saunders et al. al.,, 1991; Wiens, 1992b, 11995a; 995a; Wiens 993; Ims, 11995). 995). Moreover, Wiens et al. al.,, 11993; Moreover, how fast fast and and how how far organisms organisms move het move imposes imposes a scale on the the environment: environment: highly highly vagile animals animals integrate integrate heterogeneity over therefore perceive over broader broader scales than do sessile sessile individuals individuals and therefore the environment lter or "grain" Wiens, 11985; 985; Fahrig environment with a coarser coarser fi filter "grain" ((Wiens, Fahrig and and Palo Paloheimo, 988; Kotliar and Wiens, 990; De 99 1 ; With, 994). At the heimo, 11988; Wiens, 11990; De Roos Roos et aI., al., 11991; With, 11994). outset of of any field study or modeling modeling exercise, then, the mean mean and shape of of a species migration function responses species'' migration function determine determine the the scale(s) at which population responses to environmental environmental patchiness patchiness must must be investigated. investigated. In the tradition most metapopulation tradition of of island island biogeography biogeography theory, most metapopulation models models use interpatch migration rates major determinants patch interpatch distance distance and and migration rates as the major determinants of of patch1 994a). The 1 988) colonization probabilities (e.g., Hanski, 1994a). The Fahrig and and Paloheimo Paloheimo ((1988) simulation guration of popu simulation studies studies of of the effects effects of of the the spatial confi configuration of patches patches on population indicated that migration lation abundances abundances in a metapopulation, metapopulation, for for example, indicated migration distance, distance, rather rather than migration migration rates alone alone (or demographic demographic features features such as birth birth rate), was critically important, when interpatch important, especially especially when interpatch distances distances were great. 1 995) modeled Bachman's Bachman ' s sparrow In contrast, contrast, when Liu et al., ((1995) sparrow (Aimophila (Aimophila aestivalis) aestivalis) population population dynamics, dynamics, they found found that demographic demographic parameters parameters were more important important than mortality during during dispersal dispersal (although (although not not necessarily necessarily dispersal dispersal
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John JohnA. A. Wiens Wiens
rate rate or distance). These These differences differences may stem from differences differences in model structure, structure, but they may also refl ect basic reflect basic differences differences in the life histories of of the organisms organisms modeled. Traditional Traditional metapopulation metapopulation models usually usually do do not consider consider the the details details of of movement movement in in even even an an abstract abstract sense. sense. Movement Movement is is modeled modeled as as transition proba probabilities among cells in a grid ((Liu Liu et ai., 1995) 1 995) or et al., or movement rates rates and distances distances are ed or are distributions. Whether move are simply specifi specified are drawn drawn from frequency distributions. Whether movement ment through through the matrix matrix between between patches patches is directional directional (e.g., Fig. 2A) or follows aa diffusion, 980; diffusion, correlated correlated random random walk, walk, or or some some other other algorithm algorithm (e.g., Okubo, Okubo, 11980; Turchin, 1 989; Johnson et ai., 1 992a) is not considered, even though the differ Turchin, 1989; Johnson et al., 1992a) is not considered, even though the differences ences among among these these movement movement patterns patterns can can produce produce substantial substantial differences differences in in the the mi probability of of encountering encountering a patch in the matrix. This is especially true if if migration gration rates are low or if the number number of of individuals individuals available available to migrate migrate is quite limited (as may occur when local populations populations are are small). Movement Movement patterns patterns such such as as diffusion diffusion or or random random walks walks are are handy handy modeling modeling devices that that may have some relevance relevance to how real real organisms organisms move through through a featureless featureless matrix, but they are of of limited value (other than as neutral neutral models) in specifying specifying how individuals might respond respond to a complex landscape landscape mosaic (e.g., Fig. 2B). Conceptually, the movements movements of of individuals individuals through a landscape landscape may may be viewed as a consequence consequence of of their movements movements within individual patches patches and and 993). Within-patch their movements between patches Fig. 3A; Wiens patches ((Fig. Wiens et et ai. al.,, 11993). Within-patch movement patterns patterns vary vary among among different different patch patch types. The The probability that that an an ed time interval is a individual will encounter encounter a patch patch boundary boundary during during a specifi specified function function of of these patch-specific patch-specific movements and of of patch size and shape shape (perimeter Whether or not an individual will cross a patch (perimeter:: area area ratio). Whether patch boundary boundary upon bound upon encountering encountering it is a function both of of features features of of the boundary boundary itself ((boundet al., and of of the characteristics characteristics ai., 1987; 1 987; Wiens, 11992b) 992b) and ary "permeability"; Stamps et of of the adjoining adjoining patch patch (patch context). [This is where where another another behavior, behavior, patch patch or habitat habitat choice, choice, becomes important.] important.] Both Both costs (e.g., (e.g., predation predation risk, risk, physiological physiological stress) and benefi ts (e.g., shelter, benefits shelter, food availability, mating mating opportunities) may differ among among elements elements in a mosaic, and and movement patterns patterns within and and between patches ect these relative costs and benefits (i.e., patch quality), at least patches may refl reflect 993; Wiens, 11996a). 996a). Some simulation models of of metapopmetapop in part ((Wiens Wiens et et ai., al., 11993; 1 992; Adler ulation migration migration in in patchy patchy environments environments (e.g., (e.g., Pulliam et et ai., al., 1992; Adler and and Nuernberger, 994) vary migration costs incorporate Nuernberger, 11994) costs as a function of of distance distance or incorporate differences differences in in patch patch quality. quality. To make such an individual-based conceptualization movements conceptualization of of mosaic movements relevant to metapopulation popu metapopulation dynamics, it must be extended extended to the scale of of population Fig. 3B). matter lation rather rather than than individual individual patches patches ((Fig. 3B). In In simple simple terms, terms, this this is is aa matter of of movements and patches of shifting shifting the scale from that of patches defined by individual home home ranges ranges to to the the broader-scale broader-scale movements movements of of populations populations (i.e., (i.e., migration) migration) and and the the scale scale of of patchiness patchiness represented represented by by interactions interactions within within aa local local population population (i.e., (i.e., nodes nodes in a metapopulation). Exactly how how the translation translation from individual move movements ments to to population population distribution distribution and and interactions interactions should should be be accomplished accomplished is is one one
33
A A
(
Individual Individual
Metapopulation MetapopulationDynamics Dynamicsand ond Landscape LandscapeEcology Ecology
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B ' Population Population B
p
FIGURE33 (A) (A) Patterns Patterns of movement movement of an individual individual among among elements elements of a landscape landscape in its home FIGURE range. The movement movement pathway pathway consists consists of within-patch within-patch and between-patch between-patch components; components; both both may range. affected by the characteristics characteristics of patches patches and by their their spatial spatial configuration be affected configuration ((patch patch context). context). (B) Extension of individual individual movements movements to the the population population level. level. A local local population population may may occupy occupy patch patch i,i, Extension which individuals individuals move according according to the local local habitat heterogeneity heterogeneity within within that patch. These within which movements (characterized probability that individuals individuals will will encounter movements (characterized by a function, function, IJ,) 0;) determine the probability boundary between between patch i and patchj patch j during a given given time time interval. interval. The probability that individuals the boundary encountering the boundary will cross into into patchj, j, ' ~b0, boundary 1 ' is a function of the permeability permeability of the boundary encountering (e.g., patch patch choice). choice). Within and the behavior of the organisms organisms (e.g., Within patch j, a proportion of the dispersing individuals ((l-p) residency in the patch. Movements (0j) determine individuals l -p) may die or establish establish residency Movements within within patch patchjj (IJ ) determine probability that that the boundary boundary between between patch jj and another element the probability element in the landscape landscape (patch k) will encountered; lk ~bjkdetermines p, p, the proportion of dispersers dispersers from from patch i that move move into patch be encountered; patch k. 0 and ~bare patch-specifi patch-specific which may have density-dependent density-dependent effects Values of IJ c (as is patch density, which on movement movement and migration). migration). Developed 1 993). Developed from Wiens Wiens et al. ((1993).
of the most vexing vexing problems problems confronting confronting aa metapopulation-Iandscape of the most metapopulation-landscape synthesis. synthesis. It more general general problem problem of of translating across scales It is is part part of of the the more translating across scales in in ecology ecology (Wiens, et al., al., 1992). 1 992). (Wiens, 1989a; 1 989a; King, King, 1991; 1 99 1 ; Rastetter Rastetter et My colleagues and and II have have used used systems systems of small animals My colleagues of small animals (insects) (insects) moving moving through grassland "microlandscapes" as through grassland "microlandscapes" as experimental experimental model model systems systems (Ims ( lms et et al., al. , 1 993; Ims, Ims, 1995) 1 995) to to investigate investigate how how movements movements are affected by by mosaic mosaic structure, structure, 1993; are affected following following the the framework framework of of the the model model of of Wiens Wiens et et al. al. (1993). ( 1 993). Initial Initial studies studies of of tenebrionid beetles E l e o d e s spp.) that individuals o v e d differently tenebrionid beetles ((Eleodes spp.) indicated indicated that individuals m moved differently in in microlandscapes a few square meters differed in microlandscapes of of a few square meters that that differed in internal internal heterogeneity, heterogeneity, as as measured measured by by the the fractal fractal dimension dimension of of the the landscape landscape pattern pattern (Wiens (Wiens and and Milne, Milne, 1989). o v e m e n t alternated 1 989). M Movement alternated between between matching matching the the predictions predictions of of an an ordinary ordinary diffusion ovement diffusion model model and and those those of of anomalous anomalous diffusion diffusion depending depending on on m movement "rules," "rules," landscape landscape pattern, pattern, and and spatial spatial and and temporal temporal scales scales (Johnson (Johnson et et al., at. , 1992a). 1 992a). In particular, particular, diffusion diffusion exponents exponents changed changed significantly significantly at at spatial spatial scales scales correcorre In sponding 42 cm), cm), suggesting suggesting that that sponding to to the the size size of of vegetation vegetation patches patches (a (a radius radius of of ~= 42 the effects effects of of spatial spatial heterogeneity heterogeneity on on beetle beetle movements movements at at finer finer scales scales differed differed the fundamentally et al., al. , 1992) 1 992) demdem fundamentally from from those those at at broader broader scales. scales. Other Other work work (Crist (Crist et onstrated that structure within onstrated that variations variations in in vegetation vegetation structure within 25 25 m: m2 areas areas had had significant significant
54 54
John A. John A. Wiens Wiens
effects on beetle movements and that that these effects differed among among Eleodes Eleodes spe species. The net displacement displacement of of individuals per unit time, for for example, was greater in areas dominated dominated by bare ground and by continuous low grass cover than in more heterogeneous areas that contained cacti or shrubs, and larger beetle species exhibited exhibited greater displacements in a given habitat type than did smaller beetles. The The relative relative complexity (fractal dimension) of of the movement movement pathways, however, however, was insensitive to variation among among species or habitat types, at least at the 25 m2 m2 scale of of resolution. On the other hand, broader broader comparisons among beetles, har harlandscape mosaics revealed significant significant vester ants, and grasshoppers in the same landscape 1 995), indicating differences differences in fractal dimensions of pathways (Wiens (Wiens et al., 1995), fundamental fundamental differences differences in the ways these taxa respond to landscape heterogeneity at this scale. These studies ne, "within-patch" studies were were conducted conducted at relatively fi fine, "within-patch" scales and recorded how individual animals responded to landscape patterns. To determine how such movements might translate translate into patterns patterns of of population distribution distribution at broader spatial scales, With and Crist ((1995) 1 995) used a cell-based simulation model to project the dispersion dispersion patterns patterns of populations populations of of grasshoppers over a broader mosaic. Individuals moved within a cell of of a given habitat type according to the empirically observed movement parameters parameters for for that habitat. Movement charac characteristics changed when individuals entered cells of of a different habitat type, ac aced transition probability (this corresponds to the between-patch cording to a specifi specified between-patch component, , 05, of of Fig. 3B). The landscape mosaic was dominated dominated (65% coverage) by a single habitat type. Under certain certain specifications specifications of transition transition probabilities, a Xanthippus corallipes, corallipes, moved rapidly through this cover type. As large species, Xanthippus a consequence, consequence, it had increased patch-residence patch-residence time (and an aggregated distri distribution) in the remaining Psoloessa remaining 35% of of the landscape. landscape. A smaller species, Psoloessa delicatula, was much more sedentary and preferred delicatula, preferred a habitat comprising only 8% of of the landscape. landscape. Given its low vagility, there was a low likelihood of of individuals of of this species locating locating and aggregating within cells of of the relatively rare, pre preferred spe ferred habitat. The model simulations suggested that the distribution of of this species would not diverge from the random distribution used to initiate the simula simulations. In fact, in the field both species exhibited the general general dispersion patterns predicted predicted by the model. How do these observations and and model analyses of patch-specific patch-specific movements movements relate patch quality, boundary ef relate to the four components of of landscape landscape ecology ((patch effects, patch context, and connectivity)? The differences within-patch movement differences in within-patch patterns patterns may indicate differences differences in patch quality, but the sensitivity of of model predictions probabilities between patch indicates predictions to the value of transition probabilities patch types indicates that knowledge of within-patch within-patch movement patterns by itself is not adequate adequate to predict predict broad-scale broad-scale population distributions. Something else is needed. The most likely factors affecting the translation within-patch movements translation from individual, within-patch to population distribution over a landscape are patch patch boundary effects and the influences of patch context. If individual beetles react behaviorally to the patch influences
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Metapopulation Dynamics and and Landscape Landscape Ecology MetapopulationDynamics Ecology
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boundary another will be al boundary itself, the likelihood of of moving from one patch patch to another altered. particular characteristics of tered. If If patch patch context context is important, then then the particular of what what is beyond beyond a given patch boundary boundary will further further modify transition transition probabilities. Landscape Landscape controls controls over over movement patterns have have yet to receive receive detailed at attention in either field studies. Moreover, Moreover, all of either models models or field of these approaches approaches con consider the structure structure of of the landscape landscape mosaic to be fixed; patch dynamics in time would add another another level of of realism (and further further computational complications) to the research program. One aspect of of landscape structure that is implicit in the spatial arrangement of of mosaic elements and the transition probabilities probabilities among among them is connectivity. Landscape Landscape connectivity refers to the degree degree to which the landscape facilitates or 993). Corridors impedes movement among patches patches (Taylor (Taylor et et al. al.,, 11993). Corridors of of similar habitat 990; habitat linked together together are are thought thought to enhance enhance connectivity (Bennett, (Bennett, 11990; Hobbs, 1992), patches among among which transition probabilities Hobbs, 1 992), but dissimilar habitat patches probabilities are Through the patterns are high may also result result in high connectivity. Through patterns of of connectivity that characterize a landscape, landscape, movement pathways pathways are directed in spatially non nonrandom random manners manners (Fig. 2B), which can either increase increase or or decrease decrease the likelihood that movement among specifi specificc patches in the landscape landscape (e.g., subpopulations subpopulations in a metapopulation) metapopulation) will occur. Connectivity is related related to the coverage coverage of of a given habitat habitat type in the land landIf a continuous scape, but the relationship is strongly nonlinear. If continuous habitat is broken broken into fragments by habitat conversion, conversion, the initial effects are due primarily to the below some threshold threshold value, loss of of habitat coverage alone. As coverage coverage drops drops below however, landscapes however, the effects of of patch patch isolation begin to be more more important. In landscapes with a low proportion further decreases proportion of of suitable habitat, further decreases in coverage coverage result in a rapidly increasing distance between between habitat habitat patches patches and even even greater isolation effects (Fig. 4). For example, Andren 1 994) found that habitat loss was a good Andr6n ((1994) predictor landscapes with predictor of of fragmentation effects on birds birds and mammals in landscapes > > 30% coverage coverage of of suitable habitats, but in more highly fragmented fragmented landscapes the effects of of patch isolation and size also became important. threshold effects also emerge in simulation studies based based on percolation Such threshold theory. In simple percolation percolation models, a landscape landscape mosaic is divided divided into suitable and unsuitable habitat patches (cells) that are distributed distributed over the landscape at random, random, with a specified coverage or proportion, proportion, p, p, of of the suitable patches (Gard (Gard987, 11989). 989). Above Above some ner ner et et al. al.,, 11987, some critical critical threshold, threshold, Peril' Pcrit, cells cells of of the the suitable suitable habitat An or habitat are likely to form a continuous continuous cluster that that spans the landscape. An organism in this "percolating cluster" will be able to move or "percolate" "percolate" across ' Neill et 1 988). For the landscape; landscape; connectivity is high (O (O'Neill et ai. al.,, 1988). For a random landscape has aa in Perit has in which which organisms organisms move only to to adjacent adjacent (but not not diagonal) cells, cells, Pcrit nomandom algo value of of 0.5928. 0.5928. If If the landscape pattern is generated generated using a nonrandom algo993; With et press), the value rithm (e.g., fractal curdling; Lavorel et et ai. al.,, 11993; et ai. al.,, in press), in the the of - 0.50). Similar in Pcrit Perit occur of P Pc,.it is lower lower (0.29 (0.29-0.50). Similar reductions reductions in occur with with changes changes in crit is movement patterns patterns to allow individuals individuals to move to any adjacent cell or to cross cross
S6 56
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0% 0% Proportion of of Suitable Suitable Habitat Habitat Proportion FIGURE 44 A hypothesized hypothesized relationship between the the proportion of suitable habitat in a landscape FIGURE importance of habitat loss and patch isolation to individual individual movement or population and the relative importance availability of suitable habitat decreases, decreases. the importance of habitat loss increases dynamics. As the availability monotonically. The effects effects of patch isolation (the (the inverse of landscape connectivity) connectivity) are relatively monotonically. suitable habitat is high, but increase sharply when a connectivity threshold slight when coverage coverage of suitable percolation theory parlance). Increases Increases in individual individual vagility vagility will will move this threshold is passed (P,.,.i, (Pent in percolation values of the suitable habitat. to lower coverage values
gaps where where suitable suitable cells cells are are not not immediately adjacent ((Dale et al. al.,, 1994; Pearson gaps immediately adjacent Dale et 1 994; Pearson et 996). Field beetles moving moving through et al. al.,, 11996). Field experiments experiments with with Eleodes Eleodes beetles through random random landscapes landscapes (Wiens (Wiens et et al., al., in in press) press) indicated indicated aa threshold threshold change change in in movement movement patterns patterns when when coverage coverage of of grass grass in in aa bare-ground bare-ground matrix matrix increased increased from from 0 0 to to 20%. 20%. Changes Changes in in either either the the spatial spatial pattern pattern of of the the landscape landscape or or the the scale scale over over which which individual individual organisms organisms "perceive" "perceive" landscape landscape patterns patterns (as (as judged judged by by their their move movements) ments) can can therefore therefore produce produce high high connectivity connectivity in in aa mosaic mosaic even even when when the the favored favored habitat habitat type type occupies occupies aa relatively relatively small small proportion proportion of of the the landscape. landscape. Differences Differences in in vagility vagility among among organisms organisms (e.g., (e.g., the the grasshoppers grasshoppers studied studied by by With With and and Crist, Crist, 11995) 995) may may also also affect affect the the location location of of aa percolation percolation threshold threshold (Fig. (Fig. 4), 4), as as Fahrig Fahrig and 1 988) also and Paloheimo Paloheimo ((1988) also suggested suggested in in aa somewhat somewhat different different context. context. Details Details of of the the spatial spatial arrangement arrangement of of habitat habitat patches patches in in the the mosaic, mosaic, such such as as those those modeled modeled by 1 985) or 1 994), are by Lefkovitch Lefkovitch and and Fahrig Fahrig ((1985) or Adler Adler and and Nuernberger Nuernberger ((1994), are likely likely to to become become important important only only around around or or below below this this threshold. threshold. Most Most models models that that link link animal animal movements movements to to landscape landscape structure structure assume assume that that movement xed species movement parameters parameters are are fi fixed species traits traits and and that that migration migration can can adequately adequately be be represented represented using using average average values. values. Individuals Individuals do do vary vary in in movement movement charac characteristics, teristics, of of course, course, and and the the effects effects of of this this variation variation may may be be profound. profound. For For example, example, 1 994) found Lens Lens and and Dhondt Dhondt ((1994) found that that crested crested tit tit (Parus (Parus cristatus) cristatus) young young dispersed dispersed 11 week week later later from from small, small, isolated isolated pine pine stands stands than than did did those those in in large large pine pine forests. forests.
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Metapopulation and Landscape landscape Ecology MetapopulationDynamics Dynamicsand Ecology
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Chicks Chicks from second second broods were were also more more likely to disperse disperse into less suitable habitat fragments than were young from rst broods. Collectively, these move from fi first movement ment characteristics characteristics reduced reduced the probability that second-brood young would be integrated into winter flocks, which would affect affect their overwinter survival prob prob( 1 994) sug abilities. In another vein, the simulation studies of of Goldwasser et et al. al. (1994) suggested that that variability among among individuals could markedly increase the rate of of spread spread of of a population, even if if only a few few individuals in the population population migrated rapidly. The prospect that individual movement behavior may be facultatively adjusted to landscape landscape patterns patterns such as the interspersion or isolation of of suitable suitable 995; Fahrig and Merriam, 1 994) may further habitat patches ((Matthysen Matthysen et et al. al.,, 11995; Merriam, 1994) further complicate attempts to model migration dynamics in heterogeneous heterogeneous landscapes. Nonetheless, it is apparent between fine-scale movemove apparent that the complex interplay interplay between ment patterns, broad-scale nonlinear effects of broad-scale migration dynamics, and the nonlinear of land landscape-mosaic structure may have fundamentally important effects on the inter interpatch movements that lie at the heart of of metapopulation dynamics.
C. C. Locol Local Extinction Extinction and Recolonization Recolonization In addition addition to interpatch interpatch movement, the extinction of local populations populations in habitat patches and the subsequent recolonization of of those patches are are what drive metapopulation dynamics. Local popUlation population extinctions are often associated associated with the stochastic stochastic dynamics that that characterize characterize small populations. Deterministic Deterministic local habitat changes, however, can produce produce patch patch dynamics in the landscape landscape that that also result in the extinction of 994c). If of local populations (Thomas, 11994c). If this is the case, the local patch environment may remain unsuitable unsuitable for for some time after extinction occurs. Under these conditions, the persistence of the metapopulation depends Under of depends on how well the organisms can track the shifting spatial locations of of suitable habitat unpredictable in time patches. Because Because the location of suitable patches may be unpredictable as well as in space, how organisms move through the landscape mosaic and the scales on which they perceive environmental patchiness become all the more important. The pattern of interspersion of of suitable habitat patches through a landscape mosaic also influences extinction and colonization probabilities. The degree to which a patch is connected connected to other suitable areas or is isolated may have little direct effect on extinction, uence the immigration flow and extinction, although it may infl influence therefore determine Brown and Kodric determine the magnitude magnitude of the "rescue "rescue effect" effect" ((Brown KodricBrown, 11977). 977). Colonization, on the other hand, is clearly related related to the interplay between individual migration abilities and both both the distribution (i.e., isolation) and the connectivity of habitats in the landscape. landscape. If If fragmentation alters the land landscape so that the interspersion of of habitat patches patches no longer coincides with the migration patterns patterns of of a species, metapopulation dynamics may be disrupted. To some degree, this situation characterizes Hanski characterizes the Glanville fritillary in Finland ((Hanski et al. al.,, 11995a). et 995a).
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D. When When IsIs aa landscape LandscapeApproach ApproachNecessary? Necessary? D. few situations, landscapes, rather rather than patches in a featureless featureless In all but a few are reality. Given this, one one might might conclude that any any attempt to model or matrix, are understand understand metapopulation dynamics that that does not explicitly include landscape landscape structure would would be futile. The essence of of theory, however, however, is simplifi simplification of structure cation of reality. Good es in a way that does not violate reality too much, Good theory simplifi simplifies incorporating its essential features. In this sense, patch patch-matrix theory rep repwhile incorporating -matrix theory resents a significant improvement over over theories theories based based on spatial homogeneity homogeneity resents When can the details of of landscape landscape structure reasonably reasonably be ig ig((Wiens, Wiens, 11995a). 995a). When nored or simplifi simplified? nored ed? Green ((1994) and Paloheimo, Paloheimo, 11988, personal communi communiGreen 1 994) and Fahrig ((Fahrig Fahrig and 988, personal addressed this question question using simulation models. Green Green considered considered cation) have addressed of habitat connectivity in relation relation to population population and community per perthe effects of sistence and concluded concluded that in highly connected connected landscapes landscapes one could treat treat the entire landscape landscape as a single element (in which which case metapopulation metapopulation theory is no no entire longer very relevant). relevant). If If the landscape landscape is strongly strongly disconnected, disconnected, on the other other hand, hand, longer it may but the may be possible possible to treat treat each each element as a separate separate unit and and ignore ignore all but most basic descriptors descriptors of of patch patch structure structure (e.g., patch patch size and and separation). separation). Closer Closer the percolation percolation threshold threshold ((Fig. the other other hand, hand, the the explicit explicit spatial ar arto the Fig. 4), on the rangement of of patches patches in the landscape landscape and and the details of of individual individual movements movements rangement 's and patch patch transition transition probabilities probabilities may become become much much more more important. important. Fahrig Fahrig's and simulation analyses suggested that a landscape landscape approach approach may not not be required required when when suitable habitat habitat is abundant abundant and and widespread, widespread, when when individual movement movement distances are are large relative relative to interpatch interpatch distances distances (i.e., the "grain" "grain" of of the the envi envidistances ronment finer than than that that of when movement movement patterns do not not ronment is finer of the the organisms), organisms), when patterns do differ greatly greatly among different elements elements of of the differ among different the landscape landscape (i.e., transition transition probaproba bilities or when bilities are roughly equal equal and and high), high), or when the the habitat habitat pattern pattern is ephemeral. In most of of these situations, situations, either environment approaches approaches homogeneity or or the either the environment the organisms treat it as such. such. If organisms treat If this occurs occurs at a broad, broad, population population scale, then then it is unlikely develop. The unlikely that that metapopulation metapopulation dynamics will develop. The kind kind of of interplay bebe tween patch structure, and re retween local local patch structure, individual individual movements, movements, and and local local extinction extinction and colonization that is the essence of metapopulation metapopulation dynamics recolonization that essence of dynamics would would seem seem to to re quire a certain certain form form of one that that is in the the vicinity connectivity quire of patchiness, patchiness, one vicinity of of the the connectivity threshold analysis. Unthreshold and and does does not not meet meet the conditions conditions specified specified in Fahrig's Fahrig ' s analysis. Un der attention must der these these conditions, conditions, attention must be be given given to to the the details details of of landscape landscape strucstruc ture. ture.
V. METAPOPULATIONS, METAPOPULATIONS, LANDSCAPES, LANDSCAPES, AND CONSERVATION CONSERVATION The relevance The relevance of of metapopulation metapopulation dynamics dynamics to to conservation conservation issues issues is is treated treated in detail in many other chapters in this volume, so I will not dwell on it here. If in in many other chapters in this volume, so will not dwell on here. If
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Metapopulation Dynamics and and landscape MetapopulationDynamics LandscapeEcology Ecology
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metapopulations impli metapopulations are to be viewed viewed in a landscape landscape context, however, however, some some implications cations for for conservation conservation practice cannot cannot be ignored. The The traditional traditional focus focus of of conservation has been on reserves, reserves, and much of of the debate about number of about reserve reserve design has dealt with the size, shape, and and number of reserves. reserves. Reserves patches) in a background Reserves have have usually usually been been viewed viewed as habitat habitat islands ((patches) background matrix. Metapopulation Metapopulation theory has become important important in conservation biology because matrix tradition and because it fits neatly into this patchpatch-matrix and because because the widespread widespread occurrence 995b, occurrence of of habitat habitat fragmentation fragmentation has subdivided subdivided populations populations (Wiens, (Wiens, 11995b, 11996b), 996b), creating spatial patterns patterns that appear appear to match those those of of metapopulations. metapopulations. Metapopulation Metapopulation theory also predicts predicts stability solutions, solutions, offering offering the hope hope of of pop population persistence persistence in the face face of of local extinctions. extinctions. Habitat Habitat fragmentation, fragmentation, however, however, involves involves much much more more than than changes changes in the size and and isolation isolation of of habitat habitat patches. patches. When When a landscape landscape is fragmented, fragmented, habitats habitats are replaced by other habitats, habitats, patch patch boundaries boundaries are often often sharpened sharpened and patch patch context context changed, changed, connectivity patterns patterns are altered, and and the cost-benefi cost-benefitt contours contours of of the landscape landscape shift. Simple Simple island biogeography biogeography theory does does not not deal with such complexity of of spatial patterns, patterns, and this is one one reason reason why its value in conservation conservation efforts 982, 11984; 984; Sober6n, 992; Haila efforts is quite limited (Simberloff (Simberloff and Abele, Abele, 11982, Sober6n, 11992; et 993; and Wiens, 11995b, 995b, give other example, et al. al.,, 11993; other reasons). reasons). Island theory, for for example, predicts predicts a loss of of species species with a reduction reduction in island (patch) (patch) area-the area--the well-known well-known species -area (S-A) points above species-area ( S - A ) relationship. relationship. A scatter scatter of of points above the S - AA curve curve has has been interpreted interpreted as evidence evidence of of community community "supersaturation," "supersaturation," which which will inevi inevitably lead to a loss of relaxation"), whereas points lying much of species species ("faunal ("faunal relaxation"), whereas points much below island disturbance vol below the curve curve have have been been explained explained as results of of island disturbance (e.g., volcanic eruptions) eruptions) or extreme 1 989b). Because Because terrestrial hab extreme isolation isolation (see Wiens, Wiens, 1989b). habitat patches landscape mosaic, patches are are immersed immersed in a landscape mosaic, it seems seems more more likely that that such such scatter represents (at least in part) scatter represents part) the effects effects of of connectivity, connectivity, patch patch context, context, or or edge conditions Fig. 5). The which landscape conditions ((Fig. The specific specific ways in which landscape configuration configuration area relationships not been might affect affect speciesspecies-area relationships have have not been explored. explored. These These and and other other considerations considerations have led to challenges challenges to the "reserve "reserve men menBrussard et 992), the belief tality" ((Brussard et al. al.,, 11992), belief that that conservation conservation problems problems are are solved solved by establishing necessary, establishing reserves reserves and and ignoring ignoring the surroundings. surroundings. Reserves Reserves are are necessary, to be sure, but areas Noss and areas outside outside of of reserves may also play important important roles ((Noss and 1 99 1 ; Woinarski 992; Barrett et 1 994; Harris, 11986; 986; Saunders Saunders et et al. al.,, 1991; Woinarski et et al. al.,, 11992; et al. al.,, 1994; 1 995; Wiens, 995b, 1996b). 1 996b). For Hanski 1 994; Turner Hanski and and Thomas, Thomas, 1994; Turner et et al. al.,, 1995; Wiens, 11995b, For habitat habitat generalists or species that move move widely, management management of of landscape landscape mosaics over over large large areas areas may be essential. essential. In Australia, Australia, for for example, example, the endangered endangered Gouldian Gouldian (W oinarski et finch (Erythura (Erythura gouldiae) gouldiae) has has a limited and patchy distribution distribution (Woinarski et at. al.,, 11992). 992). Large breeding breeding populations populations still exist in several areas, areas, and these these can can be protected popUlation leaves these areas in postbreeding protected by reserves. reserves. However, However, the population postbreeding movements, with transient transient groups groups appearing appearing in widely spaced spaced (and unpredictable) unpredictable) locations locations over the landscape. Management Management by a series series of of static reserves will not not work work during during this phase, phase, when when considerable considerable mortality occurs. occurs.
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O ~ expected from expected from_~ island theory island ~ / J
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The reduced (e.g., The species-area species-area relationship. If the area of of a habitat in the landscape landscape is reduced by fragmentation), number of fragmentation), island biogeography theory predicts predicts that a new new equilibrium number of species species that is appropriate Landscape effects (e.g., connectivity, patch appropriate to the new new habitat area will be reached. reached. Landscape context), however, can reduce reduce the species loss by providing habitat refugia or increasing the likelihood that that local habitat habitat patches patches will be rapidly recolonized. On the other other hand, hand, edge effects (e.g., low boundary number boundary permeability, increased increased predation predation mortality in habitat habitat edges) edges) may reduce reduce species number below that expected from equilibrium island theory. Some of the scatter scatter of of points about reported reported speciesarea relationships may reflect the effects of species-area of such mosaic features.
The solution to such problems may be to shift from reserve management to combined with areas that receive "mosaic management," in which reserves are combined varied (and perhaps perhaps intense) human human use. If If one wishes to enhance enhance a metapopu metapopufor example, it may be necessary to manage not only lation structure in an area, for the habitat patches populations but the land patches that contain (or could contain) contain) local populations landscape features features that facilitate or impede interpatch interpatch movement as well. Too Too often, such considerations considerations are cast in terms of of corridors corridors of of like habitat habitat linking patches patches together together (e.g., the management management plan for for northern northern spotted spotted owls (Strix (Strix occidentalis occidentalis caurina); J. W. Thomas et al. al.,, 11990), landscape 990), rather than evaluating overall landscape caurina); connectivity. Proper Proper mosaic management requires that attention be given to all all of of the features features of a landscape landscape and how how they interact interact to determine the fate of of local populations populations in habitat habitat patches. I maintain that the key to accomplishing accomplishing this ob objective lies in understanding understanding how landscape landscape structure structure affects movement patterns patterns within and among patches 996b). patches (Wiens, I1996b).
VI. CONClUSIONS CONCLUSIONS The The main message of of this chapter chapter is that landscape structure may may often be an important component component of of metapopulation dynamics. Variations in patch quality in space and time, the form and permeability of of patch patch boundaries, boundaries, the composition and characteristics of of surrounding surrounding mosaic elements, and the connectivity among
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Metapopulation Dynamics and and Landscape Landscape Ecology MetapopuhtionDynamics Ecology
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landscape uence the dynamics populations and, landscape components components may all infl influence dynamics of of local populations especially, especially, the ways in which which populations populations are linked by movements of of organisms. organisms. The The synthesis synthesis of of landscape landscape ecology ecology with metapopulation metapopulation dynamics dynamics is impor important. Although I have emphasized emphasized the contributions contributions that landscape landscape ecology can can Although make make in developing developing an understanding understanding of of metapopulation metapopulation dynamics, the relation relationship between between these these disciplines disciplines should should not be one-sided. one-sided. Metapopulation Metapopulation dynamics may also contribute to the development of landscape ecology, in two ways. One contribute development of landscape emphasizing the dynamics dynamics that that occur occur in a landscape. landscape. The The spatiotemporal spatiotemporal is by emphasizing patterns patterns of of local extinctions extinctions and and patch patch recolonizations recolonizations create a shifting distribution of populations among patches. Understanding Understanding what controls controls these dynamics ad adof populations among dresses dresses issues of of spatial relationships relationships and mosaic mosaic composition composition that are at the heart of emphasis on these dynamics of landscape ecology. Moreover, Moreover, an emphasis dynamics can draw atten attention away from from the map-based map-based descriptions descriptions that characterize characterize some approaches approaches to landscape landscape ecology. The The second second way in which which metapopulation metapopulation dynamics dynamics can contribute contribute to land landscape ecology is in the area of of ecology, of theory. In contrast to many other other areas of landscape landscape ecology has developed developed rather little little theory. The lack of of theory may stem stem in part from the diverse Fig. 11), ), but it may also diverse historical historical roots of of the discipline discipline ((Fig. refl ect the complexity of landscape reflect of landscapes landscapes and their linkages. The variety of of landscape patterns is virtually unlimited, unlimited, and thus thus there is no single mosaic pattern pattern (or small set of Wiens, 1995a). 1 995a). In contrast, of patterns) patterns) about about which which theory can be generated generated ((Wiens, contrast, patch patch theory has developed developed at least in part part because "patchiness" "patchiness" can be collapsed collapsed into simple patterns patterns of of patches patches and matrix matrix (or so we believe). Further Further development development of predictive rather of landscape landscape ecology as a predictive rather than than a descriptive descriptive science science requires requires concepts patterns to their concepts or theories theories that that link landscape landscape patterns their consequences. consequences. As metapopulation metapopulation theorists theorists continue continue to add add complexity complexity and and realism realism to simple simple patch-matrix patch-matrix models, models, they they come come closer closer and and closer closer to developing developing true true mosaic mosaic models. models in enhancing understanding models. Quite beyond beyond the the value value of of such such models enhancing our our understanding of of metapopulation metapopulation dynamics, they may provide provide a wedge that landscape landscape ecolo ecologists can use to develop interactions. A linkage of develop models models of of landscape landscape interactions. of meta metapopulation theory with percolation theory might be especially fruitful fruitful (see With, With, in press). Throughout Throughout this chapter chapter I have have emphasized emphasized the importance importance of of understanding understanding movement. movement. Whether Whether or not a spatially subdivided subdivided population population functions functions as a meta metaindividuals move among population depends on how individuals among patches. patches. How individuals individuals migrate migrate is, in tum, turn, affected in a myriad myriad of of ways by landscape landscape structure. structure. Under Understanding standing these effects effects on movements movements is of of fundamental fundamental importance, importance, yet we know know very little about May and Southwood, Southwood, 11990; 990; about movement in an ecological context context ((May 1 995). Existing provide much Opdam, 99 1 ; Dunning Opdam, 11991; Dunning et al. al.,, 1995). Existing theory will not provide much help help here. here. Instead, Instead, we must must focus focus our our attention attention on well-designed well-designed empirical empirical studies studies of of how how individual individual movements movements are are affected by the explicit spatial patterning patterning of of en eninsights necesneces vironments. Such Such investigations can provide provide the information information and and insights sary to bring metapopulation metapopulation dynamics and and landscape landscape ecology together. together.
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ACKNOWLEDGMENTS ACKNOWLEDGMENTS Mcintyre, and an anonymous Diane Debinski, Mike Gilpin, Andy Hansen, Ilkka Hanski, Nancy Mclntyre, reviewer offered a wide variety of helpful comments on an initial draft of the manuscript, and con conversations with Ilkka were particularly useful in focusing my thinking about metapopulations and u.s. landscapes. My research on landscapes and spatial heterogeneity has been supported by the U.S. National Science Foundation, most recently through Grant DEB-9207010. DEB-9207010.
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II III
METAPOPULATION THEORY THEORY METAPOPULATION
The six chapters chapters in this section review much much of of the existing existing metapopulation biology, covering both both ecology and theory in metapopulation genetics and single-species and multispecies multispecies theory. Theory is developed and discussed in Part III, but with a focus on also developed particular processes rather rather than on metapopulation metapopulation dynamics in general. The fi nal chapter final chapter by Giles and Goudet Goudet in Part IV adds a useful discussion on the theory underlying genetic population differentiation in metapopulations. metapopulations. first chapters ascend from simple (Hanski) and The fi rst four chapters more complex ecological models (Gyllenberg, Hanski, and preHastings) of single species to models of competition and pre and Hassell) to models of communities communities dation ((Nee, Nee, May, and ((Holt). Holt). The The first first two two chapters chapters are are entirely entirely rewritten rewritten versions of respective chapters chapters in the the previous previous volume (Metapopulation (Metapopulation Dynamics, 11991), reader aa very concrete concrete Dynamics, 99 1 ), giving the interested reader opportunity to to judge the the kind of change change and and even progress that that has has occurred occurred in in the the past past 55 years. ' s chapter Hanski Hanski's chapter covers some some of of the the basic issues issues about about the the ecological dynamics dynamics of of single-species single-species metapopulations. metapopulations. To To start start ecological with, with, how commonly commonly do do species species persist persist in in fragmented fragmented landland-
scapes as classical metapopulations? metapopulations? As also highlighted by Harrison Harrison and Taylor in their their chapter, chapter, the answer answer is not well known known because because of of scarcity of appropriate appropriate field studies studies on large enough enough spatial scale for for long enough time. What is clear by now, however, is that some very good good examples examples of of species per permetapopulations do exist (see also Part sisting as classical metapopulations Part IV). What is the minimum habitat necessary for minimum amount amount of of suitable suitable habitat for metapopulation metapopulation survival, and what what is the minimum minimum viable meta metapopulation population size? These are likely to be controversial controversial questions, questions, like questions about the minimum viable population questions about the minimum population size in un unbroken habitats. These These are, however, however, the kinds kinds of of quantitative quantitative questions questions that ecologists will be asked, asked, and and it is our our duty to clarify not only the answers to these questions questions but also the various kinds of of uncertainties that are associated with particular particular answers answers and the risks associated associated with practical applications. applications. More More generally, ecologists ecologists will be asked to predict, in quanti quantitative terms, the dynamics dynamics of of particular particular species in particular particular fragmented fragmented landscapes, landscapes, including including the expected expected time to meta metapopulation population extinction. extinction. Hanski reviews reviews some of of the modeling modeling approaches approaches that that have been been developed developed for this purpose. purpose. Among Among other other things, these models models clearly clearly demonstrate demonstrate that that it may be very misleading population occurs misleading to assume assume that that a meta metapopulation occurs at a stochastic stochastic steady state in a rapidly changing changing landscape, landscape, a con conthat has weighty implications implications for for conservation. conservation. clusion that Patch Patch models models of of metapopulation metapopulation dynamics, such as the well-known well-known Levins Levins model, are often criticized for excessive excessive simplicity. The The chapter chapter by Gyllenberg, Gyllenberg, Hanski, Hanski, and and Hastings Hastings extends extends the deterministic deterministic single-species single-species theory to structured models, models, where the quantity of of interest interest is not just the fraction fraction of of occupied occupied patches, patches, like in the Levins model, model, but rather the dis distribution tribution of of local population population sizes. The The structured structured models models in include the effects of of birth, death, immigration immigration and emigration emigration on metapopulation metapopulation dynamics, though though still retaining retaining the abstraction abstraction of infinitely infinitely many patches and equal connectance connectance among the cations, the mathematics bebe patches. Even with these simplifi simplifications, come very complicated. predic complicated. It is encouraging that one key prediction, the possibility possibility of multiple equilibria, equilibria, stemming from the theory of of structured structured populations, populations, has been recently supported supported by a large-scale field study (Gyllenberg, (Gyllenberg, Hanski, and Hastings). Hastings). Nee, Nee, May, May, and Hassell Hassell extend the single-species single-species models models to pairs of of competitors competitors and mutualists mutualists and to predator-prey predator-prey in interactions broadly interpreted). teractions ((broadly interpreted). Theory makes makes it clear clear that the spatial structure of populations populations often often matters matters and often often makes makes it easier for for species species to coexist, which has been the main incen-
tive for for developing much of this theory in the first place (see also the chapter chapter by Harrison and and Taylor in Part I). Metapopu Metapopulation-Ievel lation-level coexistence may take striking forms, such as the emergence emergence of spatially chaotic patterns of of local abundance abundance in predator-prey models with restricted restricted movespatially explicit predator-prey move ments. A great great challenge challenge here remains remains to relate the theory to the dynamics of real metapopulations. Another Another central central theme addressed by Nee, May, and Hassell is the consequences of of habitat habitat destruction destruction on persistence persistence of single-species, single-species, competitive and predator-prey predator-prey metapopulations. metapopulations. Observing that that an analo analogous issue has for for a long time been in the center of of epidemio epidemiological theory, Nee, May, and Hassell discuss under under which cir circumstances the "eradication threshold" of a metapopulation, cumstances "eradication threshold" of metapopulation, essentially the minimum amount of of suitable habitat habitat as discussed discussed amount by Hanski, can be estimated simply by measuring the amount of equilibrium, the limiting resource for of unused habitat habitat at equilibrium, for meta metapopulation population growth. This is clearly a theme theme of of great great importance importance to conservation biologists, but also an area area where extra caution is needed in translating the theoretical results to practical rec recommendations ((Hanski). Hanski). Holt extends metapopulation models to extends the predator-prey predator-prey metapopulation chains of of three species, and to landscapes with two kinds of of habitat patches, with a possibility of habitat patches, of habitat habitat specialization. specialization. His analysis confirms that it is diffi cult to survive in sparse habitats, difficult habitats, and the species doing so are either extreme low extreme specialists ((low extinction rate, high high colonization rate) or, on the the contrary, hab habitat generalists. Species Species at higher trophic levels are are even more constrained, constrained, as the suitable suitable patches for for specific predators predators are always subsets of of patches available for for the prey (prey is gen generally absent absent in some patches). patches). In a spatial mosaic of of several habitat types, surprising patterns are possible, such as a gen generalist predator predator excluding excluding a specialist prey from particular particular hab habitself surviving on the alternative itat type and itself alternative prey in some other patch types. This outcome would be difficult to observe, predator are as both the prey and the predator are now absent from the focal habitat type! Including both complex landscapes and complex habitat ' s analyses complement communities in the same same models, Holt Holt's complement the results of of Nee, May, and Hassell and take a step toward a better understanding of better of metapopulation and metacommunity of mosaic landscapes that Wiens painted dynamics in the kind of in his chapter chapter in Part Part I. One increased habitat habitat fragmenta One of of the consequences consequences of of increased fragmentareduced potential for maintaining tion is often thought to be reduced for maintaining genetic variation in local populations and across the entire entire
metapopulation. metapopulation. The equilibrium equilibrium level and rate rate of of change change in genetic variation, measured measured for for instance instance by heterozygosity lev levels, are generally functions functions of of the effective effective size of of the popula population; hence one important important way habitat habitat fragmentation fragmentation may affect genetic variation is by changing changing the effective population population sizes. In metapopulations, metapopulations, one one may distinguish distinguish between between effective effective population respec population sizes at the local and and metapopulation metapopulation levels, respectively. Hedrick Hedrick and Gilpin explore with numerical numerical simulations simulations the effective effective metapopulation metapopulation size, taking taking as their their starting point point the Levins model with a fi n ite number of habitat patches. They Levins model finite number of habitat patches. examine how how the various various model model parameters, parameters, such as the number number of of patches, patches, population population turnover turnover rate, rate, patch patch carrying capacity and gene gene flow affect affect the effective effective sizes of of local populations populations and and the entire metapopulation. metapopulation. Consistent Consistent with theory (Barton (Barton and and Whitlock), Whitlock), they find that, under under the assumptions assumptions of of their model, model, the effective effective metapopulation metapopulation size is greatly reduced reduced by high high extinction extinction rate and and a small number number of of founders founders originating originating from from just just one one or a few existing existing populations. populations. Thus, Thus, metapopulation metapopulation dynamics per se and its key parameters, parameters, such as propagule propagule size, have have significant genetic consequences. consequences. This This theme is explored explored further further in the context of of an empirical empirical case study by Gilet Gilet and Goudet Goudet in Part IV. Hedrick Hedrick and Gilpin infer from from the generally high levels of of heterozygosity observed observed in nature nature for for allozyme markers that that metapopulation metapopulation dynamics in the form explored explored in their their model model have not been been of of overriding overriding importance importance in many species; otherwise heterozygosity levels should be much lower. lower. However, However, they caution that that increased increased habitat fragmentation fragmentation may have recently forced forced species species to conform conform to a metapopulation metapopulation structure, possibly possibly triggering triggering a course of of rapidly declining g.:: ge- netic variation. This is an argument analogous to that advanced argument analogous advanced by Hanski and by Nee, May, and Hassell Hassell in their chapters about about nonequilibrium nonequilibrium metapopulations metapopulations on their way to extinction; extinction; past habitat habitat destruction destruction may already have reduced reduced the amount amount of of suitable habitat habitat below a critical treshold, treshold, and it is only a matter of time before before the actual extinctions extinctions will occur. These conclusions ect the relatively slow time scale of conclusions refl reflect of metapop metapopulation dynamics. Barton Barton and and Whitlock Whitlock present present a comprehensive comprehensive review of of the consequences consequences of of spatial population population structure structure on the genetic composition composition of of metapopulations. metapopulations. The consequences consequences of of spatial structuring of of populations populations on adaptation adaptation and speciation have have been been a controversial controversial issue ever since Fisher and Sewall Wright Wright established established the fundamental fundamental results. In the metapopulation metapopulation con con' s shifting text, Wright Wright's shifting balance balance between between the processes processes of of random random
drift, selection, and migration is particularly intriguing. Barton and Whitlock conclude that though the shifting balance process is possible, there there are several factors which make it unlikely. Migration rate should not be too great to prevent populations from drifting to the domain of of new adaptive adaptive peaks; but migra migration rate must be sufficiently high to allow the new peaks to spread spread in the metapopulation. metapopulation. Small population size is generally beneficial for for a peak shift, but small populations are prone prone to local extinction, and generally send out fewer fewer emigrants, than large populations, which makes spreading of the new peak into the metapopulation more difficult. No grand conclusion on the shifting balance process is yet possible. The message that BarBar ton and Whitlock put forward is that the standard standard simple mea measures of of genetic population structure, structure, such as effective effective size or Fs, Fst,' are are not sufficient, but empirical studies studies should strive strive toward a much more comprehensive picture of geno of the distribution of of genotypes across populations in a metapopulation and of of the eco ecological and selective forces that that are are responsible responsible of of these these distri distributions. Studies of of population differentiation differentiation based on neutral markers have only a limited value.
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Metapopulation MetapopulationDynamics Dynamics From From Concepts Conceptsand Observations Observationsto Predictive Predictive Models Models IIkka Ilkka Hanski
I. INTRODumON INTRODUCTION The concepts of metapopulation dynamics and metapopulation persistence in fragmented landscapes have become well established in ecology during the past 5 years (Hastings and Harrison, 11994; 994; May, 11994; 994; Harrison, 11994b; 994b; Hanski, 11994b; 994b; Kareiva and Wennergren, 11995). 995). The accelerating loss and fragmentation of natural habitats ((Morris, Morris, 11995), 995), of of which most of us are personally and pain painfully aware, makes it tempting to suggest that, in an increasing number of of species, the spatial structure of populations populations is somehow somehow consequential to their their dynamics. Many studies have demonstrated that that small populations populations in small habitat fragments have a high risk of extinction (Schoener 1 987b; Kindvall and Ahl6n, Ahlen, (Schoener and Spiller, 1987b; 1 992; Hanski, 11994b); 994b); hence if 1992; if only small fragments remain, long-term persist persistence becomes necessarily a regional issue. We have now an extensive theoretical Hanski, 11985, 985, 11994a,b; 994a,b; Gilpin and Han literature on metapopulation dynamics ((Hanski, Han99 1 ; Hastings, 1991; 1 99 1 ; Gyllenberg and Hanski, 11992; 992; Hanski and Gyllenberg, ski, 11991; 1 993; Hastings and Higgins, 11994; 994; Tilman et 994; Hassell et 994; 1993; et al., al., 11994; et al., al., 11994; Durrett and Levin, 11994; 994; Hastings and Harrison, 1 994) and a large number Harrison, 1994) number of of useful empirical studies ((Harrison Harrison et 989; Nachman, 11991; 99 1 ; et at. al.,, 1988; McCauley, 11989; Harrison, 11991; 99 1 ; Sjogren, 99 1 ; Sjogren 1 994; Whitlock, 1992b; 1 992b; Thomas Sj6gren, 11991; Sj6gren Gulve, 1994; 1 994, 1995a; 1 995a; many chapters and Harrison, 11992; 992; Bengtsson, 1 993; Hanski et Bengtsson, 1993; et al., al., 1994, Metapopulation Metapopulation Biology Biology
1997 by Academic Copyright © 9 1997 Academic Press. Press, Inc. All rights of of reproduction reproduction in any form reserved. reserved.
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Ilkka Hanski HonskJ IIkka
that our understanding understanding of metapop metapopin this volume). Nonetheless, it is fair to say that ulation dynamics in real fragmented landscapes is still restricted ((Harrison Harrison and Taylor, this volume), largely because of the practical problems of conducting large spatial scale. sound empirical research at a sufficiently large publication of the previous previous volume on metapopulation metapopulation dynamics Since the publication (Gilpin and Hanski, 11991), 99 1 ) , it has become ned concept become evident that a broadly defi defined concept of of metapopulations metapopulations is needed to embrace the range of existing spatial population population structures ((Harrison, SimHarrison, 11994b; 994b; Harrison and Taylor, this volume; Hanski and Sim The classical classical metapopulation metapopulation concept of of Levins Levins ((1969a, berloff, this volume). The 1969a, number of of small and and hence extinction-prone extinction-prone local 11970), 970), which assumes a large number populations populations connected connected by not-too-much not-too-much migration, migration, is now seen as a special special case, possibly an uncommon Harrison, 11991, 99 1 , 11994b). 994b). This chapter uncommon special case ((Harrison, chapter is nonetheless focused focused on metapopulations metapopulations essentially agreeing with with the classical nonetheless concept. concept. This is for two reasons: reasons: First, it is too early early to conclude conclude that that Levins-type metapopulations are are exceptional; exceptional; a large large fraction fraction of rare and and specialized species metapopulations specialized species many lanscapes lanscapes may fall into into this category ((Hanski, Hanski and and Ham Hamin many Hanski, 11994c; 994c; Hanski mond, 1995). Second, a better better understanding understanding of of the the classical case should should enhance enhance mond, 1 995). Second, our understanding of metapopulation dynamics dynamics more more generally. our understanding of this chapter, chapter, I pose four broad broad questions: questions: In this pose four 1. commonly do species persist persist in fragmented fragmented landscapes landscapes as classical 1 . How How commonly metapopulations? This is the fundamental empirical question which I cannot cannot an anmetapopu1ations? the fundamental empirical question which swer here, here, but but I give one well-researched well-researched example example which which highlights of the swer highlights some of reasons why the answer is not better better known. known. 2. What What is the the minimum minimum amount amount of of suitable suitable habitat habitat necessary necessary for for metapop metapopulation survival, survival, and and what what is the the minimum minimum viable viable metapopulation metapopulation size? 3. Can Can we we make make quantitative quantitative predictions predictions about about the the dynamics dynamics of of particular particular metapopulations in particular metapopulations particular fragmented fragmented landscapes? landscapes? How common common are nonequilibrium nonequilibrium metapopulations, metapopulations, in which which the the rates of 4. How rates of local local extinction extinction and and recolonization recolonization are not not in balance? balance? Recognizing Recognizing the the wide wide interest interest that these these issues have have aroused aroused in conservation conservation biology biology (Western ( Western and and Pearl, 1989; 1 989; Falk Falk and and Holsinger, Holsinger, 1991; 1 99 1 ; Fiedler Fiedler and and Jain, Jain, 1992; the end four mes1 992; Harrison, Harrison, 1994b), 1 994b), I summarize, summarize, toward toward the end of of this this chapter, chapter, four mes sages stemming from the answers The final sages for for conservation conservation stemming from the answers to these these questions. questions. The final remarks are concerned remarks are concerned with with the the perennial perennial question question about about density density dependence dependence in population population dynamics. dynamics.
II. AN AN EXAMPLE EXAMPLE OF OF CLASSICAL CLASSICAL METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS WITH WITH RAMPANT RAMPANT POPULATION TURNOVER TURNOVER POPULATION One One could could argue argue that that it is futile futile to to search search for for criteria criteria by by which which metapopulametapopula tions tions of of various various kinds kinds (Hanski ( Hanski and and Simberloff, Simberloff, this this volume; volume; Harrison Harrison and and Taylor, Taylor, this this volume) volume) could could be be identified, identified, to to answer answer whether whether aa particular particular system system is a "meta"meta-
44
Metapopulation MetapopulationDynamics Dynamics
71 71
population" populations in nature population" or not. This is futile because because populations nature exhibit continuous continuous variation variation in their spatial structures structures and also because because the real issue is not not so much much to classify populations populations living in fragmented landscapes but to find find ways of of un understanding derstanding and predicting their their dynamics. This being said, it is also clear that different different approaches approaches to population population dynamics are likely to be most effective in different satisfies the the different kinds of of systems. In this spirit, I suggest that if a system satisfies 's following following four four "conditions" "conditions" then a metapopulation approach approach based based on Levins Levins's ((1969a) l 969a) original concept is likely to be helpful. I apply these conditions conditions to an example from the work work of research group of my my research group on a species of of butterfly, the Glan Glanland islands in ville fritillary Melitaea Melitaea cinxia, cinxia, which we have studied on the A ~land southwestern southwestern Finland.
Condition Condition 11 The The suitable habitat occurs occurs in discrete patches which may be occupied by mead local breeding breeding populations. populations. The habitat type suitable for for M. M. cinxia cinxia is dry meadAland islands ows, which occur as discrete and small patches patches on islands (Fig. 1), 1 ), with on/~land the mean, median, and maximum areas of 1 3 , 0.03, and 6.80 of 0. 0.13, 6.80 ha, respectively (n = 11502; 502; Hanski et ai., 1995a). 1 995a). An estimated 60of butterflies butterfties spend their et al., 6 0 - 880% 0 % of entire lifetime in the natal Hanski et 994; Kuussaari et et al., 996); natal patch ((Hanski et al. al.,, 11994; al., 11996); hence meadows have local popUlations, populations, not not just just ephemeral aggregations of of in individuals. =
Condition Condition 22 Even the largest local populations populations have a substantial risk of of extinction. If If not, then then the metapopulation would would persist simply because because of of the persistence of of the largest population(s), mainland- island population(s), and we would have an example of of mainland-island metapopulations 99 1 , 1994b). 1 994b). In M. metapopulations (which are are common in nature; nature; Harrison, Harrison, 11991, cinxia, populations in 11994 994 had had ca 500 cinxia, the largest largest local population population of of 377 extant extant populations butterflies. In this and related butterflies, butterflies, populations populations with several several hundred hundred indi indial., viduals Harrison et viduals have been been observed observed to go extinct in only a few few years years ((Harrison et al., 11991; 99 1 ; Foley, 11994; 994; Hanski 1 995a), hence Hanski et et al. al.,, 1995a), hence the large large metapopulation metapopulation in Fig. 11 has no "mainland" populations. populations.
Condition Condition 33 Habitat patches must not be too too isolated to prevent recolonization. recolonization. If they were, we would have a nonequilibrium metapopulation metapopulation heading toward toward global 1 995) con extinction. Such metapopulations metapopulations are are common; Hanski Hanski and Kuussaari ((1995) conclude that 110 0 of of the 94 resident butterfl butterflyy species in Finland represent represent the non nonAland equilibrium case due to recent loss of M. cinxia of habitat. However, However, M. cinxia on Aland islands is not one one of of them, as the mean mean nearest-neighbor nearest-neighbor distance between between suitable habitat patches is only 240 28 m, maximum 3870 1 ), and the 240 m (median 1128 3870 m; Fig. 1),
IIkko Hanski Honski Ilkka
72 72
· 10
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FIGURE land islands in southwestern habitat FIGURE 1! Map of of A ]kland southwestem Finland, showing the locations of of the habitat patches Melitaea cinxia cinxia (dots). Patches Patches that that were patches (dry meadows) suitable for for the Glanville fritillary Melitaea occupied 00 km2 occupied in late summer summer 1993 are shown by black dots. The size of the grid is 1100 km 2 (modified from 995a). from Hanski Hanski et et al al.,.. 11995a). mean, median, and maximum distances moved by migrating butterflies among habitat habitat patches in one 50-patch network were 590, 330, and 3050 m respectively 994). ((Hanski Hanski et et at. al.,, 11994).
Condition Condition44 Local populations do do not have completely synchronous dynamics. If they have, the the metapopulation metapopulation would would not persist persist for for much much longer than than the the local local pop population with the M. cinxia, cinxia, we we have have demonstrated demonstrated the smallest risk risk of of extinction. extinction. In In M. substantial asynchrony asynchrony in in the the dynamics of of populations within within an an area area of of 55 by by 55 km2 Hanski et 995a). The et al., al., 11995a). The most most recent recent results results suggest dynamics dynamics that that may may km 2 ((Hanski be be somewhat somewhat correlated correlated across across areas areas up up to to some some tens tens of of square square kilometers, kilometers, but but at at the the scale scale of of the the entire entire metapopulation metapopulation changes changes in in population population size size occur occur in in opposite opposite directions Fig. 2). directions ((Fig. 2). The The question question about about spatial spatial synchrony synchrony and and its its causes causes is is aa com complex plex one one (Thomas (Thomas and and Hanski, Hanski, this this volume), volume), but but the the point point which which II wish wish to to make make here here is is that that in in our our butterfly butterfly metapopulation metapopulation there there is is certainly certainly enough enough asynchrony asynchrony to to make make simultaneous simultaneous extinction extinction of of all all local local populations populations aa very very unlikely unlikely event event under under the the prevailing prevailing environmental environmental conditions. conditions.
44 Metapopulation Metopopuiotion Dynamics Dynamics
73 73
Aland islands from 11993 993 till FIGURE FIGURE22 Observed Observedchanges changes in the population population sizes sizes of M. M. cinxia cinxia on Aland islands from 11994. 994. The study study area area was divided divided into 2 by 2 km2 km2squares squares for the purpose purpose of this this analysis. analysis.The symbol symbol indicates indicates the the sign sign and the magnitude magnitudeof the change change in the number number of larval larval groups groups per 4 km2 km2 square square between between the 22 years years (stippled (stippled triangles, triangles, decrease; decrease; black black triangles, triangles, increase; increase; logarithmic logarithmic scale) scale) (data (data from from I.I. Hanski, Hanski, J. Pogry, P6gry, and T. Pakkala, Pakkala, unpublished). unpublished).
III. III. CLASSICAL CLASSICALMETAPOPULATION METAPOPULATIONDYNAMICS: DYNAMICS:THE THE lEVINS LEVINSMODEL MODEL The purpose and and "validity" of simple simple models in population ecology is often The misunderstood. misunderstood. Their Their purpose purpose is is not not to to replicate replicate in in the the model model as as many many details details of of real real populations populations as as possible. possible. Models Models which which do do that that are are not not simple simple and and their their pur purpose ). The pose is is different different (Section (Section V V). The purpose purpose of of simple simple models models is is to to isolate, isolate, for for aa theoretical theoretical study, study, some some feature feature of of real real populations populations that that happens happens to to be be of of interest. interest. A A simple simple model model is is not not invalid invalid just just because because all all known known real real examples examples deviate deviate in in some some respect respect from from model model assumptions; assumptions; these these differences differences may may be be immaterial immaterial for for the the purpose purpose that that the the model model was was constructed. constructed. A A simple simple model model is is defective defective if if itit fails fails to to incorporate incorporate the the critical critical variables variables and and processes processes affecting affecting the the phenomenon phenomenon under under scrutiny scrutiny or or if if itit makes makes some some critically critically unrealistic unrealistic assumptions. assumptions. 1 969a, 11970), 970), the In In this this spirit, spirit, II suggest suggest that that the the well-known well-known Levins Levins model model ((1969a, the mother mother of of all all metapopulation metapopulation models models with with population population turnover, turnover, provides provides aa valu valuable able theoretical theoretical framework framework for for studying studying systems systems such such as as shown shown in in Fig. Fig. 11 and and satisfying satisfying the the four four conditions conditions detailed detailed in in the the previous previous section. section. The The Levins Levins model model assumes assumes aa large large number number of of discrete discrete habitat habitat patches, patches, ideally ideally of of the the same same size, size, and and
74 14
IIkka Ilkka Hanski Hanski
all connected connected to each other other via migration. migration. In reality, not all populations populations are are directly directly connected restricted, but but this makes connected to each other, other, because because migration migration distances distances are are restricted, makes no behavior of unless the no important important difference difference to to the the steady-state steady-state behavior of the the model model unless the net network heterogeneous. In the Levins model, model, work of of habitat habitat patches patches is strongly strongly spatially heterogeneous. the Levins habitat habitat patches patches are are scored scored only as occupied occupied or or not, as shown shown in Fig. 11,, and and the the actual best actual sizes of of the the local populations populations are are ignored. ignored. The The model model therefore therefore applies applies best to situations situations in which which local dynamics dynamics occur occur at a fast fast time scale compared compared with with metapopulation metapopulation dynamics, either either because because the the habitat habitat patches patches are are relatively small and and hence hence local populations populations quickly reach reach the local "carrying capacity" capacity" or or because because colonization colonization rate rate is low. All extant extant popUlations populations are are assumed assumed to have have a constant constant risk risk of of extinction. extinction. The The rate rate of of colonization colonization is assumed assumed to be proportional proportional to the fraction P, and fraction of of currently occupied patches patches (sources of of colonists), colonists), denoted denoted by P, and With to the fraction fraction of of currently currently empty patches patches (targets of of colonization), colonization), 11 - P. With continuous time is given by these these assumptions, assumptions, the rate rate of of change change in P P in continuous -
dP
dP dt == cP( cP(1 l dt
- P) P) - eP eP, ,
((1) 1)
parameters, respectively. The where where c and and e are the colonization colonization and and extinction extinction parameters, The equilibrium equilibrium value value of of P P is given by
P = = 11 f>
e e. cC
(2) (2)
The fraction of habitat at equilibrium The Levins Levins model model thus thus predicts predicts that the the fraction of occupied occupied habitat equilibrium metapopulation is prepre the ratio e/c. increases with with decreasing decreasing value of of the e/c, and and the metapopulation < 1. 1 . In spite of dicted to persist persist (P (P is positive) positive) as long long as e/c e/c < of its simplicity, the Levins feature of metapopulation Levins model is most useful useful in highlighting highlighting a key feature of metapopulation dynamics: for for the metapopulation metapopulation to persist, recolonization recolonization must must occur occur at a suf sufdynamics: ficiently high increase from from high rate rate to compensate compensate for extinctions and and to allow an increase small metapopulation im metapopulation size. More More specifically, condition condition e/c < < 11,, or 11 < < cle. c/e, implies that that a local local population population surrounded surrounded by empty patches patches must must cause cause the estab estabfor the metapopmetapop lishment lishment of of at least one new new population population during during its lifetime lifetime (lIe) (l/e) for ulation to persist. persist. Equation important predictions Equation (2) leads leads to some some straightforward straightforward but but important predictions when when we we recognize recognize that, very generally generally and and not surprisingly, the risk of of population population extinction extinction decreases decreases with increasing increasing patch patch area, area, and and the probability probability of of coloniza colonization decreases Hanski, 1991, 1 99 1 , decreases with increasing increasing distance distance from from the extant extant popUlations populations ((Hanski, I1994b). 994b) . The predicts that the fraction habitat at equi The Levins Levins model model predicts fraction of of occupied occupied habitat equilibrium librium (P) (P) decreases decreases with decreasing decreasing average average size and and decreasing decreasing density of of habitat these predictions habitat patches patches in a patch patch network. network. The The results in Fig. 3 support support these predictions for 970s for M. cinxia. cinxia. This This species species went went extinct on the Finnish Finnish mainland mainland in the late 11970s and Hanski and from from many many other other regions regions in northern northern Europe Europe during during the the past past decades decades ((Hanski and Kuussaari, 11995). 995). The and Kuussaari, The most most probable probable reason reason for for these metapopulation-Ievel metapopulation-level
Metapopulation MetapopulationDynamics Dynamics
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FIGURE FIGURE33 Effects Effects of of average patch patch size and and regional density on the the fraction of of occupied patches A land islands (Fig. 11). ). The was divided into into 2 by by 2 km km22 patches in M. M. cinxia cinxia on on/~land The study area was patches (as in Fig. 2). (a) Squares patch area in the square; Squares are divided into into four four classes based on the average patch square; (b) squares number of per square (patch Note squares are divided into four four classes based based on the number of patches patches per (patch density). Note that the fraction of patches are large of occupied occupied patches patches is high in the squares squares where where patches large and where where patch patch 995a). density is high (both Hanski et (both effects are are highly significant; statistical analysis in Hanski et al., al., 11995a).
extinctions is decreased decreased density of of suitable habitat habitat patches, patches, forcing the equilib equilibrium metapopulation size to zero ). > 11). zero (e/c > The large M. cinxia cinxia metapopulation metapopulation shown in Fig. 11 is a good example of of Levins-type metapopulations. However, how representative representative is this example? example? Han Hanski and Kuussaari ((1995) 1 995) attempted to answer this question for Finnish butterflies. butterflies. By our count, 57 of of the the 94 resident Finnish species may may belong to this this category. This fi gure includes much uncertainty, though, because because the spatial population figure structures cinxia are not well known. Collecting structures of of all the 93 species apart apart from from M. cinxia the kind of of large-scale information we have collected collected for for M. cinxia cinxia (Fig. 11)) is expensive, obtaining obtaining funding for for this kind of of work is difficult, and the life cycles and and larval biologies of of most species make them much harder harder to study than than M. large spatial scale. These These are are some of the reasons why we do not cinxia on a large know how common common Levins-type metapopulations metapopulations are are in nature. Promising candidates candidates for species with Levins-type metapopulations can be found in forest insects living in small and patchy microhabitats such as dead tree trunks. trunks. One One such such example involving involving beetles specializing specializing on on dead dead aspen aspen trees trees in boreal forests is described 1994; see also described in detail by Siitonen and Martikainen ((1994; Hanski and Hammond, 995). As most insects live in forests, and as most forest Hammond, 11995). forestliving insects are more or less specialized specialized on discrete discrete microhabitats, Levins-type metapopulation metapopulation structures structures may be common in insects (see also van der Meijden Meijden and van der Veen-van Wijk, this volume). Other Daphnia water Other examples examples include include Daphnia fl eas in rock pools ((Hanski Hanski and Ranta, 11983), 983), frogs in ponds (Sj6gren, (Sjogren, 11991; 99 1 ; fleas
76 76
IIkka Ilkka Hanski HanskJ
Sjogren 994), passerine Sj6gren Gulve, 11994), passerine birds in small woodlots (nuthatch; (nuthatch; Verboom Verboom et et aI. 99 lb), 1 b), and small mammals al.,, 1199 mammals in small patches patches of of suitable habitat (pika; Smith, 11980; 980; Smith and Gilpin, this volume).
IV. SIZE IV. MINIMUM MINIMUM VIABLE VIABLEMETAPOPULATION METAPOPULATIONSIZE The minimum MVP) size has become a well-established well-established minimum viable population ((MVP) concept in population and conservation biology. MVP esti MVP is intended intended to be an estimate of of the minimum number number of individuals in a population population which has a good chance of of surviving for for some relatively long period of of time, for for instance, 95% chance of 1 00 years (Soule, 980). Though MVP of surviving for at least 100 (Soul6, 11980). MVP is difficult to apply in practice 987; Lande , 1 988b), it is a useful concept high practice (Soule, (Soul6, 11987; Lande,1988b), concept in highlighting the need for for a quantitative analysis of of the risk of of population extinction. extinction. extinction-prone In the case of of Levins-type metapopulations, metapopulations, consisting of of extinction-prone local populations, an analogous concept concept of of minimum viable metapopulation ((MVM) MVM) size may be defi ned as the minimum number popu defined number of of interacting interacting local populations necessary for Hanski et 996b). Apart for long-term persistence ((Hanski et af. al.,, 11996b). Apart from MVM, MASH) MVM, one also has to consider the minimum amount of of suitable habitat ((MASH) necessary for metapopulation persistence, persistence, because because not not all suitable habitat may may be simultaneously occupied by a metapopulation persisting in a balance between between local extinctions and recolonizations ). recolonizations (that is, P P is generally less than than 11). The original Levins model cannot be used to answer questions about MVM, MVM, because Eq. ((1) deterministic model and only applicable applicable to large networks because 1 ) is a deterministic of of habitat patches in which the stochasticity involved in local extinctions and metapop recolonizations becomes drowned by large numbers. In reality, many metapopmetapopulations may go extinct ulations live in small patch networks. Such metapopulations when all local populations happen to go extinct at the same time, even if the expected colonization and extinction rates would allow long-term persistence by Eq. ((1) l ) or by some other deterministic model. Gurney and Nisbet ((1978; 1 978; summarized summarized in Nisbet and Gurney, 1982) have analyzed analyzed a stochastic version of of the Levins model. Their Their analysis yielded the following approximation for the expected M, expected time to metapopulation extinction, extinction, T TM, (Hh/(2( l -P)) TM = T Tce M = T , Le
(HP2)/(2(1-/5)),
(3) (3)
where TL TL is the expected time to local extinction, H H is the number number of of suitable habitat patches, and P P is the fraction of of occupied patches at a stochastic steady state. nes long-term TM > 1 00 TL state. If If one one defi defines long-term metapopulation metapopulation persistence persistence as as TM > 100 TL,, Eq. (3) leads to the following condition for for reasonably large H H (Gurney and Nisbet, 11978): 978):
pJH P ~ 2: -> 3.
(4) (4)
44
Metapopulation Dynamics Metapopulation Dynamics
17 77
For example, if there there are 50 habitat habitat patches, patches, Eq. (4) says that that the colonization and extinction rates must be such that P P > > 0.42 for for the metapopulation metapopulation to persist for longer than 00 TL• P is large, than roughly 1100 TL. Assuming a good colonizer, for for which P the critical 1 0 (however, critical minimum patch number number is of the order order of of 10 (however, the approxi approximation becomes less satisfactory for H ). Empirical results for for small H). for M. M. cinxia cinxia and for other butterfl butterflyy species (Thomas and Hanski, this volume) are in broad agreement 0 - 20 small agreement with these predictions, suggesting that a minimum of of 110-20 and persistence. and well-connected well-connected habitat habitat patches patches are are needed needed for long-term long-term persistence. 1996b) have studied the stochastic Hanski et et al. al. ((1996b) stochastic Levins model numerically, numerically, incorporating such realistic realistic features as variation in patch areas areas and and the rescue rescue effect (decreased risk of extinction due to immigration; Brown and Kodric Kodric977; Hanski, 11991). 99 1 ). Figure 4 gives the predicted Brown, 11977; predicted time to metapopulation extinction in the model parameterized M. cinxia Hanski et parameterized with data on M. cinxia ((Hanski et al. al.,, 11996b). 996b). These results are in good agreement with the analytical results of of Gumey Gurney and Nisbet ((1978) 1 978) and strengthen the conclusions about about the minimum numbers of habitat patches and local populations necessary for for long-term metapopulation metapopulation lifetime combines persistence. Notice that the condition about metapopulation characteristics P, with the properties characteristics of of the species, as reflected reflected in the value of of P, properties of the landscape ((patch patch number MASH number H H).). Hence Hence the concepts concepts of of MVM and MASH cannot be applied independently. independently. Equation (3) is not very sensitive to varying assumptions about metapopulation dynamics, because the effects of these as assumptions are refl ected in the value of in reflected of P, itself a part of of the condition. For instance, making migration more restricted in space will lower the colonization rate 11 000000 cc
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and num and hence hence P, P, making making metapopulation metapopulation survival less likely. Even Even a very large large number ber of of habitat habitat patches patches is not sufficient sufficient for for metapopulation persistence if these patches patches are spread spread thinly across a large area! The The above above models include include coloni colonization - extinction stochasticity Hanski, 11991) 99 1 ) but zation-extinction stochasticity ((Hanski, but they they assume assume no no environmental environmental stochasticity stochasticity and no regional stochasticity (spatially correlated correlated environmental environmental sto stochasticity). Regional Regional stochasticity increases increases fluctuations fluctuations in metapopulation metapopulation size and decreases Hanski et aI., 11996b). 996b). However, decreases metapopulation metapopulation lifetime ((Hanski et al., However, for for re regional stochasticity stochasticity to have have a really significant effect effect the mean and the variance variance of Hanski, 11989; 989; Harrison Harrison and Quinn, 1 990). of the extinction rate rate must must be high ((Hanski, Quinn, 1990).
V. PREDlGlVE PREDICTIVEMODELS MODELSOF OF METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICS The biologist may expect The most most obvious obvious question question that that a metapopulation metapopulation biologist expect to be asked asked is whether whether some species X is likely to persist, persist, as a metapopulation, metapopulation, in some some particular particular set of of habitat habitat patches patches Y. Y. In the context context of of conservation conservation biology, biology, the set number of patches, and of of patches patches Y Y is often often a subset subset of of some some larger number of larger patches, and the the ecologist ecologist is asked asked to predict whether whether species species X, present present in the the current patch net network, re work, would would still persist persist if if some some patches patches were were removed removed or their their areas were were reduced. duced. Analytical models models of of metapopulation metapopulation dynamics, dynamics, whether whether simple simple or or more more complex Hanski, 11985, 985, 11991; 99 1 ; Hastings 989; Hastings, Hastings, 1991; 1 99 1 ; Gyl complex ((Hanski, Hastings and and Wolin, 11989; Gyllenberg et 99 l1 a; Gyllenberg and 1 992; et al. al.,, this volume; volume; Verboom Verboom et et al. al.,, 1199 and Hanski, Hanski, 1992; Hanski 993), are not helpful Hanski and and Gyllenberg, 11993), helpful in answering answering such questions, questions, be because cause these these models, models, intended intended for for examining examining the balance balance between between colonizations colonizations and extinctions incorporate specific information information about extinctions more more generally, do do not incorporate about patch patch qualities qualities and locations and hence hence cannot cannot be used to generate predictions predictions for for particular particular metapopulations. metapopulations. What What are are needed needed for for the purpose purpose of of making spe specific predictions predictions are spatially realistic realistic metapopulation metapopulation models. There There are currently three discussion of three main types of of such models [[II omit here here a discussion of spatially explicit explicit but not realistic Hanski, 11994c), 994c), such as cellular realistic approaches approaches ((Hanski, cellular automata; automata; see Caswell Caswell and 993; Durrett and 994; Nee et and Etter, 11993; and Levin, 11994; et al., al., this volume]. volume].
A. Spatially Spatially Realistic RealisticSimulation SimulationModels Models Spatially Spatially realistic simulation simulation models models generalize generalize models models of of local local dynamics dynamics to several local populations pop populations connected connected by migration. migration. Dynamics Dynamics in each local population ulation are modeled modeled separately, complemented complemented with specific assumptions assumptions about about migration. particular migration. The The model model can be linked with GIS-based GIS-based information information about about particular 994). Spatially realistic simulation models have been been landscapes (Ak�akaya, (Akqakaya, 11994). simulation models constructed Mc constructed to study the dynamics of, e.g., the spotted spotted owl in California California ((McKelvey et 993; Lahaye et 994; Lamberson 1 994; but et al. al.,, 11993; et al. al.,, 11994; Lamberson et et al., al., 1994; but see see Harrison Harrison et 993) and the dynamics of metapopulations ((Hanski Hanski et et al. 994; et al. al.,, 11993) of butterfl butterflyy metapopulations al.,, 11994; Hanski 994; Thomas volume). There There is no Hanski and and Thomas, Thomas, 11994; Thomas and and Hanski, Hanski, this volume). no limit
44 Metapopulafion Metapopulation Dynamics Dynamics
Z9 79
to the the amount amount of of "realism" "realism" incorporated incorporated in in these these models, models, but but the the realism realism comes comes to with aa cost, cost, aa large large number number of of assumptions assumptions and and parameters, parameters, which which may may be be hard hard with to verify verify and and estimate. estimate. Nonetheless, Nonetheless, spatially spatially realistic realistic simulation simulation models models may may to provide the the most most effective effective modeling modeling framework framework especially especially for for vertebrates vertebrates with with provide much often with much information information to to parameterize parameterize the the model model and and often with only few few habitat habitat patches patches and and local local populations. populations. Thomas Thomas and and Hanski (this volume) volume) discuss discuss the the apap plication of of spatially realistic realistic models models to to butterfly butterfly metapopulations metapopulations (see also HanHan plication et al., al. , 1994; 1 994; Hanski Hanski and and Thomas, Thomas, 1994). 1 994). ski et
B. State Transition Models The two two other other approaches approaches to to spatially realistic realistic metapopulation metapopulation modeling, modeling, The state transition and and incidence incidence function function models, models, are patch occupancy occupancy models models like like state transition are patch species in habitat habitat the Levins model; hence hence only the presence presence or absence absence of of a species patches is considered. transition and incidence function function models are patches considered. Both state transition are discrete practical point of of view, the fundamental fundamental discrete time stochastic models. From From the practical difference between the two two is in the the kind kind of of data data that are are needed needed for difference between for model parameterization. State transition models are parameterized with data data on observed parameterization. are parameterized rates of extinction and colonization, whereas incidence function function models can be be rates of extinction and colonization, whereas parameterized with data patterns of of patch occupancy. Pattern Pattern data parameterized data on patterns data are are generally much easier to obtain than adequate adequate data on colonization and much and extinction rates; hence the incidence incidence function function models can probably be used used more more widely. For For this hence reason, because I have have a personal incidence function function models, reason, and and because personal interest in the incidence structure and ( below). I describe their their structure and application application in greater greater detail detail (below). Sjogren 1 996) construct a state transition model Sj/Jgren Gulve and and Ray ((1996) model using lo logistic regression to estimate the dependences dependences of extinction extinction and colonization prob probpopulation size, isolation, and patch attributes. Having estimated estimated the abilities on popUlation regression parameters, parameters, metapopulation dynamics may be iterated from an arbi arbiconfiguration of patch occupancies by generating patch-specifi patch-specificc extrary starting confi guration of ex tinction and colonization probabilities probabilities in each generation. An advantage of this approach is that it is straightforward to incorporate any empirically observed effects of habitat habitat quality on extinctions and colonizations. The greatest disadvan disadvantage is that the model is parameterized parameterized with data on observed extinctions and colonizations; hence practical applications are restricted to large metapopulations with high turnover rate. The model parameters parameters can be estimated from from nonequi nonequilibrium metapopulations (which is an advantage), but the estimated extinction and colonization probabilities are sensitive to any temporal variation in these probabilities (regional stochasticity). For instance, if the extinction probabilities are estimated estimated over a time interval during during which exceptionally exceptionally many populations happened to go extinct, the model prediction would extend the exceptionally high extinction rate 1 996), Thomas rate to the future. Sjogren Sj6gren Gulve and Ray ((1996), Thomas and Jones ((1993), 1 993), and Kindvall ((1996a) 1 996a) apply a state transition model to metapopulations of a frog, a butterfly, and a bush crikect, respectively.
80 80
IIkko Ilkka Honski Hanski
C. C. Incidence Incidence Function Function Models Incidence IF) models 994a,b) are first Incidence function ((IF) models (Hanski, (Hanski, 11994a,b) are based based on a linear firstorder order Markov Markov chain in which each each habitat habitat patch patch has constant constant transition transition probabil probabilities between if patch patch i is presently between the the states of of being being empty empty or occupied. occupied. Thus Thus if empty it becomes patch-specific probability Ci becomes recolonized with a patch-specific C; in unit time (typically 11 year in practical practical applications). If If patch i is presently occupied, the population population goes extinct with a patch-specific patch-specific probability Ei in unit time. With being occupied, called the these assumptions, assumptions, the stationary probability of of patch i being given by incidence incidence of of the the species in in patch patch i, is is given by Ci L" -- Ci _.it_Ei
(5)
(5)
From 1 994a,b): From here here we proceed proceed in three steps (details in Hanski, Hanski, 1994a,b): ((1) 1 ) Specifi structure Specificc assumptions are made made about the the effects effects of of landscape landscape structure on it is assume that on the the colonization colonization and and extinction extinction probabilities. probabilities. Often Often it is realistic realistic to to assume that the proba the extinction extinction probability probability depends depends on on patch patch area area (because (because the the extinction extinction probability depends depends on population size which depends on patch patch area) but not on iso isolation. lation. A A convenient convenient functional functional form form is:
Ei=min
~--;, 1 ,
(6) (6)
where Ai is the the area of of patch patch i and and JL ~ and and x x are two parameters. In this this formulation, there is a minimum patch patch area area Ao such that that the extinction extinction probability equals equals 11 for patches patches smaller or equal to Ao. The extinction probability is related to patch area area for for convenience, convenience, because because data data on patch areas areas are easy to obtain. obtain9 The The variable of of fundamental fundamental interest interest is local population size, but it is often reasonable reasonable to assume assume that Kindvall and Ahlen, 1992) 1 992) or some other that there there exists a linear ((Kindvall and Ahl6n, other simple rela rela996c) between patch patch area and local population size; tionship Hanski et al. tionship ((Hanski al.,, 11996c) hence hence patch patch area can be used used instead. The model. The model parameters parameters can be interpreted in terms terms of of an extinction extinction model9 Assuming realistically that Assuming that extinctions are are due to environmental stochasticity, and and that that the population has a positive growth growth rate rate at low density, the value of of param parameter eter x x in Eq. (6) is related to the mean mean population growth rate rate ff and the variance 1 993; see also Foley, this volume). in in growth growth rate rate Ve as as x x = = 2f/Ve - 11 (Lande, (Lande, 1993; see also Foley, this volume). ects the The The value value of of x x thus thus refl reflects the effective effective strength of of environmental stochasticity (fiVe), large (f/Ve), large values of of x x indicating weak stochasticity. The colonization probability Ci is an increasing increasing function of of the numbers numbers of of immigrants of mainland-island mainland -island immigrants Mi arriving at patch patch i in unit time. In the case of metapopulations ((Hanski Hanski and Gyllenberg, 11993; 993; Hanski Hanski and Simberioff, Simberloff, this vol volume), ume), with a permanent permanent "mainland" "mainland" population as the sole or main source of of colonists, is colonists, aa reasonable reasonable simple simple functional functional form form is
Ci = ]3e -'~d',
(7) (7)
44
Metapopuiation Dynamics Metapopulation Dynamics
81
where island) i from and a where d; di is the distance distance ooff patch patch ((island) from the mainland, mainland, and a and and/3{3 are recolonize a little isolated isolated patch ((di d; two parameters. parameters. For For common common species, species, which which recolonize close without delay, Eq. (7) may {3 = l. close to zero) zero) without may be simplified by setting setting/3 = 1. In the case Mi is the sum individ case of of metapopulations metapopulations without without a mainland, mainland, M~ sum of of individuals originating originating from from the the surrounding surrounding extant extant populations. populations. Taking Taking into account account the the uals sizes and and distances distances of of these these populations, populations, we we may assume assume that that
4: i, jj =F
Mi = ~S~ = ~ ~ pj e-"diJAj,
(8)
j=l
for empty patches, dij where where P pjj equals equals 11 for for occupied occupied and and 0 for empty patches, d o is the distance distance between between {3 are two patches patches i and and j, j, and and a a and and/3 two parameters parameters as in Eq. Eq. (7). The The sum in Eq. (8) is denoted no interactions denoted by S Sii for for convenience. convenience. If If there there are no interactions among among the the immi immigrants grants in the establishment establishment of of a new new population, population, Ci Ci would would increase increase exponentially exponentially with Mi' M~. Often, Often, though, though, the the probability probability of of successful successful establishment establishment of of a new new population population depends depends on propagule propagule size in a nonlinear nonlinear manner manner (Schoener (Schoener and and Schoener, 983; Ebenhard, 99 1 ), and an s-shaped Schoener, 11983; Ebenhard, 11991), s-shaped increase increase in Ci Ci with increasing increasing Mi M; is better better justified, justified, M M~2 C . = C; M~ + y y2' 2' Ml + I
(9) (9)
I
where parameter (notice that when where y y is an an extra parameter (notice that when Eq. (8) is substituted substituted into Eq. (9), only only the the parameter parameter combination combination y y '' = = yl{3 y/~ can can be estimated). The The colonization colonization Pj probabilities remain constant when the pattern pattern of probabilities do not not remain constant when of patch patch occupancy occupancy (the pj values) values) changes, but but this violation of of the the assumption of of Eq. (5) is generally of of little importance Hanski, 1994a). 1994a). importance when when the metapopulation metapopulation is at a steady steady state state ((Hanski, One could could make make some some other other assumptions assumptions about about the the functional functional forms forms of of Ci One and and Ei• Ei. For For instance, instance, it is possible possible to include include in the the model model the the effects effects of of other other patch Moilanen and 997). The patch attributes attributes apart apart from from area area ((Moilanen and Hanski, 11997). The essential point point is that transformed into a parameterized parameterized model that with such such assumptions assumptions Eq. (5) is transformed model which which can be fitted to empirical empirical patch patch occupancy occupancy data. data. Assuming Assuming that that patches patches from extinction by immigration, immigration, Hanski Hanski ((1994a) arrived at may be rescued rescued from 1 994a) arrived
Ci Ji -
Ci _qt_Ei _ C i E i
1
+ IL l.~'/(SZA~[) 11 + ' I(StAf) ''
( 1 0)
(10)
where IL' = Ao0.. where/~' = ILy' /zy' for for patches patches greater greater than than A a, x, y', second step is to estimate parameters, a, (2) The The second estimate the model parameters, x, IL, ~, and and y', nonlinear maximum-likelihood using nonlinear maximum-likelihood regression or or some some other other technique. In pa parameter observed occupancies inci rameter estimation, estimation, the observed occupancies Pi Pi are regressed against the incidences Hanski, 1994a). 1 994a). Minimally, one from one dences Ji Ji ((Hanski, one needs needs the the following following data data from one metapopulation patch areas Ai their their spatial coordinates metapopulation at a stochastic stochastic steady state: patch areas A;, coordinates ' (to calculate patches at one point calculate the pair-wise distances distances di), dij), and and the state of of the patches one point ((year) year) in time time (the P pjj values). values). If If more more information information is available, it can can be used used to
82 82
IIkko Ilkko Honski Ilanski
obtain Hanski ((1994a) 1 994a) used mark obtain more more robust robust parameter parameter estimates. estimates. For For instance, instance, Hanski used m arkrecapture parameters to es recapture data data to estimate a a independently, independently, leaving only three three parameters estimate the meta timate from from occupancy occupancy data. data. The The critical critical assumption assumption at this stage stage is that that the metapopulation from population from which which the the parameter parameter values are are estimated estimated is at a stochastic stochastic steady state, that that is, that there there is no long-term long-term increasing or or decreasing decreasing trend trend in meta metapopulation population size. The The values values of of the the model model parameters parameters summarize essential information information about about gives the of extinction metapopulation processes. metapopulation processes. Thus, the the value of of/.~ gives probability of extinction J.L per per unit time in a patch patch of of unit unit size, x x gives the the rate rate of of change change in extinction extinction probability probability (and (and its inverse, inverse, expected expected time time to extinction) extinction) with with increasing increasing patch patch area, a describes the effect of distance on migration rate, y gives the colonization area, a describes effect of distance on migration y colonization efficiency, and and/3f3 is a compound compound parameter, parameter, including emigration emigration rate and and popu population density but note density ((but note that, that, with occupancy occupancy data, data, one one cannot cannot estimate y and and/3f3 independently; l 0)). independently; see Eq. ((10)). If 1 0), the values values of If one one allows allows for for the the rescue rescue effect, effect, as was was done done in Eq. Eq. ((10), of/~ J.L ' and 1 994a). To To tease and y y' cannot cannot be estimated independently (Hanski, (Hanski, 1994a). tease apart apart their turnover between between 2 or values one one may may use either either information information on on population turnover or more more l 994a); or years, as explained explained in Hanski ((1994a); or one one may may estimate (or (or guess) guess) the mini mini' mum Ao (then J.L = ( J.L ' /Ao» . The latter assumption mum patch patch area area A0 (then/~ = Ao A 6 and and y y' = = ..f x/--(~'/A6)). The latter assumption and colonization, will affect affect the predicted predicted rates rates of of extinction extinction and colonization, but but not not the Ji Ji values nor nor metapopulation metapopulation size at steady state. (3) Having Having estimated estimated the model model parameters, parameters, one one may may proceed proceed to numerically numerically iterate iterate metapopulation metapopulation dynamics dynamics in the same same or or in some some other other patch patch network network to generate quantitative quantitative predictions predictions about nonequilibrium (transient) dynamics dynamics and and nonequilibrium (transient) the stochastic steady state ((Hanski, Hanski, 11994a,b). 994a,b). This the greatest This is the step of of the greatest interest interest with with many possible possible applications. applications.
D. Tests Tests and Applications Applicationsof Incidence Incidence Function Function Models Models Perhaps Perhaps the most most direct direct test of of the model model involves involves a comparison comparison between between the the predicted predicted and and observed observed rates rates of of extinction extinction and and colonization. colonization. I was able to do that that in a long-term study of of shrew populations populations on on small islands islands in lakes in Finland Finland ((Hanski, Hanski, 11992a). 992a). Incidence were parameterized parameterized Incidence functions functions for for three three shrew shrew species were with with occupancy occupancy data data from from 68 68 islands. islands. Using the estimated estimated parameter parameter values, values, II then then predicted predicted the per-year per-year colonization colonization and extinction extinction probabilities probabilities in another another 7 islands, observed rates matched set of of 117 islands, which were were censused censused for for 5 years. The The observed matched remarkably predicted rates in all three species, which represent represent practically remarkably well the predicted three species, independent interspecific compe independent replicates replicates for for the purpose purpose of of this test (Table (Table I; interspecific competition if at all the extinction tition affects affects only little if extinction and and colonization colonization rates; rates; Peltonen Peltonen and Hanski, 99 1 ) . Using data population densities and Hanski, 11991). data on on population and the estimated estimated x x values, values, I further further inferred inferred the the relationship relationship between between the the expected expected time time to population population ex extinction probability) and population tinction (the (the inverse inverse of of the the extinction extinction probability) and the the expected expected population size conditional Notice the dramatic differences conditional on no no extinction extinction (Fig. 5). Notice differences among among the species. when assessing species. This kind kind of of information information should be be useful useful when assessing the relative
44
Metapopulation Dynamics Metapopulation Dynamics
83 83
- Island Incidence and the and TABLE TABLE II Parameter Parameter Estimates Estimatesof of aa Mainland Mainland-Island IncidenceFunction FunctionModel, Model, and the Predicted Predictedand Observed Observed per-Year per-Year Extinction Extinction and and Colonization ColonizationRates, Rates, in in Three Three Species Speciesof of Sorex SorexShrews Shrews on on Small Small Islands Islands in Peltonen and 99 1 ; Hanski, 992a)aa in lakes Lakes ((Peltonen and Hanski, Hanski, 11991; Hanski, 11992a) Predicted Predicted
Model parameters parameters Model
Observed Observed
Species Species
x x
SE
/LIe p/C
SE SE
Col Col
Ext Ext
Col Col
Ext Ext
araneus
2.30 0.91 0.46
0.68 0.24 0. 16 0.16
0.79 117.67 7.67 4.09
0.22 111.36 1 .36 11.51 .5 1
0.26 0.03 0. 18 0.18
0.04 0.28 0.53
0.20 0.05 0. 13 0.13
0.04 0.33 0.46
caecutiens minutus
Isolation varied "Isolation varied relatively relativelylittle little among among the islands; islands; hence hence the colonization colonizationprobability probability ei C i was w a s assumed assumed to be constant islands ((Hanski, Hanski, 11992a). 992a). constant for al1 all i.i. Parameters Parameterswere were estimated estimatedfrom from a single single survey survey of 68 islands To tease jJ, and e, minimum island tease apart apart the values values of of/.1, C, I assumed assumed that that the minimum island area area for occupancy, occupancy, Ao, A0, is 0.5 ha. The predicted island of 1.6 1 .6 ha, the average predicted extinction extinction probability probability was calculated calculated for an island average size of the 117 7 islands extinction rates were measured measured in a 5islands from from which which the observed observed colonization colonization and extinction rates were year study 99 1 ). study (Peltonen (Peltonen and Hanski, Hanski, 11991). a
importance small and reserves for conservation of kinds of importance of of small and large large reserves for the the conservation of different different kinds of species. species. Rapidly in Fig. Fig. Rapidly increasing increasing time time to to extinction extinction with with expected expected population population size size in values. Recalling Recalling that the value is related 55 is is associated associated with with large large x x values. that the value of of x x is related to to the the strength strength of of effective effective environmental environmental stochasticity stochasticity (above), (above), the the results results in in Fig. Fig. 55 illustrate point that that different of stochasticy stochasticy lead lead to relationships illustrate the the point different forms forms of to different different relationships between population size size (Goodman, 987; Lande, Lande, between time time to to extinction extinction and and expected expected population (Goodman, 11987; 11993). 993). In Fig. 5), well as land birds birds on oceanic islands islands ((Fig. Fig. 6), In shrews shrews ((Fig. 5), as as well as in in land on oceanic 6), there is positive relationship relationship between Following the the there is a a positive between the the x value value and and body body size. size. Following above line line of of reasoning, this suggests that small vertebrates are are more above reasoning, this suggests that small vertebrates more sensitive sensitive to to environmental ones ((Pimm, Pimm, 1991), 1 99 1 ), probably probably because because environmental stochasticity stochasticity than than large large ones small hence more vulnerable to to small individuals individuals have have small small body body reserves reserves and and are are hence more vulnerable starvation large ones ones ((Hanski, Hanski, 1992a). 1 992a). In In invertebrates, invertebrates, we we would would not expect starvation than than large not expect such between starvation time and body size; hence it it is such a a simple simple relationship relationship between starvation time and body size; hence is not not surprising Nieminen ((1996) 1 996) found found no no relationship relationship between between body surprising that that Nieminen body size size and and the the x value in herbivorous The message from here here is that the the incidence funcx value in herbivorous moths. moths. The message from is that incidence func tion models can used to interesting inferences causes tion models can be be used to draw draw interesting inferences about about the the rate rate and and causes of population extinction extinction from of the island) oc of population from knowledge knowledge of the pattern pattern of of patch patch (or (or island) occupancy. cupancy. The - island The examples examples in in Figs. Figs. 55 and and 6 6 and and in in Table Table II come come from from mainland mainland-island metapopulations, where colonization probability function of metapopulations, where the the colonization probability is is a a function of the the distance distance to Eq. (7» without permanent pop to the the mainland mainland ((Eq. (7)).. In In metapopulations metapopulations without permanent mainland mainland populations, probability has modeled with with aa more ulations, the the colonization colonization probability has to to be be modeled more complex complex expression principle remains expression like like the the one one given given by by Eq. Eq. (8), (8), but but the the principle remains the the same. same. used aa small in Fig. to para Hanski Hanski et et af. al. (( 11 9996c) 9 6 c ) used small subset subset of of the the data data shown shown in Fig. 11 to parameterize an model for Using meterize an incidence incidence function function model for the the Glanville Glanville fritillary fritillary butterfly. butterfly. Using
84 84
IIkka Ilkka Hanski Hanski 100 100
Q) EE
caecutiens
:p
araneUScaecutiens
.m
� ._.
c 0 "S 50 50 0. 0 o 0. -(D ""0 Q) -0 o Q) 0. xX W uJ
~ .
cO
�
cinereus minutus minutus
/ I
100 100
50 50
0
Expected Expected population population size size
FIGURE FIGURES5
The inverse of the per The relationship relationship between the expected expected time to population population extinction ((inverse peryear extinction extinction probability) and the expected population size (conditional on no extinction) extinction) in four species species of Sorex Sorex shrews, the three European species species in Table I, and S. cinereus, cinereus, a North American 993). The results are based on the parameter species species similar to S. caecutiens caecutiens (from Hanski, Hanski, 11993). parameter values of of an incidence incidence function model model estimated estimated from a snapshot pattern pattern of island occupancy.
2.0 2.0-
•
11.5.5
..
>< X
•
L_
E
11.0.0
• • D
-:..
I...
13. 0.5 0.5
...
• •
.
• 9
•
OQ • •
• •
9 •
•
•
• •
•
•
•
IIo· I
.,___,-L,--.----,-..,.--r--�-.__,-_, 0.0 0.0 ..L... 9 1~'00 1l's-2b 5 20 25 2's 30 a'0 35 3's 40 4'0 45 4'5 50 go 55 s's 60 e'o
FIGURE FIGURE 6 6
Body Body size size (in (in centimeters) centimeters)
The relationship vaiue of relationship between between the va~ue of parameter parameter x x in the incidence incidence function model and body size in birds on oceanic islands (reprinted with permission permission of University University of Chicago Press Press from Cook and Hanski, Cook and Hanski, 1995). 1995).
44
Metapopuiation MetapopulationDynamics Dynamics
85
1.0 1.0 0.8 0.8 Q_ 0... -o 0.6 0.6 u
> L(D Q5 PD 00 O0 (f) db _ D 0.4 0 0.4 0 0 O 00 0.2 0.2
oO~ ��
o 0 • 9
� @. � - ~ oo~p
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• 0
o o
qb � •
9 9• •
9
• g o• . o
o o
o o I
0.4 0.4
I
0.6 0.6
Predicted Predicted P P
I
0,8 0.8
11.0 .0
Comparison occupied patches M. Comparison between between the the predicted predicted and and observed observed fraction fraction of of occupied patches (P) (P) in in M. cinxia land islands. were calculated cinxia metapopulations metapopulations on on western western A ]kland islands. The The P P values values were calculated for for 4 4 by by 4 4 km2 km 2 squares. squares. Open IS habitat � 15 IS patches patches Open dots dots are are for for squares squares with with < < 15 habitat patches; patches; black black dots dots are are for for squares squares with with _> (from 996c). (from Hanski Hanski et et al., 11996c).
these parameter parameter values, we then predicted the fraction of of occupied habitat in the rest of land islands, the prediction failed, of the study area. area. For a part part of of the A Aland failed, perhaps Moilanen and Hanski, 11997; 997; Hanski because of some environmental differences differences ((Moilanen et 996c), but in most of the study area et al. al.,, 11996c), area the observed fraction of of occupied occupied patches Fig. 7). Many conservation patches matched well the predicted predicted one ((Fig. conservation applications applications do not require require quantitatively correct correct predictions, as the practical task is often to rank alternative alternative management management options in terms of their likely popUlation population dynamic consequences. appli consequences. Incidence function models should be very helpful in such applications.
VI. NONEQUllIBRIUM VI. NONEQUILIBRIUMMETAPOPULATIONS METAPOPULATIONS Traditionally, the the focus of of population dynamic modeling modeling has has been in the equilibrium equilibrium behavior of a hypothetical or some real population. Metapopulation Metapopulation modeling is no exception. Unfortunately, especially in the case of of metapopula metapopulations, it takes a long time to reach the equilibrium following any major major pertur pertur1 994). Many of bation (for an extreme example, example, see Hastings and Higgins, 1994). of the metapopulations which we care about may not have had time to reach an equi equilibrium in a rapidly changing changing landscape. landscape. In a declining patch patch network network the dis discrepancy between between the prevailing state of of the metapopulation and the equilibrium equilibrium state imposes a "debt of Hanski, 11994c; 994c; Tilman et 994), ex of extinctions" ((Hanski, et al. al.,, 11994), extinctions which are expected expected to occur occur in the course course of of time even if if the environment would not change any further. It goes without saying that this is a serious problem
86
Ilkka Honski Hanski IIkko
are typically forced to operate within a time frame too for conservationists, who are short to address any long-term consequences, however likely they may be. the only only general statement that can can be made about nonequilibrium Perhaps the that the discrepancy between the equilibrium and the existing state state dynamics is that of of a metapopulation metapopulation is likely to be greatest greatest in networks with relatively large large and because then the turnover rate and hence the rate rate of approach to isolated patches, because equilibrium are low. Extreme Extreme examples are the gradual decline of species number number land-bridge islands ((Diamond, mountaintop habitats habitats following on land-bridge Diamond, 11984) 984) and on mountaintop change ((Brown, Of greater concern, though, is the postglacial climate change Brown, 11971). 97 1 ). Of metapopulations on much smaller spatial scales may not be possibility that many metapopulations at equilibrium. example on the but butI illustrate such nonequilibrium dynamics with another example M. cinxia. cinxia. Figure Figure 8 shows the loss and increasing fragmentation of of suitable terfly M. for this species within an area of of ca 25 km22 during the past 1155-- 220 0 years. habitat for During this period, the total area area of of suitable suitable habitat declined to one-third of of its During original extent, and the number number of of distinct patches patches declined declined from 55 to 42, largely decreased grazing pressure pressure on the meadows. due to decreased Figure 9 shows the predicted predicted change change in the fraction of of occupied occupied patches patches during the past past 20 years. These These results results suggest suggest that, so far, the butterfly has tracked during of suitable habitat, apparently because the amount of of rather closely the amount of habitat and the total expected expected metapopulation metapopulation size have remained large. large. However, one should not draw draw the conclusion that the same result result would necessarily necessarily apply
G Q 0
t
yqt ' r
'b
1:3oooo 1 :30000
m A land FIGURE FIGURE 8 A map of the habitat habitat patches patches within within a 25 km km22 area in the northem northern part of the ]kland islands islands (Fig. (Fig. 1), 1 ), showing showing the presumed presumed extent extent of the suitable suitable habitat habitat for the butterfly butterfly Melitaea Melitaea cinxia ca 20 years years ago and today today (shaded) (shaded) (data (data from from Frank Frank Hering, Hering, personal personal communication). communication).
Metopopulotion MetapopulationDynomics Dynamics
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(A) Metapopulation size of ass measured measured by the the fraction fraction of of (A) Metapopulation size of the butterfly Melitaea Melitaea cinxia cinxia a occupied P in the landscape shown in Fig. 8. The occupied patches patches P landscape shown The results results were were obtained obtained by iterating iterating the the incidence parameterized with started by assuming incidence function function model model parameterized with field data. data. The The model model iteration iteration was was started assuming the (from year the patch patch network network 20 20 years years ago ago (Fig. (Fig. 8). During During a period period of of 20 20 years years (from year 300 300 to to 320 320 in in the the figure), this this network network was was reduced reduced to to its its present present size size (Fig. (Fig. 8) as as described described in in detail detail by by Hanski Hanski et et al. al. (1996b). ( 1996b). The The broken broken line gives the the expected expected (equilibrium) (equilibrium) metapopulation metapopulation size, size, whereas whereas the the contincontin uous uous line gives gives the the actual actual metapopulation metapopulation size size in the the declining declining network network (the (the lines lines give the the average average PP value value in in 200 200 replicate replicate simulations). simulations). (Middle) ( Middle) Difference Difference between between the the actual actual and and equilibrium equilibrium metameta population population sizes; sizes; (bottom) (bottom) numbers numbers of of metapopulation metapopulation extinctions extinctions in in the the 200 200 simulations simulations (no extincextinc tions for A, current patch (8) As As for A, but but now now starting starting with with the the current patch network network (Fig. (Fig. 8) and and halving halving tions in in this this case). case). (B) the area area of of each each patch patch in 20 20 years. years. Note Note that that the the equilibrium equilibrium metapopulation size drops drops to to zero, zero, but but the metapopulation size decades for most metapopulations metapopulations to to reach reach the the equilibrium equilibrium (extinction). (extinction). (Top) (Top) The The P it takes decades for most P value value in the the beginning beginning of of simulation simulation is is higher higher than than the the final value value in A A because because the the number number of of habitat habitat patches patches is et al., al., 1996b). J 996b). now smaller (reprinted with with permission permission of of University University of of Chicago from Hanski now smaller (reprinted Chicago Press Press from Hanski et
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to all all scenarios scenarios of of habitat habitat loss loss even even in in this this species. species. The The following following example example makes makes to the point. point. Let Let us us assume assume that that each each of of the the present present patches patches (Fig. (Fig. 8) 8) would would loose loose the another 50% of of its its area area in in another another 20 20 years. years. Figure Figure 99 shows shows that that such such further further loss loss another 50% of habitat habitat would would soon soon lead lead to to aa patch patch network network in in which which the the equilibrium equilibrium state state is is of metapopulation extinction. extinction. However, However, now now the the actual actual extinction extinction is predicted predicted to to metapopulation take tens tens or or even even hundreds hundreds of of years, years, because because the the last last local local populations populations to to go go extinct extinct take are typically typically the the largest ones with with the the smallest smallest risk risk of of extinction. extinction. The The inevitable inevitable are largest ones decline to to extinction extinction may may become become temporarily temporarily halted halted for for long long periods periods of of time, time, decline with the the number number of of occupied occupied patches patches fluctuating fluctuating without without any obvious trend trend (Han( Han with any obvious ski et et al., at., 1996b). 1 996b). ski
VII. FOUR FOUR CONSERVATION CONSERVATION MESSAGES MESSAGES Metapopulation Survival Survival in the Current Current Landscape landscape May Be Be Deceptive Deceptive A. Metapopulation The previous previous section section described described one message for conservation: The one important message for conservation: many lanscapes lanscapes may may have have changed changed so fast in the recent past that respective many recent past that the respective metapopulations are far from from equilibrium. In the worst case, case, the the current patch metapopulations are far current patch network is already too fragmented fragmented to support viable metapopulation, metapopulation, which is network support a viable therefore committed to extinction unless unless the loss and fragmentation of of habitat habitat is therefore and fragmentation reversed. reversed. Hanski and Kuussaari (1995) ( 1 995) estimated estimated that 1 0 of of the the 94 resident butterfly butterfly Hanski and Kuussaari that 10 94 resident species in Finland Finland are nonequilibrium metapopulation metapopulation species are presently presently represented represented by a nonequilibrium way to extinction. Generally, it is not known known how how many metapopulations on its way many metapopulations have already already reached reached the state state of of "living dead," dead," though we have little doubt that that many have. Conservationists should dismiss dismiss the false belief belief that protecting protecting the many landscape in which a species now occurs is necessarily suffi sufficient for long-term landscape cient for survival of of the species.
B. B. More More Than Than 1100 Habitat Habitat Fragments FragmentsAre Are Needed Needed Assuming Assuming that that a network of of small small habitat fragments fragments is established established for the the protection of of some some species, a natural question question to ask is how many many fragments fragments should be created/retained. message that one is not enough created/retained. The The blunt message enough is brought brought home by the fate of British butterflies on protected small reserves: tens of isolated populations of rare and endangared butterflies went extinct in 20 years, including all populations of three 992; see also Thomas 992). three species (Warren, 11992; Thomas et et ai., al., 11992). Mathematical Mathematical models models reviewed reviewed in in Section Section IV IV and and limited limited data data on on butterflies butterflies (Thomas and Hanski, Hanski, this volume) suggest suggest that that an adequate successful successful network of small habitat fragments should have a minimum of 110-15 0 - 1 5 well-connected fragments. Even this number may be insuffi cient if regional stochasticity is strong insufficient and local dynamics are strongly correlated. It is necessary to emphasize, emphasize, though, that as long as even one population survives there there is hope. Metapopulation Metapopulation decline
44
Metapopuiation MetapopulationDynamics Dynamics
89
may advance advance so slowly that that there is time to act if there is wish to act. In the case of managed of metapopulations on the brink brink of of extinction, intervention in the form of of managed recolonizations recolonizations is likely to become an increasingly necessary, and accepted, form of of management.
C. Ideal Spacing Spacing of Habitat Fragments Is a Compromise Compromise Even Even a large large number number of of small habitat fragments is no guarantee of of metapop metapoprecoloni ulation survival if the patches are located so far from each other other that recolonization and population popUlation rescue from extinction by immigration are are unlikely. A tentative practical answer answer to the question of of minimum density of of suitable habitat patches patches necessary for for long-term survival has been sought from the Levins model, Eq. ((1). 1 ). To model habitat loss, assume that fraction 11 - h of of the patches patches is per permanently because the density manently destroyed. destroyed. The The colonization rate rate becomes becomes lowered lowered because of P, and the model of empty but suitable patches patches is decreased decreased from 11 - P P to h -- P, 994; becomes ( May, 1 99 1 ; Nee, 1 994; Nee and May, 1 992; Lawton becomes (May, 1991; Nee, 1994; Nee and May, 1992; Lawton et et aI., al., 11994; Moilanen 995) Moilanen and and Hanski, 11995)
dP dP dt.· dt
- = P)) = cP(h ce(h - P - eP. ee.
(( 1111) )
At equilibrium, the fraction of of empty patches (out of of all patches, including the destroyed ones) is given by
e6' h --P *p* = - . cC =
- .
((12) 1 2)
Thus of all patches Thus the fraction of of empty patches patches out of patches remains constant as long as the metapopulation does not go extinct, which happens happens when h < < e/c. e/c. This is a seemingly very useful result, because it gives an estimate of of the critical minimum patch density from the very limited information of of the number number of of empty patches patches in still survives; no detailed knowledge in aa landscape landscape in in which which the the metapopulation metapopulation still survives; no detailed knowledge of 994). In practice, of metapopulation metapopulation dynamics dynamics is is required required (Nee, (Nee, 11994). practice, though, though, this this rule of of thumb is liable to yield an underestimate, and possibly a severe underestimate, of Hanski et al., 1996b): 1 996b): the of the critical patch density, because because of of three reasons reasons ((Hanski et al., extinction stochasticity in small patch networks (Sec rescue effect, colonizationcolonization-extinction (Section tion IV), IV), and and nonequilibrium nonequilibrium metapopulation dynamics, dynamics, when when a metapopulation metapopulation is approaching approaching the equilibrium from above (Section VI). Increased Increased patch density facilitates colonization and and is hence hence helpful, but if if habitat fragments are located located close to each each other other the the degree degree of of spatial synchrony in local dynamics may become elevated (Fig. 2), which has a negative effect on long-term survival. In theory, a row of of well-connected habitat fragments fragments might often of long-term survival than often provide provide a better better chance of than a tight cluster, but such considerations considerations are seldom practical. The The main recommendation recommendation is simply to pro provide sufficient connections their density connections among among habitat habitat fragments by maintaining maintaining their
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IIkko Ilkka Honski Hanski
at such a level that recolonization If recolonirecoloni recolonization occurs occurs within a few few generations. generations. If zation rate appears consider the appears to be worringly low, one one may have have to consider the merits of of managed managed recolonizations. recolonizations.
D. Substantial Variance in Habitat Quality Is Beneficial A major major cause of of spatial spatial synchrony in population population dynamics dynamics is spatially cor correlated weather weather effects effects (Thomas (Thomas and and Hanski, Hanski, this volume). This is not the entire entire story, interacts with attributes attributes of story, though, though, because because the effect effect of of weather weather often often interacts of meadows with habitat habitat patches. patches. For For instance, instance, in the butterfl butterflyy M. M. cinxia dry meadows with low low vegetation vegetation are generally favorable favorable for larval growth and survival, survival, but but in very very dry summers the host plants may may wither wither on the dryest meadows and larval mortality is greatly increased. reason why populations populations in large increased. Most Most likely, an important important reason habitat habitat fragments fragments have a low risk of of extinction, apart apart from the large expected expected population population size, is the greater heterogeneity heterogeneity of of habitat habitat quality in large large than than small patches. 0 gives an empirical patches. Figure Figure 110 empirical example example which which suggests suggests that that the the risk of of local extinction extinction decreases decreases with increasing increasing within-patch within-patch heterogeneity. heterogeneity. It is not not often possible possible to substantially substantially change change within-patch within-patch heterogeneity, heterogeneity, but when when multiple reserves are selected there there may be the option option of of including more more
1120 20 -
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1100 00
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FIGURE FIGURE 1100
9
.
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The relationship relationship between temporal variation variation in population size (CY) (CV) and a measure of of habitat heterogeneity in the bush cricket Metrio Metrioptera bicolor. Each symbol refers refers to one population ptera bicolor. (from Kindvall, 11996b). 996b).
44
Metapopulation Dynamics Metapopulation Dynamics
91
oorr less variation iinn habitat quality among the selected patches. patches. Though Though iitt may be tempting to aim at maximizing maximizing the "quality" "quality" of of the preserved preserved areas, there are are good reasons reasons to preserve preserve a range of of habitat qualities, to buffer buffer the metapopulation metapopulation against the adverse effects effects of of environmental environmental and regional stochasticities (Thomas (Thomas greater genetic diversity and Hanski, this volume), and possibly also to maintain greater ((Hoffman Hoffman and 99 1 ). and Parsons, Parsons, 11991).
VIII. VIII. CONClUDING CONCLUDINGREMARKS REMARKS A better better understanding understanding of of population dynamics is of of fundamental fundamental intrinsic interest ((Hassell Hassell and May, 11990) 990) as well as necessary necessary for for improved conservation conservation and management 994). Population management of of natural natural populations (Caughley and Sinclair, 11994). ecologists have made great great progress in the past decades decades using experimental, experimental, ob observational, and theoretical Price and Cappuccino, 11995), 995), but funfun theoretical approaches approaches ((Price damental questions about population regulation have remained remained controversial. Some fi eld ecologists continue field continue to resent resent the conclusion that that density-dependent density-dependent population regulation is necessary persistence of necessary for for long-term persistence of populations (den Boer, 11987, 987, 11991; 99 1 ; Wolda and Dennis, 11993), 993), and others others have doubted doubted how gen generally density-dependent 983; density-dependent regulation occurs in natural natural populations (Strong, 11983; Stiling, 11987). 987). Den 1 968, and Den Boer Boer ((1968, and later papers) papers) has has championed championed the the view that species species may persist larger than the local population thanks to the "spreading "spreading persist at a spatial scale larger of movements among of the risk" process, involving movements among asynchronously fluctuating "The consequences of local populations. "The of this spreading of of the risk in space will be a relative reduction reduction in the amplitude of of fluctuations of of animal numbers numbers in the 968). However, have entire population" population" (den Boer, 11968). However, it is simply not possible possible to have long-term persistence persistence even in a metapopulation metapopulation without some density dependence dependence in local dynamics, given that that local population population sizes are are restricted, restricted, as they always at., 1996a). 1 996a). In this respect, are, below below some maximum ( Hanski et maximum value (Hanski et al., respect, there there is no difference difference between between the dynamics of of a single population and the dynamics dynamics of of a metapopulation, Boer is metapopulation, regardless regardless of of the spreading spreading of of the risk. However, However, den den Boer correct to the the extent that the incidence incidence of density dependence dependence may be low in some incidence of persisting metapopulations, in comparison with the incidence of density depen dependence dence necessary necessary for for long-term persistence persistence of of isolated local populations. In metapopulations, metapopulations, the combination of of long persistence persistence time with little density de dependence pendence is associated with high turnover rate rate and frequent frequent local extinctions and and Hanski et 996a). Metapopulation colonizations ((Hanski et at., al., 11996a). Metapopulation persistence of of assemblages of of unstable local populations may explain some failures failures to detect statistically significant density dependence 987; Stiling, dependence in natural populations populations (den Boer, 11987; 11987; 987; Gaston 987), though a much Gaston and and Lawton, Lawton, 11987), much more important reason reason for such at., 1989; 1 989; failures is simply short runs of Hassell et of data that have been analyzed ((Hassell et al., Woiwod and Hanski, 11992). 992).
sdfsdf
Structured Structured Metapopulation Metopopulotion Models Ilkka IIkka Hanski Hanski
Gyllenberg Mats Gyllenberg Hastings Alan Hastings
USE STRUaURED I. WHY WHY USE STRUCTUREDMODELS? MODELS? mathematical model of of classical metapopulation dynamics with The simplest mathematical local population turnover is the one originally formulated by Levins ((1969a, 1 969a, 1970; 1 970; see Hanski, this this volume), dP dP
== dt dt
13P(11 - P )P) - - JLP /.tP,, f3P(
((1.1) 1.1)
where P JL iiss the extinction rate P denotes the fraction of of occupied occupied habitat patches, patches,/x f3 is the colonization rate per empty patch and per extant local population, population, and and/3 extant local population (to conform with the established notation of structured population models we have used here f3 and JL instead of c (or m) and e, respec here/3 and/x respectively, which are the usual symbols for the colonization and extinction rates in the ecological literature and which are also used elsewhere in this volume). This simple model nicely captures the key idea of a metapopulation of extinction extinctionprone local populations persisting in a balance between local extinctions and recolonizations of empty habitat patches ((Hanski, Hanski, this volume). The model pre predicts a threshold patch density necessary for long-term metapopulation persist persistence, a conclusion that is of fundamental significance for conservation ((Lande, Lande, Metapopulation MetapopulationBio/OK)' Biology
All rights Copyright Copyright © 9 1997 1997 by by Academic Academic Press. Press, Inc. All rights of of reproduction reproduction in in any any fonn form reserved. reserved.
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11987; 987; Nee 992; Hanski, 1 . 1 ) is identical Nee and and May, 11992; Hanski, this this volume). volume). Fonnally, Formally, Eq. ((1.1) identical susceptible-infected-susceptible (SIS) model model of of mathematical mathematical epiwith the susceptible - infected- susceptible (SIS) epi 99 1 ), with empty patches patches in a patch demiology (see, e.g., Anderson Anderson and May, 11991), patch network network playing the role of of susceptible susceptible individuals individuals in a population, population, and and occupied occupied patches patches corresponding corresponding to infected infected individuals. individuals. The agreement agreement between between the Levins model model and and the basic SIS SIS model model is more more than than coincidental, coincidental, as the phenomena phenomena studied studied in metapopulation metapopulation dynamics and in epidemiology epidemiology share share the same basic 994). processes 970; Levin, 11974; 974; May, 11991; 99 1 ; Lawton processes (Cohen, 11970; Lawton et et al., al., 11994). The elementary SIS model assumes homogeneous mixing of a SIS model assumes homogeneous mixing of large number number of individuals, with all infected individuals being of individuals, infected individuals being equally infectious. infectious. The Levins model 1 . 1 ) is based Hanski, this vol model Eq. ((1.1) based on similar simplifying simplifying assumptions assumptions ((Hanski, volume). unstructured in that it assumes ume). In particular, the the Levins Levins model model is unstructured assumes that that all habitat habitat patches patches and local local populations populations are identical identical in all respects. respects. This This assumption assumption involves several ecologically significant elements. First, the spatial arrangement arrangement of population exerts the same colonization of patches patches is ignored; ignored; every local population colonization pressure on each each empty patch regardless regardless of of its spatial location. This sort of of assumption assumption may be a reasonable reasonable approximation approximation in models models of of disease disease spread spread in a population population of of freely moving moving individuals, individuals, but it is less likely to be satisfactory for for habitat habitat patches xed spatial locations. patches and and local populations populations with fi fixed locations. For For the purpose purpose of of predicting predicting the equilibrium equilibrium metapopulation metapopulation size this equal-connectance equal-connectance ("mean ("mean field") assumption assumption is not badly misleading, especially if migrants migrants move move relatively long distances Durrett and Levin, 11994; 994; Caswell and Etter, 11993). 993). Not distances ((Durrett Not surpris surprisingly, if one is instead instead interested interested in the origin and maintenance maintenance of spatial patterns, the spatial arrangement Hassell et arrangement of of patches patches and populations populations becomes becomes critical ((Hassell 99 1 ; Durrett 994). al., al., 11991; Durrett and Levin, 11994). Second, Second, the the Levins model model assumes assumes that all patches patches are of of the the same size size and and quality, whereas Harrison, 11991). 99 1 ). Once whereas in nature nature this is hardly ever true ((Harrison, Once again, though, cient reason models. In the though, this is not a suffi sufficient reason to move move to more more complex models. fi rst place, first place, there are interesting interesting natural systems in which variation in patch size is not very great, for instance habitat patches instance dead dead tree tree trunks trunks that that are are habitat patches for for thousands ( Hanski and Hammond, Hammond, 11995). 995). Even if thousands of of specialist specialist insect species species (Hanski if there is substantial lessons from the substantial variation in patch patch size and quality, the qualitative qualitative lessons Levins Levins model model still apply as long as the largest patches patches are not so large that the respective immune to extinction (mainland respective local populations are effectively immune (mainlandisland metapopulations; metapopulations; Hanski Hanski and Simberloff, Simberloff, this volume). Third, Third, since all local populations populations are considered considered to be equal, equal, the Levins Levins model model ignores local dynamics, and it assumes no assumes that emigration emigration and immigration immigration have have no effect effect upon upon local dynamics. This This assumption assumption conflicts conflicts with a wide range range of of ob observations populations ((Brown Brown and 977; Hanski, servations from natural natural populations and Kodric-Brown, Kodric-Brown, 11977; Hanski, 11991; 99 1 ; Hanski 996b). In particular, particular, this simplifying assumption Hanski et et al., 11996b). assumption means means that the Levins metapopulations with the migra Levins model model is really appropriate appropriate only for for metapopulations migration rate within a relatively narrow recoloni narrow range: enough enough migration migration to allow recolonizations, Harrison, zations, but not too much much migration migration to have have an effect on local dynamics dynamics ((Harrison, 11994b). 994b). As the patch patch networks networks in nature nature come come in all shapes shapes and and sizes, it is clearly
55 Structured Structured Metapopulation Metapopulation Models Models
95 9S
desirable to to be be able able to to relax relax this this assumption. assumption. Finally, Finally, being being aa deterministic deterministic patch patch desirable model (Gilpin and Hanski, this volume), the Levins model tacitly assumes a very model (Gilpin and Hanski, this volume), the Levins model tacitly assumes a very large (effectively infi n ite) number of patches. large (effectively infinite) number of patches. There is is aa clear clear analogy analogy between between the the simplifying simplifying assumptions assumptions on on which which the the There Levins model model is is based based and and the the corresponding corresponding assumptions assumptions of of classical classical models models in in Levins popUlation ecology, ecology, such such as as the the logistic logistic equation. equation. Classical Classical population population models models population are concerned concerned with with the the total total number number (or (or density) density) of of individuals individuals in in aa population population are but neglect any differences among individuals (age, size, sex, etc.). To take these these but neglect any differences among individuals (age, size, sex, etc.). To take differences into into account account one one has has to to turn tum to to structured structured population population models, models, which which differences allow one one to to use information information about about individual individual behavior behavior to to draw draw conclusions conclusions about about allow the dynamics dynamics of of a population. population. The The book book by by Metz Metz and and Diekmann Diekmann (1986) ( 1 986) presents presents the comprehensive introduction introduction to to the the philosophy philosophy of of using using structured structured population population a comprehensive models as as well as a wealth wealth of of examples. examples. More More recently, recently, Diekmann Diekmann et et al. al. (1993a,b, ( 1 993a,b, models 1 995a,b) have have developed developed a slightly slightly different different approach approach to to structured structured population population 1995a,b) models, which we we apply apply in in this this chapter. chapter. models, which Our concept of population of of populations popUlations (for (for alteralter Our of a metapopulation is a population Hanski and Simberloff, this volume; Harrison and and Taylor, native approaches, approaches, see Hanski and Simberloff, Hanski, 1996c; 1 996c; Hastings and Harrison, 1994). 1 994). As As pointed pointed out out by this volume; Hanski, and Harrison, et al. al. ((1988, 1 988, 1989), 1 989), the theory theory of of structured structured populations populations can can be be applied Diekmann et to straightforward manner manner if if one one makes makes the analanal to metapopulations metapopulations in a relatively straightforward ogy ogy between between local popUlations populations and and individuals individuals and and between local populations and and metapopulations, metapopulations, respectively. In more detailed metapopulation metapopulation models, where for for instance dynamical dynamical changes in in patch patch quality quality are are included, included, one has has to to replace, replace, in this analogy, local popUlations populations by some other other kind of of local entities. One of of the first structured metapopulation models was presented presented by Levin and Paine ((1974, 1 974, 11975). 975). Their model was structured by the age and the size of a patch. Extinction was assumed to be age-dependent and size-dependent, but col colonization (establishment of new populations) populations) was not modeled explicitly and the effect of of migration on local dynamics was not considered. considered. Hastings and Wolin ((1989) 1 989) used a McKendrick-type model in which local populations populations are structured by age (time since colonization). They assumed that the size of a local population is a function of its age, and they could thus predict the size distribution of local populations. In this framework, it was not convenient to model the effect of migration on local dynamics, since migration does not affect the age of a popu population. Gyllenberg and Hanski ((1992) 1 992) chose local population size as the structur structuring ing variable variable and and could could thereby thereby model model explicitly explicitly the the within-patch within-patch consequences consequences of migration. Later they extended their their model to account for for variation variation in patch 1 995) have quality ((Hanski Hanski and Gyllenberg, 993). More recently, Val et Gyllenberg, 11993). et al. al. ((1995) made aa detailed analysis analysis of of the effect of migration upon local dynamics using similar similar structured structured models. models. In In this this chapter chapter we we present present aa unified unified treatment treatment of of aa large large class class of of deterministic deterministic structured structured metapopulation metapopulation models and and illustrate illustrate the the mathematical mathematical framework framework with with several nite several examples. examples. Being deterministic, deterministic, the the models models continue continue to to assume an an infi infinite number number of of patches patches and and local local populations, populations, and and the the results are are hence hence applicable to to
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large metapopulations. metapopulations. Deterministic Deterministic metapopulation metapopulation models models with with aa finite finite number number large of patches patches have have been been investigated investigated by by Levin Levin (1974), ( 1 974), Holt Holt (1985), ( 1 985), Davis Davis and and Howe Howe of et al. al. (1993, ( 1 993, 1996), 1 996), Doebeli Doebeli (1995), ( 1 995), and and ( 1 992), Hastings Hastings (1993), ( 1 993), Gyllenberg Gyllenberg et (1992), many others. others. All All these these papers papers are are concerned concerned with with the the effect effect of of migration migration on on local local many dynamics, with with a special special focus focus on how how migration migration may may synchronize and stabilize dynamics, and stabilize dynamics. On On the the other hand, these these models models ignore ignore local local extinctions extinctions and and local other hand, local dynamics. recolonizations (for (for stochastic stochastic models models of of finite metapopulations, metapopulations, see Gyllenberg Gyllenberg recolonizations and and Silvestrov, Silvestrov, 1994; 1 994; Hanski, Hanski, 1994a). 1 994a). It is practically practically impossible to to incorporate incorporate the spatial arrangement arrangement of of patches patches into into deterministic deterministic structured structured models of of the the type the treated in this this chapter. chapter. Perhaps Perhaps the the most that that can can be be done done is to to analyze analyze the the qualqual treated itative effects effects of of spatial aggregation of of habitat habitat patches (Adler and and Nfirnberger, Nurnberger, itative spatial aggregation patches (Adler 1 994). 1994). This chapter has two parts. parts. The first part part (Sections (Sections I --V) gives a nonmathenonmathe This chapter has two The first V ) gives matical description of the basic principles of modelling structured populations matical of the basic principles of modelling structured populations and shows by examples examples the the kind kind of results that that can can be be obtained such models. and of results obtained by such An empirical empirical example example illustrates the relevance of structured models. illustrates the relevance of models. The The mathmath ematical outlined in Section VI. This approach was first first used by ematical formalism is outlined al. (1993b, ( 1 993b, 1995b; 1 995b; see also Diekmann et et al., al., 1993a,c, 1 993a,c, 1995a) 1 995a) for for Diekmann et Diekmann et al. "ordinary" populations. "ordinary" structured structured populations.
II. MODELING MODELINGSTRUaURED STRUCTUREDMETAPOPULATIONS METAPOPULATIONS We consider structured We consider structured metapopulation metapopulation dynamics as the study of of the interinter processes at the local level and on the metapopulation metapopulation level level under under relation between processes the influence of the environment. We shall interpret interpret "environment" "environment" in a wide instance the fraction of of empty sense and it will often be convenient to include for instance patches among the environmental variables. Recall that in Levins' Levins'ss model the rate at which a local population gives rise to new local populations depends on the fraction of empty patches. The crucial point is that that all (nonlinear) (nonlinear) feedback takes place through the environment. By a "virgin" environment we understand understand an environment with no local populations and where the patch quality distribution distribution has has settled settled down down to to an an equilibrium. equilibrium. The most essential features of of classical classical metapopulation dynamics dynamics are are recur recurrent local extinctions and colonizations of empty patches. The Levins model is concerned with only these two processes and treats them directly at the level of the metapopulation, thus entirely ignoring local dynamics. Modeling structured metapopulations and analyzing the models take place in three steps. First one has to model mechanisms at the local level, that is, at the level of patches and local populations. In the second step one lifts the model to the level of the metapopulation by simple book-keeping, and fi nally one studies finally population dynamical phenomena phenomena at this level. Local dynamics may include both the dynamics of local popUlations populations and the dynamics of patches. Local populations grow or decline as a consequence of
55
Structured Metopopulotion Models StructuredMetapopulation Models
97 9/'
reproduction, reproduction, death, emigration, and immigration. Patches Patches may change change in size and quality, be destroyed, and and new patches patches may may be created. created. In order to model these local processes processes one has to start by specifying the basic basic local local entity corre corresponding to an individual in ordinary population var population dynamics and by choosing variables that adequately describe the local states. Here "adequate," "adequate," of of course, refers coloni to quantities that affect processes like growth, migration, extinction, extinction, and colonization. If If we, for for instance, instance, consider consider a metapopulation in a set of of patches patches of of dy dynamically changing changing quality a relevant relevant choice of of basic local entity would be a Xl ' X patch and its state would be described by the vector ((xl, x2) where Xl Xl denotes the 2 ) where quality (e.g., resource density) of of the patch and X x22 denotes the size of of the local population inhabiting it. We call the set of all conceivable conceivable local states the local local state space space and denote it by il. 11. In the example example above il f~ is (a subset of) the positive R2 . quadrant R� The following processes have to be modeled:
(i) patch quality dynamics, for for instance how do resource resource density and patch patch area change with time; change (ii) local population growth; (iii) extinction, extinction, patch destruction; destruction; (iv) colonization, patch patch formation, and production production of of dispersers, that that is, how many new basic local entities and with what what local state state at "birth" "birth" a given local entity will give rise to. When modeling these four processes one has to describe describe how they depend depend on the local state and on the environmental state. Some of of the processes, for instance the formation of new patches, may be the result of of processes independent independent of the metapopulation. If that is the case, time enters the description of patch of independent variable. formation as an independent As the metapopulation metapopulation affects its environment one has to close the loop by modeling modeling the
(v) feedback mechanism. mechanism. Next we describe in some detail how the processes (i-v) (i-v) should be modeled.
A. Colonization Without recolonization recolonization of of empty patches a metapopulation metapopulation consisting of of ex extinction-prone local populations certainly goes extinct. The foremost modeling task task is is therefore therefore to to prescribe prescribe the the colonization colonization process. process. To To answer answer some some simple simple but important questions like "will the metapopulation metapopulation persist or go extinct? extinct?"" a precise characterization characterization of of this process is sufficent. The basic idea in the present present approach is to model colonization by describing mathematically the expected cumulative number number and structure structure of of new local en entities produced in the future by a given local entity whose present produced present state state is known and when the course of of the environment is known. All this information is con-
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Mots Mats Gyllenberg Gyllenberget et 01. al.
densed rigorously densed into into aa mathematical mathematical object object called called the the colonization kernel and and rigorously defi n ed in Section VI. To give an example, suppose that colonization is modeled defined To an suppose that modeled as a two-step process: dispersers which may colonize process: local populations populations produce produce dispersers empty empty patches. patches. Then Then the the colonization colonization kernel kernel should should contain contain at at least least the the following following information: Given a local population present population with a given state (e.g., size) at present produce so and so many dispersers time, it will on average average produce dispersers in the future. Given Given a a disperser disperser having having a given given state state at at present it it will on on average colonize colonize so so and and so so many future. The future devel many empty empty patches patches in in the the future. The "so "so and and so" so" depend depend on on the the future development opment of of the environmental state, state, which for the the time time being being is considered considered to be known. known. The The populations populations on on the the patches patches colonized colonized by by aa given given local local population population will in in tum populations that regarded as "second turn give rise to new local populations that can be regarded "second generation generation offspring" offspring" of of the given ancestral popUlation. population. By applying the colonization kernel to itself itself in way to to be precise in Section VI, VI, we mathematical to in aa way be made made precise in Section we obtain obtain aa mathematical description of of this this second second generation generation or or "grandchildren" "grandchildren" to use an an analogy with ordinary popUlations. This ordinary populations. This procedure procedure can can be repeated repeated ad infinitum to to obtain obtain "great "great grandchildren," grandchildren," etc. Summing up over all generations generations we obtain what we shall At; contains call the clan kernel. The The clan kernel A~ contains information about all "descend "descendants" ants" of of a given ancestor ancestor population population with respect to the time of of colonization and and state state at colonization.
B. Patch B. Patch Dynamics, Dynamics, Population Population Growth, Growth, and Extinction Extinction The state of of the basic local entity (e.g., local population, patch) patch) changes changes The because of of popUlation population growth, changes changes in patch quality, and during its lifetime because so on. We We refer refer to all these processes processes by the common common term local state develop develop-
ment. Our Our framework allows for for stochastic stochastic local state state development. development. As our second model model ingredient ingredient we therefore choose the probability distribution of of future local states of of a local entity whose present present state is known. Here we assume assume again that that the future course of of the environment environment is known. From From this model model ingredient many many important important quantities can be computed, for for instance instance the survival probability, probability, that that is, the probability that a given local entity still exists at a given instant of of time in the future. The future state The state of of a local entity can move from its present present state to a future along local state space. Assume Assume that local entity entity has has a along different different routes routes in in the the local state space. that the the local a certain nal state. certain state state before before it it reaches reaches the fi final state. Using the the aforementioned aforementioned probability probability distribution one can compute the probability distribution of of the final final state, state, given that had this time. Summing up over interme that it it had this intermediate intermediate state state at at aa given given time. Summing up over all all intermefinal state, given the diate states states one should get the probability distribution of of the final initial state. This leads leads to a consistency relation that the second second model ingredient has has to to satisfy. The The same same argument argument also applies to to colonization; colonization; hence hence we get aa consistency relation combining the colonization kernel and the transition consistency relation combining the colonization kernel and the transition proba probabilities.
55
StructuredMetopopulation MetapopulationModels Models Structured
99
that deterministic local development development described described by differential We emphasize that equations is included in the formalism as a special case. The consistency relation for for the transition probability then reduces to the statement statement that the system of defines differential equations describing local development defi nes a dynamical system.
C. Combining Combining local Local Dynamics Dynamics and Colonization Colonization C describes the cumulative The colonization kernel introduced in Section II.A describes number and structure at time of of colonization of of "descendants" "descendants" of a given "ances "ancesnumber tral" local entity. The new local entities will develop in time and ultimately we tral" are interested in the structure of the whole metapopulation. As a first step in this are of all the "descendants" "descendants" of of a given direction we obtain a formula for the structure of "ancestor" by combining the colonization kernel with the transition transition probabilities "ancestor" entity with a given local state state at a given time of local development. Given a local entity and given the course course of of the environmental state, state, this derived derived quantity quantity yields the the expected number number of of "descendants" "descendants" with local state in a given set at a later later time. expected
Level D. The Metapopulation level The state of of the metapopulation is by definition the distribution of of local states. The derive an expression for for the distribution In Section II.C we explained how one can derive of local states of of all local entities descending descending from a given initial local entity. If of If of the metapopulation metapopulation is known, that is, if number of of the initial state of if the initial number states are are known, one obtains the state of of the meta metalocal entities and their local states population at any any future by summing summing up of all population future time simply by up the state distributions of the stemming from the the clans clans stemming the local local entities entities present present in the the initial initial metapopulation. metapopulation. Thus lifting matter Thus lifting the the model model from the local level to the the metapopulation metapopulation level is a matter of of straightforward straightforward book-keeping. book-keeping.
E. Feedback Environment Feedback to the Environment The model described so far under the that the The model has has been been described far under the assumption assumption that the environenviron mental state is a given mental state given function function of of time. In In cases where where the the environment environment can can be be instance by the the experimenter such model fully decribes decribes the the timetime controlled for for instance such a model evolution of the metapopulation. However, in natural natural metapopulations of the metapopulations the local populations populations affect affect the the environment environment and and to obtain obtain aa complete complete model model we we have have to to specify specify the the feedback feedback mechanism. mechanism. We We do do this this by by introducing introducing aa third third ingredient ingredient giving giving the the contribution contribution to to the the environmental environmental state state of of a basic basic local entity with with aa given state state when when the the environment environment has has a given state. state. The The solution solution to to the the full full model model including including feedback feedback to to the the environment environment is obtained obtained in the the following following way. One One starts starts by by choosing an an arbitrary arbitrary (continuous) (continuous) function of of time time to to describe describe the the development development of of the the environmental environmental state. state. One One then then function keeps keeps this function function fixed while while one one constructs constructs the the operators operators giving giving the the timetime evolution evolution of of the the structure structure of of the the metapopulation. metapopulation. Using Using the the third third model model ingredient ingredient
!00 1 00
Mats Gyllenberg Gyllenberg et et al. 01. Mats
one calculates calculates the the course course of of the the environment environment that that this this metapopulation metapopulation gives gives rise rise one to. to. The The function function so so obtained obtained is is then then used used to to construct construct the the structure structure of of the the metameta population and and the the procedure procedure is is repeated repeated ad ad infinitum. infinitum. At At each each iteration iteration one one obob population tains aa better better approximation approximation of of the the metapopulation metapopulation structure. structure. tains
STEADY STATES STATES AND AND METAPOPULATION METAPOPULATION EXTINCTION EXTINGION III. STEADY The important questions questions one wants to to answer answer using using mathematical mathematical metameta The most most important one wants population models models are concerned with with the the long-term long-term behavior behavior of of the the metapopumetapopu population are concerned Will the the metapoulation metapoulation persist persist or or go go extinct? extinct? Does Does the the metapopulation metapopulation lation: Will structure tend tend to to aa steady steady state state as as time time grows? grows? What What are are the the stability stability properties properties structure of of the the steady steady states? states? Can Can there there be be several several alternative alternative steady steady states, states, and and if if so, to to which steady steady state state will the the metapopulation structure actually actually converge? converge? The The anan which metapopulation structure swers to to these these questions questions depend the parameters parameters of of the the model, model, which which should should swers depend on on the reflect biologically biologically relevant properties of of the the system under consideration. Pure Pure reflect relevant properties under consideration. mathematical analysis analysis of of the the model can can thus thus lead lead to major major biological biological insights. insights. mathematical The traditional traditional approach modeling population based on on difdif The approach to modeling population dynamics dynamics is based ferential equations equations and and the the steady states are obtained by by putting all time derivaderiva ferential are obtained equal to to zero zero and and solving solving for the population population state. The The resulting resulting system system of of tives equal for the equation can be very very complicated complicated and and solving solving it is in in many many cases, cases, if if not not imposimpos equation can be least a formidable formidable task task that does not not give give any any insight insight whatsoever. One sible, at least that does whatsoever. One of the main main advantages the framework framework presented presented in this this paper allows of the advantages of of the paper is that that it allows us to to derive important biological biological quantities less on on the basis of of their their us derive important quantities more more or or less the basis interpretation example interpretation and not not as a result result of of tedious tedious formula formula manipulation. manipulation. A clear clear example is the basic reproduction number which which is of of fundamental fundamental importance importance in connec connection with existence existence of of steady steady states and and questions questions concerning concerning extinction extinction and per persistence. sistence. number of The The basic basic reproduction reproduction number number R(E) R(/~) is the expected expected number of new new local local entities entities produced produced by one one typical local entity entity during during its lifetime, when when the envi environmental value E. It is intuitively ronmental state is kept kept at the the constant constant value/~. intuitively clear clear that at equi equilibrium obtain the librium each each local local entity entity should should exactly replace replace itself. itself. We We thus thus obtain the follow following ing necessary necessary condition condition for for aa nontrivial nontrivial steady steady state: state:
R(E) e(/~) = =
11..
(A trivial steady state is a state with no local populations populations or local entities.) By a virgin environment environment we understand understand the the steady environment environment in the the ab absence of local entities. Let Eo Eo denote the virgin environment environment and define
Ro = R(Eo). The The basic basic reproduction reproduction number number Ro Ro can can be be any any nonnegative nonnegative number. number. Ro R0 is is the the expected expected number number of of new local entities produced produced in a virgin environment environment by a typical local entity during during its lifetime. If this number number is greater greater than one, then the
55 Structured Metapopulation Models StructuredMetapopulation Models
1I01 01
metapopulation metapopulation will grow grow exponentially exponentially as long long as iitt remains remains, small. If, on on the other other hand, hand, Ro R0 < < 11,, then then the the trivial steady steady state state is is locally locally stable stable and and sufficiently small metapopulations not certain metapopulations will go extinct. However, However, extinction extinction is not certain since since there the trivial there may may exist exist aa nontrivial nontrivial attractor attractor in addition addition to to the trivial one. one. The basic reproduction reproduction number number as defined defined here here is completely completely analogous analogous to The the basic reproduction used in the the reproduction ratio ratio (sometimes called called net net reproductive reproductive rate) rate) used 1 990; Heesterbeek, Heesterbeek, context Diekmann et context of of models for for infectious infectious diseases diseases ((Diekmann et al., al., 1990; For a definition discussion of of the basic reproduction within 11992). 992). For definition and discussion reproduction number number within the traditional traditional approach approach to structured structured (meta)populations (meta)populations we we refer refer to Diekmann Diekmann ((1993). 1 993).
IV. EXAMPLES EXAMPLES In this section we give some some examples which which illustrate the use of of structured structured metapopulation metapopulation models. models. A mathematically detailed analysis analysis of of these examples examples is given given in in Section Section VI. VI.
A. The The Levins LevinsModel Model The 1 . 1 ) is so simple that an explicit solution The unstructured unstructured Levins Levins model model ((1.1) solution can methods. It is, however, however, in can immediately be be written written down down using using elementary methods. instructive to consider of a structured structured model model and and use it to illustrate consider it as a special case of illustrate the preceding sections. the concepts concepts introduced introduced in in the the preceding sections. . l ) extinction In the the Levins Levins model model O (1.1) extinction is modeled modeled by prescribing prescribing a constant constant hazard /-t, which population is exponentially hazard rate rate/z, which means means that the the lifetime of of a local population distributed /-t. The population is thus thus distributed with parameter parameter/x. The expected expected lifetime of of a local population 1//z. virgin environment environment all patches and the the colonization colonization rate rate is/3. patches are empty and is {3. 1/ /-t. In a virgin It follows virgin environment follows from from the interpretation interpretation of of the parameters parameters that that in a virgin environment the expected number popUlations produced poulation number of of new new local populations produced by one one local poulation during during its lifetime equals equals {3 /3 Ro . R0 = ~. m. /z /-t We We thus thus arrive arrive at at the the well-known well-known threshold threshold condition condition
Ro=~>l I.L for for metapopulation metapopulation persistence. persistence.
B. A Simple Simple Structured B. StructuredModel Model The 1 985) but The model model presented presented in this section is essentially due due to Hanski Hanski ((1985) but we framework described we have have reformulated reformulated it it in in terms terms of of the the general general framework described in this
1102 02
Mots Gyllenberg et Mats Gyllenberg et 01. al.
chapter. xed (but chapter. We We consider consider aa fi fixed (but large) large) number number of of patches patches of of the the same same quality. quality. The The basic basic local entity is the local population, population, which can be in any of of two states states X X2, where population and x~I and and x2, where X xlI corresponds corresponds to to aa small small population and X2 x2 to to aa large large population. population. A A state state transition transition from from X x~I to X2 x2 can be due either either to population growth growth or or to immigration immigration of of individuals from other populations. We We thus explicitly assume assume that is due that immigration affects affects local dynamics. The The state transition from X2 x2 to X x~I is due to the effect populations can effect of of environmental stochasticity. Both small and large populations can go extinct as a consequence consequence of of a local disaster. The The metapopulation metapopulation state state can be represented 2 ) with m mii denoting represented by by aa vector vector (m (ml, m2) denoting the fraction of of patches patches with l, m local population 1 , 2). local population of of size size Xi xi,' (i = = 1, We We assume assume that that large large populations populations usually usually produce produce more more dispersers dispersers and and thus thus exert model exert aa higher higher colonization colonization and and immigration immigration pressure pressure than than small small ones. ones. To To model this this we we assume assume that that the the transition transition rate rate from from XI xl to to X2 x 2 equals equals 1 ')t12 -3I- a aim nt- a2m a2m2, l m 1l + 12 + Z, where describes intrinsic a2m2 the 2 describes where 11 )t12 intrinsic growth growth of of local local populations populations and and a a llmml l + + azm2 effect effect of of immigration. Here Here al eel and and a2 o~2 are are nonnegative weights. The in X2 the rate rate of The hazard hazard rate rate of of local local extinction extinction is is IL P,1l in in X XlI and and /d, x2 and and the of IL22 in is 1 partial partial disaster disaster bringing bringing aa population population from from X2 x2 to to X XlI is Yzl2 1 ' A population in state Xi is expected (l )). Note xi is expected to to colonize colonize empty empty patches patches at at aa rate rate {3i fli(1 - (ml (m~ + + m2 m2)). Note that ((1l -- (m (m l1 + -k- m m 22)))) is is the the fraction fraction of of empty patches. patches. New populations populations are are small; small; ' they they have state state X x~. I From From the interpretation interpretation of of the model parameters parameters it is clear clear that in a virgin environment (all patches patches empty) the expected expected number number of of local populations populations pro produced duced by a local population population during during its sojourn sojourn in X xlI is
{3311 l /d, 1 + -I'- 11 Y122 IL
(4. 1) (4.1)
and during during its sojourn in X2 x 2 it is J~2 21 2 [1"2 -'1- 1 ")/21 IL +
(4.2)
The The probability that the the transition from X x~I to X2 x 2 occurs occurs before extinction is
11 ")/122 /1'1l + -F 11 "Y122 IL
(4.3)
and is and the the corresponding corresponding probability probability for for the the transition transition from from x2 x 2 to to X XlI is
21 1 ')/21
21 /'2'22 + "3t- 1 ')/21 IL
(4.4)
Elementary probability considerations considerations lead to the following expression expression for the expected expected number number of of new local populations produced produced by one local population
55
Structured StructuredMetapopulation MetopopulationModels Models
1103 03
placed into a virgin environment: environment: Ro =
1
131
1 -(712//(13,1 -+- ]/12))(]/21/(~, 2 -'[- ]/21))~b['l + 712 +
(]/12/([d'l -~- ]/12))
]32
1 --(]/12/(JU, 1 q- ]/12))(]/21/(/d,2 q- ]/21))/d'2 q- ]/21 (/-/'2 -}- ]/21)]31 + ]/12]~2 I) )(J-L2 + ((JU'I "nt- 'Y12 ]/12)(/d'2 nt- 'Y2 ]/21) -- 'Y12'Y21 ]/12]/21 J-L I + 1/J-L2 )(f3dJ-L I )) + + 'Y2 ]/2,lla,a)(,Si/Ia,, + ((3'~2/~,)(]~21jU'2) 'Y12 /J-LI )(f32 /J-L2 ) m ((11 + 11 + I 1 2/J-LI /J-L2 + 'Y + 'Y2 ')#21//./,2 "}- ]/12/Jld,1 The The value of of Ro thus thus depends depends essentially on on four four parameter parameter combinations, combinations, namely increasing with f31/ f3dJ-LI and 1 2/J-LI ' and j~l//d, /~2//d,2, ]/12//./,1, and 'Y21 ]/21/P,2. is obviously obviously increasing with/31//xl and /J-L2 ' Ro is J-L I1,' f3 2 /J-L2 ' 'Y For most natural metapopulations one would assume that /~2//./,2. For most natural metapopulations one would assume that ~32/td, 2 f3 f3 ' / / 2 J-L2 2 J-L2 > that is, larger populations are to extinction and exert aa /31//xl; that is, larger populations are less vulnerable to extinction and exert f31/ ; J-L I greater then greater colonization colonization pressure pressure on empty patches patches than than small populations, populations, and and then However, if Ro also increasing increasing in R0 is is also in 'Y ]/12 but decreasing decreasing in in 'Y21 ]/21.' However, if/32//x2 us to fi find value of of Ro R0 and and in particular particular the threshold threshold criterion criterion R0 > 11 for for metapopulation persistence directly from the biological interpretation of the pa metapopulation persistence from biological interpretation of parameters. To the same To arrive arrive at the same result result using using the classical classical approach approach based based on on dif differential equations ((Hanski, Hanski, 11985) 985) one have to calculate ferential equations one would have calculate all the eigenvalues eigenvalues of a matrix matrix and and determine determine the largest largest of of them. them. The The model model treated of treated by Hanski Hanski ((1985) 1 985) is only readily calculated only two-dimensional two-dimensional and and the the eigenvalues eigenvalues can be be readily calculated but but for for higher dimensional dimensional systems the the task is extremely tedious tedious and and sometimes sometimes even even higher impossible. if there there is no no difference difference impossible. We We point point out out that that if if f3 ]32//x2 = ]31//xl, that is, if f31/J-L I ' that 2 /J-L2 = in colonization colonization capacity between between the the two two size classes, classes, then then the threshold threshold criterion criterion > I for the Levins model. Ro > > 11 reduces to the the usual condition f3 ]3/p, > 1 for the Levins model. /J-L The The nontrivial nontrivial steady steady states states can also also be found found by this approach. approach. The The details details are 1 985; see also Hastings, 1 99 1 ) found are given in Section Section VI.C.2. VI.C.2. Hanski Hanski ((1985; Hastings, 1991) found that parameter values for certain parameter values there there are two nontrivial nontrivial steady states. This simple simple for structured fundamentally different behavior than structured model model thus thus predicts predicts a fundamentally different qualitative qualitative behavior than Levins's Levins's model. model.
C. Models Models with Continuous Continuous locol Local Stote State In the previous previous section we analyzed analyzed a model model in which which a local population population biologically could be in two two different different states: "small" or or "large." In many many cases cases it is biologically more more realistic realistic to assume assume the the local local state state to be a continuous continuous variable variable representing representing for been confor instance instance the the size or age of of a local population. population9 Such models have have been con-
1104 04
Mots Mats Gyllenberg Gyllenberg et et 01. al.
o o
bifurcation bifurcation parameter parameter
FIGURE FIGURE 1I
Bifurcation diagram in the case case where impact of migration local dynamics Bifurcationdiagram where the impact migration on local dynamics is small. unstable by broken broken line. small. Stable Stable equilibria equilibria are shown shown by continuous continuous line, unstable line.
structed 1 989), Gyllenberg structed and and analyzed analyzed by by Hastings Hastings and and Wolin Wolin ((1989), Gyllenberg and and Hanski Hanski ((1992), 1 992), Hanski 1993), and 1 995). Hanski and and Gyllenberg Gyllenberg ((1993), and Val V a l eett al. ((1995). In of a In Section Section VLC.3 VI.C.3 we we give give aa detailed detailed description description of a structured structured metapopu metapopulation continuous local we shall in lation model model with with continuous local state state variable. variable. Here Here we shall only only briefl brieflyy indicate dicate what what kind kind of of behavior behavior such such a a model model can can exhibit. exhibit. From From the the parameters parameters of of the the structured structured model model one one can can derive derive aa quantity quantity that that measures impact of migration upon local dynamics. impact is is small, measures the the impact of migration upon local dynamics. If If this this impact small, the gives qualitatively Levins model the model model gives qualitatively the the same same prediction prediction as as the the Levins model (where (where the the impact nontrivial impact is is zero), zero), but but if if it it is is sufficiently sufficiently large, large, then then there there are are multiple multiple nontrivial steady steady states states for for certain certain values values of of the the colonization colonization parameter. parameter. Figures Figures 11 - 33 show show aa sample sample of of bifurcation bifurcation diagrams diagrams obtained obtained from from structured structured metapopulation metapopulation models models with is the the fracwith continuous continuous local local state. state. In In these these diagrams, diagrams, the the dependent dependent variable variable is frac-
c~
t.1
~
td r
o o 0 "~.~
,
\'"~ I
~
'..' .......;:. ....... .. - -------------------------------
FIGURE FIGURE 22
bifurcation bifurcation parameter parameter
Bifurcation diagram in the case case where where the impact migration on local local dynamics dynamics is Bifurcation diagram impact of migration large. large. Stable Stable equilibria equilibria are shown shown by continuous continuous line, unstable unstable by broken broken line. line.
55 Structured StructuredMetapopulation MetapopulotionModels Models c~ cD
f ~ , '.,9 ,9, 9, , ,
.*--4 C.; C.3 O O
td
l105 OS
---"-"'_=::_ --------------------------------_.
FIGURE3 FIGURE
bifurcation parameter parameter bifurcation
Bifurcationdiagram diagramwith largedifferences differences in in patch qualityand and large largeimpact impactof migration migration Bifurcation with large patch quality on local local dynamics dynamics ((Hanski and Gyllenberg, Gyllenberg, 11993). Stable equilibria are shown shown by by continuous line, on Hanski and 993). Stable equilibria are continuous line, unstable by by broken broken line. line. unstable
of occupied occupied habitat habitat patches patches (the size of of the metapopulation), metapopulation), which is shown shown tion of function of of the colonization parameter. parameter. The The lines in the the figure give the model as a function steady states. Note that that for some values of the colonization parameter parameter there there is steady values of only one steady steady state, which is then then necessarily stable. For For other other values, there there are two stable states separated separated by an unstable state. In these these cases cases the model has has multiple steady states. mUltiple
D. An An Empirical Example Empirical Example An example strongly suggesting the kind kind of of bifurcation shown in An example suggesting the bifurcation pattern pattern shown Figs. 2 and and hence alternative stable equilibria described and 3 and hence alternative equilibria has has been been recently recently described for the butterfly Melitaea Melitaea cinxia cinxia ((Hanski for the butterfly Hanski et et al., 1995b). 1 995b). In Finland, Finland, this this butterfly A land island occurs on than occurs on the the ~land island in the the Baltic, Baltic, in a very large large network network of of more more than 1500 1 500 habitat habitat patches patches (dry meadows) within an area area of of 3500 3500 km km22 (see Fig. 1I in Hanski, for further Hanski, this this volume; for further ecological ecological details details see see also also Thomas Thomas and and Hanski, Hanski, this et al., this volume; Hanski Hanski et al., 1995a). 1 995a). Hanski al. (1995b) the predicted Hanski et et al. ( 1 995b) tested tested the predicted bifurcation bifurcation pattern pattern by dividing dividing the the material consisting of of 524 local populations material consisting populations in 1530 1 530 habitat habitat patches patches into into 65 semisemi independent patch networks among the independent patch networks with with weak weak interaction interaction among the networks networks (for (for dede tails, see see Hanski Hanski et tails, These networks et al., al. , 1995b, 1 995b, 1996c). 1 996c). These networks vary vary in in the the number, number, sizes, sizes, and density They used and density of of habitat habitat patches. patches. They used the the occupied occupied fraction fraction of of the the pooled pooled patch area area in aa network, PA , as as the the dependent dependent variable, variable, and and the the popo patch network, denoted denoted by by PA, tential tential colonization colonization rate/3 rate f3 as the the bifurcation bifurcation parameter. parameter. The The latter latter was was measured measured aas s
e -dij~ j =
i=1
1106 06
Mats Gyllenberg et Mats Gyllenberg et 01. al.
where in km, km, Aj is the where d dijij is is the the distance distance between between patches patches i and and j j in Aj is the area area of of patch patch j j in population size; in ha ha (square (square root root transformation transformation was was used used to to scale scale patch patch area area to to population size; 996c), and {3 thus Hanski et al. al. 11996c), and n n is is the the number number of of patches. patches./3 thus defined defined is is the the average average Hanski et value of expected immigration calculated on value of the the expected immigration rate rate to to the the patches, patches, calculated on the the assump assumption because (3 has tion that that all all patches patches are are occupied occupied ((because/3 has to to measure measure the the potential, potential, not not actual, rate the sizes actual, rate of of colonization). colonization). The The expected expected immigration immigration rate rate depends depends on on the sizes and Hanski, 11994a). 994a). and distances distances of of local local populations populations from from the the focal focal habitat habitat patch patch ((Hanski, The square root transformation in calculating {3 is used to spread the data The square root transformation in calculating/3 is used to spread the data points points more nition of {3 simply more evenly evenly along along the the x-axis x-axis in in Fig. Fig. 4. 4. This This defi definition of/3 simply says says that that with with larger larger and and less less isolated isolated habitat habitat patches patches within within aa region region the the potential potential colonization colonization rate recapture results rate is is higher, higher, which which accords accords with with direct direct markmark-recapture results on on butterfly butterfly movements ((Hanski Hanski et 994). et al., 11994). movements The {3 and Fig. 4) 4) resembles greatly the the The observed observed relationship relationship between between/3 and PA PA ((Fig. resembles greatly predicted bifurcation Fig. 2). 2). Note networks with with large {3 were were predicted bifurcation pattern pattern ((Fig. Note that that all all networks large/3 practically with {3 in in the the upper upper "branch" in increased with/3 "branch" in practically fully fully occupied, occupied, and and that that PA increased Fig. A values strikingly PA values is is strikingly Fig. 4, 4, as as predicted predicted by by the the model. model. The The distribution distribution of of the the P 995b), supporting alternative bimodal Fig. 4, bimodal ((Fig. 4, Hanski Hanski et et al., al., 11995b), supporting the the hypothesis hypothesis of of alternative stable existing metapopulations less than stable equilibria. equilibria. The The existing metapopulations with with less than ca ca 70% 70% of of the the hab habitat occupied are two stable itat occupied are predicted predicted to to be be in in transit transit toward toward one one of of the the two stable equilibria equilibria Infrequent long-distance long-distance migration migration prevents prevents permanent permanent metapopulation metapopulation ((Fig. Fig. 4). 4). Infrequent
10
Ql '0.
o.e
g
o.e
'0
:J U
<1l
05
04
�
01
c 0
-=
<1l
0.3
•
02
i
·
... . • ••
• •
• •
•
-• •• • •• • •
00
-0. 1
FIGURE FIGURE 44
,
• •• ·
07
:E � '0
• ., ..,.
.-
09
0
•
•
• •
• •
• 2
potential colonization rate
• 3
Empirical with at least least 5 patches, Empiricalresult result for 65 semi-independent semi-independentpatch patch networks networks with patches, all isolated 95), sizes isolated by at least least I1 km from from other other patches. patches. These These networks networks vary vary in the number number (5 .. . . 95), sizes 0.56 hal (average (average area area 0.02 0.02 . . .. . 0.56 ha) and density density of patches patches (total (total area area covered covered by the network network 611 km2). Each point point on the graph graph refers refers to one network. network. The y-axis y-axis gives gives the fraction fraction of 11.9 .9 . .. .. 6 km2). Each pooled measure of potential pooled area area in the network network that that was occupied, occupied, and the x-axis x-axis has a measure potential colonization colonization rate (see (see text; ai., 11995b). 995b). et al., text; more more details details in Hanski Hanski et
55 Structured Structured Metapopulation Metapopulation Models Models
107 1 07
extinction in in these these semi-isolated semi-isolated patch patch networks networks (see (see Fig. Fig. 1I in in Hanski, Hanski, this this volvol extinction ume). ume).
v. DISCUSSION DISCUSSION V. Intuitively, understanding understanding the the role role of of local local population population dynamics dynamics (other (other than than Intuitively, just local local extinction) extinction) in in metapopulation metapopulation dynamics dynamics requires requires the the use use of of aa structured structured just metapopulation model. model. For For example, example, how how does does the the rate rate of of growth growth of of local local poppop metapopulation ulations (relative (relative to to the the time time scale scale of of migration) migration) affect affect metapopulation metapopulation persistpersist ulations ence? Virtually Virtually all questions questions related related to to genetic genetic structure structure require require the the use use of of strucstruc ence? tured metapopulation metapopulation models models (Hastings ( Hastings and and Harrison, Harrison, 1994). 1 994). Ultimately, Ultimately, the the tured utility of of structured structured metapopulation metapopulation models models will will be be determined determined by by whether whether they they utility provide "better" "better" answers answers to to these these and and other other biological biological questions questions than than the the unstrucunstruc provide tured models. tured models. In this this paper brought forth forth by paper we have have presented presented a framework, framework, originally brought et al. al. ((1993b, l 993b, 1995b; 1 995b; see also Diekmann et et al., al., 1993ac, 1 993ac, 1995a), 1 995a), that that Diekmann et Diekmann also Diekmann allows us to treat treat a large variety of of structured structured metapopulation metapopulation models models in a unified unified allows fashion. Within Within this framework, framework, we have have shown shown how how to find find steady steady states and fashion. states and fundamental biological biological questions persistence and and exex how to answer answer fundamental questions concerning persistence tinction number plays tinction of of metapopulations. metapopulations. Here Here the the notion notion of of basic basic reproduction reproduction number plays a crucial role. It was defi ned in a purely defined purely mathematical mathematical way, but its clear clear and and intuitive intuitive biological biological interpretation interpretation as the expected number number of of new local entities entities produced produced by a typical local entity during during its lifetime lifetime made made it possible possible to find steady states states and and criteria criteria for persistence persistence directly on the basis basis of of the biological biological interpretation of of the model model parameters. This is a clear advantage compared compared with the traditional approach to structured metapopulation metapopulation models models as reviewed for instance 1 994). instance by by Hastings Hastings and and Harrison Harrison ((1994). The The most most fundamental fundamental difference between the unstructured unstructured and and structured metapopulation metapopulation models models is that with the structured models there can be multiple equilibria. The possible presence presence of multiple equilibria suggests that that caution caution is metapopulation models to make inferences for for conservation. required when using metapopulation Simple metapopulation metapopulation models have been used to make arguments about about the 994). effect of habitat destruction on the persistence of species (Tilman et al., 11994). Yet, these arguments are usually based on models with a single stable equilibrium. As we we have shown, shown, structured metapopulation models can can have aa much more more complex Figs. 2 and 3). What complex bifurcation structure with multiple equilibria ((Figs. What this means in practice practice is that that unlike the simpler models where where equilibria equilibria change smoothly as the environment changes, changes, the structured metapopulation metapopulation models can exhibit more drastic threshold threshold effects: small changes in the environment environment can cause potentially y potentially large large changes changes in in the the equilibrium equilibrium levels of of the the species. The The butterfl butterfly example which we we described in in Section Section IV.D IV.D strongly strongly suggests suggests that that this this is is not not idle speculation; speculation; multiple multiple equilibria equilibria are are likely likely to to occur in in real real metapopulations metapopulations with with substantial substantial migration migration between between populations. populations.
108 1 08
Mats Gyllenberg Gyllenberg et et al. 01. Mats
We have have shown shown how how the the simple simple Levins Levins model model can can be be expressed expressed within within the the We framework we we have have developed developed here. here. Thus, Thus, itit is is not not the the framework, framework, but but different different framework biological assumptions assumptions which which lead lead to to the the presence presence of of multiple multiple equilibria equilibria and and biological thresholds. With With this this potentially potentially large large difference difference in in the the behavior behavior of of the the models, models, thresholds. can we make make any generalizations concerning concerning the circumstances that that lead lead to to threshthresh can we any generalizations the circumstances 1 985; Gyllenberg Gyllenberg olds? Using Using the the results results developed developed here here and and elsewhere elsewhere (Hanski, ( Hanski, 1985; olds? 1 992; Hanski Hanski and and Zhang, Zhang, 1993; 1 993; Val Val et et al., al. , 1995), 1 995), itit appears appears that that the the and Hanski, Hanski, 1992; and key ingredient ingredient for the presence presence of of multiple mUltiple equilibria equilibria is that that immigrants immigrants affect affect the the key for the local dynamics dynamics and and extinction extinction of of existing existing populations. populations. From From the the perspective perspective of of local individual fitness, fitness, multiple multiple equilibria equilibria at at the the metapopulation metapopulation level level are generated individual are generated by immigrants immigrants to to empty empty patches patches or or very very small populations popUlations having having smaller smaller longlong by term fitness than than immigrants immigrants arriving arriving at at somewhat somewhat larger larger populations populations (Gyllenberg (Gyllenberg term and Hanski, Hanski, 1992). 1 992). At At very very high high local densities, densities, negative negative density dependence dependence may may and again reduce reduce fitness. fitness. again We conclude by emphasizing emphasizing that role of of habitat habitat fragfrag We conclude by that conclusions conclusions about about the the role al. , 1994) 1 994) mentation and on ecological ecological communities communities (e.g., (e.g., Tilman Tilman et mentation and destruction destruction on et al., should be be based based on framework of of structured structured metapopulation models. As As the the should on the framework metapopulation models. amount amount of of suitable habitat habitat is reduced, reduced, the the system may may undergo undergo a bifurcation bifurcation rere moving the the equilibrium equilibrium which which allows the the persistence persistence of species. The The develdevel moving of the species. opment of these more complex models models does does lead lead to to one one major major problem when one one opment of these more complex problem when wants to to apply the models models to to answer answer practical questions in conservation--there conservation -there wants apply the practical questions proliferation of of parameters. parameters. This This is a problem problem for for which which we we do do not not have have a is a proliferation general answer. general answer.
THE MATHEMATICAL VI. THE MATHEMATICALFORMALISM FORMALISM A. A. Modeling Modeling 1. 1. Colonization Colonization
Consider Consider a basic basic local entity which which at time t has state state x x E E fl. 1"~. Suppose Suppose that units later later (that is, at real time t + + s) it gives gives rise to a new of s time units new local entity of state y E E fl. f~. We We then then call (s, y) E ~ R+ X x fl 12 the colonization colonization coordinates coordinates of of this this local entity. We ne the colonization We now now defi define colonization kernel kernel A as follows: For For each each x)E R+ X fl (t, x)E f~ and and each each subset subset A A of of R R X x O. 1~.
AE(t, x)(A) = = expected number number of of new local entities entities with with colonization colonization AE (t, x)(A) coordinates in A A produced produced by a local entity which at time t coordinates x, given the course of of the environment environment E. E. had state x,
This definition is perhaps perhaps most easily understood understood when when we take A A = =
x w o9 with w o9 C C O. f~. AE(t, x)([O, x)([0, s) X w o9)) gives the expected cumulative cumulative num num[0, s) X
ber of of new local entities with state at colonization colonization in w o9 produced produced by a local entity x during during s units of of time since the reference reference time t. Notice that that we do with state x not condition condition on survival of of the "ancestor" entity and thus AE implicitly takes
55
Structured StructuredMetopopulotion MetapopulationModels Models
1109 09
extinction meta extinction into into account. account. A£ AE is is the the basic basic ingredient ingredient of of our our general general structured structured metapopulation population model. The The populations populations on the patches patches colonized colonized by a given local population population will in turn give rise to new new local populations populations that that can be regarded regarded as "second "second generation generation offspring" define AkE(t, A�(t, x)(A) x)(A) offspring" of of the given ancestor. It therefore makes makes sense to define exactly rst gen exactly as as A£(t, Ae(t, x)(A) x)(A) but but for for the the kth kth generation generation rather rather than than for for direct direct (fi (first generation) l )th generation eration) "offspring." "offspring." Since the the (k + + 1)th generation offspring offspring are are direct direct offspring offspring of requires that that of the the kth kth generation, generation, consistency consistency requires
i
I A� A~e+'(t,x)(A) = ~+ (t, x)(A ) = JR
where
.+ •x "
A(t )(A _ T) AHt, x)(d'T" X dO, A(t + + 'T, ~', � ~)(A_~)Ake(t,x)(d~ • d~),
(6. 1) (6.1)
(0", � (0" + A_~ = {{(or, ~)) E ~ R+ 1"~:: (or + 'T, ~', 0 ~) E E A A}. A _T = }. R + •X 0 ancestor Since the time coordinate coordinate in A A measures time relative to time tt when the ancestor had had state x, this this translation translation in in the the time time direction direction is is necessary. necessary. 1 ) shows A£o Regarding Regarding the Formula Formula (6. (6.1) shows that that one one obtains obtains A� A~ by by iteration from from Ae. the right-hand l ) as kernels A£ right-hand side side of of (6. (6.1) as the the definition definition of of aa "product" "product" of of the kernels Ae and and A�, A~, A� A~ is indeed the the kth power power of of A£. AE. Summing Summing over all "generations," "generations," we define the clan clan kernel kernel oc
A~ = ~] A~.
(6.2)
k=l
2. Local Local State State Development Development
We We model local state state development and and survival of of a basic basic local entity as a Markov process with extinction and/or and/or patch patch destruction as transition to an ab absorbing state. As As the second model ingredient we therefore therefore choose the transition s)(w) of probabilities probabilities u£(t, uE(t, x x;; s)(w) of this process. Specifically,
uu£(t, E(t, x; s)(w) s)(~o) = = probability that that a basic basic local entity entity which has state x at , time time tt still exists exists s time time units later later and and then then has has aa state in in wC C 0, ~ , given given the the course course of of the the environment environment E. The The case case of of deterministic local local development is included included as a special case. case. In this X(t, x; s) s) satisfying this case case u£(t, uE(t, x; s) is a measure measure concentrated concentrated at the the point point X(t, satisfying an an ordinary differential differential equation equation with with initial condition
a0 m X(t, = g(X(t, g(X(t, x; s), E(s» E(s)),, X(t, x; s) = Os as X(t, x; t) t) = = x x.. X(t, Here when the the local local state state iiss x Here g(x, g(x, ee)) iiss the the rate rate at at which which the the local local state state changes changes when and the environmental state has the value e. The (t, x; s) ::-= u£(t, s)(O) is the survival The total probability mass mass fiF ~(t, ue(t, x; s)(~) survival probprob-
!!1 1 00
Mats Gyllenberg Gyllenberg et et al. al. Mats
ability. It It satisfies satisfies the the initial initial value value problem problem ability. 0a £(s» ~(t, CZF (t, x; CZF (t, x; x; s), s), E(s)) -- ~(t, x; s), s) , = -Ix(X(t, x; s) s) = J.L(X(t, x; as Os -
CZF (t, x; x; t) t) = = 1, 1, ~(t, where J.L(x, e) e) is is the the extinction extinction rate. rate. where/z(x, Being transition transition probabilities probabilities of of aa Markov Markov process process the the measures measures uE(t, uE(t, x; x; s) s) Being have to to satisfy satisfy the the C Chapman equation have h a p m a n --Kolmogorov K o l m o g o r o v equation
In
�; ss - or)(oj) u E(t, x; uE(t, x; s)(w) s)(w) = uE(t + ue(t, x; or)(d~) u)(d� ) + ~r, u, ~; u) (w) ue(t, = f , ue(t -
(6.3) (6.3)
Q. Equation for all all t, all 00 -< Equation (6.3) for t, all all w C (6.3) is is aa consistency consistency relation. relation. :::; o" u -:::; s, s, and and all C l'l. It has has the the following following interpretation: interpretation: Consider Consider aa basic basic local local entity entity with with state state xx and and It time units units later.The later. The probability probability that that its its state state belongs belongs to to w assume state s�c s time assume it has has state �; ss - o')(w). at time t + u)(w). Summing Summing up up over over all possible possible intermeinterme at time ue(t ++ o', u, ~:; + ss is uE(t has to the probability probability of of transition transition from from xx to to w w as as time time diate states states s�c one diate one has to get get the to t + Equation (6.3) (6.3) expresses expresses exactly exactly this. this. The The same same argument elapses from t to elapses from argument + s. Equation holds for colonization, too, too, and and therefore therefore we we get get another consistency relation relation holds for colonization, another consistency combining uu and and A: A: combining -
Q» AE(t, xx)(A) ([0, s) s) •X [l)) x)(A n A (t, x)(A Ae(t, ) ( A ) == A(t, f') ([0,
In
+ AE(t + s, ~)(A_~)uE(t, s)(d� ) . AE(t + � )(A _ s )UE(t, x; s)(d~). + f Jn
(6.4) (6.4)
3. Combining Combining Local Dynamics Dynamics and Colonization Colonization Consider Consider aa basic basic local local entity entity with with state state x x at time We describe describe by time t. We by u'f.; expected size and whole clan basic local local entities uCe(t, the expected and structure structure of of the the whole clan of of basic entities (t, x; s) the descending We descending from from the the given given ancestor ancestor and and including including the the ancestor ancestor in in the the clan. clan. We thus thus define define
u'f.; (t, x; s)(w) u~(t, s)(oo) = = uE(t, ue(t, x; s)(w) s)(w)
+ +
uE f( ue(t(t + + n )[0, s)xlI J t 0, s)x
o', �, ~:, s - O ~:)(oJ)A~(t, x)(dor X d� d~).) . u, (w) A'f.;(t, x)(du -
(6.5) (6.5)
The rst term The fi first term on the right-hand right-hand side of of (6.5) gives the probability probability that that the ances ancestor tor'' s state state is in w o) at time t + + s and and the latter latter term term describes the the expected expected structure structure of emphasize that of descendants descendants at at time time t + + s. We We emphasize that A'f.; A~ was was constructed constructed from from A AEE in in Eq. (6.2) so u'f.; u~ defined defined by (6.5) can can be regarded regarded as constructed constructed from from our our basic basic ingredients E and ingredients A AE and uu E. E. from the interpretation interpretation that that u'f.; u~ should should satisfy the the Chapman Chapman- KKol olIt is clear from mogorov mogorov equation equation and that A'f.; A~ and and u'f.; u~ have have to satisfy a corresponding corresponding consistency consistency 1 995b) proved relation. relation. And And indeed, indeed, Diekmann Diekmann et al. ((1995b) proved that that Eqs. Eqs. (6.3) (6.3) and and (6.4) hold hold with with UE u E and and A AE replaced by by u'f.; u~ and and A'f.;, A~, respectively. respectively. E replaced
55 Structured Structured Metapopulation Metopopulotion Models Models
!1 I1 I1
4. The The Metapopulation Metapopulation Level Level 4. The state state of of the the metapopulation metapopulation is is by by definition definition the the distribution distribution of oflocal local states. states . The m on o n the the local local state state space space More precisely, precisely, the the metapopulation metapopulation state state is i s aa measure measure m More n such such that that for for every every to e 12, n, m(to) m( w) gives gives the the (expected) (expected) number number of of basic basic local local 1~ wC If for for instance instance the the local local entity entity is is aa patch patch and and the the entities with with state state in in the the set set to. entities w. If (Xl ' x2), where x, Xl denotes denotes patch patch quality quality and and x2 denotes the the local state state is is aa vector vector (x,, local X2 denotes X2), where size of of the the local local population population inhabiting inhabiting the the patch, patch, then then m([s m([gl ' r/l] 711 ] •X [so2, [g2' */2]) 712]) is is size the number number of of all all patches patches with with quality quality in in the the range range ~:1 gl -< ::5 x, X l -< ::5 771 supporting local local the 71 1 supporting populations with with size size in in the the range range ~:2 ::5 r/2. ::5 x2 X2 -< g2 --populations 712. We now now define define the colonization operators operators Ve VE and the next next state state opop We the colonization and VtV~ and and the VE and and U~ Vt- as as follows: follows: erators Ue erators
L =I L I
AE(T, x)([0, (VE (t + T)m)(w) = f~a AE('r, w) m(dx) , x)([O, t) •X to)m(dx), (VE(t ~', 'r)m)(to)= + T,
(6.6) (6.6)
At- (T, x)([0, (VHt + T)m)( w) = f , A~('r, w) m(dx) , x)([O, t) xX to)m(dx), (V~(t ~', ~')m)(to) + T,
(6.7) (6.7)
m(dg ) , UE(T, ~; g; t)(to) t)(w) m(d~), (UE (t + T)m)(w) == f ue(~', (UE(t ~', ~')m)(to) + T,
(6.8) (6.8)
((U~(t VHt + + T, ~', T)m)(w) ~')m)(to) = = f u u~(~', ~; t)(w) t)(to) m(dg m(d~).) . t- (T, g; n
(6.9) (6.9)
n
In assume that the state the metapopulation metapopulation is is given given at In these these formulae formulae we we assume that the state m of of the at time (t + time T. ~'. The The measure measure VE VE(t + T, ~', T)m ~')m gives gives the expected expected cumulative cumulative amount amount and and distribution distribution with respect respect to state at colonization colonization of of new new local entities entities produced produced by metapopulation in the time interval [T, by this initial metapopulation [~, T~" + + t). V Vce(t + T, ~', T)m ~')m has has Ht + the the same same interpretation interpretation but but it it takes takes the the whole whole clan clan into into account. account. V Ue(t + T, ~', T) ~') tells E (t + us n" during us where where the the local local entities entities of of the the initial initial metapopulation metapopulation "have "have moved moved in in f~" during tt time units since the initial time (t + time units since the initial time T. ~-. VtU~(t + T, ~', T)m ~')m is the solution of of the metapop metapopulation model. It represents represents the expected expected state of the metapopulation at time tt + given the state m at time T. + T~"given the state m at time ~-. If E of If the the environment environment E E is is aa constant constant function function/~ of time, time, then then both both At Ae (t, x) and and Ut u~ (t, X; x; s) are independent independent of of t. It follows that the same is true of of At(t, A~(t, x) and t (t + u£(t, - (6.9) also of Vt (t + u~(t, X; x; s) and hence by (6.6) (6.6)-(6.9) Ve(t + s, t), Vt V~ (t + + s, t), V Ue(t + and In particular, the one parameter families s, t), t), and V U~(t + s, s, t). t). In particular, the one parameter families s, £(t +
Tt (s) = t (t + Te(s) = V Ue(t + s, t), t),
Tt(s) = V c(t + + s, s , tt), ) T~(s) = Uet(t
Wt (s) = We(s) = Vt(t Ve(t + + s, s, t), t),
W£(s) W~(s) = = V£(t v~(t + + s, s, t) t),,
(6.10) (6. 1 0) (6.11) (6. 1 1) (6.12) (6. 1 2) (6.13) (6. 1 3)
are ned and are well-defi well-defined and give give aa complete complete description description of of the the time time evolution evolution of of the the metapopulation metapopulation in in the the special special case case of of aa constant constant environment. environment.
1 12 !12
Mats Gyllenberg GyUenberg et et al. 01. Mats 5. Feedback Feedback and and the the Full Full Model Model 5.
We denote denote by by y(x, rex, e) e) the the contribution contribution to to the the environmental environmental state state of of aa basic basic We when the the environment environment has has state state e. e. Consider Consider aa metapopmetapop local entity entity with with state state xx when local ulation m at at time time 0. O. As As explained explained in in Section Section VI.A.4, VI.A.4, itit will will have have state state ulation with with state state m
f a u~(O, ~:; t)(.)m(d~) at time t. Adding Adding all all the the contributions contributions of of all all the the local entities entities we we obtain obtain the the total total at contribution contribution
E I fay(
rex, E(t» f a u~e(O' uE (O, ~; g; t)(dx)m(d~ t)(dx)m(dO, x, E(t)) ), n
which is fed fed to to the the environment environment through through aa possibly possibly nonlinear nonlinear function function F. F. We We thus thus which arrive at the following following equation: arrive equation:
(I
)
In
u £(O, g; t)(dX)m(dO ) 9. E(t) = F rex, E(t» E(t)=F(fnY(x,E(t))fau~(O,~;t)(dx)m(d~) n
(6. 1 4) (6.14)
Let us recapitulate recapitulate the the situation. The The ingredients of the model are are A AE, Let ingredients of the model e, uuE, e, and r, where where Ae AE and the consistency consistency relations relations (6.3) and (6.4). (6.4). The The and y, ue satisfy satisfy the (6.3) and and UE model consists consists of of Eqs. (6.2), and (6.14), (6. 1 4), which which we we repeat repeat here: model (6.2), (6.5) (6.5) and ~c
(6.15) (6. 1 5)
A~ - ~ A } , k=l
(t, x; s)(w) s)(w) = U£(t, u~(t, x; s)(to) = uE ue(t, s)(to) + +
E(t) E(t) = =F
r s)• uE(t ++ a, AE (t, x)(do" x)(da X fo, o', g, ~c, ss - O(w) ~)(w) A~(t, • dg d~),) , )[0.
(I
n
s)xn
rex, y(x, E(t» E(t))
E
u£(O, ue( 0 , g; ~; t)(dX)m(dO t)(dx)m(d~)
))
.
(6.16) 1 6) (6. (6.17) (6. l 7)
In the 1 5 ) - (6. 1 7 ) m is the given the model model (6. (6.15)-(6.17) given initial initial state state (at time 0) of of the meta metapopulation. 1 5 ) - (6. l 7) we under population. By a solution solution of of the initial value problem (6. (6.15)-(6.17) we understand stand
In
(O, g; U~(t, O)m = = ~a uE u~(O, ~; t)( t)(.)m(d~), = VE(t, S(t)m :"= · )m(dO ,
(6.18) (6. 1 8)
1 5) - (6. 1 7). In practice the solution is found where E solves (6. (6.15)-(6.17). found by the method of of successive successive approximations: approximations." One starts by guessing the course course E E of of the environ environ1 5 ) and substitutes ment. Using this initial guess one constructs constructs AE A~ according according to (6. (6.15) substitutes it into the right-hand 1 6). The uE right-hand side of of (6. (6.16). u~ obtained obtained in this way is substituted
55 Structured Structured Metapopulation Metopopulotion Models Models
113 1 13
into (6.17) (6. 1 7) and and one one considers considers this this relation relation as as the the definition definition of of aa new new approximation approximation into E. Repeating Repeating this this procedure procedure over over and and over over again again one one gets gets aa of the the environment environment E. of sequence of of environmental environmental functions functions which which under under suitable suitable conditions conditions converges converges sequence to some some function function E. E. The The solution solution of of the the full full model model is is then then given given by by (6.18) (6. 1 8) with with to E on on the the right-hand right-hand side. side. the limit limit function function E the
B. Steady States steady state state is is by by definition definition aa solution solution that that does does not not change change with with time. time. So So AA steady constant function/~ function E satisfying steady state state consists consists of of a constant (6. 1 5 ) - (6. 1 7) and and aa satisfying (6.15)-(6.17) aa steady metapopulation state state m m such such that that metapopulation S(t)m = = m for for all t -> � 0. S(t)m O. where S S iiss defined defined by by (6.18). (6. 1 8). where At first determining the steady states would be a diffi cult At first sight sight it seems seems like determining the steady states would difficult task in the the present present approach with the traditional approach based on on task approach as compared compared with approach based differential equations, where the the steady-state at least least formally formally the the differential equations, where steady-state condition condition is at simple Am = some nonlinear nonlinear operator A. Nonetheless, Nonetheless, we we show show that simple Am = 0 for for some operator A. that ap appearance and that the present present approach approach is close close to biology and allows pearance is deceptive, deceptive, and that the and allows for for intuitive intuitive and and helpful helpful interpretations. interpretations. First First we notice that that a solution solution with m = = 0 for all t � >- 0 is always always a steady E satisfi F(O). Then es (6. 1 5) state. To define E = To see this simply define/~ = F(0). Then obviously obviously/~ satisfies (6.15)state iiss called trivial, and iitt corresponds (6.17) S(t)O = = O0.. This steady state corresponds to (6. 1 7) and S(t)O metapopulation metapopulation extinction. Next we consider consider nontrivial nontrivial steady steady states. We We assume assume E for E, that that that the the environment environment is is kept kept constant constant at at/~, that is is E(t) = =/~ for all all tt � -> O. 0. The The operator (oo), which 1 2) is is given operator WE We(~), which according according to to (6. (6.12) given by by
In
(6. 19) (6.19)
E) = R(/~) = 11.. R(
(6.20) (6.20)
(WE(oo)m)(w) E (x)([O, (0 (We(~)m)(w) = = f~ A Ae(x)([0, ~)) X x w)m(dx) og)m(dx),,
contains all information about the state at colonization of of the entire next "gen "generation" produced by the initial metapopulation m. Intuitively it is clear that that at a nontrivial equilibrium each local entity should on average exactly replace itself. l 995b), we make this idea precise by defining the Following Diekmann et al. ((1995b), E) as basic reproduction number R( R(E) as the the spectral spectral radius radius of of the the operator operator WE(oo). We(~). E) is Positivity arguments that are are usually biologically self-evident self-evident ensure that R( R(/~) an an eigenvalue eigenvalue of of WE(oo) We(w) which which is is simple. simple. E) is The The eigenvector eigenvector corresponding corresponding to to the the eigenvalue eigenvalue R( R(/~) is denoted denoted by by bE be and and E) is the expected it it gives gives the the stable stable distribution distribution of of states states at at colonization. colonization. R( R(/~) expected number number of new local entities entities produced produced by a "typical" (that (that is, sampled randomly from from bE) be) local local entity entity during during its its lifetime. lifetime. We We thus thus arrive arrive at at the the following following necessary necessary condition condition for for aa nontrivial nontrivial steady steady state: state:
1II1 44
Mots Mots Gyllenberg Gyllenberget et 01. al.
Let E satisfy (6.20) and Let/~ and consider consider at a certain instant of of time time a collection of of newborn local r time local entities distributed according according to bE bE.. ~" time units later their state distribution will be TE( r)bE . Notice that since the environment is now constant Te(~')be. we use the time invariant version (6. l 0) of (6.10) of the next state operator. Since the with time measure if! distribution distribution bE be of of states states at at colonization colonization does does not not change change with time the the measure th defined by
rh =
f0 c Te ( z) b -edz
(6.2 1) (6.21)
satisfi es satisfies
S(t)m = (t)if! for all t � = T� T~(t)th -> O 0.. For a rigorous proof 1 995b). So proof of of this result we again refer refer to Diekmann Diekmann et al. ((1995b). the E) isis aa steady 1 7 ) holds. Taking (6.2 1 ) into the pair pair (m, (m,/~) steady state state provided provided Eq. Eq. (6. (6.17) Taking (6.21) into account account we we see see that that the the condition condition takes takes the the form form
E=F(fz
y(x,E) fo=Te(~-)bed~-)(dx)).
(6.22) (6.22)
Being Being an eigenvector eigenvector corresponding corresponding to a simple eigenvalue, eigenvalue, bE be is of of course course de determined only up to an arbitrary multiplicative constant. constant. If If this constant constant can be chosen E of chosen such such that that (6.22) holds, holds, then then the the solution solution/~ of (6.20) and and the the corresponding corresponding if! ned by (6.2 1 ) indeed form If F th defi defined (6.21) form a steady state. If F is linear linear the arbitrary constant can always be adjusted such that (6.22) holds holds so in that case (6.20) is also a sufficient sufficient condition for for a steady state. However, in general the equilibrium en environment has to be determined from both (6.20) and (6.2 1 ). (6.21).
C. C. Examples 1. The The Levins Levins Model Model In In the the Levins model model the the basic local entity entity is is the the local population, and and since since the model is unstructured have the same unstructured all populations populations have same state. state. The The local state space fl ~ thus consists consists of of a single point. In particular, particular, A and u do not depend depend on x. patches, we choose x. Since the colonization rate rate depends depends on the fraction fraction of of empty patches, choose this number this fraction fraction as as the the environmental environmental state state and and denote denote it it by E(t). The The expected expected number of of new local populations (divided by the total number number of of patches) produced by a population population extant at time t during during the time interval [t, t + + s) s) given the course course of of the the environment environment is is thus thus
AE(t)([O, Ae(t)([0, s)) s)) = =
ftf t+s+s t
(3E( r)e-ILTdr. flE(T)e-~'*dr.
For For time-independent time-independent environments environments the the corresponding corresponding operator operator WE(s) is simply simply
55
Structured Metapopulation Models StructuredMetapopulation Models
1!!5 1S
the number the number W t ( s ) = ~E,
f0Se_~,r
= /3 E'(1 - e-~) tx
considered R. In particular, considered as a linear operator on the one-dimensional space R.
- = W~,(~) -- f3 R(E') 13 E ~,. R(E) = Wt;(oo) . /x J.L =
-
1 ; that is, all empty, and we arrive The virgin environment is is/~ = 1; that is, all patches patches are are empty, and we arrive at at £ = the well-known threshold condition 13 R0 = '--- > 1
(6.23) (6.23)
for nontrivial equilib for metapopulation metapopulation persistence. persistence. When When (6.23) holds, holds, the the unique unique nontrivial equilibrium is obtained from rium is obtained from - = f3 - = 1. R(/~) R(E) = /3/~ E = l. /z J.L -
2. 2. A A Simple S i m p l e SStructured t r u c t u r e d Model Model
In In this this section section we we show show in in detail detail how how the the model model treated treated in Section Section IV.B can can be put into the present present framework. We We use the same notation as in Section IV.B. The The metapopulation state is a measure m m on the local state space space n 12 = = {XI } which can m l , m2) m2 ) with mi {xl, X x2} can be be represented represented by by aa vector ((ml, m i denoting denoting the the 2 ' fraction of of patches patches with local population of of size Xi xi, (i = = 1, 1 , 2). ' We We choose choose
E1 = celml + o~2m2
(6.24) (6.24)
and
E2 = 1 -
(ml + m2)
(6.25) (6.25)
as environmental variables. Note Note that E E I1 describes describes the the effect effect of of immigration upon local patches. The local population population growth growth and and that that E E22 is is simply the the fraction fraction of of empty empty patches. The transition is thus EI and popUlation in in state transition rate rate from from XI xl to to X x22 is thus 'Y1 Yl22 + + E1 and aa population state Xi xi is is expected expected to f3i E to colonize colonize empty empty patches patches at at aa rate rate/3; E2. 2• Using Using these environmental variables and and the assumption assumption stated in Section IV.B it it is is aa straightforward straightforward exercise exercise to to write write down down explicit expressions expressions for for A AeE and uE ue.• Since we are mainly interested in metapopulation extinction and and steady states we shall only calculate A tt; = = ([0, (0 oo)• to) for for constant constant environments environments ) X w) £ = (£1 ' £2 )· The The expected expected number number of of local populations populations produced produced by a local population
1!i1 66
Mats Gyllenberg et 01. Mats Gyllenberg et al.
during during its sojourn sojourn in X x~I is
/3,t72
(6.26) (6.26)
~'LI -~" ")/12 -+- /~l
and and during during its sojourn sojourn in X x:z it is
z £z /32/?2 /3 21 + IL /'s z + 1' Y21
(6.27) (6.27)
The probability that extinction is The probability that the the transition transition from from xXl1 to X x22 occurs occurs before before extinction P tEl l = = P
Yl2 nt- £1 /~1 IZ + I' EI IZ + IL J['Ll1 + + I' 'Yl2 + J~l
(6.28) (6.28)
= -=-= -
and Xl is and the the corresponding corresponding probability probability for for the the transition transition from from X x 22 to to Xl
I'Z "Y21I
(6.29) (6.29)
q = ~ . q IL 1"s2 + + 1'2 ~/211
Consider Consider a local local population population which which initially initially (at (at time time 0) has has state state X xl.I ' The The probability probability that that it will enter enter state state Xz x2 exactly exactly n times times is n - I1((11 - Pt, q) P~l q"Ptl q PEl q)
(6.30) (6.30)
and expected number and the the expected number of of entries entries in X x2z is thus thus
(6. 31) (6.31)
pt,
I1 - PE1 Pt, q q"
Similarly, Similarly, the the expected expected number number of of sojourns sojourns in state state X XlI is 1
(6.32) (6.32)
11 - PEl P t l qq "
Combining 1 ) and nds the Combining (6.26), (6.26), (6.27), (6.27), (6.3 (6.31) and (6.32) (6.32) one one fi finds the expected expected number number of of new populations populations produced produced by by a local local population population initially initially in state state XI Xl during new during its entire E (O, Xl) Xl) can entire lifetime. lifetime. Since Since all new new populations populations are are small small (have (have state state X xlI )) A At(0, can conveniently be conveniently be represented represented as a vector vector with with zero zero second second component: component:
(-
At(0, Xl)([0, ~) • .)
= =
I1
/31E'2 /31£Z
�
I1 - P tPElq l q IL tXll + + I'IZ ")/12 + "~- E II
+
Pt,
/32E'2 )
1 - P t ~ q ~[-L2-~- Y21 .
(6.33) (6.33)
0
.
In analogous way In a completely completely analogous way we we obtain obtain
AE(O, x At(0, x2)([0, ~) X • .)) z)([O, co) q -
/31/~2
1
1 - Pt~q I,zl + Y12 + E1 + 1 - P t l
0
/32E2 ].z2 +
")/21
) .
(6.34) (6.34)
Structured StructuredMetopopulotion MetapopulationModels Models
55
1!17 17
The rep The corresponding operator operator W W~(~) t(oo) at the metapopulation level can thus be represented 6.33) and resented by a 2 by 2 matrix matrix with the vectors ((6.33) and (6.34) columns. The The (6.34) as columns. spectral oo) spectral radius radius (largest eigenvalue) eigenvalue) of of W We(~) is t(
R(E')
= R( £ ) =
f3 ' £2 /3,E'2
11
_
11 - P eP. t~ ,qq JL la.~ + 'Y'2 712 + "Jr- E E1, , +
+
+
f3 P~:, /32E'2 PE, 2£2 ~ . JL 11 -pP ~ tq, q ~22 + + 'Y2' Y21
(6.35)
The populations is given The virgin environment environment corresponding corresponding to no local populations given by b y /£~ , = = 0, £ 0, E22 = = 11 and and thus thus R R o0 = =
1
/3~ f3 ....!.'_ .!...
_ _ _ _ _ _ _ _ _ _ _ _ _
+
»( 'Y2 J(JLz2 + 11 - ("II (TlZ/(/.L -1- 'Y'2 ~12))(~21/(~/, -] 'Y21 ~21))» JL ]J'l, -31- 'Y12 "Y12 2 I(JL ,1 +
(~'~2/(~ + ~2))
+ 1 -
(y12/(iJ,
1 +
/32
~/12))(~21/(1J, 2 + ~21)) /LL2 -31- ~21
(]J'l + "YI2)(IJ'2 + ~/21) "~- T12")/21 +
/JL 2 )(f3 , /JL , ) + ((1l + 'Y2, yz~//z2)(/3,/~) + ((7,2//z~)(/32//.L2) 'Y l 2 IJL , )(f3 2 /JL 2 ) + I 11 + 'Y21 JL 1 2 IJL , "yZl//J,22 + -+- 'Y "Y12/~I From depends essentially From this expression expression for for Ro R0 we see that that the value of of Ro depends essentially on IJL four f3 1 IJL I , f3 /JL , ' " and JJL 2 ' R Ro0 is is four parameter parameter combinations, combinations, namely namely/3~//z~, ~2/}[.L2, "Y12//~1, and "y21//./,2. 'Y2 2 2 'Y 2 ' For most natural metapopulations one f3 2 /JL 2. f3 , /JL , and obviously increasing increasing in in/3~//z~ and ~2/]d, For most natural metapopulations one 2 populations are vulnerable would that f3 2 1JL 2 > would assume assume that/32//t./,2 > f3 /31//}L/,I larger populations are less vulnerable 1 IJL , ;; that is, larger to extinctions patches than extinctions and and exert exert a greater greater colonization colonization pressure on empty empty patches small is also small populations, populations, and and then then Ro R0 is also increasing increasing in in 'Y'2 T12 but decreasing decreasing in 'Y21 Tzl.' 2/JL22 < /JL " decreasing 2 and increasing in However, if if f3 j32/~J, < f3 /31/]J, 1, then Ro is decreasing in 'Y12 and increasing Y21. 'Y' 'Y , 21 ' This We emphasize that the This result is in concordance concordance with biological biological intuition. intuition. We approach nd the value of approach employed employed in this paper paper allowed allowed us to fi find of Ro and and in partic particpersistence directly from ular the the threshold criterion criterion Ro R0 � --> 11 for for metapopulation metapopulation persistence from the biological biological interpretation interpretation of of the parameters. parameters. To arrive at the same same result result using the classical 1 985) one would would classical approach approach based based on differential differential equations equations (Hanski, (Hanski, 1985) have have to calculate calculate all the eigenvalues eigenvalues of of a matrix matrix and and determine determine the largest largest of of them. them. 1 985) is only two-dimensional The The model model treated treated by Hanski Hanski ((1985) two-dimensional and and this can can be readily done but but for for higher higher dimensional dimensional systems the task is extremely extremely tedious and and I JLI , if there sometimes if f3 2 1JL 2 = sometimes even impossible. impossible. We We point out that if/~2//.L2 -'- f3 ]31//.L1, that is, if there , is no difference between the two two size classes, difference in colonization colonization capacity capacity between classes, then then the condition f31JL > Levins threshold criterion criterion Ro R0 > > 11 reduces reduces to the usual condition/3//z > 11 for for the Levins model. £ 1 £ 2 )' The We = ((El,/~2). We now now proceed proceed to look at nontrivial nontrivial steady steady states states £ E --The ei ei' f; ) has the form genvector genvector corresponding corresponding to the the eigenvalue eigenvalue R( R(E)
be= (0)"
(6.36) (6.36)
1II1 88
Mats Mats Gyllenberg Gyllenberget et 01. al.
The The interpretation interpretation of of (6.36) (6.36) is simply simply that that all populations populations in newly newly colonized colonized now a straightforward I ' It is now patches patches belong belong to size class X Xl. straightforward task to calculate calculate the the (6.2 1 ) . We emphasize that to steady ) from steady metapopulation metapopulation state state m rh = = (ml (rhl,' m rh2) from (6.21). We emphasize that 2 apply this formula the next formula one one need need not not evaluate evaluate the next state operator operator TE T~ since since only its can be integral integral from from ° 0 to 00 ~ is relevant relevant and and this this can be calculated calculated directly. The The result is
c,(0)
/'/7/2
C2
C2
(6.37) (6.37)
T12
where where + J-L I' Cl = Y21 'Y21 + -~- J-L2 /'L2 + -'~ Y12 ')/12 ++ E E1I -t/'/'1, CI =
I J-L 2 ' C2 = = ((')/12 "[- E E1)/-z2 "[- Y21J-LI ')/21]J'l + "~- J-L ~LL1/./'2. c2 Y12 + I ) J-L2 + Substituting Substituting (6.37) (6.37) into (6.24) (6.24) and and (6.25) (6.25) w wee obtain obtain together together with the condition condition R(E) , E2 , R(/~) = = 11,, with R(E) R(/~) given given by (6.35), (6.35), three three equations equations in three three unknown unknown (EI (/~1,/~2, 99 1 ) found found that that for certain parameter 1 985; see also Hastings, and and a). Hanski ((1985; Hastings, 11991) for certain parameter values values there are are two two nontrivial nontrivial steady states. This This simple structured structured model model thus ' s model. predicts predicts a fundamentally fundamentally different different qualitative behavior behavior than than Levins Levins's model. 3. 3. A Model Model with with Continuous Continuous Local Local State State In this section described in terms terms of of section we shall analyze a model model originally described differential Local population 1 992). Local differential equations equations by Gyllenberg Gyllenberg and and Hanski Hanski ((1992). population size is considered modeled explicitly. considered as as a continuous continuous variable variable and and dispersion dispersion is modeled In this model, model, there there are are two two basic basic local local entities, a local local population population and and a disperser. population is denoted denoted by xX and disperser. The The size of of a local population and it has the range range [0, 00). ~). The The state of of a disperser disperser is denoted denoted by the the symbol d. d. The The local state space space is d } X [0, 00). populations have have an thus thus n 1~ = = {{d} ~). We We assume assume that that local local populations an intrinsic den denbirths and A local sity-dependent growth growth rate rate g(x) g(x) which is due due to local births and deaths. deaths. A population produces dispersers rate y(x). population with state X x E E [0, 00) ~) produces dispersers (emigrants) (emigrants) at a rate ~x). Dispersers Dispersers enter enter a patch patch at a rate rate a. a. If If this patch patch happens happens to be empty empty then then the disperser disperser dies dies immediately. If, on on the other other hand, it is occupied occupied the the disperser disperser immigrates into popUlation into the existing local population. population. The The net net growth growth of of a local population is therefore therefore modeled modeled by the following following ordinary ordinary differential differential equation: equation: dx dx dt = = g(x) g(x) - y(x) y(x) + + aE aE1I (t). (t). dt
(6.38) (6.38)
Here patch at XE, (t, x; Here E E1I (t) denotes denotes the the number number of of dispersers dispersers per per patch at time time t. Let Let Xe, x; ss)) be the solution solution of of (6.38) (6.38) at time t + given the value value x x E E [0, 00) oo) at time tt.. Dis Dis+ s given persers rate v) or persers disappear disappear either either because because they die (at a rate or because because they enter enter a distributed with papa patch. The The lifetime of of a disperser disperser is therefore therefore exponentially exponentially distributed rameter rameter a a + + v. Local populations populations go extinct extinct as a result result of of local "disasters," "disasters," which which we we assume assume to occur occur at the density-dependent density-dependent rate rate/x(x). follows that that J-L (x). It follows X;. sS)) = UE UE,! ((t,t, x,
{
)) d~') d'T) SXE,U,X:S) (XE, (t, f exp( e x p ( - Ii! f~ J-L tx(XE; (t, x x;; 'T ~')) 6xE,, ....., )S) U "d [ eexp x p(( - ((aa + + v v)s) 6d -
-
if if x x E E [0, [0, 00), oo), if if x X= = d. d.
(6.39) (6.39)
55
Structured StructuredMetopopulotion MetapopulationModels Models
1!!9 19
Here denotes the point mass Here 8x ~x denotes mass concentrated concentrated at x. The model includes two forms local The model includes two forms of of reproduction reproduction by the local entities: entities: local populations populations produce produce dispersers dispersers (emigration) (emigration) and and dispersers dispersers produce produce new new local local populations populations (colonization). (colonization). We We model model colonization colonization in the the spirit of of the the Levins Levins model patch model by assuming assuming that that the the rate rate at which which a disperser disperser colonizes colonizes an empty empty patch is proportional (3) to the number proportional (with constant constant/3) number of of empty patches. We We emphasize emphasize that arriving at an that in this this model model colonization colonization is not not the result of of one one disperser disperser arriving an empty patch patch (as mentioned mentioned above above such such dispersers dispersers are are assumed assumed to suffer suffer sudden sudden death) colonize empty patches patches for death) but but dispersers dispersers colonize for instance instance by producing producing offspring offspring that can disperser itself itself does not not enter the can initiate new new local populations. populations. The The disperser enter the colonized may thus colonized patch patch but but continues continues its life as a disperser. disperser. One One disperser disperser may thus very well colonize true colo colonize several patches during during its lifetime. We We realize that that the true colonization real systems, but have nization mechanism mechanism may may be very very different different in many real but we we have chosen patches which which do nature. A chosen this this model model since since it allows for for empty empty patches do occur occur in in nature. A not model very similar similar to ours ours with a more more realistic colonization colonization mechanism but but not analyzed by V Val 1 995). allowing for for empty patches patches has been been analyzed a l eett al. ((1995). Denoting popu Denoting the fraction fraction of of empty patches patches by by E E22 and and assuming assuming that the population of colonized patch of a newly newly colonized patch has size ° 0 we can now now write down down the colo colonization nization kernel kernel as follows: follows:
{
.
A E (t, x)([O, Ae(t, x)([O, s) s) X x .)) ftJ y(X El (t, x; u» exp( - fg /L (XE1 (t, x; r» ~ [0, co ~),) , du 8 = ~f~T(XEI(t,x; or)) exp(--fgtx(Xel(t,x; r)) dr) dr) do" t~d if x E uE2( u)du 80 if i f xx = = d. [ f ftJ ~ / 3{3 eexp( x p (- (a (a + v» v))trEz(~r)&r 60
=
(6.40) (6.40)
We E(t, x)([0, x)([O, s) X We observe observe that that the range range of of A Ae(t, • .9)) is the two-dimensional two-dimensional subspace subspace Mb spanned 80 and ects the fact that new new local entities spanned by 60 and 8d, 6a, which which refl reflects fact that entities can be in either of of two two local states: states: d d in case case of of a disperser disperser and and 0 in case case of of a local local popu popueither lation. W £ (00) we need only to look lation. It follows follows that that to find the eigenvalues eigenvalues of of We(oo) we need look at its restriction rst component dispersers restriction to Mb. The The fi first component of of an element element in Mb refers refers to dispersers and and the second second to local populations. populations. £ I' £ W£ (00) to Mb Let £ E = = ((/)l, E'2) be a constant constant environment. environment. The The restriction restriction of of W~(~) 2 ) be can be represented can represented as a 2 X • 2 matrix
(
0 %(/)~)) /3~e2/(a + v) 0 '
(6.41) (6.4 1)
where where
J
F "€, d = ~ (£ E1) = ~ y(x) V(x) !/I qJ£1 ~, (x) dx, dx, !/I (x) = £ I (X) qJ~l =
11
_
exp
g(x) E I exp g ( x ) -- y(x) 3,(x) + + a ~E1
(_ Jo(x -
/L(O
_
( r - Y-~i + aE c~E'lI g(g) y(O +
dg
)
. "
The 1 ) have have clear important biological biological interinter The elements elements of of the matrix matrix (6.4 (6.41) clear and and important produced by pretations. ( £ I ) is the pretations. "€, %(/)1) the expected expected number number of of dispersers dispersers produced by aa local pop population ulation in a newly colonized colonized patch patch during during its lifetime. The The element element in the lower lower
1120 20
Mots Gyllenberg et Mots Gyllenberg et 01. al.
left comer comer is the the expected expected number number of of patches patches colonized colonized by a disperser. disperser. The The matrix root of (6.41) two eigenvalues: eigenvalues: + + and and - the square square root of the product product of of the nonzero nonzero (6.4 1 ) has two elements E) is thus elements of of the matrix. The The spectral radius radius R( R(E) thus not a dominant dominant eigenvalue eigenvalue and generation and there will be no convergence convergence toward toward a stable stable distribution distribution at the the generation level. This is exactly as it should be: local populations produce dispersers populations produce dispersers and and vice versa, versa, and and thus if the initial metapopulation metapopulation consists consists entirely of of dispersers dispersers (or of of local local populations), populations), every second second generation generation will consist entirely entirely of of dis dispersers persers and and every every second second of of local local populations. populations. However, However, since the the lifetime of of local local entities entities is distributed, distributed, the the metapopulation metapopulation will converge converge toward toward a stable stable distribution in distribution in real time. The form The condition (6.20) for for aa nontrivial steady steady state state now now takes takes the form L'2c~(E'I)
~--
1.
ce+t,
(6.42) (6.42)
Recalling interpretation of 1 ), we infer Recalling the the interpretation of the elements elements of of the matrix (6.4 (6.41), infer that that Eq. (6.42) (6.42) states states that that at steady steady state state every local local population population exactly exactly replaces replaces itself. 2» be an eigenvector Let (b( l), b( R(E), that (b (~), b (2)) eigenvector corresponding corresponding to R(E), that is, to the the positive positive 1 ). Then eigenvalue eigenvalue of of matrix (6.4 (6.41). Then
b(2) = / / 3 / 7f3E 2 / 2( /( ~ _� + + v) 1,) b( b(l) 2) = b( l). 'g (E ) ~(E1 I )
(6.43) (6.43)
2» denote Let Again the first com Let m rh = = (m(l), (rh ~), m( /~/(2)) denote the the steady metapopulation metapopulation state. Again the first com1) ponent ponent refers refers to dispersers dispersers and and the second second to local local populations. populations. Applying Applying (6.2 (6.21) we fi nd find
11 b (l) l) = Ill (1) -- -~ b (l) m( centp a + v
(6.44) (6.44)
rh(Z)(dx) = q% (x)dx b (2).
(6.45) (6.45)
Using nitions of Using the the defi definitions of E E~l and and E Ez, which have have been been stated stated verbally above above and and 2 , which which which mathematically have the form form (6.22), (6.22), we obtain _ 11 E b (l) E1I = = -~ b~l) centp a + v
(6.46) (6.46)
/)2 = 1
(6.47) (6.47)
-
/(/~1)
b(2),
where
I ( , ) = f q, (x)dx is the the expected expected life-time of of a local local population. population. Equation Equation (6.46) (6.46) is an analog analog of of the well-known well-known relation relation in epidemiology: the prevalence prevalence of of a disease equals equals the incidence incidence rate rate times the average average duration duration of of the disease. disease.
StructuredMetapopulation MetapopulationModels Models 55 Structured
1121 21
equations in The system (6.42), (6.43), (6.46) and (6.47) iiss a system ooff four equations 2) one obtains l ) and b ((2) unknowns. Solving for for b ((1) four unknowns.
I
(l ) = JJ;2 , = f3 /3 f y(x)!/JE, "y(X)~t~1(x) (X) dxE dxE,11E2, bb(1)
(6.48) (6.48)
2) = bb ((2) E 22 - f3E fl/~ I1/~
(6.49) (6.49)
(6.49) into (6.47) one gets and inserting (6.49)
11 E I E 2 /(EI )· - / ~E22 = = f3 /3/~/~2/(/~).
(6.50) (6.50)
Eliminating/~2 (6.42) and (6.50) one finds the relation Eliminating E 2 from (6.42)
(
I f3 /3 __ Cf,(E ~(~)I ) -- E ~?it(~?l) I /(E I » c ~+ +vv a
)= =
11..
(6.5 ( 6 . 5 1 )1 )
Once/~ (6.51),/~2 obtained from (6.50) (6.50) and fi finally Once E I has been solved from (6.5 1 ), E 2 iiss obtained nally the equilibrium equilibrium size distribution distribution of of local populations populations from from (6.45). (6.45). The The virgin virgin environment environment is given by E E1I = = 0, 0, E E22 == 11 and and thus thus the the trivial steady state state corresponding corresponding to metapopulation metapopulation extinction extinction is stable stable as long long as steady f3 R = __ Cf, %(0) < 1. R62 = ce + -+- vp (0) < 1 . a
(6.52) (6.52)
11 __ %'(0) < leO), l(0), gae q+- pv Cf,'(0) <
(6.53) (6.53)
we see that nontrivial Since/~ = 0 satisfi satisfies (6.51) = 11 we that the the branch branch of of nontrivial Since EI = es (6.5 1 ) if if Ro = steady states given by (6.5 1 ) bifurcates bifurcates from from the trivial solution at Ro = (6.51) = 11.. The The bifurcation can be both both supercritical supercritical and and subcritical. By differentiating differentiating (6.5 (6.51) bifurcation 1) respect to the bifurcation bifurcation parameter parameter one can can find out which which case case implicitly with respect find out occurs. this, let us take/3f3 as bifurcation occurs. To To illustrate illustrate this, us take bifurcation parameter. parameter. Now, Now, if if
then supercritical (Fig. (Fig. 1), then the bifurcation bifurcation is supercritical 1 ), and and if if the the reverse reverse inequality inequality holds holds in in (6.53), then then the bifurcation The situation (6.53), bifurcation is subcritical subcritical (Fig. ( Fig. 2). The situation described described in Fig. 11 is qualitatively qualitatively identical identical to prediction of whereas the to the the prediction of the the Levins Levins model, model, whereas the pattern effect of consequence of of the the effect of migration migration upon upon local dynamics. dynamics. pattern in Fig. 2 is a consequence To phenomenon better, better, let To understand understand this this phenomenon let us us have have a closer closer look look at the the condition condition (6.53). The derivative %'(0) of how fast the derivative Cf, '(0) is a measure measure of how fast the number number of of dispersers dispersers (6.53). The produced by by aa local population popUlation increases increases as as the the number number of of dispersers dispersers increases increases produced from can be be from zero. zero. As As zero zero dispersers dispersers have have no no effect effect upon upon local local dynamics, dynamics, %'(0) Cf, ' (0) can interpreted as as a measure measure of of the the impact impact of of migration migration on on local local dynamics. dynamics. If If this this interpreted impact impact is small, small, the the model model gives gives qualitatively qualitatively the the same same prediction prediction as as the the Levins Levins model than the model (where (where the the impact impact is zero), zero), but but if if it is is sufficiently sufficiently large large (larger ( larger than the v) l(0), 1(0), then then there there are are multiple multiple nontrivial nontrivial steady states for for certain certain threshold (a (a + threshold + v) values values of of the the colonization colonization parameter/3. parameter f3. Hanski Hanski and and Gyllenberg Gyllenberg (1993) ( 1 993) considered considered an an extension extension of of the the above above model, model, in which which each each patch patch was was assumed assumed to to have have aa fixed fixed quality quality affecting affecting local local dynamics dynamics
1122 22
Mots Mats Gyllenberg Gyllenberget et 01. al.
as well as extinction and and colonization. colonization. This quality may be for for instance patch size. This model can be written in the general framework described here. The The basic local entity is either a disperser disperser or an occupied patch structured by patch quality and the size of population. The analysis of of a local population. of Hanski and Gyllenberg much richer structure than the one one described described ((1993) 1 993) showed that this model has a much here with the possibility of of bifurcations bifurcations shown in Fig. 3 among others.
6 II
Two-Species Metapopulation Metopopulotion Two-Species Models Sean Nee
Robert M. May May
Hassell Michael P. Hassell
I.I. INTRODUGION INTRODUCTION In this chapter chapter we primarily discuss two-species two-species metapopulation models al although, though, for for some topics, it is natural natural also to refer refer to results results for for single-species single-species models models and and models with more more than than two species, and and we shall do so. In Section Section II, we generalize ski, this volume) generalize the single-species single-species Levins model (Han (Hanski, volume) to include include the the three three simplest simplest ecological ecological relationships relationships between between two two species species coexisting coexisting as as metapopulations: competition, predation, and mutualism. In Section III we focus primarily on predator-prey predator-prey relationships relationships in spatially explicit metapopulation models. The The types of models discussed discussed in this this chapter have been studied over the years from from a large variety of perspectives and and interests, but in order order to present a thematically ed discussion, thematically unifi unified discussion, we we will will describe describe these these models models from from the the point point of of view of of the the consequences consequences of of changes changes in in the the amount amount of of suitable suitable habitat habitat on on the the abundances abundances of of the the species and, and, ultimately, ultimately, on on their their persistence. persistence. This This question question is is not not only only topical, topical, but but one one which which we we believe believe will will become become increasingly increasingly prominent prominent in in metapopulation metapopulation studies. We We will see see that, that, for for very very simple models models of of each each of of the the three three relationships, relationships, the the consequences consequences are are surprising surprising indeed. indeed. Devastation Devastation on on the the scale scale of of the the rain rain forests, forests, for for example, example, has has the the obvious obvious consequence consequence that that vast vast numbers numbers of of species species will will be be extinguished extinguished as as their their habitat habitat is is destroyed destroyed in in its its entirety. entirety. More More subtle subtle effects effects arise arise in in less less extreme extreme circumstances, circumstances,
Merapopulation MetapopulationBiology Biology
Copyright Copyright ©9 1997 1997by by Academic Academic Press, Press, Inc. Inc. All Allrights rights of of reproduction reproduction inin any anyform form reserved. reserved.
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Sean Sean Nee Nee et et 01. al.
such North such as the the management management of of the the old growth forests of of the the American American Pacifi Pacificc Northow of west or the ebb and fl flow of habitat types in Europe as a result of of changing policies on policy on agricultural agricultural subsidies. subsidies. The The U.S. U.S. government government has has recently recently abandoned abandoned its its policy of "protect one, abandon ten," whereby national parks and nature reserves remain of "protect one, abandon ten," whereby national parks and nature reserves remain as isolated islands 995). islands in a sea of of unlimited environmental devastation (Babbitt, 11995). The maintain diverse The new new goal goal of of landscape landscape management management is to to maintain diverse communities communities and and the habitat destruction the quality quality of of habitat habitat patches that still still remain. remain. In the the models models of of habitat destruction that we will study in this chapter, we deliberately assume that that patches patches that that remain remain are are of of the the same same quality quality as as before. Hence, we ignore ignore such such phenomena phenomena as as "edge "edge effects" inevitable consequences of destruction effects" and other inevitable consequences of destruction and fragmentation fragmentation which are biologically important (e.g., Robinson et 1 995; Wiens et al. al.,, 1995; Wiens this volume). We absence of We will will see see that that changes changes in in the the amount amount of of habitat habitat per per se, se, even in in the the absence of any other other effects, effects, can can have have surprising consequences. consequences. The abundance persistence of abundance and and persistence of many species species may be largely largely controlled controlled by a single limiting resource of suitable resource which is often often related related to the amount amount of suitable habitat terri habitat that that is is available. available. For For example, example, we can can imagine imagine the the abundance abundance of of aa territorial terri torial species species as as being determined by by the the availability availability of of suitable breeding territories, the abundance of of a predator determined determined by prey availability, or the abun abundance dance of of aa disease disease organism determined by by the fraction of of aa host population that is unvaccinated. interact with unvaccinated. Many other things, such as life-history factors, will interact resource resource availability to generate generate the actual abundance, abundance, but such factors can be treated as given, constants, while the availability of resource is altered of the limiting resource by changing the amount of of habitat. defined species, an important insight to emerge emerge from from meta metaFor such simply defi ned species, unable to persist, even in the presence population analysis is that a species may be unable presence of of suitable habitat, if local extinction rates are greater greater than than colonization rates (Hanski, this volume). This was fi rst realized first realized in epidemiology, which is the most well-developed metapopulation theory t h e o r y-a - - a host is a "patch" "patch" of of suitable habitat for a disease disease organism, infection is "colonization," host recovery or death is "ex "extinction." A cornerstone of Kermack and of epidemiology is the threshold theory of of Kermack McKendrick 1 927), which demonstrates of a minimum number of of McKendrick ((1927), demonstrates the existence of susceptible disease to susceptible individuals required for for aa disease to achieve achieve an an epidemic epidemic outbreak in a community. Furthermore, Furthermore, epidemiology has long known that it is not necessary necessary to to destroy destroy all all the the habitat habitat of of aa species in order order to to eradicate eradicate it (vaccination (vaccination pro programs grams are are wanton wanton acts of of environmental vandalism from from the point of of view of of a disease disease organism). Smallpox would still exist outside a containment containment laboratory in Atlanta if if it was necessary to achieve the impossibility of OO% vaccination of l100% Ross coverage to eradicate eradicate it. Contrary Contrary to the prevailing opinion of of the time, Ross ((1909) 1 909) demonstrated necessary to demonstrated that it was was not not necessary to eliminate eliminate mosquitoes mosquitoes entirely in in order order to eradicate malaria but that, that, instead, instead, there was a threshold ratio of of mosquito density to human human density below which malaria malaria could not persist persist (Heesterbeek, (Heesterbeek, 11992). 992). In a pathbreaking pathbreaking metapopulation metapopulation analysis of of the Northern Northern spotted spotted owl, Lande 1 988a) deduced planned level of Lande ((1988a) deduced that the the planned of destruction of of breeding breeding territo-
66
Two-Species Metopopulotion Models Models Two-SpeciesMetapopulation
1125 25
ries, the patches in his model, would entirely eradicate eradicate the owl, although the advocates of the plan thought that enough territories would be left to maintain a viable population_ population. The amount amount of habitat habitat destruction destruction that that can result in the eradication of of a species can be surprisingly small, and the estimation of "eradication thresholds" is clearly a valuable goaL goal. For a single-species metapopulation, the Levins model suggests a simple estimate estimate of the minimum number of of patches required for for the metapopulation 995; Hanski et 1 996b): metapopulation to persist (Kareiva and Wennergren, Wennergren, 11995; et ai. al.,, 1996b): it is simply the number number of patches which which are observed to be unoccupied when the system is at a dynamical equilibrium equilibrium between colonization and extinction. Inspired by epidemiological arguments 1 99 1 ), Lawton et arguments (Anderson (Anderson and May, 1991), et ai. al. ((1994) 1 994) presented a simple and 1 994) ob and general general derivation of of this result. Nee ((1994) observed that that the result could be biologically generalized generalized further: a simple estimate of of the eradication eradication threshold for for a population, or metapopulation, is the unused amount of patches of suitable habitat of its limiting resource at equilibrium ((patches habitat being the limiting resource for a metapopulation). estimate provides a starting point for for the development of a deeper deeper Such an estimate understanding by crystallizing in its derivation some important important assumptions which can be relaxed for for further analysis. For example, an important important assumption of the metapopulation, metapopulation, is that the individual members of the population, or patches of affect each other only indirectly, through the consumption of the limiting resource resource (see Section II.B ILB for for a discussion of of this in the particular particular context of of predator predatorrelationships). Biologically, this assumption can be violated in many ways, prey relationships). with important implications implications for the estimate estimate of the eradication threshold: threshold: Lande ((1987), 1 987), for example, l 996b) studies the example, studies the Allee effect and Hanski ((1996b) "rescue "rescue effect." effect." In Section ILB, II.B, we acknowledge the implications implications of the possible ratio dependence dependence of of trophic trophic interactions interactions on the eradication threshold threshold estimate. The simple estimate estimate is also based on an entirely deterministic deterministic model, and the 1 988a) and Hanski ((1996b). 1 996b). implications of stochasticity are discussed by Lande ((1988a) 1 994) discuss how several Frankly pilfering epidemiological work, Lawton et et at. al. ((1994) other factors which may be of of real-life importance render render the simple estimate either an over- or an underestimate. underestimate. It is possible that the simple estimate, treated treated as a rough empirical estimate, estimate, may be very useful in situations where our knowledge is grossly insufficient insufficient for an estimate of all the life history parameters parameters and spatial complexities complexities required for a detailed analysis (Anderson and May, 1991; 1 99 1 ; Lawton et 1 994). However, et ai. al.,, 1994). then there there arises the general general question of of how actually to determine determine the unused amount of limiting resource resource or suitable habitat, habitat, and an innovative approach approach to 1 996). this question is presented presented by Doncaster et et ai. al. ((1996). The The assumption that the metapopulation metapopulation is actually at equilibrium equilibrium when we come to estimate the eradication threshold is clearly clearly of great importance (Lande, 11988a; 988a; Lawton et at., 1994; 1 994; Hanski, 11996b). 996b). However, it is of et al., of historic interest to observe that the fi rst to suggest in the epidemiological context that the unused first
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Sean Nee Nee et Sean et 01. al.
amount amount of of limiting resource can can be used to estimate estimate the the eradication threshold 1 970), who was considering appears appears to have been been Smith ((1970), considering the nonequilibrium nonequilibrium situation of of episodic episodic epidemics. epidemics. He He observed that if, after an an epidemic of of yellow fever, for for example, example, has has died died out one one then then observes observes that, say, 20% of of the the population is still susceptible, susceptible, then this suggests that the disease could not establish itself in a population that that consisted consisted of of only 20% susceptible susceptible individuals. individuals. The The subsequent subsequent development of of the subject subject begins begins from equilibrium arguments arguments appropriate appropriate for for 99 1 ). endemic diseases diseases (Anderson (Anderson and May, 11991). As consequences As we we will will see, see, habitat habitat destruction destruction may may have have qualitatively qualitatively new new consequences when in an when we we come come to to consider consider species species which which are are enmeshed enmeshed in an intricate intricate web web of of ecological relationships, one itself to caricaturing abun ecological relationships, one which does not not lend itself caricaturing the abundance We will dance of of the species species as as being determined determined by by aa single single limiting resource. We will see see that that habitat habitat destruction destruction may may actually actually be be of of benefit to to inferior inferior competitors competitors and and that small reductions in the amount of of suitable habitat may catastrophically con consign mutualist associations just associations to oblivion. The analysis of of two-species models models is just the rst step the fi first step in developing an understanding understanding of of the the consequences consequences of of habitat change for for the tangled tangled bank bank of of Nature. change The (Han The models models of of Section Section II II retain the the Levins formalism formalism of of patch patch models (Hanski, this volume), so we do not explicitly model local dynamics dynamics (see Gyllenberg Gyllenberg et al. al.,, this volume), nor do we have have an explicit spatial structure structure (see Hanski, this volume). Levins volume). Local Local populations populations do do not not have have stable stable equilibria but, but, as as in in the Levins model, model, they they go go extinct extinct at at constant constant rates. rates. Furthermore, Furthermore, asynchronous asynchronous patch patch dy dynamics namics are are "built in" to the the structure of of the models. In all the models models studied studied in this section, for those those values values of of the colonization and and extinction parameters parameters for which a nontrivial metapopulation equilibrium exists, this equilibrium is at least locally stable. Therefore, Therefore, these more complex, two-species two-species metapopulations, like their single-species single-species counterparts, counterparts, can persist even though local extinctions are are inevitable. Because of of the lack of of explicit local dynamics and and the built-in asyn asynchrony, these asking questions about possible inter these models models are are not suitable for for asking possible interactions between local dynamics dynamics and metapopulation dynamics. In Section III we turn tum to models models which have have both explicit spatial structure and explicit explicit local patch dynamics. Questions Questions of of possible possible interactions between may now We fi rst ask: does does local and metapopulation dynamics dynamics may now come come to the fore. We first it make popUlation equilibria make any difference difference to the local stability of of population equilibria if they are connected migration? As we connected to other other populations, at at the same equilibria, equilibria, by migration? we will will see, the answer is no for single-species single-species the answer no for a broad broad class class of of models, and this this is true for as well as for multispecies metapopulation for multispecies metapopulation models. We We then then go on to discuss the the prey systems with unstable rich spatial spatial phenomena phenomena exhibited exhibited by predatorpredator-prey unstable local dynamics and will see that, in this case, the effects dynamics and effects of of habitat habitat destruction destruction on metapopulation persistence persistence depend depend on the dynamic dynamic geometry geometry of of the system. The The models of Section II allow us to investigate amount of of models of investigate changes changes only in the the total total amount habitat. investigate the habitat. The The spatially spatially explicit explicit models models of of Section Section III allow allow us to investigate the consequences consequences of of different different patterns patterns of of habitat habitat destruction. destruction.
66 Two-Species Two-SpeciesMetapopulation MetapopulationModels Models
1127' 27
II. TWO-SPECIES TWO-SPECIESPATCH PATCHMODElS MODELS II. The The two-species two-species patch patch models models we we will will study in in this section section all have the the same same structure. structure. At At any any particular particular time, patches patches of of suitable suitable habitat habitat are are either either empty empty or or occupied occupied by one one or the other, other, or both, of the species. The The rates at at which one species colonizes uenced by the colonizes other patches, and goes locally extinct, extinct, are infl influenced presence or or absence of the other other species. species. We alter alter the the total number of of patches in the the network network and and observe observe the the consequences consequences for for equilibrium equilibrium species species abundances. abundances.
A. Competition Competition A. Consider two competing species utilizing the same habitat (resource) patches. One species is competitively superior to the other, and we make the simplifying assumption assumption that that the superior superior competitor, species A, completely excludes the in inferior competitor, species B, from patches which which it occupies. occupies. The The inferior inferior com competitor can nevertheless nevertheless coexist with the superior superior species at the landscape landscape level because because it it has has either either aa higher higher colonization colonization rate r a t e- - iitt is is aa "fugitive" "fugitive" or or "weed" "weed"-or a lower lower local extinction rate, or both. both. Denote Denote the proportion proportion of of empty empty patches patches in a landscape landscape by x, patches patches occupied by species A by y, and patches patches occupied occupied species B by z. z. Species Species A A and and B have have colonization colonization rate rate parameters CA and and CB Cs parameters CA by species respectively, and generalization of of the and extinction extinction parameters parameters eA eA and and eB' es. A A simple generalization Levins model model incorporating incorporating our our assumptions assumptions is (Nee and and May, 11992) 992) dx dx dt dt
--- __ C A X y -~ e A y
- - CBXZ +
dy
cAy(x + Z) z) dt = = C AY(X + -- eeAY' AY,
dt
eSZ,
(1) (1)
dz m
dt -
cBzx
eBz
CAZy.
Because the the inferior inferior competitor competitor iiss "invisible" "invisible" to to the the superior superior species, species, the the Because dynamics of of the the superior superior competitor competitor are are described described by by the the standard standard Levins Levins model model dynamics 1 . As As patches patches are de (Hanski, this this volume). volume). In In the the pristine pristine world, world, xx + (Hanski, are de+ yY ++ zz == 1. + yy + = h, where where hh is the the fraction fraction of of suitable suitable patches patches that that remain remain stroyed, xx + stroyed, + z = in in the the landscape. landscape. It It is is easy easy to to mentally mentally switch switch between between numbers numbers and and proportions, proportions, and one one can can describe describe the the conclusions conclusions of of this, this, and and subsequent subsequent models, models, using using the the and is the the number number of of patches patches in in the the pristine pristine words interchangeably. interchangeably. If, If, for for example, example, No is words state, then then xNo xNo is is the the number number of of patches patches which which are are empty, empty, although although not not destroyed, destroyed, state, yNo is is the the number number of of patches patches occupied occupied by by species species A, A, which which are, are, necessarily, necessarily, not not yNo destroyed, and h, the the system system is is actually actually two-dimentwo-dimen and so so on. on. Because Because xx ++ yy ++ zz == h, destroyed, sional, as as are are the the others others we we will will study study below, below, but but we we write write down down all all three three equations equations sional, for for completeness. completeness.
1128 28
Sean Nee Sean Nee et et al. al.
The nontrivial equilibrium 1) equilibrium solution of of this this system, found found by setting Eqs. Eqs. ((1) equal to zero and solving, is 1 X*
= -- (hc A CB
y* = h
e A -k-
e~),
eA
(2)
CA ' 2*
--
z* =
ee AA((CCA A + nt- C c BB)) - - e e--9-B A B - hh cCA cC BB ' CAC CC BB CACBB -
-
-
where where the the asterisk asterisk denotes denotes the the proportion of of patches patches at equilibrium equilibrium (there is, of of z* = course, course, always the the trivial trivial solution solution x x** = = h, y y** = = z* - 0). Feasible equilibria are Feasible equilibria persists in the landscape globally stable. If If CCA/eA > 11,, the superior superior competitor competitor persists landscape AleA > and a necessary condition condition for for the inferior inferior competitor competitor to exist is CB
CA
eB
eA
-->--.
(3)
To unoccupied by either To understand understand this expression, expression, imagine imagine a pristine pristine world unoccupied either species species until a local population population of of the the inferior inferior competitor competitor is established established on one of of the patches. The left-hand left-hand side of of expression (3) is, simply, the average average number number of extinc of new new patches patches that that this patch patch would would colonize colonize before before experiencing experiencing local extinction; in the language inferior competitor. language of of epidemiology, it is the "Ro" of of the inferior competitor. The right-hand Ro of Hence, a necessary right-hand side is the R0 of the superior superior competitor. competitor. Hence, necessary condition condition for for the inferior competitor competitor to exist exist is that it have a higher RD R0.. It is worth emphasizing emphasizing that inferior inferior competitors competitors do not require a higher higher colonization colonization rate to persist 1 992). persist and this is discussed discussed further further in Nee and May ((1992). The The effect effect of of habitat habitat destruction, destruction, modeled modeled by lowering lowering h, on the the equilibrium equilibrium abundances Even though abundances of of the two species is illustrated illustrated in Fig. 11.. Even though the inferior inferior competitor patches, nevertheless nevertheless competitor can persist only by virtue virtue of of colonizing colonizing empty patches, the effect effect of of habitat habitat destruction destruction is to increase increase the abundance abundance of of the inferior inferior com comresult of of the decrease decrease in the abundance abundance of of the superior superior competitor competitor petitor: this is a result across com across the landscape. landscape. Hence, Hence, in this model, model, habitat habitat destruction destruction results results in the competitive petitive release release of of inferior inferior competitors. competitors. Once Once the superior superior competitor competitor has disap disappeared peared (at the eradication eradication threshold threshold level of of destruction, destruction, h = = eeA/CA), the abundance abundance A icA ), the of of the inferior inferior competitor then starts to decline decline with increasing destruction. Hence, Hence, simply changing changing the the amount amount of of suitable suitable habitat, habitat, without without any any changes changes in the the quality quality of of the the remaining remaining habitat, habitat, is expected expected to increase increase the the regional abundance abundance of of "weedy" species, to the extent "weedy" extent that that inferior inferior competitors competitors persist persist by virtue virtue of of a higher colonization rate. The same qualitative qualitative result is observed in a spatially explicit explicit analog analog of of this model, model, in which which colonization colonization is purely local (Dytham, 11995), 995), and ingredients" and in a model model which which incorporates incorporates the many many other other "real "real life life ingredients"
66
Two'Species Two-SpeciesMetopopulotion MetapopulationModels Models
1129 29
1 -
:fl 0.75 tl) 0.75 t£: u u � C\I t~ 0. '~
0.5-
0
~....empty
~. 0.25 superior 0
o0
0.25 0.25
i':"r: .....................".... 0.5 0.5
I 0.75 0.75
""~ 1
increasing l -h) increasing habitat habitat destruction destruction ((l-h)
The I ), with and eeB/CB The equilibria equilibria of of competition competition model model ((1), with eA eA = = e eB, e A/CA = = 0.5, 0.5, and = B , eA/cA B/cB = 0.25. It equilibria as levels of of habitat destruction; 0.25. It is is natural natural to to view view these these equilibria as functions functions of of increasing increasing levels habitat destruction; hence hence we we plot plot them them as as functions functions of of 1I - h. h.
FIGURE FIGURI: 1|
' s ((1994a) which 1 994a) "incidence function" approach meta which are are contained contained in Hanski Hanski's "incidence function" approach to metapopulation models (Moilanen 995). population (Moilanen and Hanski, 11995). This model, model, as all the others others we will describe, describe, has multiple interpretations. interpretations. pools For For example, example, interested interested in the co-occurence co-occurence of of Daphnia Daphnia water water fleas in rock rock pools on islands, 1 983) studied islands, Hanski Hanski and and Ranta Ranta ((1983) studied a three-species three-species competition competition model model in which, in our but instead instead describes describes our terms, h does does not reflect habitat habitat destruction destruction but the different islands of For the Daphnia Daphnia different numbers numbers of of rock rock pools on islands of different different sizes. For 1 99 1 ). system, see also Bengtsson Bengtsson ((1991). important to keep keep in mind mind that we are are making making statements statements about equilibIt is important about equilib rium rium abundances. abundances. Destroying Destroying habitat habitat does does not instantaneously instantaneously exert exert its effects, effects, just just as a population population whose whose per-capita per-capita growth growth rate is reduced reduced below below one one does does not not instantaneously instantaneously go go extinct, extinct, but but dwindles dwindles to oblivion oblivion at a rate determined determined by its life-history parameters. cient parameters. So, for example, example, a level of of habitat habitat destruction destruction suffi sufficient to ultimately eliminate eliminate both both competitors competitors may seem to have have no no visible effect effect at all over the observational observational time span span of of individual individual observers observers and and scores scores of of contem contem1 996b). This porary species may actually be the "living dead" ski et al. dead" (Han (Hanski al.,, 1996b). This fact has recently been given the vivid vivid tag "the extinction extinction debt," debt," reflecting reflecting a growing growing has interest of systems interest in "transient" "transient" dynamics, i.e., the behavior behavior and and appearance appearance of from their their equilibria equilibria (Tilman (Tilman et al., 11994). Such transient transient phenomena phenomena can 994), Such away from be very long-lasting, and bear long-lasting, especially in spatially structured structured systems, and bear no re resemblance Higgins, 11994). 994). semblance to the ultimate ultimate equilibrium equilibrium state (e.g" (e.g., Hastings Hastings and and Higgins, However, important to emphasize that what However, even even in simple systems systems it is important emphasize that what we obob-
1130 30
Seon Nee et Sean Nee et 01. al.
serve of the serve today, or even this decade, decade, may may not not give a true picture of the next next century, even if there are no further further habitat alterations (Heywood et 1 994; Hanski et et al., al., 1994; et metapop al., 11996b). 996b). Furthermore, studies of al., of coral reefs, an important class of of metapopulations, have ciently frequently to have indicated that major major disturbances disturbances occur occur suffi sufficiently prevent prevent the systems ever attaining attaining their equilibrium compositions (Tanner (Tanner et et al. al.,, 11994). 994). The The model model we have have just just described is the the simplest simplest extension of of the the Levins Levins model model to incorporate incorporate competitive relationships. relationships. Related models models of of varying de degrees of of complexity have been studied over over the years and we will briefly describe a few of of these. The purpose purpose of of this discussion discussion is to illustrate the range range of of questions questions that can be addressed, and principles that and the range of of principles that can be illustrated, by such models. Their Their ultimate origin lies in the easy impression impression that that one can form that in terrestrial plant and marine communities there do not seem to be enough niche niche dimensions Gause ' s dimensions to allow the large large number number of of species we observe observe to satisfy Gause's exclusion exclusion principle. This impression may or may not be mistaken (e.g., Knowlton Knowlton and Jackson, 994; Silvertown, 11987) 987) but, in any case, it draws Jackson, 11994; draws attention attention to other other important factors potentially mediating coexistence. Hutchinson 1 95 1 ), without Hutchinson ((1951), an explicit quantitative model, drew attention attention to the fact that species can coexist, even while using the same resource, resource, if they differ in their their competitive and and mi mi1 994) has gration abilities. Recently, Tilman ((1994) has emphasised this point point in the the context of 1 95 1 ), with plants in mind, analyzed a of terrestrial plant communities. Skellam ((1951), quantitative model of of competing species coexisting as metapopulations and gener generHom alized the analysis to a landscape with patches of of different quality, as did Horn ' s paper and MacArthur 1 972). (Skellam MacArthur ((1972). (Skellam's paper also analyzed what are now called "edge ' s analysis was motivated effects" and sink" systems ! ) Skellam and "source"source-sink" systems!) Skellam's motivated by what appears to have been been a topical question at the time, namely why some plant 1 974) used a two species seem to thrive in "unpromising "unpromising situations." Slatkin ((1974) twospecies metapopulation model to examine whether whether there there is a metapopulation metapopulation an analog of Lotka - Volterra model of the "priority effect" that one observes in the simple Lotka-Volterra of rst" on a landscape of two competing species, whereby the species to "arrive fi first" 995). However, However, excludes excludes the other. There There is not, in his model (but see Hanski, 11995). the analysis of of Hom Horn and and MacArthur, MacArthur, which allowed for for different different types of of habitat of alternative, stable communities depending patches, did discover discover the possibility of 99 1 ). Slatkin also enquired on initial abundances abundances (see also Case, 11991). enquired into the effects of of changing changing the extinction extinction rate parameters parameters on the abundances abundances of of the competitors, inspired by ideas of of predator-mediated predator-mediated coexistence, and found found that, indeed, ele elevating vating the patch extinction rate could allow allow coexistence which which was not not previously possible. Hastings 1 980) asked the question in a multispecies Hastings ((1980) the same same question multispecies version of of ' s model, inspired by speculation Slatkin main Slatkin's speculation about about the the role of of disturbance disturbance in maintaining coral reef reef diversity. He obtained obtained the interesting result that, as the local extinction rate rate is changed, the number number of of species that can coexist does not change change in a simple fashion, fashion, but can rise and fall several times. [The otherwise excellent 1 994) contains one error: model ((1) 1 ) is not based review by Hastings and Harrison Harrison ((1994) based 1 980).] Most recently, multispecies models have been studied from on Hastings ((1980).]
66
Two-Species Two-SpeciesMetapopulation MetapopulafionModels Models
1131 31
the point ooff view of the evolution ooff virulence and of species diversity (Nowak and May, 11994; al., 11994). and 994; Tilman et ai., 994).
B. Predation B. circumstances, because of of either overexploitation overexploitation or unstable unstable local In some circumstances, predators may drive local populations populations of their dynamics (Section III), specialist predators prey extinct and, consequently, themselves as well. Nevertheless, Nevertheless, the two species can persist as metapopulations metapopulations over a landscape landscape as long as local dynamics are not of such metapopulations, metapopulations, under undersynchronous. We now examine a simple model of predator-prey relationship relationship has a broad biological meaning in instanding that the predator-prey relationship, plantplant-herbivore, host-paracluding, in addition to the conventional relationship, herbivore, host-para host-parasitoid relationships. relationships. In the absence of of the predator, patches of site, and host-parasitoid ev, which may be very small. prey ("victims") suffer local extinction at a rate ev, Predators Predators can colonize colonize only patches patches containing containing prey, prey, and and patches patches containing containing both predators and prey go extinct extinct at a rate eep. Colonization parameters of of prey-only predators p• Colonization patches and patches patches containing containing both predators predators and prey are CCv and cCp, patches v and p , respectively. Denoting the the proportion proportion of of empty empty patches patches by x, x, prey-only patches patches by y, y, and Denoting patches patches containing containing both both predators predators and prey by z, z, a simple simple model of this system model of is (May, 11994): 994): dx dt - ev y + epZ
CvXy.
dt = CvXy - CpyZ
-
-
evy,
(4) (4)
dz dt
=
CpyZ -
epZ.
Notice that assume that prey in predator/prey not z) do do not Notice that we assume that prey predator/prey patches patches (fraction (fraction z) colonize empty patches. the analysis colonize empty patches. Relaxing Relaxing this this assumption assumption would would complicate complicate the analysis without introducing any interesting interesting new without introducing new features. features. This system has has the the following This following nontrivial nontrivial equilibrium equilibrium solution solution * x** = = h - - y* y --z *z,* ,
x
Y* = ep,
(5) (5)
Cp
Cv C v "3t- Cp
(h_ep_e~). Cp
Feasible Feasible equilibria equilibria are are globally globally stable. stable. The The effect effect of of habitat habitat destruction destruction on on the abundances abundances of of predators predators and and prey prey is illustrated illustrated in in Fig. Fig. 2. The The first thing thing we we the notice is that that habitat habitat destruction destruction has has no no effect effect on on the the equilibrium equilibrium number number of of the the notice prey-only prey-only patches patches until until it reaches reaches such a level level as as to to extinguish extinguish the the predators predators
1 32 132
Sean Nee Nee etet el. a!. Seen 1 -
to 00.75 .75t..c u O
�
..., ro 0. q.... . - .. 0o ff) to t-c o O 4''::; ~ u ro t~
00.5 .5-
•
"
.
�.�:.:>:
� "' '' '' ...~. .................... .~,~ • .................... 1(.................. ••••••••••••••••••~. ,.
� 00.25 .25-
q t.. .-
••
�mpty '-,,empty •
prey prey
.
•
. ••
""-,,,
•
•
•
•
".........
y
..«
_______
........ .
. "-......... . .. .
.
4•
•
. .. . . . .. . .. .. .
"..
•
• ~'"~ ••
•
,4
••
~%. • • •
O ;-----�I.-��-r ------� ----� I I "|• 0.25 0o 0.25 0.5 0.75 1 0.75 0.5 ""~
•
increasing habitat habitat destruction ( l -h) increasing destruction (1-h) fIGURE 22 The equilibria of FIGURE of predation model (4). (4), shown as functions of of increasing levels of of habitat combinations ep/Cp epicp = = 0.35, ev/C,, ejc, = = 0.2, and cv/Cp cjcp = = 0.5. destruction, 1I -- h, for parameter combinations
entirely. predator abundance abundance declines increasing habitat habitat destruc entirely. Second, Second, the the predator declines with with increasing destruction in in spite spite of of the the fact fact that is no no change change in in the the prey prey abundance. tion that there there is abundance. The equilibrium abundance The result result that that habitat habitat destruction destruction does does not not affect affect the the equilibrium abundance of prey, as persist, is of prey, as long long as as predators predators can can still still persist, is not not some some uninteresting uninteresting oddity oddity generated is, in generated by by the the simplicity simplicity of of the the model model but but is, in fact, fact, true true of of aa very very broad broad class class of of models. models. The The following following result result is is well well known known in in many many ecological ecological contexts, contexts, such such as as the the effect effect of of prey prey productivity productivity on on predator-prey predator-prey abundances abundances (e.g., (e.g., Arditi Arditi and and Ginzburg, 989), and Ginzburg, 11989), and we we simply simply rephrase rephrase it it here here for for the the context context of of habitat habitat de destruction. struction. Consider Consider the the following following general general model model for for predator-prey, predator-prey, p p --vv, , dynam dynamics: ics: dv dv -- = = F(v, F ( v , p, p , K,.), K,.),
dt
(6) (6)
dp dp = = pG(v,.). pG(v,.). --
dt
This prey dynamics This can can be be construed construed as as either either aa model model of of predatorpredator-prey dynamics in in aa single, single, homogenously homogenously mixed mixed population population or or aa model model of of predator-prey predator-prey patch patch dynamics, dynamics, such such as as we we have have just just considered. considered. K K is is the the carrying carrying capacity capacity of of the the prey, prey, i.e., i.e., the the abundance abundance the the prey, prey, or or prey prey patches, patches, would would achieve achieve in in the the absence absence of of predation. predation. The The dot dot denotes denotes other other parameters. parameters. K K may may itself itself be be aa function function of of other other parameters, parameters, 's choice depending depending on on one one's choice of of model. model. The The functions functions F F and and G G can can be be chosen chosen in in accord accord with with any any model model of of functional functional responses, responses, migration migration regimes, regimes, life-histories, life-histories, and and so so on. on. Indicating Indicating equilibrium equilibrium abundances abundances with with an an asterisk, asterisk, equilibrium equilibrium (v*, (v*, pp*) *) satisfi es G(v*,.) * , p*, satisfies G(v*,.) = - 00 and and F(v F(v*, p*, K,.) K,.) = - O. 0. Since Since v* v* is is determined determined by by the the fifirst rst of * , leavof these these equations, equations, changing changing K, K, by by habitat habitat destruction, destruction, only only affects affects pp*, leav-
66
Two·Species Two-SpeciesMetopopulotion MetapopulationModels Models
1133 33
ing v* unaffected. unaffected. The only restrictive assumption is that the predator' predator ' s per capita growth rate, G, is independent independent of of overall predator predator density, i.e., the population growth model is "prey-dependent" (e.g., Arditi and Ginzburg, 11989). 989). If If the function G(v,.) is replaced by G(v, p,.), p,.), so the the predators predators are now "interferential" "interferential" rather than "laissez-faire" 976b), the argument breaks down and equilibrium "laissez-faire" (May, 11976b), equilibrium prey abundances principle, be abundances are a r e affected affected by habitat destruction in ways that may, in principle, very complex. Biologically, laissez-faire laissez-faire predators affect each other only indi indirectly, through their depletion of of prey, or prey patches. Although prey-dependent prey-dependent models models of laissez-faire laissez-faire predators have have dominated theoretical 1 989) have argued that a class of models theoretical ecology, Arditi and Ginzburg ((1989) of of interferential predation may be generally superior. superior. In particular, particular, they advocate "ratio-dependent" "ratio-dependent" models, in which v and p p enter the general function G above as functions of theoretical and of their their ratio vip. v/p. This suggestion is based on both theoretical empirical lively controversy (see McCarthy empirical arguments arguments and and is is the the subject subject of of lively controversy (see McCarthy et et al. al.,, 11995, 995, and references references therein). An entirely analagous, but uncontroversial, dis distinction between between prey-dependent and ratio-dependent ratio-dependent models exists in epidemi epidemiology: the former are considered considered appropriate appropriate for, for for example, example, aerosol aerosol transmitted transmitted diseases such as measles, whereas the latter are more suitable for for sexually trans transmitted and vector-borne disease (e.g., Thrall et 993). et al. al.,, 11993). Whether Whether predators predators (or predator predator patches in metapopulation models) are lais laissez-faire or interferential hab interferential has important implications implications for for the consequences consequences of of habitat change on abundance and persistence. To illustrate this, we will see that the eradication threshold of the predators can, in principle, principle, be readily estimated estimated if they are laissez-faire, interferential. It follows from laissez-faire, but not if they are interferential. from the general arguments following model (6) that if the carrying capacity of of the prey, K, is reduced by habitat destruction to the abundance abundance of prey patches that we see today in the presence of predation, then the predator population will go extinct. This follows from the fact fact that the prey equilibrium is then the same as the carrying capacity of the landscape hence, no predators. We mhence, We can check this general result with the simple model (4). In the absence of of predators, the system reduces to a Levins model for the prey, which informs us that the carrying capacity of the landscape is h h - eJcv ev/Cv.• To find the eradication eradication threshold of of the predators, i.e., metapopulation, the above that value of of h, h, herad herad ,, which extinguishes the predator predator metapopulation, arguments lead us to the equation -
h erad
ev
=
v*
=
Cv
ep
(7) (7)
Cp '
which we solve find solve to find e V
herad __ ep + - - , Cp cv
(8) (8)
in agreement with Eq. (5c). As observed in the laissez the Introduction, as long as organisms in general are laissezfaire, affecting each other other only indirectly indirectly through through their "consumption" "consumption" of of a "lim-
1134 34
Sean Nee Nee et et 01. al. Sean
resource," such as their occupation of suitable breeding breeding territories, or, in this iting resource," case, prey patches, then the simplest estimate of their eradication threshold is just unused amount amount of that that limiting resource resource (Anderson (Anderson and and May, 11991; the unused 99 1 ; Lawton et al. al.,, 11994; et 994; Nee, 11994). 994).
C. Mutualism C. construct a simple and and illustrative model model of of mutualism, mutualism, we imagine imagine two two To construct of which can survive for some time on a habitat patch, patch, but requires requires species, one of other species for for its migration into new new patches, whereas whereas the other other species the other depends on the presence presence of of the first one for both both survival and and reproduction. reproduction. One One depends biological biological relationship relationship of of this type is the one between between a plant plant species species and and a spe spedisperser or pollinator. pollinator. A less well-known well-known inspiration for the model cialist seed disperser is the coviruses of 977). These RNA of plants (Bruening, 11977). RNA viruses get their their name name from the fact fact that that no no single virus virus particle particle contains contains all the information from information necessary necessary for a complete complete cycle of of infection. infection. For For example, example, there there are two tobacco for tobacco rattle virus virus particles, a long long and and a short short one. The The long particle particle carries the the gene gene for for the the repli repliparticles, case, while the the short short particle particle carries carries the the gene gene for for both both particles particles'' coat coat protein. protein. The The case, long particle's particle's RNA RNA can multiply in a plant plant on its own, but but ultimately it requires requires presence of of the short short particle for for encapsulation transmission to a new new the presence encapsulation and transmission Nee and and Maynard Maynard Smith Smith ((1990) argue that that this strange strange state state of of affairs affairs arose plant. Nee 1990) argue arose process of of mutual mutual parasitism. by a process To provide provide a mnemonic mnemonic for for the the subscripts, subscripts, we will refer refer to plants plants and and dis disTo persers. As As before, before, x refers refers to the the proportion proportion of of empty patches, patches, y refers refers to the persers. proportion and z to the proportion of of patches patches occupied occupied by by the plant only, only, and the proportion proportion of of patches occupied by both plant and and disperser. The subscripts and d refer refer to patches occupied by both plant disperser. The subscripts p and these latter two two patch patch types, respectively. respectively. Our Our assumptions assumptions lead lead to to the the model model ddx x edz + edZ d t -= eeppyy +
dt
- CpZX, cp zx,
dy dt = CpZX-
epy -
(9) (9)
CdZY,
dz dt -
CdZY
This the nontrivial This has has the nontrivial solution solution
edT""
- -
x* x * == hh - y *y- z* * , z* , y , = eA, Cd
z* z * ==-
�'(( 2
(1 ( 1 O) 0)
- )
aa- +± x/a 3 J a22 - 4 /4f3
,
66 0.3 0.3
td} ID
Two-Species Metopopulotion Models Two-SpeciesMetapopuhtion Models
1135 35
-
>,,
tO Q" O 14-. O ~r
0 .1 5 o.15
J S (f \
-
xX
xX
.o "~ tO t-" tL_ ~ . ~_
o 0
0.8 0.8
0.4 0.4 fraction fraction of patches with both plant and disperser disperser (z)
Trajectories with cp Trajectories in the phase phase space space of model model (9) with Cp = = Cd cd = " - 4 ,4, ed ed = - - 00.5, . 5 , ep e p= 11.5, .5, and h = = 0.8. 0.8. The symbol symbol X X marks marks the locations locations of the two equilibria: equilibria: the one on the left left is a saddle locally stable. saddle point point and the one on the right right is locally stable.
FIGURE FIGUR[ 3 3
where where
d(cp + a = h - eed(cp + Cd cd)) a = h , cpe CpCd d _
((11) 11)
] 3 - eped
CpCd The eradication threshold, The eradication threshold, h herad is found found to to be be erad ,, is ed(CP + Cd) herad
-----
CpCd
+ 2
. / e p ed
.
~ / CpCd
((12) 1 2)
patches Above Above h herad there are are two two equilibria equilibria that that differ differ in in the the abundance abundance of of patches erad ', there * . Local Local stability stability analysis analysis shows shows that with plants and with both both plants and dispersers, dispersers, zz*. that the the larger larger of smaller one saddle point. point. of the the two two is is a a stable stable equilibrium, equilibrium, whereas whereas the the smaller one is is a a saddle Figure Figure 33 illustrates illustrates the the trajectories trajectories of of the the system system in in the the vicinity vicinity of of these these two two points points for values. (There is also, for one one particular particular choice choice of of parameter parameter values. (There is also, of of course, course, the the trivial trivial solution solution x x** = = h h,, y y** = = zz** = = 0.) 0.) Figure Figure 4 4 shows shows the the equilibria equilibria as as functions functions of of increasing increasing habitat habitat destruction. destruction. When close to there When habitat habitat destruction destruction approaches approaches close to the the eradication eradication threshold, threshold, there remains aa large across the remains large metapopulation metapopulation of of mutualists mutualists across the landscape landscape at at the the stable stable equilibrium. two equilibria ned in equilibrium. However, However, at at the the eradication eradication threshold, threshold, the the two equilibria defi defined in Eqs. 1 0) collide Eqs. ((10) collide and and annihilate annihilate each each other, other, leaving leaving only only the the trivial trivial equilibrium equilibrium
1136 36
Sean Nee Sean Nee et et al. al.
08]
0.8
0.6 0.6-
Z** 0.4 0.4-
Z
0.2 0.2 ..........
............
.................
. ... ... •...•.... . . ...
/
00 4--I ...................i................................. -,'"'"~! I I -.----.====r===� 0.2 0.1 0.3 0.5 o0 0.1 0.2 0.3 0.4 0.5 0.4
increasing habitat l -h) habitat destruction destruction ((1-h)
increasing levels levels The two equilibria equilibria zz** of mutualism mutualism model model (9), (9), shown shown as functions functions of increasing of habitat destruction, line is the stable destruction, 11 - h. The solid solid line stable equilibrium. equilibrium.The colonization colonizationand extinction extinction 125, up to the eradication parameters parameters are the same same as in Fig. Fig. 3. yy** (not (not shown) shown) remains remains at 0. 0.125, eradication threshold threshold level of destruction. = h. level destruction. Thereafter, Thereafter, y* = = zz** = = 0 and x* =
FIGURE FIGURE 44
-
of example of what is is described described in in mathe of metapopulation metapopulation extinction. extinction. This This is is an an example of what mathematics as the conservation conservation context. context. To matics as aa "catastrophe," "catastrophe," an an appropriate appropriate term term in in the To describe result more viable association association of mutualists living living describe this this result more vividly, vividly, a a perfectly perfectly viable of mutualists in by the in great great abundance abundance across across aa large large region region can can be be completely completely destroyed destroyed by the construction construction of of just just one one more more shopping shopping mall. mall. This This is is vivid, vivid, but but not not entirely entirely real realistic. system was close to then chance istic. If If the the system was that that close to the the eradication eradication threshold, threshold, then chance effects, effects, not included included in in the the simple simple deterministic deterministic model, model, and and the the existence existence of of the the saddle saddle not point to the persistence of the mutualists, point would would combine combine to to create create a a serious serious threat threat to the persistence of the mutualists, rendering if chance patch rendering the the system system vulnerable vulnerable to to extinction extinction if chance events events move move the the patch abundances into region of Fig. 33 which sweeps the above-threshold metapopmetapop abundances into aa region of Fig. which sweeps the above-threshold ulation alternative stable ulation to to the the alternative stable state state of of oblivion. oblivion. A the A sudden, sudden, large large catastrophic catastrophic change change in in the the fate fate of of the the metapopulation metapopulation as as the result of amount of habitat may seem like like a result of aa tiny tiny change change in in the the amount of suitable suitable habitat may seem a peculiar peculiar and this result result by by and unfamiliar unfamiliar outcome. outcome. One One can can become become more more comfortable comfortable with with this considering the simpler and more familiar considering the same same phenomenon phenomenon in in a a simpler and more familiar context. context. Con Consider population dynamics dynamics in single population population unun sider aa simple simple model model of of population in which which a a single dergoes logistic growth dergoes logistic growth described described by by the the two two parameters parameters rr and and K, K, the the per-capita per-capita the stable growth respectively. As growth rate rate and and the the carrying carrying capacity, capacity, respectively. As long long as as rr > > 11,, the stable below 11 equilibrium K. However, However, lower equilibrium is is aa population population of of size size K. lower rr ever ever so so slightly slightly below and only equilibrium and the the only equilibrium is is extinction. extinction. Armstrong 1 987) noted were metapopulation of Armstrong ((1987) noted that that although although there there were metapopulation models models of competition competition and and predation, predation, there there were were none none of of mutualism, mutualism, and and he he constructed constructed model (9) (9) with ll the Hastings and 1 989) also also studied studied a model with h = = I1 to to fi fill the gap. gap. Hastings and Wolin Wolin ((1989) a mutualism conclude that metapopulation systems systems always have mutualism model model to to conclude that mutualist mutualist metapopulation always have
66
Two·Species Metapopulation Models Two-SpeciesMetapopulation Models
1137 37
a stable equilibrium, equilibrium, which they claim contrasts with mutualism mutualism models that do not incorporate spatial structure. A more more general general review of of models, including metapopulations of of humans and schistosomes, with two alternative alternative stable states, metapopulations "thresholds" and "breakpoints," 1 977). For "breakpoints," can can be found in May May ((1977). For a specific ex example, of of alternative alternative stable states, see Gyllenberg et et al. al. (this (this volume).
III. SPATIAllY SPATIALLYEXPLICIT EXPLICIT METAPOPULATIONS METAPOPULATIONS In this this section we will emphasize emphasize the qualitatively new new features features that can can arise in metapopulation metapopulation models that are spatially explicit. We do this quite generally for for a variety of of interactions, interactions, including single-species, single-species, competing species, and predatorprey systems, but we will dwell in more detail on the latter. Most of predator-prey the models models assume that that the habitat takes takes the form of a grid or lattice of of "cells" "cells" containing local local populations with discrete generations (for a discussion of various spatially explicit metapopulation metapopulation models, see Hanski and SimberIoff, Simberloff, this vol volume). Demographic parameters defi n e population growth within, and migration Demographic define growth of individuals between, cells. Such models with discrete time and space but concon 992, tinuous population state have been dubbed "coupled map lattices" (Kaneko, 11992, 11993; 993; Soh� 1 992) and, because So16 et et at. al.,, 1992) because of of their complexity, have mainly been explored by numerical simulations. In all the examples below, the following rules apply. Within each generation there are two distinct phases: ((1) 1 ) a period when the local population reproduces reproduces and matures, and (2) a distinct migration stage when some mixing between local populations occurs. Reproduction can thus oc occur cur following, following, or or prior prior to, to, migration, migration, but but not not at at the the same same time time [this [this would would require require careful formulation in a model to keep track of those individuals that migrated et ai. al.,, 11995)]. 995)]. and those that remained developing within the patch (Hassell et The models in this section have explicit local dynamics. They are, therefore, popu suitable for for the study of of the effects of migration on the dynamics of of local populations in a metapopulation. metapopulation. This is where we begin.
A. local Local Stability Although Although most of of the theory theory of of population population dynamics has concentrated concentrated on isolated isolated populations with no interchange interchange of individuals between other populations populations in the region, this does not mean that that spatial spatial dynamics have been neglected neglected in population ecology; far from it. However, the emphasis has been primarily on the of individuals within a single patchy habitat. It effects of the spatial distribution of is implicitly assumed in this work that that the dispersing stages mix thoroughly before redistribution within the habitat according to specified behavioral or statistical rules. This implies that the individuals are able to disperse widely across the entire habitat which has, in turn, tum, implications for for the spatial scale that is appropriate appropriate for the study. The general conclusion from this body of work, whether whether it involves single species (e.g., de Jong, 11979; 979; Hassell and May, 11985), 985), competing species
1 38 138
Seon Nee Nee et et al. 01. Sean
(e.g., Shorrocks Shorrocks et et al., al., 1979; 1 979; Atkinson Atkinson and and Shorrocks, Shorrocks, 1981; 1 98 1 ; de de Jong, Jong, 1981; 1 98 1 ; HanHan (e.g., ski, 1981, 1 98 1 , 1983; 1 983; Ives Ives and and May, May, 1985), 1 985), predator-prey predator-prey interactions interactions (e.g., (e.g., Hassell Hassell ski, May, et al., al., 1990; 1 990; Hassell Hassell et et al, ai, 1991b; 1 99 1 b; May, 1973; 1 973; Chesson Chesson and and Murdoch, Murdoch, 1986; 1 986; Pacala Pacala et Rohani et et al., al., 1994), 1 994), or or disease-host disease -host interactions, interactions, is is that that spatial spatial variation variation in in the the Rohani risk of of mortality mortality enhances enhances population population stability. stability. Because Because of of the the assumptions assumptions made made risk about migration, migration, no no patch patch or or grouping grouping of of individuals individuals can can have have any any degree degree of of about independent temporal temporal dynamics dynamics from from generation generation to to generation. generation. By By shifting shifting our our independent spatial scale scale upward upward to to that that of of aa metapopulation, metapopulation, asynchronous asynchronous dynamics dynamics become become spatial possibility. aa possibility. We commence commence with with a very simple case case of of a single single species species reproducing reproducing and and We very simple for resources resources in a metapopulation. metapopulation. The The environment environment is made made up up of of competing for uniform, discrete habitats or patches arranged in a regular grid in which live local uniform, discrete habitats or patches arranged a regular grid which live local populations of of herbivoros herbivoros insects. insects. The The insects insects have have discrete discrete generations, and in in populations generations, and each generation generation some some of of the the adult adult females females disperse disperse to to neighboring neighboring populations. popUlations. each Following the migration stage, stage, the the females females oviposit oviposit and and the the larvae larvae that that subsesubse Following function of of the the density within their their quently emerge emerge compete compete for resources resources as a function density within population. Such a patchy environment can be conveniently modeled as a local population. patchy environment conveniently modeled two-dimensional arena which the the local populations populations are distributed among among a two-dimensional arena in which are distributed square grid of of cells. cells. In each generation generation there is a migration phase, in which which a square fraction of of the adult insects insects leave the patch which they emerged emerged and move move fraction patch from which to to neighboring neighboring cells. We We first first focus focus on on the the conditions conditions for for stability stability in in such such a system, system, when all the local populations populations move to a common, stable stable equilibrium, equilibrium, and and then in the following section examine examine the more complex dynamics that can occur when the unstable. the local local populations populations are are unstable. habitat We first assume that that the local population dynamics within a single habitat are based on a familiar single-species model for intraspecifi intraspecificc competition,
= ANO AN(1 + + aN aN)) --b, N 'I = b,
( (~3) 3)
where N 'I and N are the population sizes iin n successive generations, A nite A iiss the fi finite rate of increase and a and b are constants defining the density dependent survival 976; de Jong, 11979). 979). The stability properties of (Hassell, 11975; 975; Hassell et al., 11976; Eq. ((13) 1 3) depend solely on the parameters b and and A A and are are described in Hassell ((1975). 1 975). We now ask the question "To what extent are are these stability properties altered if the model is extended to a metapopulation by linking a number of these local populations, all with identical parameters, by limited migration?" In aa recent paper, Bascompte and Sole 1 994) have explored such So16 ((1994) such a meta metapopulation model 1 3) model in which local populations with dynamics dynamics described by by Eq. ((13) are are linked by diffusive migration to to their four nearest neighbors. Their Their results are are surprising: surprising: as migration rate is increased, the dynamics become increasingly un unstable, stable, and and thus increasingly diverge diverge from those those of the nonspatial, homogeneous model 1 3). This model ((13). This is is counterintuitive, counterintuitive, since since one one would would expect expect increasing increasing migration migration to to link more more effectively effectively the the separate separate local populations populations and and so so bring bring the the properties properties of of the the spatially spatially structured structured and and homogeneous homogeneous models models closer closer together together (Ruxton, (Ruxton, 995). The 11994; 994; Hassell Hassell et et al. al.,, 11995). The explanation explanation lies lies in in the the biologically biologically implausible implausible
66 Two-Species Two-SpeciesMetapopulotion MetapopulationModels Models
1139 39
way way that that Bascompte Bascompte and and Sole So16 formulated formulated migration migration within within their their model. model. Couched Couched as diffusion equation, as aa discrete discrete analog analog of of aa reactionreaction-diffusion equation, their their model model fails fails properly properly to segregate segregate the the processes processes of of survival and and migration, migration, and and as as aa result, the the same same to individuals may may simultaneously simultaneously fail fail to to survive survive and and yet yet disperse disperse (Hassell (Hassell et et at. al.,, 11995). 995). If this problem is avoided avoided by by segregating competition and migration (for (for example, larvae that compete for for resources and and adults that disperse), disperse), completely different conclusions can be drawn: drawn: spatial structure structure now now has no no effect effect on on the stability cally, if stability properties properties of of the the system. system. More More specifi specifically, if we we assume assume periodic periodic bound boundJ-t of the emerging adults ary conditions and migration migration of the form that that a fraction fraction/x ary within a habitat habitat disperse by moving with equal probability to one of the eight surrounding habitats, and hence a fraction 11 - J-t ~ remain behind, it can be shown analytically that that the local stability boundaries of this metapopulation are are identical et at., 1 996). This is true for all with those for a single local population (Rohani for et al., 1996). migration rates, J-t, and for rates,/x, for all grid sizes. It does not depend on the number or location of the habitats to which the dispersing individuals individuals move; all that that is re required is for the patterns of migration to be the same for all cells. Indeed, the result is also independent of the details of the within-habitat density dependence provided it takes the form f(N f(N).). This result-that resultmthat the metapopulation and local populations have the same stability properties-is properties m is much broader. It applies equally to comparable predator-prey interactions equally to comparable models models for for interspecifi interspecificc and and predator-prey interactions et at., al., 11996). (Rohani et 996). For a broad class class of of models, therefore, therefore, the the introduction of of spatial structure of the systems. This result makes has no affect on the local stability properties of sense intuitively, equilibrium, sense intuitively, provided that that the the environment environment is is uniform, uniform, so so that that at at equilibrium, populations have have the same density. Thus, equilibrium, migration to Thus, at equilibrium, all local populations and from local populations populations is in balance balance and does not alter alter the equilibrium equilibrium prop properties of populations. A number number of of factors will, of of course, course, confound confound this this of the local populations. conclusion. Most obviously, a spatially heterogeneous heterogeneous environment environment is simple conclusion. bound to introduce different dynamics dependent dependent on the the nature the spatial introduce different nature of of the unevenness. However, even in homogeneous environments, moving to a metameta unevenness. population may may change change the the stability properties properties under under some some conditions. Vance Vance population ( 1 984) has discussed discussed aa range range of of single-species single-species models in which which migration migration between between (1984) habitats stabilizes, but but occasionally destabilizes, destabilizes, the the population population as a whole; habitats often often stabilizes, Reeve (1988) ( 1 988) has has shown for host-parasitoid models models that that stability stability is reduced reduced by by Reeve for host-parasitoid the interaction interaction between migration and and density density dependent dependent host host rates rates of of increase; increase; the between migration and Hastings Hastings (1992) ( 1 992) has has explored explored age-structured age-structured metapopulations metapopulations where where strong strong and levels of of density density dependence dependence and and asymmetric asymmetric migration migration between between age-classes age-classes is levels destabilizing. destabilizing. Another class class of of studies studies is, is, in in aa limited limited sense, sense, the the converse converse of of those those discussed discussed Another above and and examines examines whether whether immigration immigration has has aa stabilizing stabilizing effect effect on on unstable unstable local local above population dynamics, dynamics, in in particular, particular, chaotic chaotic dynamics. dynamics. The The general general conclusion conclusion is is population that immigration immigration readily readily turns turns chaos chaos into into periodic periodic dynamics dynamics (e.g., (e.g., Gonzalez-AnGonzalez-An that dujar and and Perry, Perry, 1995; 1 995; Hastings, Hastings, 1993). 1 993). It It is is now now understood understood that that chaos chaos is is often often dujar -
1140 40
Sean Nee Nee et et al. al. Sean
doua structurally unstable feature of dynamical systems that follow the period dou bling route to chaos, easily abolished via period doubling reversals in the face of perturbations like immigration (Stone, 11993), although it may be a more robust perturbations 993), although feature of systems that approach chaos by other routes (Rohani and Miramontes, 11995). 995).
B. Complex ComplexSpatial Spatial Dynamics Dynamics B. metapopulaMoving beyond the region of local stability, spatially explicit metapopula tions may show strikingly novel dynamics. Broadly, such metapopulations are characterized by unstable local populations tending to fluctuate asynchronously, characterized and and by the metapopulation as a whole tending to persist persist much much more more readily than docuin the comparable spatially homogeneous model. Such behavior has been docu for metapopulations of single species (Bascompte and and Solt\ So16, 11994; 994; Hassell mented for et al. al.,, 11995), for competing competing species (So16 et al. al., , 11992; et al. al.,, 1994) et 995), for (Sole et 992; Halley et 1 994) and for various predatorprey systems (Taylor, 11988, 988, 11990, 990, 1991). 1 99 1 ). Here, we concen predator-prey conceninteraction, between between hosts and parasitoids, for which trate on just one kind of interaction, et al. al.,, 1991a, 1 99 1 a, 1994; 1 994; these dynamics have been thoroughly displayed (Hassell et et al. al., , 1992). 1 992). Comins et host-parasitoid systems are characterized characterized by the adult female para paraInsect host-parasitoid "searching" stage and laying their eggs on, in, or near sitoids being the only "searching" near the hosts that they encounter; these hosts are then subsequently killed by the feeding (Askew, 11971; feature of of having reproduction reproduction defi defined larvae (Askew, 97 1 ; Godfray, 1994). 1 994). This feature ned host-parasitoid associations particularly simple and directly by parasitism makes host-parasitoid convenient models of of predator-prey predator- prey systems in general. general. We general model for the interaction interaction between an insect host We begin begin with a general model for between an and and its specialist specialist parasitoid parasitoid in a completely homogeneous homogeneous environment environment (Hassell, (Hassell, 1978), 1 978), N N'' == ANf(P)
=
PP'' = cN[1 cN[ 1 -- f(P)], f(P)],
((14) 1 4)
where where N', N ' , P', P', N N and and P P are the host and and parasitoid parasitoid populations, respectively, respectively, in successive successive generations, generations, A A is the host rate of of increase, increase, as before, before, f(P) f(P) represents represents of a host escaping parasitism parasitism (assumed here, for the probability of (assumed here, for simplicity, only to to depend depend on parasitoid parasitoid density) and and c is the average average number number of of adult adult female female parasitoids parasitoids emerging emerging from from a parasitized parasitized host host (henceforth (henceforth assumed assumed to to be be one). The The dynamics dynamics of of this model model have been explored explored using using a wide range range of of expressions expressions for the different different parameters parameters (Hassell, (Hassell, 1978). 1 978). The The model model will be be unstable unstable unless unless for the (1) sufficiently nonrandomly ( l ) the the parasitoids parasitoids attack attack hosts hosts sufficiently nonrandomly (Pacala (Pacala et et al., al., 1990; 1 990; Hassell Hassell et et al., al., 1991 1 99 1 b), b), (2) A A is sufficiently sufficiently density density dependent dependent (e.g., (e.g., Beddington Beddington et et al., al. , 1975; 1 975; May May et et al., al., 1981), 1 98 1 ), or or (3) c is density density dependent dependent (e.g., (e.g., Hassell, Hassell, 1980; 1 980; Hassell et al., al., 1983). 1 983). Hassell et To extend extend this this to to a metapopulation, metapopulation, we we assume assume the the same same environment environment as To
66
Two-Species Metapopulation Models Two-SpeciesMetapopulation Models
1141 41
before, parasitoids_ IInn each before, but but now now the local populations populations of of hosts are attacked attacked by parasitoids. each generation generation the dynamics dynamics consist consist of of two phases. First, a reproduction-and-parasit reproduction-and-parasitism phase phase in which which hosts and parasitoids parasitoids interact interact within individual individual patches patches ac according to Eqs. ((14). 1 4). The phase where of The second phase is a migration phase where a fraction fraction of the emerging J.LN ' and a fraction female parasitoids, emerging adult adult hosts, hosts,/XN, fraction of of emerging emerging adult adult female parasitoids, J.L immediate neighboring neighboring /Zp, each patch patch redistribute redistribute themselves themselves to the the eight eight immediate p , in each patches. patches. This migration migration is assumed assumed to be spatially symmetrical symmetrical for for both species, species, but the previous host-parhost-par the fraction fraction of of dispersers dispersers is species-dependent. species-dependent. In most previous asitoid studies, studies, these dispersing dispersing individuals individuals have have been been distributed distributed over all other other patches according to some specifi e d behavioral or statistical rules (e.g., Hassell, patches according specified behavioral statistical rules Hassell, 11978; 978; Chesson 986; Pacala 1 990; Hassell 1 99 1lb). b). Chesson and and Murdoch, Murdoch, 11986; Pacala et et al., al., 1990; Hassell et et al. al.,, 199 Here, Here, however, however, rather rather than entering entering a "pool "pool"" for such global migration migration (i.e., (i.e., a fornicatorium in the sky), the dispersing hosts and parasitoids move outglobal fornicatorium move out ward, using the nearest neighbors neighbors the same same rule rule as above, to colonize colonize equally the eight eight nearest of of the patch patch from from which they emerged emerged (slightly different different assumptions may be necessary along re along the boundaries boundaries depending on whether whether cyclic, absorbing, absorbing, or reflective boundary effect on boundary conditions conditions are used; the the choice choice of of condition condition has little effect the outcome, outcome, provided provided the the arenas arenas are are not not very small small).). The whole whole system is thus described described by by the following following set of of equations: equations:
N; Ni = -- lJ(PJ Ji f(P~) Q O~;
= -
cJD cJ~[1 - f (I(PJ] P~)]
= =
cl; cJ~ - cN; cN~
l; Ji = = A[N;( A[Ni(I1 - J.LN ~ u )) + + J.LN l-tu{Ni}] { N; I]
) (( 1155)
P ;( 1 J.Lp) + PI; = = Q Q;(1 -/-re) + J.L Id.PlQi}. p{ Q;I .
hosts in patch Here Here N; N; and l; Ji are, respectively, respectively, the adult and juvenile juvenile hosts patch i, Q Q ;i is emerging parasitoids parasitoids in patch patch i, and P; P; is the postmigration postmigration population population the newly emerging of of parasitoids parasitoids in patch patch i which which search search for host larvae larvae to parasitize. parasitize. The The curly brackets brackets represent represent incoming incoming individuals, obtained obtained as appropriate appropriate sums over the relevant patches. The The function function for parasitism is given by the unstable Nicholson Nicholson and and Bailey term, term, f(P f(P,) = exp(aP exp(aP,), isolated population population is unstable unstable with with I ), so a single isolated I) = rapidly rapidly expanding expanding oscillations oscillations although although the the metapopulation metapopulation as a whole may be persistent. persistent. Persistence Persistence in this model is associated associated with some some striking spatial spatial patterns patterns of of local population chaos," "spi"spi population abundances, abundances, which which have been been labeled labeled as "spatial "spatial chaos," rals," 1 99 1 b; Comins 1 992). Figure rals," and "crystal lattices" lattices" (Hassell et et al. al.,, 1991 Comins et et al., 1992). Figure 5 shows the approximate approximate boundaries for for these these different different patterns patterns in relation relation to the host and parasitoid parasitoid migration migration rates and for for a chosen chosen value of of A and an arena arena width width A and of population den of n = = 30. The The spiral spiral structures structures are are characterized characterized by the local population deneither direction around almost sities forming spiral waves waves which which rotate rotate in either direction around almost im immobile mobile focal points. The The phase-space phase-space dynamics dynamics of of each each patch patch form form a close ap approximation fixed track, track, even though These proximation to a fixed though no exact repetition occurs. occurs. These patterns position and patterns are are apparently apparently chaotic, since since the position and number number of of focal points vary
1142 42
Seon Nee Sean Nee et et 01. al. CRYSTAL LATIICE'
CHAOS
0.8 0.6
J1 p
SPIRALS
0.4 0.2
\ 0.2
0.4
J1 N
0.6
0.8
Dependence f-LN and/xp and f-Lp for width of 30 and Dependence of of the persistent persistent spatial spatial pattern pattern on on/Xy for arena arena width of 30 and A A= = 2. The The boundaries boundaries are obtained by simulation simulation and are approximate. approximate. The The single single hatched area area
FIGURE FIGURES5
indicates marked as "hard-to-start indicates the the region in which the spatial pattern pattern is chaotic; chaotic; the region marked "hard-to-start spirals" spirals" represents pattern is unlikely to be established represents parameter parameter combinations combinations for for which which the the persistent persistent spiral pattern established by starting starting the the simulation simulation with with a single single nonempty nonempty patch. patch. Spirals Spirals may may be established established in these these cases cases by 00 generations. starting with with aa lower lower f-LN starting /XN and and increasing increasing it it after after 50 50 to to 1100 generations. Metapopulation Metapopulation extinction extinction f-LNN or f-Lp; this area imperceptible in the figures (after occurs occurs for for some some combinations combinations with very very small small/x or/xp; area is imperceptible the figures (after et al., al., 11991a). Hassell et Hassell 99 1 a).
slowly with time time in nonrepeating nonrepeating patterns. The The combined combined metapopulation exhibits what what appear appear to be stable stable limit cycles (Fig. 6a). Spatial Spatial chaos chaos is characterized characterized by the host and and parasitoid population population densities densities fluctuating from patch-to-patch patch-to-patch with no long-term long-term spatial organization. Randomly Randomly oriented oriented wave fronts are observed, but each each persists only briefly. The total metapopulation generally remains within narrow narrow bounds, but occasional large excursions are observed (Fig. 6b). Despite the lack of indefinitely of recognizable structure, structure, the populations appear appear to coexist indefinitely (as long as the arena arena is sufficiently large). Finally, the rather rather extreme extreme combination of of very low host migration and very high parasitoid migration gives persistent persistent crystal lattice-like lattice-like structures, structures, in which relatively high density patches patches occur occur at a spacing of of approximately two grid units, and the metapopulation as a whole is stable (Fig. 6c). The The entire metapopulation may may go extinct in this this model model for several reasons. First, the total area area may be too small (see next section). Second, the starting conditions example, conditions for for the simulation may be unfavorable unfavorable for for persistence. persistence. For example, in the region described as "hard to start spirals" in Fig. 5, persistence is impossible described "hard start persistence if the simulations are are started from a single nonempty cell. Once Once the populations
66 Two-Species Two-SpeciesMetapopulation MetapopulotionModels Models
fI)
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a) ((a)
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,
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1143 43
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200 200
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generations Generations
300 300
FIGURE FIGURE 66 Typical time series of average population size corresponding corresponding to the three classes of spatial behavior (bl chaotic behavior shown in Fig. 5. (al (a) Stable cycles where where spatial spirals occur, occur, (b) chaotic population dynamics where spatial chaos chaos occurs, occurs, and (cl (c) a stable equilibrium where where the "crystal lattice-like" pattern is observed observed (after (after Hassell et et al., spatial pattern al., 11991a). 99 1 al.
initiated, however, however, by simultaneous simultaneous colonization colonization of of many cells, cells, persistence persistence are initiated, metapopulation extinction extinction may may arise arise from from intrinsic intrinsic dynamic dynamic always occurs. occurs. Third, Third, metapopulation instability. This This region region is small and restricted to to parameter parameter combicombi instability. and in Fig. 5 is restricted nations in which which either/ZN either J.LN or J.Lp is very small. note that that persistence persistence nations or/Zp small. Finally, we note remains possible possible for wide range the host host rate of increase increase (A), ranging remains for a wide range of of values of of the rate of (A), ranging from close close to unity to very large. The are to favor from The principal principal effects effects of of increasing A favor A the formation formation of of spirals spirals (rather (rather than than spatial chaos) chaos) at low low host host migration migration rates rates and and the to to reduce reduce the the spatial spatial scale scale of of the the persisting persisting spirals. spirals.
C. C Habitat Habitat Destruction Destruction and and Spatial Spatial Dynamics Dynamics The metapopulations metapopulations described described in in the the previous previous subsection subsection persist persist readily readily for for The wide range range of of host host and and parasitoid parasitoid demographic demographic parameters. parameters. An An important important adad aa wide ditional requirement requirement was was aa sufficiently sufficiently large large number number of of local local populations populations (e.g., (e.g., ditional = 30). 30). Any Any reduction reduction of of the the grid grid size size (Comins (Comins et et al., al. , 1992) 1 992) or or grid side side length length -grid fragmentation of of the the habitat habitat (Hassell (Hassell et et al., al. , 1993) 1 993) runs runs the the risk risk of of disrupting disrupting the the fragmentation dynamics of of the the metapopulation metapopulation as as aa whole, whole, either either by by reducing reducing the the number number of of dynamics local populations populations below below aa critical critical level level required required for for the the combined combined metapopulation metapopulation local to persist persist or or by by interfering interfering with with the the migration migration required required to to link link the the unstable unstable local local to populations. populations. Habitat Habitat destruction destruction has has generally generally the the dual dual effect effect of of reducing reducing the the amount amount of of
Sean Nee Seen Nee et et al. ol.
1144 44
0.8 0.6 0.4 0.2 c.. 4 �._o 0.8 �~ 0.6 t-c:
ic:
'0 0
� ._z- 0.4 .IQ :is
2l e e 0.2 O.
1 2 1 5 20 25 30 0.1 ���
6 10 �
12
0.8 0.6
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15 20 25 30 n) Side Side length length ((n) FIGURE numbers of patches patches in a square grid of side FIGURE 77 Extinction Extinctionprobability probability in relation relation to the numbers square grid = 0.89, 0.89, A Ex length n and the fraction fraction of hosts hosts migrating migrating to neighboring neighboring patches (fLN) (/ZN) (fLp (/Zp = A= = 2). Extinction is measured as the proportion replicates failing to persist over proportion of 50 replicates over 2000 generations. generations. Each replicate is started by setting third patch from setting a nonzero population population density density in only only the third from the left in the row. The same same 50 pairs pairs of initial used for all the parameter top row. initial host host and parasitoid parasitoid densities are used combinations. numeric underflow underflow (densities less than than about 1 0-45); howhow combinations. Local Local extinction extinction occurs occurs by numeric (densities less about 10-45); ever, the results are robust when when local extinction extinction thresholds for both host and parasitoid are modeled modeled explicitly (after Hassell et al., 11991a). 99 I a). (after Hassell et al.,
habitat between habitat fragments. habitat and and restricting restricting the the opportunities opportunities for for migration migration between habitat fragments. Within the Within the the hosth o s t - pparasitoid a r a s i t o i d metapopulation metapopulation outlined outlined above, above, it it is is clear clear that that the probability long-term persistence persistence decreases number of patches is reduced probability of of long-term decreases as as the the number of patches is reduced (Fig. 7). on the the characteristic (Fig. 7). The The extent extent of of this this effect effect depends depends to to aa large large extent extent on characteristic spatial dynamics. Thus spatial scale scale of of the the dynamics. Thus with with parameter parameter combinations combinations producing producing "crys "crystal tal lattice" lattice" patterns, patterns, the the overall overall populations populations can can persist persist in in a a stable stable interaction interaction even even with in which large with very very small small grids grids of of n n = = 2. 2. At At the the other other extreme, extreme, interactions interactions in which largescale sizes, while scale spirals spirals occur occur are are especially especially vulnerable vulnerable to to shrinking shrinking grid grid sizes, while inter interactions producing producing chaotic two. This This actions chaotic spatial spatial patterns patterns are are intermediate intermediate between between the the two. trend also clear clear from Fig. 7, 7, which which represents represents a slice across Fig. 55.. Within Within the the trend is is also from Fig. a slice across Fig. region of chaos all lengths of I S and whereas region of chaos all interactions interactions persist persist for for side side lengths of 15 and above, above, whereas within the producing spirals extinction increases within the region region producing spirals the the probability probability of of extinction increases so so that that with = 0.8 extinctions oc with the the relatively relatively large large spirals spirals generated generated with with JLN /ZN = 0.8 some some extinctions occurred 30 and was no persistence at curred for for all all simulations simulations with with n n < < 30 and there there was no persistence at all all with with nn < S. Failure is thus associated with < I15. Failure to to persist persist in in small small arenas arenas is thus associated with insufficient insufficient space space in in which which to to fit fit a a self-maintaining self-maintaining pattern. pattern. These These general general trends trends remain remain true true for for increased migration distances (Comins (Comins et different different values values of of A A and and also also for for increased migration distances et al. al., , 11992). 992). Regions partially subdivided inhospitable Regions of of suitable suitable habitat habitat may may become become partially subdivided by by inhospitable corridors Mankind ' s ever-increasever-increas corridors that that restrict restrict movement movement within within the the overall overall area. area. Mankind's ing which they ing network network of of roads roads must must have have this this effect effect on on the the habitats habitats through through which they
66
Two·Species Metapopulation Models Two-SpeciesMetapopulation Models
1145 45
plunge. plunge. By pushing ecological ecological communities communities closer to the limits limits of of their range, climate change produce similar effects change is also likely to produce effects in which habitats habitats shrink in a patchy way, leaving pockets with limited limited connections connections for for the species species within within them. To illustrate possible effects illustrate the possible effects of of such disturbance, disturbance, let us modify modify the environment patches explored imposing barriers of environment of of 30 X x 30 patches explored above by imposing of one patch movement between patch width width with varying numbers numbers of of "gaps" for for movement between the subareas subareas (Hassell et 993). Two conclusions stand out. First, as noted noted above, interac et al. al.,, 11993). Two conclusions interactions with characteristic most characteristic spatial dynamics on a large scale are by far far the most filling easily disrupted. disrupted. For example, example, an interaction interaction with a single persisting persisting spiral filling bisects the habitat, the 30 X x 30 arena arena always becomes becomes extinct when when a barrier barrier bisects habitat, while interactions interactions with small-scale spirals or chaotic chaotic spatial dynamics can persist persist much much more easily as the habitat is disrupted. disrupted. In short, habitat habitat subdivision subdivision affects affects species influenced by the characteristic species persistence persistence in a way that that is strongly influenced characteristic scale of of the spatial dynamics.
D. Multispecies MultispeciesSystems Systems The two-species parasitoid models of two-species hosthost-parasitoid of the previous previous sections lay bare some interesting dynamical properties of Important questions of metapopulations. metapopulations. Important questions remain, remain, however, however, in our our understanding understanding of of how metapopulation metapopulation dynamics dynamics may affect community structure (Holt, this volume). volume). Here, we examine the specific specific case of influence the coexistence three-species host of how how spatial processes processes may influence coexistence of of three-species hostparasitoid systems 994; H. N. Comins parasitoid systems (Hassell (Hassell et et al. al.,, 11994; Comins and and M. P. Hassell, un unpublished) 1 5), in which published) that are are straightforward straightforward extensions extensions of of Eqs. ((15), which the the third species may be another host, another hyperparasitoid. another parasitoid, parasitoid, or a hyperparasitoid. The results are very similar for the different different systems: a third third species can coexist stably within the spatial dynamics dynamics (spiral waves waves or chaos) generated generated by an existing host- parasitoid interaction, interaction, provided provided that it is relatively existing two-species, two-species, host-parasitoid sessile compared compared to its competitor. competitor. Coexistence Coexistence thus depends depends upon upon a kind of of fugitive 1 95 1 ; Levins Levins and Culver, 97 1 ; Horn and Mac fugitive coexistence coexistence (Hutchinson, (Hutchinson, 1951; Culver, 11971; MacArthur, 97 1 ; Hanski and Ranta, 1 983; Nee and May, 1992, 1 992, 11994; 994; Hanski Hanski and Arthur, 11971; Ranta, 1983; and Zhang, 993). For example, two-parasitoid-one-host system, coexistence Zhang, 11993). example, in the two-parasitoid-one-host coexistence occurs different migration migration occurs most most easily when when the two parasitoid parasitoid species have very different rates, provided that low migration is matched matched by high within-patch within-patch searching searching efficiency, and vice versa. Similarly, in the case of two-host- one-parasitoid inin of two-host-one-parasitoid teractions, teractions, coexistence coexistence occurs readily when the two host host species species have have very dif different migration either the larger migration rates and the relatively immobile species species has either rate of of local-population local-population increase or is less susceptible susceptible to parasitism. parasitism. Finally, in the host - parasitoid - hyperparasitoid system, coexistence host-parasitoid-hyperparasitoid coexistence demands demands that the hy hyperparasitoid parasitoid and a much much perparasitoid has a higher higher searching searching efficiency than the parasitoid lower migration hyperparasitoid should higher searching lower migration rate. That That the hyperparasitoid should have the higher searching efficiency is also in accord nonspatial models models of host accord with the conclusions conclusions from from nonspatial of hostparasitoid - hyperparasitoid interaction interaction (Beddington Hammond, 1977; 1 977; Has parasitoid-hyperparasitoid (Beddington and and Hammond, Hassell, 11979; 979; May and 1 98 1 ). and Hassell, 1981). An An interesting additional additional point is that coexistence coexistence in these models tends to
1146 46
Sean Nee Seen Nee et et al. oil. a a
b b
c c
FIGURE distribution (with (with linear scales) of (a) hosts, hosts, (b) highly FIGURE 88 Maps Maps of the spatial spatial density density distribution linear scales) dispersive paras ito ids, and (c) relatively two parasitoids, relatively sedentary sedentary parasitoids, parasitoids, in a snapshot snapshot from from a one-host and two= 0.5, 0.05. The three parasitoid simulation with parasitoid simulation with AA = 2, /-LN /ZN = 0.5, /-L /Zp~ = 0.5, /-Lp, /Zp2 = -- 0.05. three grids grids should be men menPI = tally superimposed to perceive the relationships relationships between between the densities of the three three species. species. Spiral foci exist at the ends of the "mountain figure (excluding "mountain ranges" in the left-hand figure (excluding ends at the edges of the grid). In the time evolution evolution of the system system the "mountain "mountain ridges" are the peaks peaks of population density continuous motion. peaks or foci, remain in almost almost exactly waves and are thus thus in continuous motion. The peaks foci, by contrast, remain the same place, for indefinitely times (after Hassell et al., 11994, 994, where same place, indefinitely long long times et al., where further further details details are given). =
be be associated associated with with some some degree degree of of self-organizing self-organizing spatial spatial separation separation between between the the competing competing species. species. This This is is best best seen seen when when the the spatial spatial dynamics dynamics show show clear clear spirals. spirals. In the In the the case case of of two two competing competing parasitoids parasitoids with with very very different different migration migration rates, rates, the relatively tends to ned to relatively immobile immobile species species tends to be be confi confined to the the central central foci foci of of the the spirals, spirals, where where it it is is the the most most abundant abundant species, species, and and the the highly highly dispersive dispersive species species occupies occupies the remainder remainder of of the the "trailing "trailing arm" arm" of of the the spirals, spirals, as as shown shown in Fig. 88 (H. N. Comins Comins the in Fig. (H. N. and and M. M. P. P. Hassell, Hassell, unpublished). unpublished). Since Since the the foci foci of of the the spirals spirals are are relatively relatively static static in less mobile mobile species occur only in these these models, models, the the less species appears appears to to occur only in in isolated, isolated, small small "islands" "islands" within within the the habitat, habitat, much much as as if if these these were were pockets pockets of of favorable favorable habitat. habitat. As less divergent divergent between As the the migration migration rates rates become become less between the the species, species, the the niche niche of of the dispersive species the less less dispersive species spreads spreads further further into into the the arm arm of of the the spirals. spirals. Such Such spatial spatial segregation species, purely segregation of of the the competing competing species, purely as as aa consequence consequence of of the the dynamics, dynamics, is is an an intriguing intriguing property property of of these these spatial spatial models. models.
IV. CONClUSION CONCLUSION Theoretical Theoretical studies studies of of spatially spatially distributed distributed populations populations with with restricted restricted migra migration between have revealed for populations populations to to tion between patches patches have revealed a a fundamental fundamental tendency tendency for become into spiral local abundance. become spatially spatially organized organized into spiral or or chaotic chaotic patterns patterns of of local abundance. Spirals are cycles in population size time, while Spirals are associated associated with with cycles in average average population size over over time, while chaotic condition chaotic patterns patterns lead lead to to time time series series that that are are also also chaotic. chaotic. A A necessary necessary condition for dynamics, but with very for these these patterns patterns are are unstable unstable local local dynamics, but the the results results persist persist with very low low host host rates rates of of increase, increase, with with very very low low migration migration rates, rates, and and even even if if aa small small minority minority of of adults adults disperse disperse much much more more widely. widely. Several Several interesting interesting features features follow follow on on from from these these patterns, patterns, such such as as ((1) the spatial spatial segregation segregation of of competing competing species species 1 ) the
66
Two-Species Two-SpeciesMetopopulotion MetapopulationModels Models
114Z 47
described in the previous section and (2) the relative non-invasibility of of popula populations showing spiral waves (Boerlijst et 993). Theory et ai., al., 11993). Theory is far ahead ahead of of exper experiment and observation in this instance. instance. While direct observation of of these kinds of of spatial dynamics dynamics in the field presents presents enormous logistical problems, problems, it may be possible to determine determine properties properties of the population density density distributions which which are diagnostic of of spirals or spatial chaos (for example, example, particular particular patterns patterns of of delayed covariance). covariance). Work Work of of this kind would be most welcome to facilitate facilitate bridging bridging the gap between theory theory and and empirical empirical results.
sdfsdf
From From Metopopulotion Metapopulation DDynamics ynomics to Communit Community Structure y Structure Some Some Consequences Consequencesof Spatial Spatial Heterogeneity Heterogeneity Robert D. Holt
NTRODUCTION I. IINTRODUCTION The most fundamental fundamental structural properties of of a local community are the member species and the pattern of of their number and relative relative abundances of of its member dynamical interactions ((Roughgarden Roughgarden and Diamond, 11986). 986). The history of of com community ecology largely revolves around variations on a small number number of of perennial perennial 1 ) the relationship between species diversity and environenviron themes, including: ((1) mental heterogeneity (e.g., resource resource diversity or disturbance disturbance regimes; Chesson, 11986; 986; Huston, 11994), 994), (2) the implications of of direct and and indirect indirect interactions interactions for community structure (e.g., dynamical constraints on food chain length; Pimm, 11982; 982; Schoener, 993; Wootton, 11994), 994), and Schoener, 11993; and (3) historical contingency, such as multiple stable states (e.g., priority effects effects in competition). A consideration consideration of of spatial spatial dynamics can enrich enrich all these traditional themes in community ecology. The insight that local colonizations and extinctions de determine rst articulated 960s in the theory termine local community structure structure was fi first articulated in the 11960s of MacArthur and Wilson, 11967). 967). This seminal work was of island biogeography ((MacArthur soon complemented interactions of complemented by analyses of of the effects on species species interactions of patch 1 97 1 ; dynamics and spatial fluxes in mosaic landscapes (e.g., Levins Levins and Culver, 1971; Horn and MacArthur, 972; Levin, 11974; 974; Whittaker and Levin, 1977; 1 977; Holt, 1985), 1 985), MacArthur, 11972; Metapopulalion Metapopulation Biology Biology Copyright © 997 by Academic Press, Press. Inc. All rights of 9 11997 of reproduction in any form reserved.
1149 49
1150 SO
Robert Holt Robert D. D. Holt
a line of of thinking thinking which in recent recent years has crystallized into a rich body of of theory under 1 99 1 ). under the rubric of of "metapopulation "metapopulation dynamics" dynamics" (Gilpin and Hanski, Hanski, 1991). If If a "metapopulation" "metapopulation" is defined to be a set of of local popUlations populations coupled coupled by dispersal Hanski, 11991), 99 1 ), a "metacommunity" dispersal ((Hanski, "metacommunity" may be defined defined simply as a set of of local local communities communities in different different locations, locations, coupled coupled by dispersal dispersal of of one one or more of of their constituent 99 1 , p. 9). At present, constituent members members (Gilpin and Hanski, Hanski, 11991, present, there there is an explosion explosion of of interest in the consequences consequences of of spatial dynamics for for single-species single-species dynamics Hanski, this volume), interactions dynamics ((Hanski, interactions between between species species (e.g., Bengtsson, Bengtsson, al., 11994; 994; Kareiva Wennergren, 11995; 995; Nee ai., this volvol 11991; 99 1 ; Hassell et et al., Kareiva and and Wennergren, Nee et et al., ume), and, more more broadly, the structure of of entire entire ecological communities communities (e.g., Case, 11991; 99 1 ; Nee 992; Tilman, 994; Caswell 993; Holt, Nee and May, May, 11992; Tilman, 11994; Caswell and Cohen, Cohen, 11993; Holt, 11993). 993). My aim in this chapter pertinent chapter is not not to to provide provide a synoptic overview overview of of all pertinent work metapopulation dynamics for community structure. work on the implications implications of of metapopulation structure. Instead, in Instead, I use variants variants of of standard standard metapopulation metapopulation model model to examine several interlinked terlinked questions in community ecology which which have not to date been been examined examined in depth, 1 ) How heterogeneity depth, but but deserve deserve further further attention: attention: ((1) How does landscape landscape heterogeneity influence the composition of of local communities? communities? (2) Can metapopulation metapopulation dynam dynamics constrain food chains? constrain food food web web structure, structure, for for instance the average average length of of food chains? (3) When When do indirect indirect interactions interactions constrain constrain community community membership membership at the level of landscapes? In this chapter, of standard of entire landscapes? chapter, I use straightforward straightforward extensions extensions of standard metapopulation 99 1 ) to examine focus metapopulation models models (e.g., Hanski, 11991) examine these questions. questions. My focus development and and the the articulation articulation of of hypotheses hypotheses which which warrant warrant em emis on theory development pirical pirical scrutiny.
II. OF LANDSCAPE ETEROGENEITY ON ON LOCAL II. EFFEGS EFFECTSOF LANDSCAPEHHETEROGENEITY LOCALCOMMUNITY COMMUNITYCOMPOSITION COMPOSITION Imagine a landscape landscape that has has been been colonized colonized over over an evolutionary time scale from a larger rst a noninteractive noninteractive com larger species pool. For For simplicity, I consider consider fi first community (i.e., no interspecific interspecific competition competition or predation) predation) and and examine examine the influence influence of of heterogeneity heterogeneity at the landscape landscape level on local community community structure. structure. Most meta metapopulation population models models to date have have assumed assumed that the patches patches comprising comprising the meta metapopulation 1 972) and population are are physically homogeneous homogeneous [though [though Hom Horn and and MacArthur MacArthur ((1972) Hanski 1 992b, 11995) 995) do consider inter Hanski ((1992b, consider habitat heterogeneity in the context context of of interspecies species competition]. competition]. Yet, in practice, practice, large areas almost almost always encompass encompass spa spa1 98 1 ; Holt, 11992). 992). Such tially heterogeneous heterogeneous local conditions conditions (Williamson, (Williamson, 1981; Such re regional heterogeneity can influence influence local community community structure structure in a variety of of ways, particularly if if rarer species are are considered. considered. Species abundance abundance distributions distributions typically reveal that that a substantial substantial fraction fraction of of 994). Surveys species in local communities communities consists of of rare species (Gaston, (Gaston, 11994). conducted conducted at mUltiple multiple sites, replicated replicated over over time, often often show that that many many rare spe species display a pattern of of local extinctions extinctions and and recolonizations. recolonizations. For For instance, instance, in the the Eastern 1 98 1 , pp. Eastern Wood Wood study of of a bird community community discussed discussed by Williamson Williamson ((1981, 93 - 100), 28 of recorded species 93-100), of 44 44 recorded species went went locally extinct at least once once in the the 26
77
Consequences Consequencesof Spatial Spatial Heterogeneity Heterogeneity
1! S511
years of of the study. For For some of of these these species, species, "extinctions" "extinctions" may be be recorded recorded experi because the site provided only a small sample drawn drawn from populations populations experiencing the landscape at a coarser coarser spatial scale (1. (J. Bengtsson, Bengtsson, personal commu communication). nication). For For other other species, the Eastern Eastern Wood Wood population population may may be part part of of a clas classical metapopulation, metapopulation, in which which a balance balance between between colonization colonization and and extinction extinction across across the landscape landscape permits permits regional regional persistence, persistence, despite despite the ephemeral ephemeral occur occurrence Hanski, this volume). rence of of populations populations in local communities communities ((Hanski, volume). However, 1 994b; see also Harrison However, Harrison Harrison ((1994b; Harrison and and Taylor, this this volume; volume; Schoener, 11991) argued that species species which which in a particular particular patch patch network network show show Schoener, 99 1 ) has argued frequent frequent extinctions extinctions and and colonizations, colonizations, may may actually have have a few few persistent persistent pop poppersistence. Moreover, ulations, which which permit permit overall overall persistence. Moreover, a local population that never never goes extinct extinct may nonetheless nonetheless prove prove to be a sink population, population, maintained maintained by a regular ow of regular fl flow of individuals individuals from from self-sustaining self-sustaining source populations populations (Shmida (Shmida and Ellner, 984; Holt, 985, 1993; 1 993; Pulliam, 988; S. Hubbell, Hubbell, personal Ellner, 11984; Holt, 11985, Pulliam, 11988; personal communi communication). cation). Local species species richness richness thus thus may be enhanced enhanced if, at the landscape landscape scale, habitat habitat long heterogeneity provides provides each each species species with some some habitat habitat patches permitting longterm persistence. persistence. Guaranteed Guaranteed local survival survival in some some habitats habitats allows allows a species species to be present range present (e.g. (e.g.,, as rare transients transients or sink populations) populations) over over a much much broader broader range of useful to consider meta of habitats. To examine examine this effect effect in more more detail, detail, it is useful consider a metapopUlation model that incorporates population incorporates habitat habitat heterogeneity in colonization colonization and and ex extinction rates. rates.
A. A Metapopulation Metapopulation Model Model for a Heterogeneous Heterogeneouslandscape Landscape Assume patches, of Assume that that the landscape landscape consists of of a large large number number of of habitat habitat patches, of For simplicity, consider which which a fraction fraction H H are are suitable suitable for for the community. community. For simplicity, I will consider that that just just three three habitat habitat types are are present: present: patches patches of of two two distinct distinct habitat habitat types, potentially potentially occupied occupied by species in the community, community, embedded embedded in a third third matrix matrix habitat, unsuitable unsuitable for for any of of them. Let Let hi h/be of habitat habitat patches of be the fraction fraction of patches of 1 . Some species hab type i. Necessarily, h h i1 + -k- h h 2 2 = -- H H � --< 1. species in in the community may be habitat specialists on just just habitat I1,, others specialists on habitat 2, and yet others may be habitat generalists, able to use both habitat habitat types (possibly to different degrees). patches of p;i denote denote the fraction f r a c t i o n of of habitat habitat patches of type i occupied occupied by a focal focal Let P species. species. The The total total occupancy occupancy of of this species species over over the entire entire landscape landscape is P p = = Let ei be the rate of extinction of the focal species in patches of habitat · PlI + + P P2. P 2 Let eg be the rate of extinction of the focal species in patches of habitat type i and patches due to emigration from and ci cijj the rate rate of of colonization colonization of of type i patches from patches following model dynamics of patches of of type type jj (i, jj = = 11,, 2). The The following model describes describes dynamics of the the total metapopulation: metapopulation:
dp dp~i
=- - (( CCIl l PIPI l dt dt d dp~ P2
=- - (( cC221 PI PI l dt dt
+ P I ) - ee~IPl I -k- C1 Clzpz)(hl 2P2 )(hl --Pl) -
+ -k-
-
cc22P2)(h - e e2P2 22P2 )(h22 --PP22)) 2 P2
((1) 1)
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RobertD.D. Holt Holt Robert
In In aa metapopulation metapopulation with with homogeneous homogeneous habitat habitat patches, patches, colonization colonization and and extinction extinction rates rates should should be be independent independent of of patch patch type, type, so so cij co = = Cc and and ej ei = = e. e. Model Model then reduces reduces to to the the usual usual form, form, dp/dt dp/dt = = cp(H c p ( H - - p) p) - ep e p (Levins, (Levins, 11969a; ((1) 1 ) then 969a; Hanski, 11991, this volume). Hanski, 99 1 , this Let Let us us consider consider first first aa species species specialized specialized to to just just habitat habitat i. By By assumption, assumption, this species species cannot cannot occupy occupy habitat jj at at all, all, hence hence Pj pj = = O. 0. When When rare, the species increases per occupied increases at at aa rate rate ((per occupied patch) patch) of o f ccjjhj iihi - ej ei and and equilibrates equilibrates with with aa fraction fraction p ne the 15 = = hj hi - eJcjj ei/cii of of the the landscape landscape occupied. occupied. If If we we defi define the "conditional "conditional incidence" incidence" li of of species ii to be the probability probability that that it occupies occupies a patch, given ((Holt, Holt, 11993) 993) Ij that that the patch patch is suitable suitable for for it, then at equilibrium Ij li = = 11 - eJhjcjj ei/hicii. . The species will persist in the landscape (without repeated invasion from an external external source pool) only if cjjhj ciihi > > ej ei.. Assume Assume that that in in the the regional regional source source pool, pool, species species specializing specializing to to the the two two habitats are equally common, and that each ensemble of habitat specialists can be described with the same bivariate frequency distribution of colonization and extinction rates. Further, assume that in the landscape habitat I1 is the commoner, i.e., hI h~ > > hh2. rarer habitat (habitat type 2) should have fewer 2 • It follows that the rarer species that are habitat specialists. Moreover, those specialist species which are present should on average have lower overall occupancies, compared with species specialized to the commoner habitat. If species have incidences, but in their If two two species have equal equal conditional conditional incidences, but differ differ in their habitat habitat specialization, must have have a specialization, the the species species specializing specializing in in the the rarer rarer habitat habitat must a higher higher colonization rate rate or a lower extinction rate. Extinction and and colonization rates reflect reflect many aspects of individual and population ecology, such as life history responses to disturbance, disturbance, temporal temporal dynamics in local popUlation population abundance, abundance, re respecialization, resistance resistance to resident predators, and so forth. Hence, Hence, there there source specialization, differences in entire suites of of ecologically relevant traits should be systematic systematic differences between ensembles species specialized ensembles of of species specialized to rare, rare, as opposed to common, habitats. habitats. To go further further with this line of of reasoning, reasoning, one one would need need to specify specify statistical statistical To parameters e and and Cc among among specialist species species in the species species distributions for for the parameters pool. This This would would be be an an interesting interesting exercise, exercise, but but at at the the present premature, pool. present juncture juncture premature, the paucity paucity of of data data on these these parameters parameters at at the the level of of entire entire guilds guilds or or given the communities. The The above above theoretical theoretical results results provide provide testable testable hypotheses hypotheses for future communities. for future comparative community community studies studies of of rare rare and and common common habitats. habitats. I now now turn tum to to a comparative and generalists. generalists. comparision comparision of of habitat habitat specialists specialists and
B. Habitat Habitat Specialists Specialists and Generalists The cross-habitat cross-habitat colonization colonization terms terms in in model model (1) ( 1 ) represent represent aa kind kind of of landland The "mutualism": the the incidence incidence of of aa species species in in one one habitat habitat type type may may be be enhanced enhanced scape "mutualism": scape because the the species species is is present present in in another another habitat habitat as as well. well. If If aa species species can can colonize colonize because patches of of aa second second habitat habitat type, type, without without reducing reducing its its rate rate of of colonization colonization of of patches patches of of the the first first habitat habitat type, type, itit obviously obviously should should be be able able to to persist persist better better in in aa patches
Consequences Consequencesof Spatial Spatial Heterogeneily Heterogeneity
77
1153 53
heterogeneous landscape. landscape. Moreover, a habitat generalist may be a member member of of the the local local community community in a particular habitat type, though this species species would disappear disappear in a homogeneous homogeneous landscape landscape consisting entirely of of just that that habitat type. The The above above model model permits permits aa closer closer analysis analysis of of these these effects. effects. To To determine whether whether or not a species species can persist, one examines examines its rate of of increase when when it is rare rare (i.e., at low occupancy). If If a species increases increases when rare, increase it it will persist, persist, whereas whereas if if it it decreases decreases when when rare, rare, it it is vulnerable vulnerable to to extinction. extinction. When When aa species species is is rare rare across across both both habitat habitat types, types, we we can can approximate approximate the the above above model with a pair of of linear linear differential equations. The The initial growth growth rate rate of of the species when rare is given by the dominant dominant eigenvalue eigenvalue of of this simpler simpler model, species
AA(h,, (h " h2) h2)
2
= (A , = �89[A [A~, + + A2 A2 + + .J ~/(A, - A2 A2)) 2 + + 4 4 C'2c2 Cl2C2~hlh2], l h , h2 ] ,
(2) (2)
where where Ai = c i i h i -
(3) (3)
ei
is the the rate of of metapopulation metapopulation growth when when habitat type type i alone is available in the landscape. landscape. If xed and increases with h2. If h h~, is is fi fixed and C'2C2' c~2c2~ >> 0, 0, A h2. Thus, the the ability ability of of aa species species A increases to utilize a second habitat may facilitate its persistence in a heterogeneous land to utilize a second habitat may facilitate its persistence in a heterogeneous landscape. parameter Consider Consider the the special special case case of of CCllC22 - - Cl2C2 C12C21. This constraint constraint on on parameter l l C22 = l . This values could arise arise in in two two biologically biologically distinct distinct ways, each each quite plausible plausible in in dif different circumstances: circumstances:
11.. Colonization could be determined determined entirely by the site site of of colonization (i.e., = the presence presence or particular - - C'2 C12 and and C22 c22 = = C'2 C 1 2 ). ) . For For instance, instance, the or absence absence of of aa particular mortality factor, say a natural influence the likelihood of natural enemy, could influence of local If the natural natural enemy is found predictably in some habitats, but not colonization. If others, this should lead to spatial heterogeneity in colonization rates. 2. Colonization rates of of empty patches patches could be determined determined entirely by the C l l = czl, C2" and For example, site dispersers (i.e., site of of origination origination for for dispersers (i.e., Cl~ and C22 c22 - " C'2 C 1 2 ). ) . For example, the the two habitat habitat types could differ differ in the local average average abundances abundances achieved by a species. If If individuals individuals emigrate emigrate at a constant per per capita rate, the habitat type with larger populations populations will exert a disproportionate effect effect on the colonization of of empty larger patches. patches. Clll Cl
=
=
In defining the In this this special special case, case, the the combination combination of of parameters parameters defining the sign sign of of the growth rate when the species species is rare is given by the following expression:
c22h2 e2
Cllhl el
G - - ~ + ~ .
When When G < < 11,, the metapopulation declines declines toward toward extinction; conversely, when when G > > 11,, the metapopulation grows when it is scarce scarce in the landscape. landscape. If If each each ciihJe; Gih~/e~ > > 11,, the the species species could could persist persist in in either either habitat habitat alone. alone. If If each each Gih~/ei < < I1,, but but G > > 11,, a species can persist persist in the entire entire landscape, landscape, even even though though c;;hJe;
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Robert Robert D. D. Holt Holt
it cannot 1 ), for for cannot persist in any any single single habitat habitat alone. At At equilibrium equilibrium in model model ((1), habitat habitat i 0 = [ciiP~ ( h i -
p?
-
e;)] + cijp* (hi - Pi* ).
(4) (4)
If O. The The bracketed bracketed If the cross-habitat cross-habitat parameters parameters cij are positive, positive, both both p p*i > > 0. larger P term term on on the the left left of of (4) (4) is is zero zero at at P Pii = = hi h i -- eJcii ei/cii and and is is negative negative for for larger Pi.i' Because negative; Because the right-hand right-hand term term is positive, positive, the bracketed bracketed term term must must be be negative; because hence hence p p*i > > P P ii.' Thus, Thus, aa species species'' incidence in in one one habitat habitat type type is is enhanced enhanced because permits a of spatial spatial coupling coupling with the second second habitat habitat type. Habitat Habitat generalization generalization permits of species habitat, given species to be present present with a higher higher than expected incidence in each each habitat, given colonization colonization and and extinction extinction rates rates for for each each habitat habitat in isolation. isolation. Model 1 ) illustrates Model ((1) illustrates how how "spillover" "spillover" between between habitats habitats can can enrich enrich local com comHolt and 992), munities. Assume Assume that that habitat habitat 2 is a "black-hole" "black-hole" sink ((Holt and Gaines, Gaines, 11992), which which can can be colonized colonized but does does not not provide provide colonists colonists for for either habitat habitat type (for (for concreteness, concreteness, one one can imagine imagine that that popUlation population densities densities are very low in habitat 2, so these popUlations populations provide negligible negligible sources for for colonists). colonists). Hence, Hence, Cc~1 I and and C2 CI2 O. The 12 = c2~1 > > 0, but but C22 c22 = = C12 = - - 0. The incidence incidence of of the the species species in in habitat 2 is 12 i i i = pi Ih2 = which increases with pt. If H is then p , + c2I l(e2 p*/h2 = c2 c2~*/(e2 + c2,p*), which increases with p*. If H is fixed, then p* = IP P ) H , which decreases linearly with h2 • Hence, the incidence ( prob eJc H h 2 e l / C l l , I I which decreases h2. (probh2 ability of increases in habitat 2, of occurrence, occurrence, per per patch) of of this spillover spillover species species increases the less habitat type, relative frequency of habitat type less frequent frequent is this habitat relative to the frequency of the habitat that metapopulation. that actually sustains a viable metapopulation. Species rates in their preferred hab Species with high high colonization colonization or or low extinction extinction rates their preferred habitat should should exhibit a high high occupancy occupancy in this habitat habitat and and can can secondarily secondarily have have a high high incidence incidence in habitats habitats where where they cannot cannot persist. persist. Such Such spillover spillover effects effects should should be most involving in particular particular those those species most noticeable noticeable in rare rare habitats, habitats, involving species with high high occupancy occupancy in frequent frequent habitat habitat types. In some some circumstances, circumstances, utilizing a second second habitat habitat may may permit permit a species species to persist persist in a landscape landscape even even if if there there is no no colonization colonization among among patches patches of of the second second habitat habitat type. For For instance, instance, imagine that that patches patches of of habitat habitat 2 are are overdispersed, overdispersed, sufficiently far the species cannot far apart apart that that C22 c22 = - - 00, , and and furthermore furthermore that that the cannot persist in habitat habitat 11 alone. The The condition condition for for such such a species to persist persist in the entire land landscape scape is is
Cllhl Cllhl hlhzClzC21 < 1< + ~ . el el ele2
(5) (5)
The The right-hand right-hand inequality is always always met met if if e2 e2 is sufficiently small. Sparse Sparse habitats habitats with with low local extinction extinction rates rates can have have a large large effect on the the overall overall persistence persistence of of a species, even even if if the geometry geometry of of the landscape landscape does does not permit permit such such habitats habitats to sustain the species on their their own. In effect, colonization colonization of of sparse sparse but provides a kind "spatial storage but low-extinction low-extinction habitat habitat patches patches provides kind of of"spatial storage effect" effect" (Holt, (Holt, 11992), 992), amplifying colonization colonization rates overall in the the more more widespread widespread habitat.
77
Consequences of Heterogeneily Consequences of Spano SpatialI Heterogeneity
1155 55
This test This two-habitat metapopulation model model leads to several interesting and testable conclusions. In a heterogeneous landscape: conclusions. landscape: 11.. Habitat Habitat specialists will be disproportionately disproportionately common common in those habitats habitats that are most common in the landscape. landscape. because they can 2. Some generalists generalists may persist persist in the the landscape landscape precisely precisely because can exploit a range range of of habitat habitat types. 3. Species which which can persist persist in one habitat habitat can thereby thereby incidentally incidentally occupy other communities. This spillover effect other habitats, habitats, enriching enriching those those local communities. effect should be particularly defining the community membership in sparser particularly important important in defining sparser habitats habitats and be characterized commoner habitats. characterized by species species with with high occupancies occupancies in commoner habitats. rates, or 4. Specialists on on rare rare habitats habitats should should have have unusually low low extinction extinction rates, or high landscape high colonization colonization rates, rates, relative relative to the entire entire ensemble ensemble of of species in the landscape (including habitat generalists). im (including both specialists specialists on common common habitats habitats and habitat generalists). This This imfactors plies a systematic bias bias at the community community level in entire entire suites suites of of ecological ecological factors correlated correlated with local extinction or colonization rates. The The above model model deliberately deliberately ignored ignored species species interactions. interactions. Yet, Yet, in practice, practice, habitat habitat suitability for for a given species and its local colonization colonization and and extinction extinction rates rates may be largely largely determined determined by interactions interactions with other other species. Several Several au authors homo thors have have considered considered metapopulation metapopulation models for for species interactions interactions in homoexamined com geneous geneous landscapes landscapes (e.g., see Nee Nee et et al. al.,, this volume) and and have have examined competitive interactions interactions in heterogeneous heterogeneous metapopulations metapopulations (Hom (Horn and MacArthur, MacArthur, 11972; 972; Hanski, 11992b). 992b). In the consider some implications the remainder remainder of of this paper, paper, I consider implications of metapopulation, using of trophic interactions interactions in a heterogeneous heterogeneous metapopulation, using natural natural exten extensions of of the above model. model.
III. FOOD CHAINS III. METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICSOF OF FOOD CHAINS The simplest specialist predator simplest trophic interaction interaction is the the one one between between a specialist predator and its prey, and the the simplest simplest food food web web is an unbranched unbranched chain of of trophic specialists. specialists. Here I first first consider consider a metapopulation metapopulation model model for for a three-level three-level food chain. chain. A food food Here chain describes describes a set of of tight sequential sequential dependencies dependencies among among species. species. In many many chain circumstances, circumstances, it is reasonable reasonable to expect expect that that such sequential sequential trophic trophic dependency dependency will lead lead to nested nested distributional distributional patterns, in which which a given species will be nec necessarily absent 1 993, absent in a patch patch if its required required prey population population is absent absent (Holt, (Holt, 1993, 11995). 995). Let the state of ed by the food chain of a patch patch be identifi identified the length length of of the the food chain it contains, contains, such that 1 " a patch that "0" "0" denotes denotes an empty patch, patch, ""1" patch with with just the basal basal prey species, species, "2" patch "2" a patch patch with both both the basal basal prey prey and and an intermediate intermediate predator, predator, and and "3" a patch with both both these plus plus a top predator. predator. The The fraction fraction of of patches found found in state i is the food chain denoted by Pi p;.' We We assume that the basal species species in the chain is a habitat habitat specialist specialist and that its required habitat habitat occupies occupies a fraction fraction hh < < 11 of of available available
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Robert Robert D. D. Holt Holt
patches landscape. The following model -extinction patches in the landscape. model describes describes colonization colonization-extinction dynamics metapopulation: dynamics in this metapopulation: Npl
dt
-- (co~p~ + co~p2'
+
ColP3")(h - P~
-- (cl2P2 4- c ' ( 2 P 3 ) P l + c~ p2
dt p3
dt
-- P2
-- P3)
+ e21P2 -
elopl,
(6)
-- (cl2P2 + C'[2P3)Pl -- (e20 + e 2 1 ) P 2 + e32P3 -- r
= c23P3P2 -- (e30 + e31 + e32)P3.
For For clarity, the order order of of the subscripts subscripts for for the the colonization colonization and extinction extinction coef coefficients indicates the direction ow among ficients direction of of fl flow among states states (read them from from left to right). right). Thus, Thus, the ccij's denote the rate rate at which which colonization colonization transforms transforms patches patches from state u 's denote i to state j; j; the e;/s eij's likewise likewise set the rates rates of of extinction, extinction, changing changing patches from state i to state state j. j. In the basal parameter C� basal prey equation, equation, the parameter c~I arises arises because because empty patches patches can be colonized colonized by prey prey originating from patches patches with both both the basal prey prey and and the intermediate intermediate predator. predator. Likewise, C� c0'I~denotes denotes colonization colonization of of empty patches patches by basal prey emigrating predators emigrating from patches patches with both both the intermediate intermediate and and top predators (as well as the basal prey), and C'{ c'~2 describes colonization colonization of of prey patches by 2 describes intermediate intermediate predators predators dispersing dispersing from patches patches with the full food food chain. chain. If If these these parameters parameters are are positive, positive, colonization colonization dynamics dynamics at lower lower trophic levels levels involves habitat 1 ) (although such heterohetero habitat heterogeneity, heterogeneity, comparable comparable in spirit to model model ((1) geneity is not a fixed landscape feature, feature, but but instead instead emerges as a dynamical dynamical feature feature of interactions). of the trophic trophic interactions). The most important important assumption assumption made made in the the above above model model is that the the food chain 982), and that if chain builds builds up via sequential sequential colonization colonization (see, e.g., Glasser, Glasser, 11982), a prey population population goes extinct extinct in a patch, patch, so does any predator directly or indi indirectly supported nes of premises, a wide supported by that that prey. Within Within the confi confines of these key premises, wide range range of of assumptions assumptions about about local local dynamics dynamics can be embodied embodied in the model. model. It is useful useful to examine examine the the properties properties of of this model by building building it up up from its base. The The basal species, species, on its own, satisfies the standard standard metapopulation metapopulation model model dpl
dt h
= Colpj(h - P l )
- elOPl.
p; = ok'o l ' The At equilibrium, equilibrium, p'~ = h - ee r~o/Co~. The basal prey species species persists persists provided -
h > elo
e ro . CCO1 Ol
h>~.
This This inequality also ensures ensures that the basal basal species increases increases when when rare.
(7)
Consequences Spotial Heterogeneily Consequencesof Spatial Heterogeneity
77
1157 57
A. Two Trophic Trophic Levels Levels When When the intermediate intermediate predator predator is also present, present, the model model takes the the form form
dpl == ((co~p~ CO I PI + P2 ) + c C oIP ' o l P2z ))((hh - P P lI - P2) dt dt
dp,
I
--
Cl2P2Pl
+
2 dt
=
Cl2P2Pl
--
e21P2 -- e loPl
(8)
(e20 + e21)P2.
The The model model resembles resembles a standard standard predator-model, predator-model, but but with with the crucial dif difference ference that that the predator predator patches patches also contain contain prey and and can can therefore therefore contribute contribute rate of of generation generation of of new new prey prey patches, either by prey colonization colonization of of empty to the rate patches, either patches, patches, or by by predator predator extinctions extinctions unaccompanied unaccompanied by by prey extinctions. extinctions. For For sim simplicity, we will assume only if assume that that the predator predator goes goes extinct extinct locally only if the prey prey also goes the former former effect. goes extinct extinct (i.e., (i.e., ee21 = 0), so here here I consider consider only the 21 = There predator may have on its prey There are are two two kinds kinds of of effects effects a specialist specialist predator may have prey in this model: it may may alter the prey prey extinction extinction rate, rate, or or it may change change the rate rate of of prey colonization colonization of of empty patches. patches. In general, these effects effects could could be either positive positive or negative: negative: 1. Biogeographic Biogeographic "Donor "Donor Control" Control"
Some Some predators predators may may have have negligible negligible effects effects on local prey prey dynamics dynamics and and so = ee:0 are are unlikely to alter prey prey colonization colonization or or extinction extinction rates, rates, i.e., eel0 and 10 = 20 and 'I This (DeAngelis, 11992) 992) in a spatial = C� C~l. This is "donor "donor control control"" (DeAngelis, spatial context: context: prey Cc01 Ol = dynamics may constrain predator, without constrain the the distribution distribution of of the predator, without reciprocal effects predator on its prey. by the predator 2. Increased Increased Prey Prey Extinction Extinction
The the literature The scenario scenario that that has has received received by far far the most most attention attention in the literature on prey interactions predators reduce prey abunabun local predatorpredator-prey interactions is the one one in which predators dances dances so greatly that both both populations populations face face enhanced enhanced extinction extinction risks (i.e., eel0 < lO < ee20) 975; Taylor, 11991; 99 1 ; Hassell eett al., 1 992). Even Gilpin, 11975; a l . , 1992). Even in the absence absence 20 ) (e.g., Gilpin, of of any effect effect of of the predator predator on prey abundance abundance in "typical "typical"" years, the predator predator may during episodes disturbance, reduced reduced may heighten the risk of of prey prey extinction extinction during episodes of of disturbance, prey prey resources, resources, or or extreme climatic events. events. Even in predator-prey predator-prey models with with stable equilibia bounded bounded well away from from zero, following following large perturbations perturbations there there can can be transient phases phases at low densities, densities, greatly increasing increasing the likelihood likelihood of of local extinction unpublished results). extinction for for both both species species (R. D. Holt, unpublished results). 3. 3. Decreased Decreased Prey Prey Extinction Extinction In a wide wide range range of of circumstances, circumstances, predators predators can reduce reduce the magnitude magnitude of of fluctuations 973b; Rosenzweig, Rosenzweig, 11973) 973) or fluctuations in prey abundances abundances (May, 11973b; or even even in-
1 58 158
Robert D. D. Holt Holt Robert
crease average average prey prey abundances abundances (Abrams, (Abrams, 1992). 1 992). For For instance, instance, if if prey prey respond respond crease behaviorally to to predators predators by by reduced reduced exploitation exploitation of of their their own own resources, resources, overover behaviorally et al. al. (1985) ( 1985) exploitation may may be be less less likely likely in in the the presence presence of of aa predator. predator. Sih Sih et exploitation reported aa surprising surprising number number of of cases cases in in which which removal removal of of aa predator predator led led to to aa reported decrease in in the the abundance abundance of of the the focal focal prey. prey. Many Many of of these these cases cases seem seem to to involve involve decrease indirect interactions interactions in in multispecies multispecies assemblages (e.g., competitive competitive interactions interactions indirect assemblages (e.g., among prey, prey, held held in check check by by generalist generalist predators), predators), but but it it is is not not clear clear that that all all do. do. among In cases where aa predator predator enhances enhances the the mean mean abundance abundance or or reduces reduces the temporal In cases where the temporal variability of of its its prey, prey, it it is is conceivable conceivable that that ee 10 e 20 ' variability J O >> e2o.
4. Decreased Decreased Prey Prey Colonization Colonization 4. If local local prey prey densities densities are reduced by by predation, predation, the the flux flux of of dispersers dispersers If are greatly greatly reduced available for colonizing empty empty patches patches is is likely to to be be reduced reduced and, and, hence, hence, we we available for colonizing COl > C� could expect that could expect that c01 > c~.I '
5. Increased Prey Prey Colonization Colonization 5. Increased If predators predators increase above, then predators may may If increase local prey density, as noted noted above, then predators also indirectly facilitate facilitate prey if prey differentially also indirectly prey colonization. Alternatively, if prey differentially disperse in response response to perceived perceived increases increases in the local risk risk of of predation, of disperse predation, rates rates of prey emigration may be higher higher from from patches patches with predators predators than from patches patches prey emigration may than from without predators. In such cases, one might expect expect that C� I << c01. C� I ' without predators. In such cases, one might that c~ In general, are more general, I suspect suspect that scenarios scenarios 1, 1 , 2, and and 4 are more likely than than either either 3 or 55.. In comparisons among important to to keep keep in mind mind or among systems, systems, however, it is important the potential for "counterintuitive" effects effects of of predators predators on prey prey extinction or or colcol the for "counterintuitive" onization rates. onization rates. As noted noted above, when the prey occurs occurs alone it equilibrates equilibrates at p'~ = h h -p; = 0, or increase when that CI The predator predator can can increase when rare rare provided provided that c~2pm > 0, or ee Jlo/Co~. O /col ' The 2P I -- ee2o 20 >
20 > ee Jl_O__qo+ + ~ee2o . • hh > CCOl I2 O l CC12
(9) (9)
This 1 994) notes, the This simple simple result has has several implications. First, as May ((1994) requirement predator in a metapopulation metapopulation is more requirement for for persistence of of a specialist specialist predator of its prey (compare conditions stringent than the requirement for for the persistence persistence of persist in rare habitats, (9) and (7). Specialist predators are are not likely to persist habitats, unless unless they have very high colonization rates or very low extinction rates. Predators which increase the extinction rate of their prey are particularly unlikely to persist in in rare rare habitats. habitats. As a limiting case, consider a donor-controlled system (i.e., ee20 = eJ e~0) in O ) in 20 = which the predator colonization rate rate is a times that of of its prey. In this case, the predator can increase when rare provided a a > > ((11 - 1)11, I ) / I , where I denotes the equilibrial incidence of the prey when alone. A predator specializing specializing on a prey with low incidence (i.e., I < << 11)) must have a much higher colonization rate than that of of its prey. Hence, in a heterogeneous landscape, landscape, specialist predator-prey predator-prey
77
Consequences Spanal Heterogeneity Consequencesof Spatial Heterogeneity
1159 S9
interactions should occur widespread habitat types. occur disproportionately in the more more widespread Those Those specialist interactions interactions which do occur in rare habitats habitats should involve pred predators which either have unusually high colonization rates, rateS, or little effect effect on prey extinction rates, or involve prey which themselves themselves are habitat habitat generalists. generalists. The The above conclusions conclusions were drawn from the the condition for the the predator predator to increase increase when when rare, rare, aa condition condition which which provides provides aa criterion criterion for for robust robust persistence persistence face of of large large perturbations perturbations in the fraction of of patches patches occupied occupied by either either in the face species. The above model also defines a joint equilibrium equilibrium for for the predator predator and prey with both present present at positive occupancies, occupancies, as follows (recall we are assuming assuming that = 0 that ee21 ): 2 l = 0):
p 20/c 1 2 P*� = = ee2o/Cl2,
'
p *� = = 89 + .J x/AA 22 + 4 B], B], p �[A ± where
( ) p* p� ( e,0 h - p� - )
A == =- hh - p * A -
,
B == CO l B---col---7 col COl
1
CO ( 1I + + Co__l__~+ +.l C 'ol, CO 2 f
h-p*
-
((10) 1 0)
12 Ccl2"] C 'ol, / ' CO "
e lO
-
Coll / CO
.
If < 0, hence aa joint equilibrium If both both A < < 0 0 and and B B < 0, then then pp*� < < 0, 0, and and hence equilibrium with both both species species present in positive numbers numbers does not exist. If If B > > 0, 0, then then the positive branch in the above solution leads leads to a unique positive equilibrium. The condition that that B > > 0 0 is equivalent equivalent to the condition for for invasion by the predator, predator, when the prey prey is is at at aa predator-free predator-free equilibrium. equilibrium. The The condition for for B B > > 0 0 can can be be expressed expressed invades, the system settles into a unique unique p*� < < p; P'I-. Thus, Thus, when the predator predator invades, as p equilibrium in which the prey is reduced reduced to a lower occupancy than than when alone. If when rare. However, a joint equilib If B < < 0, 0, the predator predator cannot cannot increase increase when equilibrium rium may nonetheless nonetheless exist if A > > 0, 0, (the larger branch branch in the above above solution). When When this occurs, occurs, the system exhibits exhibits alternative, alternative, locally stable stable states, one with, and pred and one without, the specialist predator. Moreover, Moreover, the equilibrium with the the predator present present present has the prey at a higher occupancy than when the prey is present alone. However, bl < I O • For However, this this outcome outcome is is impossible if if Cc~l < CO c0~, and ee20 > eelo. For alter alterl ' and 20 > prey system, the predator native equilibria equilibria to exist in this predatorpredator-prey predator must either either enhance As enhance the prey colonization rate, or reduce reduce the prey extinction rate, or both. As noted noted above, above, there there are are reasonable reasonable circumstances circumstances leading leading to to such such counterintuitive counterintuitive effects effects of of predation predation on prey dynamics. When When such effects effects are are present, present, it is feasible feasible for the metacommunity metacommunity to exist in alternative stable states.
B. Three Three Trophic Trophic Levels Levels Let Let us us then then return return to to the the full full food food chain chain model, Eq. (6). (6). Rather Rather than than attempt attempt aa full full analysis analysis of of this model here, here, II will will simply touch touch on on some some interesting limiting limiting
1160 60
RobertD.D. Holt Holt Robert
donor-controlled at each level and where pred predcases. Consider a system which is donor-controlled face extinction only when its prey goes extinct, but predators do ators in a patch face not c, e30 e3 11 =not affect affect prey prey extinction extinction rates rates (i.e., (i.e., CO! Col = = C� Col e30 = e20 == ' I == -=c, = e20 = ee, , and ande3 ezl1 = = 0). Without the top predator, the intermediate predator has an equilibrial e2 occupancy of of occupancy
ee ee Clll cC el The top predator predator invades if p� p* > > e/ e/c23, c23 , or p� p * ==h h -
-
- .
ee ee ee hh >>- +-m++ - + ~ CC Cl2 C23 C 12 C23
((11) 1 1)
predator shuts down prey emigration Alternatively, if the intermediate predator (i.e., (i.e., C� C~lI = = 0), 0), and and e2l e2~ = = e31 e3~ = = e32 e32 = = 0, 0 , the the top top predator predator increases increases when when rare rare provided provided h > el___~0+ C01
e2_.__0.0-4-
C12
+ C01
.
((12) 1 2)
\ C23 fl'
For both both special cases, the condition for for invasion by the top predator For predator is more more stringent than than the condition condition for for invasion predator (compare (compare invasion by the intermediate predator conditions ((11) or ((12) (4)).. A habitat rare to sustain the intermediate conditions 1 1 ) or 1 2) to (4» habitat that is too too rare intermediate predator will not contain contain the top top predator, predator, either. However, However, more more common common habitats predator habitats able to sustain a specialist intermediate predator, predator, but not may be able not a similarly specialized top predator. predator. Thus, Thus, if if there are constraints constraints on species species'' colonization colonization specialized abilities, food chains trophic specialists are are not abilities, long long food chains composed composed of of trophic not likely to to charchar acterize rare rare habitats (see also Schoener, Schoener, 1989). 1 989). The emerges from The basic basic conclusion conclusion that that emerges from this this model model is that that metapopulation metapopulation dynamics can can constrain of specialist chains, particularly particularly in hetconstrain the length length of specialist food food chains, het erogeneous where the erogeneous landscapes landscapes where the basal basal species is specialized specialized to a rare habitat. habitat. Trophic specialization specialization on such automatically forces habitat specialization Trophic such species automatically forces habitat specialization on species of of higher trophic rank, thereby experience experience all the on higher trophic rank, which which thereby the spatial spatial concon straints on on the distribution of compounded by additional straints the distribution of the the lower-ranked lower-ranked species, species, compounded additional limitations of of their their own own (Holt, 1993). 1 993). The full model alternative stable The model can can admit admit alternative stable landscape landscape states. For For instance, instance, a top top predator predator may may be be able able to to stabilize stabilize an an intrinsically intrinsically unstable unstable interaction interaction between between an an intermediate intermediate predator predator and and its its own own prey prey (May, (May, 1973b; 1 973b; Rosenzweig, Rosenzweig, 1973), 1 973), thus thus extinction for patches extinction rates rates may may be be low low for patches with with the the full full chain. chain. If If the the landscape landscape initially has has all all species in in all patches, patches, then then it it may may persist persist in in this this state state because because of of low extinction extinction rates. rates. However, However, if if the the system system starts with just just the the intermediate intermediate low starts with predator predator and and its own own prey, prey, the the intermediate intermediate predator predator may may go go extinct extinct because because of of highly cannot highly unstable unstable local local dynamics. dynamics. In In this this case, case, obviously obviously the the top top predator predator cannot invade, prey is absent. landscape may invade, because because its its own own prey absent. Thus, Thus, the the landscape may either either have have just just the prey the entire the prey alone alone or or the entire food food chain. chain.
77
Consequences Consequencesof Spotiol Spatial Heterogeneity Heterogeneity
1161 61
IV. APPARENT APPARENTCOMPETITION COMPETITIONIN METACOMMUNITIES METACOMMUNITIES Food Food chains are are useful useful starting points for examining examining the implications implications of of meta metapopulation dynamics for for community structure, structure, but most natural food webs are are much more complex, because because there are typically multiple species species on each each trophic 99 1 ). I will next level and complex linkage patterns patterns across levels (e.g., Polis, 11991). examine the potential for for strong indirect indirect interactions interactions arising in metapopulations, metapopulations, constraining constraining species membership membership in local communities. In standard standard food web models, species richness richness at intermediate intermediate trophic trophic levels is often limited by a com combination of of two mechanisms: mechanisms: exploitative compettion (via effects effects of of species at these levels on abundances abundances of of lower trophic trophic levels) and apparent apparent competition (via effects 1 982). effects on the abundance abundance of of higher trophic trophic levels) (Pimm, 1982). Consider a landscape of patches of two habitat types, each containing a single landscape of patches of two habitat habitat specialist. A generalist generalist predator predator which can exploit both prey species in the two habitats patches of habitats is present. If If predators predators can colonize patches of both habitat habitat types from patches patches of of either either type, the dynamics of of the two prey species species are are indirectly linked. If If predators predators increase increase prey extinction rates or depress prey colonization rates, it may be possible for other for one prey species to exclude indirectly the other species, in effect by providing a reservoir popu reservoir maintaining a resident predator population (Holt and Lawton, 11994) 994) To explore the potential for apparent landscape context apparent competition competition in a landscape consider the following model, which splices the forms of of model ((1) 1 ) (metapopu (metapopulation dynamics in a heterogeneous heterogeneous landscape) landscape) and model (7) (predator-prey (predator-prey metapopulation dynamics):
dPp lr
dt dt
== cClPl(hl - P ~P r -- qr ql)) - ee lrPr pl - p lP( cr (lcl qr rqr l r P r (h r -
+ +
Cc12q2) r2q2 )
dP2
p2 = Pic2rqr + -- c C2Pi z p z (hh 22 -- P2 P 2 - - q2 q2)) - e e 22P2 P 2 - - P2(czlql + C c22q2) 22q2 ) dt dt
((13) 1 3)
dql dt
= p l ( c l l q l + c12q2) -- e lqql
dq2 dt
- - P 2 ( c 2 1 q l q- r
-- e2qq2-
patches of Here, the P Pii are the fraction fraction of of patches of type i occupied occupied by by prey i, and the are the fraction fraction of of such patches occupied by both this prey and the generalist generalist qqii are
predator. standard metapopu predator. In the absence absence of of the predator, predator, each prey obeys a standard metapopulation model, in which Ccii and ei e i are respectively the colonization and extinction parameters parameters of the prey species specialized specialized to habitat i (which occupies a fraction hi of patches of the landscape). landscape). The The quantities quantities cij cij scale the rate rate of of colonization of of prey patches 1 ). of of type i by generalist generalist predators predators dispersing from type jj patches patches (as in model 1). The predator habitat-specific rate predator and prey go extinct in each each habitat at a habitat-specific rate eiq• eiq.
1162 62
Robert Robert D. D. Holt Holt
As in the food chain chain model model (6), we assume assume here here sequential sequential colonization, colonization, so that predators predators do not not colonize colonize a patch patch until it is occupied occupied by a suitable suitable prey prey 1 3) deals predator-prey population. However, model ((13) deals with only a subset of of the predator-prey particular, prey colonization occurs only interactions feasible feasible in model (6). In particular, colonization occurs from from patches patches in which which the predator predator is absent. absent. (Pennitting (Permitting colonization colonization from patches with with predators predators would make make an already parameter-rich parameter-rich model model even even more more complicated, consideration of complicated, so I defer defer until future future work work consideration of such more general general models.) Predators extinctions. Predators over-exploit their prey, coupling coupling predator predator to prey extinctions. When prey i is at equilibrium and alone, it occupies a fraction When prey is at equilibrium and alone, it occupies a fraction Pi = hi hi- Pi = eJ e i / ccii of of the the landscape. The predator, predator, when rare rare and invading invading a landscape landscape with prey prey i only only present present aa equilibrium, equilibrium, grows grows at at an an instantaneous instantaneous rate rate Ai i~ i = = Cii c i i PPi -i eiq• eiq. If If both both prey are are present present at equilibrium, equilibrium, then then expression expression (2) defines the initial growth rate function of rate of of the predator predator population, population, as a function of its growth rate rate Ai Ai in each each habitat, the consequences habitat, considered considered separately. All the the above above conclusions conclusions about the consequences of of habitat habitat generalization generalization on persistence persistence and and equilibrial equilibrial incidence incidence carry over over to a trophic generalist habitats, including generalist that encounters encounters different different prey in different different habitats, including prey prey species species unable unable to sustain sustain the predator predator population population by themselves, themselves, and and so forth. pos However, the present present system is dynamically much much more more complex complex than than was pos1 ), because sible in model ((1), because the prey have have their their own own colonizationextinction dy dy' s dynamics, namics, namics, constraining the predator predator's dynamics, and the predator predator can in tum turn drive prey extinctions. extinctions. Consider predator are present at their Consider a system system in which which prey prey 11 and and the the predator are present their respective respective equilibrial equilibrial occupancies: occupancies:
PP*l�
= =
ee l qlq � ~, , q CClI I1 q *
= =
Cl(hl -- e e l lqicl q / C l l )l ) -- e elj cl(h l -=----'----'-"--'-'--CCII + -3t- C ClI II
Prey species species 2 can can increase increase when when rare provided provided
1 @2 P2 dt
-- c2h 2 -
e2 -
c 2 1 q *1 >
O.
The The analogous analogous equilibrial equilibrial occupancies occupancies and and criterion criterion for for invasion invasion by species species 11 are transposing the indices indices 11 and 2 in the above above expressions. expressions. The prey given by transposing species species may may coexist coexist at the landscape level if if both both invasion invasion criteria are are satisfied satisfied simultaneously. simultaneously. The The resident resident prey prey indirectly reduces reduces the rate of of invasion invasion by a second second prey species, can invade patches species, because it sustains sustains a predator predator metapopulation metapopulation which can patches once once they contain contain the invading invading prey. This indirect indirect inhibitory effect, effect, called called appar apparent 977), arises co ent competition competition (Holt, 11977), arises even though the the two prey species species never cocompetition raises the pos occur within within any given habitat patch. Such apparent apparent competition possibility of of exclusion exclusion due due to shared predation predation in a metacommunity. metacommunity. A limiting case of potential for of the above model model suffices to illustrate illustrate the potential exclusion exclusion by apparent apparent competition. competition. For simplicity, assume assume that there are no solo prey prey extinctions extinctions (i.e., ei ei = - - 00), ) , that that the predator predator colonizes colonizes much much more rapidly rapidly than
77
Consequences of Spatial Spotiol Heterogeneity Consequences Heterogeneity
1163 63
it goes extinct, and that the predator colonizes the two habitats habitats indiscriminately (i.e., cij cij == = cq). Cq). With these assumptions, the criterion for for invasion by prey 2, 2, given that prey 1 I occurs occurs at equilibrium with the the predator, is approximately
c2h2 c2h 2 cClhl 1h 1
cq Cq
- > ---
cC11 + -~- cq Cq
Similarly, the criterion for for invasion by prey 11 is
+ cq cq Cq
C C2 + Cq
cr 2h2 c1h Clh 11
--- > -
If 2), both inequalities If cq Cq � << Ci c i (i = = I1,, 2), inequalities will usually be satisfied, and hence the two prey species should be able to coexist in the landscape. By contrast, when Cq » >> Ci ci (i = = 1, 2), then one of of the two inequalities will not hold. In this case, cq 1 , 2), the can increase the prey prey species species with with higher higher cihi cihican increase when when rare rare and and the the other other prey prey species species is common, whereas whereas the alternative alternative prey cannot reciprocally increase when rare. The model shows that given a predator which is both a habitat habitat generalist and a trophic generalist, habitats may generalist, alternative prey species specialized specialized to different habitats apparent compe indirectly interact interact via predator predator colonization of prey patchespatches wapparent competition (Holt, 11977, 977, 11984; 984; Holt and Lawton, 11994) 994) at the landscape If such landscape level. If predators are effective colonizers and can induce local prey extinctions, one prey species restricted restricted to the community in one habitat habitat can indirectly exclude another another prey species in a different local community. The potential for prey exclusion via metacommunity dynamics raises an in interesting methodological dilemma. Given such exclusion, a survey of of seemingly suitable but empty habitat - generhabitat patches patches will not reveal reveal the cause of of absence absencemgener alist predators, which which can colonize only after the missing prey has invaded. The usual sort of descriptive surveys may completely miss the dynamical cause for species exclusion landscape. exclusion from a heterogenous landscape. A criterion for for dominance in apparent competition is given by the compound parameter parameter cihi• cihi. Prey species with a low value of of this quantity are particularly vulnerable to exclusion by shared predation. Prey specialized specialized to rare rare habitats (low h) hi) are are more more likely to be excluded excluded by predators sustained by prey prey inhabiting more widespread widespread habitats. habitats. Likewise, prey species which are poor colonists (low c) c;) are more prone to exclusion exclusion by apparent apparent competition. A low ci ci may reflect either poor individual dispersal abilities or low local prey population sizes. I have 984) analyzed a one-predator, have previously (Holt, 11984) one-predator, two-prey species model in which each prey was specialized specialized to a different different habitat. habitat. This model explicitly tracks abundances 1 3» and assumes density abundances in each habitat (unlike ((13)) densityindependent sink population structure). independent predator predator dispersal (leading to a sourcesource-sink Such dispersal permits prey to experience experience apparent competition, competition, despite habitat habitat segregation. The prey species with lower intrinsic growth rate is vulnerable vulnerable to exclusion by the alternative prey, and the likelihood of of such exclusion increases with increasing predator predator dispersal.
1! 64 64
Robert Robert o. D. Holt Holt
These These earlier earlier results are consistent consistent with the conclusions conclusions drawn drawn above above for for shared shared predation predation in a metacommunity. Given Given low low inherent inherent extinction extinction rates, the "intrinsic metapopulation is its rate "intrinsic growth growth rate" rate" of of a prey prey metapopulation rate of of colonization, colonization, which is cihi. c i h i . This This compound compound parameter parameter determines determines prey community composition, composition, just just as the usual within-patch usual intrinsic growth growth rate does does in determining determining dominance dominance in within-patch apparent 984; Holt 993). apparent competition competition (Holt, 11984; Holt and and Lawton, Lawton, 11993).
CONClUSIONS CONCLUSIONS Classical metapopulation metapopulation theory theory assumes assumes that landscapes landscapes are comprised comprised of of a large Most models large number number of of patches patches available for for colonization. colonization. Most models assume that the patches are physically homogeneous. homogeneous. Yet in natural natural landscapes, metapopulations metapopulations are are likely to span span a wide wide range range of of local environmental environmental conditions. conditions. In this chapter, chapter, I have po have used used variants of of the Levins Levins metapopulation metapopulation model model to examine examine some some potential consequences consequences for for community community structure structure of of habitat habitat heterogeneity. heterogeneity. These These theoretical theoretical results suggest suggest that sparse sparse habitats habitats in a heterogeneous heterogeneous land landscape scape are likely to sustain sustain a biased biased array of of species, including including habitat habitat specialists specialists with unusually unusually high high colonization colonization or low extinction extinction rates and and habitat habitat generalists generalists sustained sustained via spillover from more more abundant abundant habitats. Trophic cation of Trophic specialization specialization leads to a kind kind of of magnifi magnification of these these effects, so that each persist each additional additional level must must satisfy increasingly stringent stringent criteria for for persistence. One One broad broad implication implication of of this result is that that metacommunity metacommunity dynamics au automatically tends tends to constrain constrain food food chain chain length. length. Trophic Trophic generalization generalization leads to an avenue avenue for indirect interactions interactions among among alternative alternative prey species. If If alternative prey species are are habitat habitat specialists, but a predator predator is a habitat habitat generalist, predator predator colonization colonization can can couple the dynamics dynamics of of these these prey prey species. This gives rise to apparent apparent competition competition at the metacommunity level, which which in some some circumstances circumstances can can lead to the exclusion exclusion of of prey species that that are poor poor colonists, colonists, or are are specialized specialized to rare rare habitat habitat types. The The ideas ideas presented presented here here provide provide a first pass pass through through the the potential potential implica implications tions of of habitat heterogeneity heterogeneity for for metacommunity metacommunity dynamics dynamics and and structure. structure. One One promising promising direction direction for for future future work work will be be in developing developing spatially spatially explicit explicit models models (Kareiva and and Wennergren Wennergren 11995; Nee et et al. al.,, this volume) volume) with limited dispersal dispersal (Kareiva 995; Nee and and various various patterns patterns of of spatial heterogeneity. heterogeneity. My My expectation expectation though, though, is that that the the general conclusions reached general conclusions reached here here will prove prove robust. robust.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS I thank reviews of the manuscript thank Ilkka Ilkka Hanski Hanski and Jan Bengtsson Bengtssonfor very very thoughtful thoughtful reviews manuscript and the National National Science Science Foundation Foundation for financial financial support. support.
8
Genetic Effective Size of a Metapopulation Philip W. Hedrick
Michael f.E. Gilpin
Population structure has long been been recognized recognized as having a major influence influence Population on the of genetic genetic variation has been been the topic topic of of exten the maintenance maintenance and and loss of variation and and has extensive research in population genetics (e.g., Wright, 11978; 978; Slatkin, 1985, 1 985, 1987). 1 987). of the impact of population structure on genetic variation Generally, investigation of subpopuhas assumed that subpopulation sizes remain constant over time, i.e., subpopu lations do not go extinct. It has been shown that if a population exhibits meta metapopulation dynamics, i.e., patches in which subpopulations exist become unoc unoccupied because of local extinction, that many of of the generalizations generalizations of of earlier studies of population structure 1 977; Maruyama and Ki structure may not hold (Slatkin, 1977; Kimura, 980; Wade mura, 11980; Wade and and McCauley, 1988; Gilpin, 1991). 1 99 1 ). The amount of genetic variation in a population is generally determined using the measure of both measure heterozygosity because because of both its biological importance (individuals are are either heterozygotes or homozygotes) and the extensive theory that predicts heterozygosity levels due to various evolutionary factors. In the present context, we are concerned with two different aspects of heterozygosity: the average average level distriof heterozygosity in a subpopulation or a metapopulation and the spatial distri bution of heterozygosity due to the structure of the population. Generally, the steady-state values of these heterozygosity values are of interest to evolutionary genetics, while changes, particularly losses, in heterozygosity are of particular importance to conservation biology. Metapopulation Metapopulation Biology Biology Copyright © All rights orm reserved. Copyright 9 1997 1997 by Academic Academic Press, Inc. All rights of of reproduction reproduction in in any any fform reserved.
1165 65
1166 66
PhilipW. W. Hedrick Hedrickand and Michael MichaelE.E. Gilpin Gilpin Philip
Both Both the the steady-state steady-state levels levels and and changes changes in in heterozygosity heterozygosity are are governed governed by by the the effective effective size size of of the the population. population. In In general, general, the the effective effective population population size size cor corrects rects census (or (or breeding) breeding) population population number number to to account account for for aa variety variety of of (mainly (mainly demographic) demographic) real real world world considerations considerations such such as as the the sex sex ratio ratio of of breeding breeding individ individand life history characteristics (e.g., Lande and and BarrowcIough, Barrowclough, 11987; Cabauals and 987; Caba I1ero, 994). The llero, 11994). The effective population size is usually calculated for for a group of individuals with given particular particular demographic demographic properties properties in which there there is random random mating (although other mating structures have also been examined, CabaIlero, Caballero, 11994). 994). The The effective population size size is generally defined as the size of of an ideal population that results in a given variance in allele frequency or amount of in inbreeding. However, because of our interest in the level of genetic variation, we will estimate the effective population size in an ideal population that results in a given loss of heterozygosity (this has been termed the eigenvalue effective pop population size by Ewens, 11989). 989). This effective size can be estimated either for subpopulations of a metapopulation or for total metapopulation composed of a group of subpopulations. Population dynamics similar to that in a theoretical metapopulation in natural populations are not uncommon (Harrison and Taylor, this volume) and there do appear to be particular instances in which habitats are fragmented that metapopmetapop ulation dynamics is an appropriate description of the population structure structure at a 995b; Thomas and Hanski, this volume). As regional level (e.g., Hanski et et al. al.,, 11995b; a result, there has been increasing interest in the impact impact of of metapopulation struc structure on genetic variation in endangered species and other organisms that exist in either of of natural or human causation (Hastings extremely fragmented habitats, either and Harrison, 11994). and 994).
I. AN AN EXAMPLE EXAMPLE Before examining examining the the specific effects effects of of metapopulation metapopulation dynamics dynamics on on effeceffec Before tive population population size, size, it is useful to to give give an an heuristic heuristic example example to demonstrate demonstrate how how metapopulation dynamics dynamics can can influence influence the the maintenance maintenance of of genetic variation. variation. GilGil metapopulation pin (1991) ( 1 99 1 ) gave simulation example in which which he he assumed assumed that that there there are are pin gave a simple simple simulation example in three subpopulations subpopulations or or patches patches in the metapopulation, metapopulation, each with with an effective effective three (and census) census) population population size size of of 500. 500. All AIl the the subpopulations subpopulations were were initiated initiated with with aa (and high high level level of of heterozygosity. The important important sequence sequence of of events events in in this this simulation simulation starts starts in in generation generation 48 48 The (see Fig. Fig. 1) 1 ) when when patch patch 2 goes goes extinct extinct and and is is recolonized recolonized from from patch patch 3 with with aa (see consequent reduction reduction in in heterozygosity. This This loss loss occurs occurs because because it it is is assumed assumed consequent that recolonization recolonization is is by by only only two two individuals, individuals, e.g., e.g., aa fertilized fertilized female. female. The The next next that significant event event is is when when empty empty patch patch 11 is is recolonized recolonized from from patch patch 22 with with aa founder founder significant population having having no no genetic genetic variation. variation. Finally, FinaIly, when when patch patch 22 goes goes extinct extinct in in population
Genetic Genetic Effective Effective Size Size Of of aa Metapopulafion Metapopulation
88
167 1 67
PATCH PATCH 1
HIGH H H HIGH
aI
2
HIGH H H HIGH
3
HIGH H H HIGH
00 I
FIGURE ]1 FIGURE
++ ,
LOW H H LOW
++
J
,4I
+
,~l . . ,
H == 0O H
i i '
: LOW LOW H H
,
II
20 20
40 40 I
60 60 I
GENERATION GENERATION
.
II
. ' '
,
. , . .
,t;' . H=0 H=O I , , .
, .
H= O ,t+ H=0 80 80 I
_
100 1 00 I
The level level of of heterozygosity heterozygosity (H) (H) over over time time in in aa simulation simulation of of aa population population existing existing in in The three patches patches (after (after Gilpin, 1 99 1 ). The short vertical vertical bars bars on on the the right-hand right-hand end end of of horizontal horizontal lines lines three Gilpin, 1991). The short indicate extinctions in aa patch and the the arrows arrows indicate indicate recolonization. recolonization. indicate extinctions in patch and
generation 71, 7 1 , the metapopulation metapopu1ation has has no variation although although there are still 500 500 generation no variation individuals remaining remaining in patch patch 1. 1 . All All of of these individuals individuals can be traced traced back back to to individuals can be some 1 . Gilpin 199 1 ) termed this some individuals individuals in patch 3 before before generation generation 551. Gilpin ((1991) this pro procoal cess through which metapopulation dynamics reduces reduces genetic variation the coalesence of metapopu1ation, i.e., the loss of of the metapopulation, of genetic variation being traced traced back to a few individuals that are the ancestors present ancestors of of all the individuals in the present metapopulation. metapopulation. This example illustrates an extreme case in which the loss of of genetic variation in the metapopu1ation metapopulation can be dramatically lower than that expected from a pop population the size of the average census number in the system. Gilpin ((1991) 199 1 ) found that in general the most dramatic lowering of of genetic variation occurred for for ex extinction and and recolonization values at which the average number number of occupied patches was low enough that that the metapopu1ation metapopulation itself was in danger danger of extinction. However, genetic variation may be of secondary interest in metapopulations with high extinction expectation so we will examine metapopulations that include more patches and with a balance of local extinction and recolonization rates that makes the the likelihood likelihood of of extinction extinction low. low.
II. II. METHODS METHODS One One methodical approach used used to examine examine metapopulations in a population genetics 1 977), Maruyama 1 980), Wade genetics context context is is that that of of Slatkin Slatkin ((1977), Maruyama and and Kimura Kimura ((1980), Wade and 1 988), Ewens 1 989), and and McCauley ((1988), Ewens ((1989), and Barton Barton and and Whitlock (this volume). volume). These 1 970) These authors authors use use an an infinite infinite (or (or finite) finite) patch, patch, spatially spatially implicit, implicit, Levins Levins ((1970)
1168 68
Philip Hedrick and and Michael Michael E. E. Gilpin Philip W. Hedrick Gilpin
metapopulation structure in which there is instant recolonization of of empty patches. Thus, Thus, some constant constant fraction of of the local populations go extinct each each generation, all of which are immediately recolonized by some number of colonists of are recolonized number of which then, during during the time step (or over time, Barton and and Whitlock, Whitlock, this volume), grow grow up to the the local carrying capacity. Our Our model, on the other other hand, is expanded expanded from the earlier earlier approach approach of of Gilpin ((1991) 1 99 1 ) as introduced above and 1 977) and and differs differs from the the approaches approaches of of Slatkin ((1977) patches, each of of others in several ways. First, our model has a finite number of of patches, which can support a local population, but which but which can be empty for a number of of time steps (see examples examples in Figs. 11 and and 2). Second, we decouple decouple gene flow from the the number 1 988). Third, number of of colonists, a possibility suggested by Wade Wade and and McCauley McCauley ((1988). the time that is governed by the the colonization the time that aa local local population population remains remains extinct extinct is governed both both by colonization probability and also by the number of extant source patches. Finally, we examine probability and also by the number of extant source patches. Finally, we examine the influence influence of of metapopulation dynamics on genetic genetic variation separate separate from ge genetic drift within patches infinite population size within a patch. patches by assuming assuming an infinite patch. While the previous approaches approaches can be approximated analytically, our our approach approach appears to be tractable tractable only using computer simulation (however, see Whitlock appears and 996). Further, and Barton, 11996). Further, while while the the general general behavior behavior of of the the two two approaches approaches are are similar, in some cases appear to yield quantitatively different cases they appear different answers. Because of parameters parameters that Because there there are are a number number of that can influence influence the effective effective population metapopulation, and nature of population size size of of aa metapopulation, and because because of of the the complicated complicated nature of the interaction model, the interaction of of stochastic stochastic processes processes within within and and between between patches patches in in our our model, understand the process of of heterozygosity loss we will use computer computer simulation to understand and estimate estimate the effective effective size in a metapopulation. We We will check check the simulations through the use of analytical approximations for the behavior of through of approximations for of single patches. patches. Our approach will be to assume Our approach assume some standard standard conditions conditions and and then then sequentially sequentially alter these these various parameters parameters and and assumptions assumptions to determine their effects. effects.
Description of Parameters Parameters A. Description First, let us assume that the metapopulation is divided up into Np Np patches, patches, each of patch, random assumed. of local population population size K. Within Within each each patch, random mating mating is assumed. We We will examine examine the the changes changes in in heterozygosity heterozygosity for for aa single single locus locus with with two alleles, both of of which have an initial frequency frequency of of 0.5 in all patches. patches. both 1 970) we assume Following Levins ((1970) assume a colonization rate rate (probability) of of c and extinction rate e and based on occupancy of all other patches in the meta and extinction rate and based on occupancy of all other patches in the metapopUlation. patches in the meta population. The The variable p p** gives the observed fraction of of patches metapopUlation that We assume population that are are occupied at at any any one one time. We assume that the the actual actual proba probac* = cp* so bility of of colonization colonization to to an an unoccupied unoccupied patch patch is c* = cp* so that that if if some some of of the the patches are not occupied, occupied, the rate of of colonization is lowered because because the pool of of potential potential colonizers colonizers is is reduced. reduced. For For cc** < < e, the the metapopulation metapopulation will go go extinct. extinct. For nite number patches, extinction is possible even even For a metapopulation metapopulation with a fi finite number of of patches, for for cc** > > e, much in the the same way that that a small population population can go extinct from demographic demographic stochasticity even with the individual birth rate greater greater than the
88
Genetic Metapopulafion GeneticEffective Effective Size Size of of aa Metapopulation
1169 69
individual death rate. Note that this is a spatially unstructured unstructured model, essentially equivalent to the original model of 1 970). of Levins ((1970). NtI founders randomly chosen from a A patch is assumed to be colonized by N given occupied patch, termed the propagule-pool model or randomly chosen from all occupied patches, patches, called the migrant-pool model (Slatkin, 1977). After reco reco1 977). After lonization, Ionization, it is assumed that the subpopulation expands expands in one generation to its generation are randomly carrying capacity, K. The individuals in the following generation drawn from the parental allele frequency pool to simulate genetic drift within a patch. After uence of After evaluating evaluating the infl influence of genetic drift in local populations, populations, to de determine the impact impact of of metapopulation dynamics independent independent of of genetic drift within a patch, we will assume that the number number of of individuals within a patch is infi nite. When the infinite. the population size size is assumed to be infinite within the the patch, then there is no change in allele frequency from from genetic drift and the only change change When there is gene flow, flow, within a patch occurs from gene flow when it is present. When each generation a proportion m of the individuals in a given patch patch comes from another given occupied patch, making the total amount of gene flow into a patch patches. per generation, mN;, mN*, where N; N* is the number number of of occupied occupied patches.
B. Estimation Estimation of Effective Effective Metapopulation Size Size and Other Values To estimate estimate the the effective effective population size, Ne, the the relationship relationship which which gives the the change in heterozygosity between consecutive generations,
-( ( -1))
1 H , , + !1 = H Ht+ H ,t 11 - 22~V~ Ne ' -
-
"
((la) 1 a)
is used whereH where HtI andH andH,+l I+ ! are the mean heterozygosities over all occupied patches and over replicate computer simulations in two consecutive generations t and ft + 985). Ne + 11,, respectively (e.g., Hedrick, 11985). N e is is the effective population size that results in the given amount of loss of heterozygosity between the two generations. Therefore, an estimate of the effective population size is Therefore,
=
H H,
, N Nee = 2(H 2(H,, - H O , +, +l ) !" )
(( l1 bb))
If � wee assume assume that the average heterozygosity within a subpopulation (or patch) patch) is H Hs, of the average effective subpopulation size is s , then the estimate of H ,(s) Ht(s~ = 2(H,~s)Ne(s~ = 2(H H, + ! (s) ,(s) - H,+l~s~)"
Ne(S)
(2a)
(2a)
Likewise if H metapopulation (calcu HvT is the average heterozygosity in the total total metapopulation (calculated from the global allele frequency in the metapopulation), metapopulation), then the estimated estimated for the metapopulation is effective population size for
Ht~ Ne~ = 2(H,~ - H,+~)"
(2b) (2b)
1170 70
Philip W. Hedrick Hedrick and and Michael Michael E. E. Gilpin Philip Gilpin
To for To estimate estimate heterozygosity, heterozygosity, aa given given metapopulation metapopulation simulation simulation was was run run for 21e The first 2/e + + 25 25 generations. generations. The first 21e 2/e generations generations were used used to allow the metapop metapopulation dynamics dynamics to to become become stabilized stabilized (the (the expectation expectation is is that that approximately approximately 90% more extinctions during this period, period, 90% of of the patches patches would would have have had had one one or or more extinctions during of the last 25 pairs of - ((1I - ee)(2/e)), and the heterozygosities heterozygosities of of consecutive consecutive gen gen)<2/e) , and 11 erations used in estimation. A runs demonstrated demonstrated erations were were used in the the estimation. A number number of of preliminary preliminary runs that the decay of period (see also Fig. 3) of heterozygosity had had stabilized stabilized for for this period and of and yet yet there there was was still still enough enough heterozygosity heterozygosity remaining remaining to to give give an an estimate estimate of the size. For the effective effective population population size. For each each parameter parameter set, set, the the mean mean heterozygosity heterozygosity in in aa given 000 independent given generation generation was was the the result result of of 11000 independent replicate replicate simulations simulations (ex (excluding generations generations in which which individual individual simulations simulations had had metapopulation metapopulation hetero heterocluding zygosity values of of zero). From From these these heterozygosities, heterozygosities, 25 effective population population 25) were size values values (generations (generations 21e 2/e + + I1,, 21e 2/e + + 2, 2 . . . . . . 21e 2/e + + 25) were calculated c~ilculated and and sizes for were averaged averaged to to give give the the estimate estimate of of the the effective effective population population sizes for the the these were patches and and the the metapopulation, metapopulation, making making each each an an average average of of approximately approximately 25,000 25,000 values. values. In In addition, addition, the the extent extent of of diversity diversity among among the the subpopulations subpopulations was was measured measured using (Nei, 987). using (Nei, 11987). -
FST =
HT -- Hs . Hr
(3) (3)
The S T was period for The average average value value of of F Fsx was still still changing changing (increasing) (increasing) during during the the period for which which the heterozygosity heterozygosity and and effective effective population population sizes sizes were calculated calculated so its value was was used used only to determine determine the relative relative differences between between the effects of of value various various parameters. parameters. At metapopulation with nite number patches, an At any any one one time time in in aa metapopulation with an an infi infinite number of of patches, an average the patches patches are expected to be occupied occupied (e.g., average of o f pp = = 11 - e/e e/c of of the are expected to be (e.g., Levins, Levins, 11970). 970). Because here have Because the metapopulations metapopulations that that we are are examining examining here have a finite number number and number of of patches, patches, there there is is stochastic stochastic variation variation in in the the number and fraction fraction of of occupied in a occupied patches. patches. The The average average or or expected expected census census number number of of individuals individuals in a metapopulation nite number metapopulation with a fi finite number of of patches patches should should be approximately approximately
N' = pKNp.
(4) (4)
Because number of Because our our colonization rate is not not constant constant but but is a function function of of the number of patches patches occupied occupied and and we we assume assume that that extinction extinction and and colonization colonization occur occur consec consecutively utively and and not not simultaneously, simultaneously, we we have have actually actually calculated calculated the the average average census census number, number, N. To To determine determine the difference difference between between the census census number number and and the ef efNe(T)IN fective popUlation size metapopulation, we calculate the the ratio fective population size of of the the metapopulation, we can can calculate ratio Ne(T~/N which unity if values are if the the which should should be be close close to to unity if the the two two values are similar similar and and near near zero zero if effective size is is much much lower the census number. effective popUlation population size lower than than the census number. = 0.05, As As aa standard standard set set of of parameters, parameters, it it was was assumed that that ec = = 0.2, 0.2, e = 0.05, and and = 110. Given these these colonization and and extinction rates, and with this number number of of Np = 0. Given rates, and patches, empirical estimates estimates we we found found that that slightly more than 70% of patches, from from empirical slightly more than 70% of the the patches occupied on time and the probability probability of metapatches were were occupied on average average at at any any one one time and the of meta-
88 Genetic GeneticEffective EffectiveSize Sizeofof ao Metopopu!otion Metapopulation
1171 71
0.2%).
population extinction, extinction, all all patches patches being being unoccupied, unoccupied, over over the the fifirst generations population rst 550 0 generations was quite quite small small (approximately (approximately 0.2%). was The The effective effective population population size size within within aa subpopulation subpopulation can can be be estimated estimated using using an an analytical analytical approximation approximation to to check check the the simulations simulations in in the the following following manner. manner. The The mean mean number number of of generations generations for for aa turnover, turnover, an an extinction, extinction, and and aa recoloni recolonization for for aa given given subpopulation, subpopulation, is is approximately approximately zation
11 11
(t = = -- + + - - 11,' e cc
(lie)
(5) (5)
1
the expected expected time to to a subpopulation extinction (lIe) (I/e) plus plus the expected expected which is the time to aa recolonization (l/c) minus 1 (because (because recolonization can can occur occur in the after extinction has occurred). The heterozygosity same generation immediately after after (t generations is then after
(
_
H ,, ==HH 0 0 11 H
N
1 )(1 1 )'-1 f 2N ' _
_
1
1-
2N
_
(6a)
,
where N is the number number of of individuals in the subpopulation (progeny are drawn randomly so that the effective size within a patch should be equal to the census census size). Given that that the expected expected effective effective population population size within the the subpopulation subpopulation is N'~s~, then
Ht = By substitution, then then
N'
=
N'e~s~ = e(5)
1
1)t
(6b) (6b)
. . 2Ne(s>
1/2 - [(1 1/(2Nf))( 1 - (l/(2N)))t-lP'(
------
1/2
11 - [(1 - 1/(2Nyl)(1 - (1/(21~))t-1] l/t"
(7) (7)
III. III. RESULTS RESULTS There There are are several several factors factors that that should should influence influence the the effective effective size size of of aa metameta popUlation, namely, namely, the the carrying carrying capacity capacity of of aa patch, patch, the the rates rates of of extinction extinction and and population, colonization, the the number number and and source source of of founders, founders, the the number number of of local local patches, patches, colonization, and the the rate rate of of gene gene flow flow between between patches. patches. After After briefly briefly discussing discussing several several exex and amples amples of of the the general general patterns patterns of of the the results, results, we we will will investigate investigate these these parameters parameters individually individually in in aa sensitivity sensitivity analysis analysis to to determine determine what what impact impact they they have have on on the the effective effective population population size size and and its its relationship relationship to to the the census census number. number.
A. A. General General Patterns Patterns Before Before we we discuss discuss the the influence influence of of particular particular parameters, parameters, itit isis useful useful to to visualize the the general general pattern pattern of of the the results. results. As As an an introduction introduction to to the the general general visualize pattern pattern of of results results found found in in the the following following simulations, simulations, Fig. Fig. 2 gives gives aa graphical graphical
2
1172 72
Philip W. Hedrick Hedrick and and Michael Michael E.E. Gilpin Philip Gilpin
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FIGURE22 The T h e allele frequencies, represented represented by by the the amount a m o u n t of of shading shading in in the patches FIGURE allele frequencies, in aa box, box, in the 10 10 patches of a a metapopulation m e t a p o p u l a t i o n over over 50 5 0 generations. generations. Gaps Gaps in in a a horizontal horizontal series series of of boxes boxes represent represent uunoccupied noccupied of patches. simulations, cr = 0.05. The parameters are K = patches. In In all all three three simulations, = 0.2 0.2 and and e e = = 0.05. The other other parameters are (A) (A) K = 00 ~,, m rn = = Nfs = 0.0, and K = 50, 50, m 0.02, and NfI == 1; I ; and (C) K 50, m 1 , and 0.0, and Nt NI = = I1;; (B) (B) K rn = = 0.02, and N and (C) K = = 50, rn = = 0. 0.1, and N = 5. 5. =
summary ini summary from from three representative representative simulations. simulations. In each case, there are 10 initially occupied patches, patches, all having having an an initial frequency of of 0.5 for for both alleles and and having having colonization colonization and and extinction extinction probabilities probabilities of of 0.2 and and 0.05, respectively. respectively. The The simulation simulation runs runs for 50 generations generations from left to right with with occupied occupied patches patches shown numbers of shown by boxes and extinct patches patches indicated indicated by gaps of of varying varying numbers of generations. generations. After rst few 0 patches After the fi first few generations, approximately approximately 7 of of the 110 patches are are oc occupied cupied at any one time, as expected. expected. The allele frequency frequency in a local popu population, indicated uenced differ indicated by the fraction of of the box that that is shaded, shaded, is infl influenced differentially here by genetic drift, gene flow, and colonization. colonization. In these examples, examples, at the the end end of of the simulation simulation the middle middle example example is nearly fixed fixed for for one allele
88 Genetic Genetic Effective Effective Size Size of of aa Metapopulafion Metapopulation
173 1 73
while while the the top top one one is is split split between between subpopulations subpopulations that that have have high high frequencies frequencies of of one one allele allele or or the the other, other, and and the the bottom bottom simulation simulation has has most most subpopulations subpopulations polymorphic polymorphic for for the the two two alleles. alleles. These These results results are are generally generally consistent consistent with with the the expectations expectations from from higher higher gene gene flow flow in in the the bottom bottom example, example, the the lower lower number number of of founders founders in in the the middle middle one, one, and and the the infinite infinite carrying carrying capacity capacity in in the the top top simulation. simulation. Because Because we we are are not not assuming assuming that that there there is is some some force force to to supplement supplement or sustain sustain heterozygosity, heterozygosity, such such as as mutation, mutation, balancing balancing selection, selection, or or external external gene gene or flow inputs, inputs, genetic genetic variation variation in in the the metapopulation metapopulation will will ultimately ultimately fall fall to to flow zero. Trivially, Trivially, itit falls falls to to zero zero if if the the metapopulation metapopulation goes goes extinct. extinct. However, However, we we zero. have carried carried out out most most of of our our simulations simulations with with values values of of c = 4e 4e and and have Np Np = 10, 1 0, so so that that the the probability probability of of metapopulation metapopulation extinction extinction is is very very low low in in the the short term. term. short Figure 33 illustrates illustrates the the pattern pattern of of H H in in the the metapopulation metapopulation and and FST FST over over 50 50 Figure Np = 10, 1 0, c = 0.1, 0. 1 , ee == 0.025, 0.025, K K = 50, 50, m m = = 0.02, 0.02, and and replicate simulations simulations with with Np replicate N 2/e or NtI = 1. 1 . Over Over the the first first 11 - (1 ( 1 - e) e)2/e or 90 90 generations generations of of the the simulation, simulation, the the distridistri bution of of both both H H and and FsaFST widens widens and and then then appears to reach reach aa quasi-steady quasi-steady state. state. bution appears to In the the last last half half of of the the generations, generations, both both H H and and FsT FST seem seem more more or or less less uniformly uniformly In spread between some upper value. However, the average average level level of of hethet spread between zero zero and and some upper value. However, the erozygosity is is declining characteristic of of the the parameters of the the simulation. simulation. erozygosity declining at at aa rate rate characteristic parameters of There is is also leakage each into the the absorbing absorbing state of There also a a probability probability of of leakage each generation generation into state of H 0 and and FST Fs-r = 0 0 in which the the amount the metapopulation H = 0 in which amount of of variation variation in in the metapopulation is is zero. zero.
c=
=
=
=
=
-
c=
=
-
=
Heterozygosity
FIGURE FIGURE 33
Generation
1 80
The The distribution distribution of of 50 50 replicate replicate simulations simulations of of metapopulation metapopulation for H H (top) (top) and and FS Fs-r T (bottom) O patches. I , ee = (bottom) when when there there are are I10 patches, K = 50. 50, cc = = O. 0.1, = 0.025. 0.025, m m == 0.02. 0.02, and and NfI = = I1.. The The simulation 80 generations. simulation was was run run for for 1180 generations,twice twicethe the approximate approximatetime timefor for the the initially initiallycolonized colonizedpatches. patches, to to have have aa 90% 90% probability probability of of extinction. extinction. =
1174 74
Philip W. Hedrick Hedrick and and Michael Michael E. E. Gilpin Philip Gilpin
B. Carrying Carrying Capacity Capacity of a Patch Patch First, let us examine the effect of of different numbers numbers of of individuals or carrying capacity, K, in the patches, a value that is central to the conclusions conclusions of of Gilpin ((1991). 1 99 1 ). Table Table I gives the estimated effective population population size for for different patch sizes ranging from 25 to 00. ~. Notice that for for the smallest patch size, the effective population metapopulation is 38.2, 2 1 .9% of population size for the the whole whole metapopulation 21.9% of the average census census reduced to 16.4 1 6.4 number, number, N. The effective population population size within a patch is even reduced from the number in a the carrying capacity value value of of 25. Even Even for for this small census number patch, the effective population of of the metapopulation metapopulation is about about the census number number for for one and a half half occupied patches. Because, on average, average, approximately 70% of of the number of the patches patches were were occupied, occupied, an actual census census of of the the number of patches occupied occupied and the effective population population size within a patch patch would would give a very distorted picture picture of of the actual actual effective population population size of of the metapopulation. metapopulation. As the patch size increases, the effective size of of the metapopulation does not increase number increase very much and asymptotes fairly quickly; it is only 64.2 when when the number in the patch is infinite. In other words, even though though the census number number within a patch may be very large, this has very little influence influence on the loss of of heterozygosity in the metapopulation (and the estimated effective metapopulation size) which is primarily determined by the metapopulation dynamics. Notice also that the esti estimated effective population population size within a patch patch is about about 40 40 to 50% of of that that in the the mated total metapopulation. In the last column, the values of ST are of F FST are given for for the different different K values. For For the smaller carrying capacities, the values are extremely high, reflecting the inin fluence of of genetic drift drift within the patches patches as well as the extinction-recolonization extinction-recolonization process. ects the differentiation that occurs occurs only by the exex process. For K K = = 00, ~, FST refl reflects tinction-recolonization tinction-recolonization process and this alone can generate generate substantial differen differentiation among patches. The second column of of Table I gives the expected analytical values for the subpopulation subpopulation effective size from from expression expression (7). The observed simulations are very close to the analytical expectations for for effective population size values, sup supporting porting the validity of of the methodology used here. In the sixth column of of Table Table J, I, the expected census number is given using expression (4). The The observed average census fth column, is about 5% lower census number number for for the simulations, given in the fi fifth for for all levels of of the carrying capacity, reflecting reflecting the lower lower colonization rates rates re resulting 00% patch occupancy. suiting from from less than 1100%
C. C. Rates Rates of Colonization Colonization and Extinction Extinction For number of For metapopulation metapopulation existence when when there there are an infinite number of patches, the rate of When there of colonization must be larger than the rate of of extinction. When there are are finite number of of patches, the rate rate of of colonization must be several times that of of a fi nite number the extinction rate rate for long-term long-term metapopulation metapopulation existence (the exact ratio ratio is
88
Genetic GeneticEffective Effective Size Size of of aa Metapopulation Metapopulation
1175 75
The and for The Estimated Estimated Effective Effective Population Population Size Size within within aa Patch, Patch, Nem N~(sl,, and for the the Total Total Metapopulation, Metapopulation, Ne Ne(r~, for Different Different Local Local Carrying Carrying Capacities Capacities (K) (K) Given Given That That cc = = 0.2, 0.2, ( T), for ee = - 0.05, 0.05, and and Np Np = l1 0Oaa
TABLE II TABLE
=
K K
25 50 1100 00 200 00
Se(s) N e(s)
N N'(s) ;(s)
N Se(r) e(T)
N N
N N''
N N Se(e(T)I r)/N
FsT FST
116.4 6.4 22.4 22.4 30.4 36.0 40.7 40.7
1 6.2 16.2 23.4 23.4 30.0 35.0 35.0 4 1 .9 41.9
38.2 38.2 46.4 46.4 64.2 64.2 66.7 66.7 76.6 76.6
1174.4 74.4 345.9 345.9 690.2 690.2 11392.9 392.9 00 ~
1187.5 87.5 375 375 750 750 11500 500 00 ~
0.2 19 0.219 0. 1 34 0.134 0.093 0.093 0.048 0.048 0.000 0.000
0.678 0.678 0.5 19 0.519 0.424 0.424 0.370 0.370 0.31 0.3122
T gives a FS FST gives the diversity diversity ooff aIlele allele frequencies frequencies over over the subpopulations subpopulations and N N iiss the average average observed observed census N' are the approximate values given census population size size for for the metapopulation. N;( N'e(s) approximate values given by s ) and N' expressions expressions (7) (7) and (4). (4).
a
higher higher when when the absolute absolute levels levels of of these these parameters parameters is higher higher and and the the number number of of patches patches is smaller). As As a result, we we have have calculated calculated effective effective population population sizes sizes for for an array of metapopulation existence of colonization colonization and extinction extinction rates that permit metapopulation existence for for aa substantial substantial time time period period (Table (Table II). II). First, notice notice that the rate of of colonization colonization has has very little influence influence on the effective for the total metapopulation, metapopulation, effective population population sizes, sizes, either within within a patch or for
The The Estimated Estimated Effective Effective Population Population Size Size within within aa Patch, Patch, Ne Ne(s~, (s) , the the Metapopulation Metapopulation Effective Effective Size, Size, Ne(T), Ne(r), and and the the Average Average Observed Observed Occupancy Rates Range of Occupancy Rates (p*) (p*) of of aa Patch Patch for for aa Range of Extinction Extinction and and Colonization Colonization Rates Rates When When KK = ~ and and Np Np - 1100
TABLE TABLE IIII
= 00
=
e c C
0.025 0.025
0.05 0.05
0.1
0.2 0.2
N Ne(s) e(s )
0.8 0.8 0.4 0.4 0.2 0.2 0. 0.11
80.3 80.3 74.6 74.6 881.2 1 .2 87.6 87.6
40.0 40.0 38.6 38.6 40.7 40.7
119.9 9.9 20.6 20.6
9.8 9.8
N Ne(T) e(T)
0.8 0.8 0.4 0.4 0.2 0.2 0. 0.11
205 .1 205.1 1187.1 87 . 1 1186.1 86. 1 1179.4 79.4
98.4 98.4 84.8 84.8 76.6 76.6
4 1 .9 41.9 35.3 35.3
14.4 14.4
p** p
0.8 0.8 0.4 0.4 0.2 0.2 0. 0.11
0.989 0.989 0.952 0.952 0.8 81 0.881 0.723 0.723
0.976 0.976 0.893 0.893 0.732 0.732
0.936 0.936 0.75 0.7511
0.794 0.794
1 76 176
Philip W. Hedrick Hedrick and Michael Michael E. Gilpin Gilpin Philip
with levels levels between between 0.1 0. 1 and and 0.8 0.8 having having nearly nearly the the same same effective effective size. size. On On the the with other hand, hand, as as the the extinction extinction rate rate increases, increases, both both effective effective population population sizes sizes are are other greatly reduced. reduced. For For example example with with cc == 0.8, 0.8, the the effective effective metapopulation metapopulation size size is is greatly increases from from 0.025 0.025 to to 0.2. 0.2. reduced from from 205.1 205. 1 to to 14.4 1 4.4 as as ee increases reduced In the bottom of of the the table, table, the the observed observed occupancy occupancy rates rates for for aa patch patch are are given. given. In the bottom On the the diagonals which have have approximately approximately the the same same occupancy occupancy rates, as the the On diagonals which rates, as absolute values and extinction extinction rates absolute values of of the the colonization colonization and rates are are increased, increased, the the effective effective population is greatly greatly decreased. decreased. For For example, example, with with occupancy occupancy rates rates around around 75%, 75%, population the the effective effective metapopulation metapopulation size size is is reduced reduced from from 179.4 1 79.4 to to 14.4 1 4.4 as as the the absolute absolute values of of the the colonization colonization and and extinction extinction rates rates increase increase eightfold. eightfold. The The explanation explanation values for these these results results is is that that the the level level of of extinction extinction governs governs the the number number of of turnovers turnovers for (extinction rate of (extinction and and then then recolonization) recolonization) because because the the rate of colonization colonization is high high relrel to that that of of extinction. extinction. This This high high turnover turnover results results in in aa large large reduction reduction in in hethet ative to erozygosity and and consequently consequently aa reduction reduction in in the the effective effective population population size. size. erozygosity
Number of Founders D. Number Founders Recolonization of of an empty patch in aa metapopulation metapopulation model generally is Recolonization empty patch model generally assumed to assumed to occur occur from from a limited limited number number of of founder founder individuals. individuals. Above Above we we have have assumed that that they they were were all drawn randomly from from a given patch in the the propro assumed drawn randomly patch [as in pagule-pool model model of of Slatkin Slatkin (1977)]. ( 1 977)] . On On the the other other hand, hand, the the founders founders can can be be pagule-pool drawn randomly drawn randomly from from all occupied occupied patches patches [similar [similar to to the the migrant-pool migrant-pool model model of of Slatkin (1977)]. Table III, when Slatkin ( 1 977)]. As As shown shown in Table when the the number number of of founders founders is small small for for the propagule-pool propagule-pool model, model, then then the effective effective population population size is greatly reduced reduced the = 50) 50) or or infinite. infinite. For example, when when whether the the subpopulation subpopulation is is finite finite (here (here K K= whether For example, the fourfold from the number number of of founders founders is is increased increased fourfold from 2 2 to 8, c = = 0.2, 0.2, e = = 0.05, 0.05, and and the subpopulation increased subpopulation size is infinite, the effective metapopulation metapopulation size is increased subpopulation is 50, the approximately fourfold from 76.6 76.6 to 292.7. 292.7. When When the subpopulation increase in the number number of of founders founders has has a substantial substantial influence increase influence but but has a much much founders from 2 to 8 result result smaller effect. For For example, increasing increasing the number number of of founders in 63% increase, increase, 46.4 46.4 to 75.5, 75.5, in effective metapopulation metapopulation for for c = = 0.2 and and e = = 0.05. 0.05. Increasing the also reduces ST , particularly Increasing the number number of of founders founders also reduces the the value value of of F FST, particularly when K For both combinations of colonization and extinction rates given when = ~. For combinations of colonization and given here when the carrying capacity in infinite, a fourfold increase in the here when the carrying capacity in infinite, a fourfold increase in the number number of of founders ST ' founders results results in in aa nearly nearly fourfold fourfold reduction reduction in in F Fsa-. Estimated sizes and given for Estimated effective effective sizes and FST FsT are are given for the the migrant-pool migrant-pool model model in in Table Table III III where where the the Nf NI values values are are indicated indicated by by an an asterisk. asterisk. When When the the founders founders are are drawn from the migrant pool and other parameters are equivalent, the effective drawn population ST values lower than population sizes are larger larger and the F Fs~than for the propagule-pool propagule-pool model. For model. For example, example, when when two two founders founders are are drawn drawn randomly randomly from from all all occupied occupied patches patches rather than than from one patch patch and the carrying capacity capacity is infinite, the meta metapopulation effective 1 2.6 rather 50% larger. population effective size size is is 1112.6 rather 76.6, 76.6, about about 50% larger.
= 00.
88 Genetic Genetic Effective Effective Size Size ofof ea Metopopul(]fion Metapopulation
177 1 77
The Estimated Estimated Effective Effective Population Population Sizes Sizes and and Fsx FSf for for Different Different TABLE TABLE III III The Founder Numbers, Numbers, Nr, Nfl when when tip Np == 10, 1 0, cc == 0.2 0.2 and and ee == 0.05 0.05 or or cc == Founder 0.4 and and ee == 0.1, 0.1 , and and KK == 50 50 or or ~a ooa 0.4 cC
ee
K K
0.2
0.05
50
Nj N/
Ne(s) Ne(s)
Ne(T) Ne(r)
FST FST
2 4 8
22.4 3 1 .5 31.5 38.9
46.4 59.9 75.5
0.519 0.5 1 9 0.400 0.347
2 4 8
40.7 8 1 .9 81.9 1 67.9 167.9
76.6 152.7 1 52.7 292.7
0.3 1 2 0.312 0 . 1 66 0.166 0.082
oc
2* 4* 8*
78.9 1 73. 1 173.1 408.3
1 12.6 112.6 222.9 487.9
0.21 7 0.217 0.096 0.046
50
2 4 8
1 5.3 15.3 22.6 32.8
27.0 37.5 52.2
0.474 0.323 0.238
2 4 8
20.6 4 1 .3 41.3 78.8
35.3 64.2 122.1 1 22 . 1
0.354 0. 1 90 0.190 0.G98 0.098
2* 4* 8*
46. 1 46.1 105.0 204.7
59.9 123.0 1 23.0 232.3
0.228 0.101 0. 1 0 1 0.048
00
0.4
0. 1 0.1
00
U
If NIt is indicated by an asterisk, then founders are randomly drawn from "If occupied patches.
E. E. Numbers Numbers of Subpopulations Subpopulations Table IV IV gives the effective effective population population sizes when the the number n u m b e r of of subpopuTable gives the sizes when subpopu lations ranges ranges from from 5 5 to to 40. 40. For For all all different different numbers numbers of of subpopulations, subpopulations, the the lations effective and remains remains at effective size size within within a a subpopulation subpopulation does does not not change change and at approxi approximately 40. The mately 40. The effective effective size size of of the the metapopulation metapopulation is is approximately approximately doubled doubled as as the is doubled doubled from 0 to to 40 the number n u m b e r of of subpopulations subpopulations is from 110 to 20 20 and and from from 20 20 to 40 and and increases increases from from 57.2 57.2 to to 306. 306.11 as as the the number n u m b e r of of subpopulations subpopulations is is increased increased eight eightfold fold from from 55 to to 40. 40. As As a a result, result, the the ratio ratio of of the the effective effective subpopulation subpopulation size size to to the the effective less as effective metapopulation metapopulation size size bbecomes e c o m e s less as the the number n u m b e r of of subpopulations subpopulations in increases. subpopulations increases creases. The The diversity diversity among among subpopulations increases as as the the number n u m b e r of of subpo subpopulations pulations increases. increases. This This suggests suggests that that with with smaller smaller numbers numbers of of patches patches in in the the metapopulation metapopulation they they have have more more similar similar allele allele frequencies frequencies because because of of aa higher higher connectedness while metapopulation some connectedness while with with more more patches patches in in the the metapopulation some patches patches are are quite each other. quite unconnected unconnected to to each other.
1178 78
Philip W. Hedrick Hedrick and and Michael Philip Michael E.E. Gilpin Gilpin TABLE TABLE IV IV The The Estimated Estimated Effective Effective Population Population Sizes Sizes for for Different Different Numbers Numbers of of Subpopulations Subpopulations When When K K == 00, ~, Cc = = 0.2, and e = 0.05 0.2, and - 0.05 N N pv
N N,(s) e(s)
N N,(r) e(T)
N N,(s)/N,(r) e(s/Ne(T)
5 110 0 20 40 40
43.3 40.7 39.4 39.4 39.5 39.5
57.2 76.6 1148.8 48.8 306. 306.11
0.756 0.5 31 0.531 0.265 0. 1 24 0.124
FST FST 0. 1 66 0.166 0.3 12 0.312 0.386 0.41 0.4188
F. Rate Rate of Gene Gene Flow Flow We ow between subpopula We have assumed assumed until now that that there there is no gene gene fl flow subpopulations have tions and and that that the the subpopulations subpopulations are are only only connected connected because because patches patches that that have become become extinct are are recolonized recolonized from other other patches. patches. Table Table V gives the estimated estimated effective flow is increased effective population sizes when the level of of gene flow increased from 0.0 to 0.02 metapopulation size 0.02 from each occupied occupied patch. First, notice that the effective effective metapopulation 1 9.7 as m increases. is increased increased from 76.6 to 2 219.7 increases. This This result is somewhat counter counterintuitive because it is generally assumed that that a strongly subdivided population, population, one with low rates rates of of gene gene flow, would retain retain more more overall genetic genetic variation. In fact is low, level of ST is is the the largest ST declines declines fact and and as as expected, expected, when when m m is low, the the level of F FST largest and and F FST as m increases. increases. In addition, the average average effective effective subpopulation size is nearly nearly as large as the effective basis effective metapopulation size as m gets to 0.005 or larger. The The basis for these these results results is that that the effective effective metapopulation size is being driven driven by the metapopulation dynamics ow does somewhat dynamics making making it very low and and gene gene fl flow somewhat over overof heterozygosity by restoring genetic genetic variation into patches patches that come the loss of have lost genetic genetic variation. However, because all the the subpopulations subpopulations now are
TABLE TABLE V V The The Estimated Estimated Effective Effective Population Population Sizes Sizes for for Different Different Levels Gene Flow Patches When When K Levels of of Gene Flow between between Patches K= - 00, ~, Cc = - 0.2, 0.2, 0 e= and Np - 0.05, 0.05, and lip = - 110 m
m
N Ne(s) e(s)
N Ne(r) e(T)
N N~(s)/N~(r~ e(s/Ne(T)
0.00 0.00 0.001 25 0.00125 0.0025 0.0025 0.005 0.01 0.02 0.02
40.7 40.7 66.5 66.5 88.9 1125.8 25.8 1156.7 56.7 2 17.7 217.7
76.6 76.6 95. 95.11 1115.7 1 5.7 140. 140.11 1172.4 72.4 2 1 9.7 219.7
0.53 0.5311 0.699 0.699 0.769 0.769 0.898 0.909 0.909 0.991
FsT FST 0.3 12 0.312 0.224 0.224 0. 1 67 0.167 0. 1 14 0.114 0.069 0.069 0.040 0.040
88
Genefic GeneticEffecfive Effective Size Size of of aa Metapopulafion Metapopulation
1179 79
flow, the rate of connected by the gene flow, of loss of of heterozygosity is the same for for the total metapopulation and the separate subpopulations. subpopulations.
IV. CONCLUSIONS CONCLUSIONS We have have determined the the relationship relationship between between effective size size of of a metapopu metapopulation and the parameters that govern the dynamics of the metapopulation: the parameters of number number of of patches, the local extinction and recolonization recolonization rates, the local carrying capacity, the number number of of founders founders that that recolonize recolonize a metapopulation, metapopulation, and and the rate of ow between of gene fl flow between extant patches. Prediction of of the effective metapopulation understanding of of the retention size, based based on these parameters, parameters, may allow understanding retention of of heterozygosity in spatially structured populations populations and should should be of of great great value in conservation conservation planning. planning. We have investigated in a sensitivity analysis the influence of these param of these parameters on the effective metapopulation size and the relationship relationship of of the effective size to the census number. of our number. Under Under the assumptions assumptions of our basic model as the the carrying capacity capacity is increased, the effective metapopulation size is increased increased but asymptotes at a value less than 00 even than 1100 even when when the carrying carrying size in each patch is assumed to nite. In this case, the census to be infi infinite. census number number may may not not be at at all related to metapopulation to the the effective effective size because the the effective effective size is governed by metapopulation dynamics. Because the census size increases with an increase increase in K, the ratio of of the effective metapopulation metapopulation size to the census number number drops drops quickly as the census census size increases. 199 1 ) when he commented increases. This point point was emphasized emphasized by Gilpin ((1991) "that the ability of of a metapopulation to retain genetic variation, which may be defi ned as proportional defined proportional to its so-called so-called effective population population size . . . can be one one to two orders of of magnitude lower than than the maximum total total number number of of individuals individuals in the system." Obviously the size of of this effect can be large, large, but to be one one order order 00 ((probably probably high for of of magnitude lower, K must be around around 1100 for many vertebrates but low for for many invertebrates or plants). plants). This This effective subpopulation subpopulation size could could in theory be determined by genetic estimates of 989) of effective population population size [changes [changes in allele frequency (Waples, (Waples, 11989) or 1 99 1 )] but or the the amount amount of of linkage linkage disequilibrium disequilibrium (Waples, (Waples, 1991)] but would would not not be ob obvious from demographic estimates of of effective population population size. In other other words, it would be difficult to assess the effective size of of the metapopulation by determin determining the effective size within several patches patches because the metapopulation effective size is governed governed by the extinction and and recolonization recolonization dynamics. The main factors determining the low effective metapopulation size are are the the rate recolonizing rate of of extinction extinction of of patches patches and the number number and type of of founders founders recolonizing empty patches. increased if patches. Overall, the effective metapopulation metapopulation size is increased if the rate of of turnover, of extinction (or the rate of turnover, extinction and subsequent subsequent recolonization) recolonization) is reduced, of founders founders reduced, the number number of of local populations populations increased, the numbers numbers of
1180 80
Philip W. Hedrick Hedrick and and Michael Michael E.E. Gilpin Philip Gilpin
is increased, increased. The increase in effective increased, and and the the rate of of gene flow is increased. The increase effective size with an increase ow is unexpected but is due to gene fl ow countering increase in gene fl flow unexpected but flow the influence extinction -recolonization dynamics which causes the metapopumetapopu extinction-recolonization lation to coalesce at a heterozygosity of zero. Gene flow acts to restore of Gene restore variation to patches with zero heterozygosity, thereby reducing the rate of patches thereby reducing of heterozygosity effective metapopulation size. loss and and increasing increasing the estimate of of effective As when the As expected expected intuitively, intuitively, the the effective effective metapopulation metapopulation size size is larger larger when the founders than for Whit founders follow the migrant-pool model than for the propagule-pool model. Whit1 990) generalized lock and McCauley ((1990) generalized these these two models to allow a proportion of the founders founders from the migrant-pool and of and a proportion from the propagule-pool and found found intermediate combinations more more closely mimicked the migrant-pool if there were spatial structure results. On On the other other hand, if there were structure in the metapopulation with higher probability of colonization then the with aa higher probability of colonization from from closer subpopulations, then the founders because most founders founders may more closely approach approach the propagule model because founders McCauley, would come from one or a few few neighboring neighboring subpopulations (Wade (Wade and McCauley, 11988). 988). There make it somewhat There are are several several aspects aspects of of the the model which which would would make somewhat more more realistic or factors that that we we did here. First, First, rather realistic or other other factors did not not discuss discuss here. rather than than allowing allowing the population to to carrying some the population to grow grow to carrying capacity, capacity, it it can can be be allowed allowed to to grow grow at at some rate When this is done, the the effective effective size of the population population rate to the the carrying carrying capacity. When of the because more is even lower than than what what is given in Table Table I because more genetic genetic drift occurs occurs during during the growth growth period period than than at at carrying carrying capacity. capacity. Second, the the carrying carrying capacity of partitioned between between different of the metapopulation could be be kept kept constant constant but but partitioned different of patches (a contrast similar populations or numbers of similar to the single, or few, large populations many small populations When more patches populations idea idea for for reserves). When patches are present present given given the same same total total carrying capacity, capacity, the the effective effective metapopulation size is somewhat somewhat lower than individuals are patches. than if the same number number of of individuals are spread spread across a few few patches. Finally, several other factors, parameters in space or factors, such variation of of the parameters or time, correlation correlation in in space space or time time of of rates rates of of extinction extinction or or colonization, and and extinction extinction being being a function function of of the genetic genetic constitution, constitution, were not not investigated. investigated. In traditional of population structure, traditional studies studies of structure, there there has been an attempt attempt to summarize using the number of summarize the the general general impact impact of of population population structure structure using the number of mi migrants per generation generation parameter. parameter. In general, number grants between between subpopulations subpopulations per general, if if the number of migrants is greater than unity, then then there structuring found found while if if of migrants there is little sub substructuring the number number of of migrants migrants is greater greater than than unity, then then the subpopulations subpopulations may greatly differ differ in allele frequencies. frequencies. On On the other hand, hand, with a metapopulation structure structure the amount of flow does of population differ the amount of gene gene flow does not not give the the complete picture of differentiation. For example, even flow, a metapopulation metapopulation can even with substantial substantial gene gene flow, have because of of the frequent have high high diversity over patches patches because the extensive extensive impact impact of of frequent extinction and recolonization. Overall, a metapopulation in which all local populations populations suffer suffer extinction and and recolonization tends reduce greatly with the pop recolonization tends to to reduce greatly genetic genetic variation, variation, with the effective effective population size being being dramatically lower than than what one would estimate estimate from the
88
Genetic GeneticEffective Effective Size Size of of aa Metapopuiation Metapopulation
1181 81
average census number number or the sum Sum of of the effective population sizes within each patch. We We see two two major major lessons from these findings, one one for for evolutionary biology and one for perspective given the for conservation conservation biology. First, in an evolutionary perspective observed molecular markers observed high levels of of heterozygosity for for allozymes and other other molecular in most species, it appears unlikely that most species spend their entire evolu evolutionary history history in a Levinesque Levinesque metapopulation metapopulation configuration. This is in general support support of of the observational observational conclusion of of Harrison and Taylor (this volume). On the other other hand, the results from from our our simulations might explain the difference difference between the inferred inferred effective population sizes and local population numbers numbers of of well studied, high density species, such as Drosophila, Drosophila, which seem to have local and higher than and regional densities several orders orders of of magnitude magnitude higher than their effective popUlation size inferred from the level of genetic variation. population of genetic land Second, with increasing anthropogenic anthropogenic fragmentation of of the natural natural landscape, it may be that unnatural metapopulations are being formed that very well might be subjected to the forces similar to those those modeled modeled in our simulations. Because many human impacts are are fairly recent, recent, these fragmented fragmented populations populations might be in the initial phase phase of of heterozygosity heterozygosity loss. In particular, particular, conservation conservation biologists should be cognizant cognizant of of the genetic implications of of metapopulation dy dynamics and realize that heterozygosity may be lost much much faster than predicted predicted from either census numbers population size. numbers or traditional estimates of of effective population As an example of 1 989) and of an application to conservation, conservation, Pimm et et al. al. ((1989) and 1 99 1 ) have suggested that metapopulation dynamics, rather pop Gilpin ((1991) rather than than a pop' Brien et ulation bottleneck (O 983), may have been important (O'Brien et al. al.,, 11983), important in reducing reducing the level of of genetic variation in the cheetah. The The relatively normal levels of of genetic variation recently found satellites and microsatellites in the cheetah ((MeMe found for mini minisatellites ' Brien, 11993, notti-Raymond 993, 11995) 995) appear notti-Raymond and O O'Brien, appear to be consistent with regener regeneration of Menotti of variation at these highly mutable loci since the bottleneck ((MenottiRaymond and O 'Brien, 11993). 993). On the other higher level of O'Brien, other hand, the higher of variation for for these more highly mutable loci is also consistent consistent with equilibrium levels predicted by neutrality theory given a low effective metapopulation size as de determined from the Hedrick, 11996). 996). the present study ((Hedrick,
ACKNOWLEDGMENTS ACKNOWLEDGMENTS We We appreciate appreciate the comments commentsof Nick Nick Barton Barton and and Ilkka Ilkka Hanski Hanski on an earlier earlier draft draft of the manu manuscript. P.W.H. was NSF. script. P.W.H. was partially partiallysupported supported for this this research research by by NSF.
sdfsdf
9
The Evolution Evolution of Metapopulations Metapopulations N. H. Barton
Michael C. C. Whitlock
I.I. INTRODUCTION I NTRODUmON Natural simplest models models in ways Natural populations populations differ differ from from the simplest ways which which can sig significantly affect their their evolution. evolution. Real Real populations populations are are rarely all of of the same size; nifi cantly affect the same the rates time; some rates of of migration migration into and and out out of of populations populations vary vary in space space and and time; populations established, while populations flucfluc populations go extinct, and new new ones are are established, while all populations tuate in size. Furthermore, not like those Furthermore, the genetic genetic properties properties of of real species are are not assumed wide variety of assumed in simple models. Alleles Alleles are are exposed exposed to a wide of selection, selection, mutation mutation rarely creates creates novel novel genotypes genotypes with each each mutation mutation event, event, generations generations metapopu overlap, and and environments environments vary from place place to place. Evolution Evolution in a metapopulation predictions of lation can be substantially substantially different different from from the predictions of single-population single-population models models of models and, and, indeed, indeed, very different different from from the the simplest simplest models of subdivided subdivided spe species. cies. Most species species inhabit a wide wide geographic range and and are subject to random random drift populations. Nevertheless, Nevertheless, spatial subdivision and stochastic stochastic per perin small local popUlations. turbations do not necessarily have a significant significant effect effect on evolution. Species Species might adapt as new alleles spread through through the regions in which they are favored; they might split into new, reproductively isolated, isolated, species if adaptations adaptations established established in different places tum turn out to be incompatible incompatible with each other. other. This is essentially the view held by R. A. Fisher 1 930); as we show below, it implies that Fisher ((1930); that population population Metapopulation MetapopulationBiology Biology Copyright © 1997 by Academic Press, Inc. All rights of 9 1997 of reproduction reproduction in any fonn tbrm reserved.
1183 83
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N. H. Barton and Michael Michael C. N.H. Barton and C. Whitlock Whitlock
structure 1 932) structure and gene interactions can be ignored. In contrast, contrast, Sewall Wright Wright ((1932) argued that a "shifting "shifting balance" balance" between random random drift and natural selection is necessary Mayr ((1942) 1 942) necessary to to establish establish new new combinations combinations of of genes. genes. In In aa similar similar vein, vein, Mayr argued argued that that gene gene flow prevents prevents divergence divergence in the the main main part of of a species species'' range and random shifts founder populations populations and that that speciation speciation depends on on random shifts in isolated founder (Mayr, 982; Provine, 986). (Mayr, 11982; Provine, 11986). In these contrasting In this this chapter, chapter, we we summarize summarize the the present present state state of of these contrasting views, views, by asking what effect population structure might have on adaptation and speci by asking what effect population structure might have on adaptation and speciation. Neutral Neutral variation is irrelevant irrelevant to these these processes, processes, and and so we must understand understand the effects effects of of migration and random random drift drift on selected genes. genes. To To date, most theory makes makes the unrealistic assumption that species species are divided into local populations populations of of constant constant size and with constant constant migration migration rates. We We pay particular particular attention attention to the the consequences consequences of of random random changes changes in in population population structure structure and, and, in in particular, particular, to local extinctions and recolonizations. We We will see that that if if all genes genes affect affect fitness fitness in the same way, regardless regardless of of the geographic location or genetic genetic background, background, then population structure amplifies random structure has no qualitative effect: it merely amplifies random drift cacy of drift and hence interferes interferes with the effi efficacy of selection. Population structure structure has fundamental effect if if there there are are interactions between between genes genes (i.e., dominance, dominance, a more fundamental epistasis, epistasis, or or frequency frequency dependence, dependence, allowing allowing alternative alternative "adaptive "adaptive peaks") peaks") or or if if the environment environment is heterogeneous (so that that the the local local population can can adapt adapt to local conditions). conditions). The The crucial crucial issue, issue, then, then, is is whether whether interactions interactions between genes and and with the local environment are extensive extensive enough enough for evolution to depend strongly on population structure. on population structure. There genes There is a close analogy between the subdivision of of a population of of genes across space and and across across diverse genetic genetic backgrounds. backgrounds. A gene can find itself itself as associated with a genetic background background which is at a selective advantage advantage and and will thereby increase by "hitch-hiking." Similarly, a gene itself in an area in gene can find find itself which the local local population is expanding expanding and and hence hence can gain an advantage through "spatial hitch-hiking." hitch-hiking." Recombination Recombination transfers transfers genes between genetic back backgrounds, just just as migration moves genes from from place to place; both processes processes reduce the random tness induced by the geographic or genetic background. random variation in fi fitness Also, genes genes can be adapted place in in which which they they find find themselves themselves and Also, can be adapted both both to to the the place and to the genetic background background in which they are embedded. embedded. At present, considerable effort effort is being devoted devoted to the study study of of hitch-hiking; many many of of the the techniques and and concepts in this area carry over to understanding effects of of spatial population concepts understanding the effects structure Barton, 1994, 1 994, 11995). 995). structure (cf. (cf. Barton, We the effects of population population subdivision We begin begin by by summarizing summarizing the effects of subdivision on on neutral neutral variation, variation, a relatively straightforward body of of theory. We We then then consider the sim simplest kinds kinds of of natural selection, selection, in which alleles are either either uniformly favorable favorable or deleterious; again, the results results are are straightforward. Population structure structure has more complex effects favored in different places; we show effects when different different alleles are favored different places; ow and population turnover impede adaptation how gene fl flow adaptation to a heterogeneous heterogeneous environment. Finally, we consider consider the most most complex case, in which genes genes interact interact with each each other, so that that populations populations can evolve toward toward alternative stable states. states.
99
The Evolution Evolutionof Metapopulations Metapopulations The
1185 8S
effects of of gene interaction interaction in a structured structured population are crucial to an unThe effects un ' s ((1931, adaptation via Wright 1 93 1 , 11932) 932) "shifting derstanding of of both both speciation speciation and adaptation Wright's "shifting balance" balance" process.
II. NEUTRAL NEUTRAL VARIATION VARIATION IN METAPOPULATIONS METAPOPULATIONS A. Models Models with Stable Stable local Local Populations Populations Sewall Wright popula Wright established established many of of the basic results results of of single-locus single-locus population genetic theory (Wright, 11931, 93 1 , 11932). 932). In particular, model of particular, he described a model of multiple multiple populations populations which which has come come to be known known as the "island "island model," model," a series of population of identical populations populations of of size N N individuals, individuals, with a fraction fraction m of of each population emigrating and being replaced by immigrants drawn at random from the common emigrating being replaced immigrants drawn random from common pool of of migrants. Such a metapopulation metapopulation will eventually reach reach an equilibrium equilibrium genetic variance variance among among local populations, populations, expressed expressed by the standardized standardized coeffi coefficient cient FST FST:: F F SST T
1
= ---
~"
4Nm 4Nm + + 1
9
((1) 1)
(F ST is, for (FsT for a two-allele two-allele system, the variance among among populations populations in allele fre frefrequency in the entire quency divided divided by p( p(11 - p), where p p is the allele frequency metapopulation.)) metapopulation. Other Other traditional traditional descriptions descriptions of of population structure structure include stepping-stone stepping-stone models (Kimura and Weiss, 1 964; Maruyama, 1 97 1 , 1 972), where models (Kimura Weiss, 1964; Maruyama, 1971, 1972), where populations populations exchange migrants only with adjacent exchange migrants adjacent populations. populations. These These models models approach approach the geographic metapopulations, where where local populations usually do geographic structure structure of of real metapopulations, not not exchange exchange migrants migrants equally with all others others (Hanski, this volume; see Maru Maruyama, 11971, 97 1 , 11972; 972; Hartl 989; Slatkin, 1993, 1 993, for Hartl and Clark, Clark, 11989; for more more details). details). -
B. Models Models with Extinction/Recolonization Extinction/Recolonization Many partitioning of variance in Many of of the same same factors which which contribute contribute to the partitioning of variance the island which control island model model are are also those those which control the the evolution evolution of of variance variance among among metapopulations re metapopulations with unstable unstable local dynamics and hence hence extinctions extinctions and recolonizations. increases the variance among colonizations. Essentially, small population population size increases populations population size allows populations and migration migration decreases decreases it. Small population allows genetic drift (and therefore therefore more more variance variance among among populations), populations), but because this scales with the reciprocal between population reciprocal of of population population size, there there is a nonlinearity nonlinearity between population size and the amount Thus, variable amount of of variance variance created created among among populations. populations. Thus, variable population population sizes increase increase the amount amount of of genetic differentiation differentiation among among populations populations relative relative to the case number of individuals were more case where where the same number of individuals more evenly distributed distributed among populations populations (see Whitlock, 1992a). among 1 992a). Extinction populations to Extinction of of local local populations populations allows allows the the variance among among populations
1186 86
N. H. Barton and Michael Michael C. N.H. Barton and C. Whiriock Whitlock
increase 977; Wade Wade and McCauley, 1988; 1 988; WhitWhit increase in most circumstances circumstances (Slatkin, 11977; 990; McCauley 1991, 1 99 1 , 1993; 1 993; Whitlock et 993; Wade Wade lock and McCauley, 11990; et at., al., 11993; et 994), because extinction usually implies that new populations are being et at., al., 11994), formed somewhere, either either in the same habitat habitat patches or elsewhere. elsewhere. Colonization Colonization is associated with founder effect; the size of of a new new local population is unlikely to be as large as the equilibrium size of of a local population. It is possible for extinction and colonization to reduce differentiation differentiation among populations, but only if genes move predominantly when new populations populations are formed. Extinction and recolonization will increase increase the genetic differentiation differentiation among populations if k k< <
2Nm 2Nm
--
I1 --~ b
11 +t 2 2''
(2)
where k k is the the number of of individuals which colonize new new patches, N is the the size of of extant extant populations, m is the migration rate among among populations, and 4~represents represents the probability that two colonizing genes come from the same source population (Whitlock and McCauley, 11990). 990). Note that that in most cases this condition is met; unless the process of of colonization colonization is substantially substantially different from the process of of migration migration among extant extant populations, kk will be approximately approximately equal to Nm N m (both of of which represent represent the number number of of individuals coming into a patch in one gener generation). The genetic differentiation of several species has been studied in relation to local extinctions 1 993; Giles and Goudet, this volume). extinctions (see McCauley, 1993; Similarly, whenever different populations have have different demographic demographic pa parameters, the the variance variance in frequency of neutral genes genes across populations is affected. affected. rameters, When When populations split, the variance variance among among them is increased, increased, because because of of the at., 11981; 98 1 ; Whitlock, 11994). 994). drift allowed by reduced reduced population sizes (Smouse et et al., When When the the populations split along along family lines, the the effect effect is greatly enhanced enhanced at., 11981; 98 1 ; Whitlock, 11994). 994). The population structure (Smouse et et al., structure of of the Yano Yanomama 98 1 ) clearly demonstrates mama people (Smouse et et at. al.,, 11981) demonstrates this effect. Migration decreases decreases the variance among among populations, by mixing mixing and homog homogenizing gene frequencies. The pattern of of migration is critically important: important: spatially variable migration migration patterns patterns result in a much higher degree degree of population differ differentiation than the mean migration rate would indicate (Whitlock, 11992b). 992b). This effect arises there arises because because some demes will have low immigration rates rates and can therefore fore drift drift especially strongly. Drift and migration affect population structure in a very nonlinear way, and hence mean differentiation differentiation is enhanced enhanced by the inclusion of of relatively isolated populations. As will be seen below, the effects of of demo demographic be graphic variability are are even stronger when evolution depends depends on interactions between genes. An important special special case of of spatial variation in migration rates is a type of of metapopulation metapopulation where where some populations consistently send out more migrants migrants than they receive and others receive more migrants than they put out (a "source-sink" "source-sink" metapopulation; Hanski and and Simberloff, this volume). The sinks would have smaller smaller (or zero) population sizes without migration migration from the sources. The dif-
99
The The Evolution Evolutionof of Metopopulotions Metapopulations
118/' 87
ferentiation among sink populations is likely to be much greater than that among source populations, but sink populations are not likely to diverge much from the source populations nearby (see Whitlock, 11996). 996). The main implication implication from these models is that, while populations in nature can express large degrees of variance as usually measured, measured, this may in fact fact mean very little in terms of of the evolutionary potential of the metapopulation. If the sink populations are contributing much to the variance among populations, but have a low probability of persisting to con contribute to future future generations, then the evolutionary events which occur in such sinks have little relevance. (See the sections below on local adaptations adaptations and the shifting shifting balance.) balance.) Temporal variability of of migration rate is less important than the the above factors in affecting the differentiation differentiation among popUlations populations in neutral genes. The effective migration rate is decreased decreased by a factor equal to half half the temporal variance in migration rate (me (m e = ---- iii m - - (iii ( / ~ 22 + -t- (T�)/2), tr2m)/2), which will be a fairly small effect even with quite large coefficients of 977; Nagylaki, 11979; 979; of variation (Sved and and Latter, 11977; Whitlock, 11992b). 992b). However, temporal variation can can become very important in predicting the evolutionary consequences consequences of of population structure for selected genes (see Moore and Tonsor, 11994, 994, and below).
C. C. The Effective Effective Size of a Metapopulation Metapopulation One of of the most important important evolutionary properties properties of of a species is its variance effective size. This is the size of popUlation in which drift of an ideal population drift causes the genetic variance to decline decline at the same rate as in the population in question. (An "ideal" "ideal" population is one in which individuals mate at random random and have equal chances chances of of reproductive reproductive success). The effective effective size helps predict the ability of of the population to fi x favorable mutations, the probability of xation of deleterious fix of fi fixation alleles, the amount amount of standing variation available to selection for for adaptation, adaptation, and other important important parameters parameters (see, for for example, the discussion in Crow and Kimura, ef11970). 970). A species which is subdivided into local populations has a different ef fective size than an undivided species with the same number of of individuals (Slat (Slat977; Maruyama and Kimura, 11980; 980; Chesser et 1 993; Whitlock and kin, 11977; et al. al.,, 1993; Barton, 11995); 995); therefore it is of of interest to discover the specific implications of of population subdivision for for effective size. In variance effective population In general, three three factors affect variance population size, and and each each of of them can be expressed in many ways. These are are the census population size (i.e., the total number metapopulation), the variance number of of individuals in the metapopulation), variance in re reproductive success among individuals (which can be caused by unequal unequal sex ratios, variance variance in any fitness component, etc.), and nonrandom mating. Furthermore, correlations between these factors are important; correlation across generations in reproductive success of of related individuals reduces effective effective size. Subdivision also affects the the ecological ecological properties of of a species. The The total num number of of individuals in a subdivided species is likely to be much less than than that that in
1!tl8 88
N. H. Borton and Michael Michael C. N.H. Barton and C. Whitlock Whitlock
an undivided habitat destruction undivided species. Subdivision Subdivision is often often the result of of habitat destruction and fragmentation, which which reduces reduces the resource resource available available to support support the species. species. Morefragmentation, More than large ones, so that over, smaller smaller populations populations have have a higher higher rate rate of of extinction than many part of many small habitat fragments fragments are occupied occupied only part of the time (Hanski, (Hanski, this rst principle volume). By the fi first principle above, above, as the census census population population size is reduced, reduced, the effective reduced. effective popUlation population size is also reduced. Second, population subdivision can suc Second, population subdivision can affect affect the variance variance in reproductive reproductive sucHigher variance cess. Higher variance in reproductive reproductive success translates translates into a lower lower effective effective population contributing much much to population size, essentially because because some some individuals individuals are are contributing the next generation and there is hence a higher chance that a random next generation and there hence higher chance that random pair pair of of alleles will be identical. island model, population size identical. In the island model, with constant constant local local population and metapopulation is actually increased and migration migration rate, rate, the the effective effective size of of the the metapopulation actually increased relative relative to an undivided undivided population population of of the same census census size, by a factor factor of of ST)' This reproductive success 11/(1 /( I - F FST). This is because because in the standard standard island model, model, the reproductive of xed. Hence, individuals within popUlations of each each population is fi fixed. Hence, the fitnesses fitnesses of of individuals within populations are become more are negatively negatively correlated. correlated. Thus, Thus, as the alleles within populations populations become more be closely related, the variance variance in average average reproductive reproductive success of of each each allele beif each comes smaller, and and the effective effective metapopulation metapopulation size grows. grows. In contrast, contrast, if each popUlation fitness, as for population is allowed allowed to have have a different different mean mean fitness, for example example in the natural natural case case when when local extinctions are allowed, the variance variance in reproductive reproductive individuals into local local poppop success can can be greatly greatly increased increased by the clumping clumping of of individuals effective size of whole metapopulation ulations, and and the effective of the whole metapopulation can can be greatly greatly de decreased (Hedrick and more general model of creased (Hedrick and Gilpin, this volume). volume). A more general model of effective effective size is complex Whitlock and 996) but can illustrated with a complex (see Whitlock and Barton, Barton, 11996) can be illustrated If the census size of populations is N, but but their output special case. If of all populations output varies, varies, then the overall effective size, Ne, then the overall effective Are, is -
Nil
N~
-
-
(1 + Vs(R))(1 - FST) + 2NFsTVs(R)
o
(3) (3)
Here tness among Here there there are are n local populations, populations, Vs(R) V~(R) is the the variance variance in fi fitness among popu populations, 969; lations, and and FST is the the standardized standardized variance variance across across populations populations (Wright, (Wright, 11969; Weir 984). Each Weir and and Cockerham, Cockerham, 11984). Each population population is assumed assumed to have have the the same size, size, and 996, de and migration migration is not not geographically geographically structured structured (Whitlock (Whitlock and and Barton, Barton, 11996, describe population size scribe more more general general cases). cases). FST increases increases with with decreasing decreasing local local population and ST is large and variance in reproduc and decreasing decreasing migration migration rate; rate; if if F FST and the the variance reproducpopulations is small, then tive success success among among populations then subdivision subdivision increases increases effective popUlations vary in fitness, then much less size. On On the other other hand, hand, if if populations then Ne will be much than it would would be without without subdivision subdivision (see Fig. 11 for for an example example with with local local ex extinction). tinction). Finally, Ne is affected the equation affected by nonrandom nonrandom mating. mating. In the equation above, above, the the degree degree to which which mating mating is nonrandom nonrandom due due to population population subdivision subdivision is described described by F ST' Local mating depending on FST. mating can increase increase or or decrease decrease effective size, depending on the the distribution distribution of of reproductive reproductive success. success.
99
1189 89
The The Evolution Evolution of of Metapopulations Metopopulotions
11.0.0
0.8
�
o
o. 0.~
Ne
nN nN 0.4'
9 "~,o
"~ ~ : ~,.. , , .....
o. 0.~
.
~
II e l m I i i
I 9 mmwm
.:. -:.:.::::
_ m ~
~
~
_L anm •• _ NAaman umom• • 9 • • ••• -
- � ...
o.ot0.0 ----.---"""T'"---, 0.0 0.0
FIGURE FIGURE 1I
"
I
0. 0.11
Extinction Extinction rate rate
,
u
0.2 0.2
Increasing Increasinglocal local extinction extinction rates rates reduce reduce the effective effective population population size size of the species. species.
= 1 . The upper continuous ( = = 50, k = = 2, and m m= = 0. 0.1. continuous line line is for unrelated unrelated colonists colonists (4) = 0), the dotted dotted = identical colonists ( = = 1). I ). (From = 0.5, 0.5, and the dashed dashed line line is for genetically genetically identical colonists (~ (From Whitlock Whitlock 1 996, Fig. and Barton Barton ((1996, Fig. 2).)
N
line line is for ~
III. SElEGION SELECTIONIN METAPOPULATIONS METAPOPULATIONS A. The Establishment Establishmentof Favorable Favorable Alleles Alleles The involves two The establishment establishment of of aa new new and and favorable favorable allele allele involves two stages: stages: first, first, the around its the increase increase from from a a single single mutant mutant copy copy to to appreciable appreciable frequency frequency around its orig original inal location, location, and and second, second, its its spread spread through through the the whole whole population. population. Even Even if if a a new new mutant most copies copies will mutant has has an an appreciable appreciable advantage, advantage, it it is is unlikely unlikely to to be be fixed: fixed: most will be be lost lost by by chance chance in in the the first first few few generations. generations. With With panmixis, panmixis, the the probability probability of of fixation fixation is is just just twice twice the the selective selective advantage advantage of of the the heterozygote heterozygote (2s, (2s, for for s « << I1;; Fisher, 922; Haldane, 927). Remarkably, this probability probability is is not not affected Fisher, 11922; Haldane, 11927). Remarkably, this affected by by spatial spatial subdivision, subdivision, provided provided that that population population sizes sizes are are constant constant and and there there is is sym symmetric 1 970). Hence, Hence, the species accu metric migration migration (Maruyama, (Maruyama, 1970). the rate rate at at which which aa species accuN mulates alleles is mulates favorable favorable alleles is not not affected affected by by subdivision subdivision and and equals equals 4Ns, where where N is the whole By analogy, analogy, the the probability probability of fix is the the effective effective size size of of the whole population. population. By of fixation linked to the gene in question, ation is is not not altered altered by by balancing balancing selection selection at at genes genes linked to the gene in question, provided isms do provided that that the the balanced balanced polymorph polymorphisms do not not fluctuate fluctuate in in frequency. frequency. Random introduce an an additional Random extinctions extinctions and and recolonizations recolonizations introduce additional source source of of random hence reduce reduce the probability: an an advantageous random drift drift and and hence the fixation fixation probability: advantageous allele allele may be be lost lost when the population population in in which goes extinct. extinct. In In the the island island may when the which it it arises arises goes model, common pool, fixation model, where where migrants migrants are are drawn drawn at at aa rate rate m m from from a a common pool, the the fixation
1190 90
N. H. Barton and Michael Michael C. N.H. Bartonand C. Whitlock Whitlock
probability can Iizuka, 11991; 99 1 ; Barton, Barton, 11993). 993). If If colo probability can be be calculated calculated (Tachida (Tachida and and Iizuka, colonists populations, the be reduced nists are are drawn drawn from from many many populations, the fixation fixation probability probability cannot cannot be reduced by if colonists from aa by more more than than half, half, to to ss instead instead of of 2s. 2s. However, However, if colonists are are derived derived from single be if inbreeding at foundation of single ancestral ancestral gene gene (as (as may may be if there there is is severe severe inbreeding at foundation of new have a 2). The new populations), populations), extinction extinction can can have a much much greater greater effect effect (Fig. (Fig. 2). The chance chance that weakly favored allele will will be xed is from 2s to 2s/[(1 2s/[( l + that a a weakly favored allele be fi fixed is reduced reduced from 2s to + A/m)( A is is the the rate rate of the deme size (assumed A/m)(1l + + 2NA)] 2NA)],, where where A of extinction extinction and and N N the deme size (assumed to 993). Thus, if the rate of extinction is to be be constant; constant; Barton, Barton, 11993). Thus, if the rate of extinction is higher higher than than the the rate � m), m), and than random random drift (A � reduc rate of of migration migration (A (A >> and faster faster than drift (A >> 1/2N), 1/2N), the the reduction be severe; severe; a species consisting wiII accumulate favorable tion can can be a species consisting of of n n demes demes will accumulate favorable mutations which is is independent mutations at at aa rate rate 2snp.,m/A 2snkem/A22,, which independent of of deme deme size size and and inversely inversely proportional rate. proportional to to the the square square of of the the extinction extinction rate. The fixation probability probability is is not not the as its The effect effect of of extinction extinction on on fixation the same same as its effect metapopulation (NeiN A + effect on on the the effective effective size size of of the the metapopulation (Ne/N = = ((11 + + 2N 2NA + 4Nm)!(4N(m 977), and extinction could could interfere 4Nm)/(4N(m + A)); A)); Slatkin, Slatkin, 11977), and so so extinction interfere substantially substantially with with adaptation adaptation without without having having an an appreciable appreciable effect effect on on the the diversity diversity of of neutral neutral markers small). However, However, abundant markers (provided (provided that that local local population population sizes sizes are are very very small). abundant species be limited Adaptation to to species may may not not be limited by by the the establishment establishment of of new new mutations. mutations. Adaptation changed depend on appre changed circumstances circumstances may may depend on alleles alleles that that previously previously occurred occurred at at appreciable with quantitative high heritability the ciable frequency, frequency, as as with quantitative traits traits of of high heritability and and as as may may be be the case industrial melanism 98 1 ). Even case for for industrial melanism in in the the peppered peppered moth moth (Lees, (Lees, 11981). Even if if new new mutations negligible delay. mutations are are required, required, these these may may occur occur so so often often as as to to cause cause negligible delay. The rate rate of of mutation mutation to to any any new new protein protein differing differing by by one one amino-acid amino-acid is is = ~ 110 -9, The 0 -9, and so so all all such such mutations mutations occur occur at at least least every every generation generation if if the the whole whole population population and is larger larger than than = ~ 1109. For example, example, warfarin warfarin resistance resistance in rats arose arose independently 09• For in rats independently is at twice in 1 0 years poison (Drum(Drumat least least twice in Britain Britain within within 10 years of of the the application application of of this this poison 2s
I .11,e" ' •.
' . •t:J. ........ .
4Nm = O.Ol ~ 4Nm=0.01 t. 3 4Nm = 0.1 O. 1 o 0 ~ , .......... r............ r............ [] 4Nm = 11 110 0 8 4 6 2 0 o AIm g/m FIGURE ]:I~URI: 2 ~ For a given ratio between extinction and migration (AIm), (A/m), the fixation fixation probability de decreases as migration and extinction rates increase. The solid solid curve gives the limit of low migration limit of weak and extinction rates rates (2sl( (2s/(1l + AIm)), A/m)), while the dotted curves give the limit weak selection l + 2NA)]). (2s/[( (2s/[(1l + + Alm)( A/m)(1 2NA)]). The symbols symbols give exact numerical numerical solutions. Colonists are derived from a single gene; (From Fig. Fig. 2 of Barton ((1993).) 1 993).) gene; 4Ns 4Ns = I1.. (From
".. ....Q..
=
- . - - . ...
. - . - - . � .. . - .... . .... . . . . . . .
99
The Evolution of The Evolution of Metopopulotions Metapopulations
1191 91
mond, 11966). 966). Thus, the loss of of favorable alleles caused by random extinction in a metapopulation cant long-term consequences. metapopulation may not have signifi significant consequences. Once an allele is established established in large numbers, numbers, it is almost certain certain to spread through the whole population. In the island model, it will increase exponentially, where p is its average as p/ p / qq = ~ exp( exp(tt s(l s(1 -- FST» FST)), where/S average frequency, s its selective ' advantage, ST advantage, and t the time. The The rate rate of of increase increase is reduced by a factor 11 -- F FST because selection can act only on that fraction of of genetic variance variance that is found within populations popUlations (see below). This result applies even if population sizes vary randomly, as in the usual metapopulation models; since FST is usually small, subdivision is unlikely to delay the spread spread of of favorable alleles by much. When migration distances are restricted, the way in which an allele spreads depends depends critically on the pattern Mollison, 11977). 977). If the number number of pattern of of migration migration ((Mollison, of long longrange range migrants migrants falls away exponentially or faster, faster, then the allele spreads spreads as a J2S, where a is the standard smooth wave, with speed a o'xf~, o-is standard deviation of of distance distance between Fisher, 1937). 1 937). However, if migration is more lep1ep between parent and offspring ((Fisher, tokurtic than this, spread is faster and messier: single genes jump ahead and establish themselves, while others leapfrog past (see Shaw, 11995). 995). While luck is needed to observe this process as it occurs, it may leave a trace trace in the pattern of of secondary contact. If two popUlations meet each other in smooth populations advance advance to meet waves, they will become become separated by a sharp sharp genetic boundary, but if they spread through sporadic advances, advances, they will eventually be demarcated demarcated by a broad cline of variable shape ((Nichols Nichols and Hewitt, Hewitt, 11994). 994).
B. Deleterious Mutations In higher organisms, the net rate rate of of mutation to deleterious deleterious alleles is high enough to cause a substantial "mutation load" ((Kondrashov, Kondrashov, 1988). 1 988). In a sexual population, this manifests manifests itself itself in three three ways. First, when an equilibrium is reached reached between between mutation and selection, mean fitness is reduced. If the effects of of mutations are multiplicative, fitness is reduced by a factor eexp( x p (- U U),), where where U is the the total rate of of mutation across the the diploid genome. If If additional mutations mutations reduce fi tness by a larger factor ("synergistic epistasis"), Kon fitness epistasis"), this load may be reduced ((Kondrashov, 11984). 984). Second, mutations may xed in the whole may be fi fixed whole population. population. This This process process may lead to a catastrophic melt catastrophic decline in fitness in small populations populations ("mutational ("mutational melt993), but may also destroy adaptations down"; Lynch et et al., al., 11993), adaptations based on very weakly selected alleles in even large popUlations. populations. For example, example, codon usage is biased toward ciently translated codon, but the bias is not complete; toward the most effi efficiently this suggests that Bulmer, that the less favorable codon has often been fixed by drift ((Bulmer, 11991). 99 1 ). Finally, deleterious deleterious mutations interfere interfere with selection at linked loci. In an asexual population, population, weakly favored alleles can be fixed only if they arise in the genetic background carrying no deleterious mutations, which greatly reduces their chances 994). Even in a sexual population, however, deleterious mutachances (Peck, 11994).
1192 92
N.H. Bartonand and Michael MichaelC.C. Whitlock Whitlock N. H. Barton 110 0
2D
E
11D D
~9 Island
0.1
0.01 0.01
FIGURE F]GM~I: 3 ~
,
i
,
,
,
,
,
4Ns
0.1 0.1
,
,
1
The mutation and migration migration (f,Llm)";,, plotted against If the The critical ratio ratio between mutation (~/m)crit, plotted against 4Ns. If mutation rate rises above above this value, the deleterious deleterious allele is fixed. Each of three for (from (from mutation rate Each set of three lines is for bottom bottom to top) top) f,Lls tx/s = = 0.2, 0.2, 0. 0.1, 1 , 0.05. 0.05. Values for for the island model model are calculated calculated analytically, analytically, while while those for for oneone- and and two-dimensional two-dimensional stepping stepping stone stone models models are are taken taken from from simulations simulations of of 401 poputhose popu of 3311 X • 3311 populations populations (in 2D), each each with 2N = = 20 haploid individuals. lations (in I1D) D) or of haploid individuals.
tions can can substantially substantially reduce reduce fixation fixation probabilities (Charlesworth, 1994a; Barton, tions probabilities (Charlesworth, 1 994a; Barton, 11995). 995). Does spatial spatial subdivision subdivision accentuate accentuate these these various various deleterious deleterious effects effects of of mu muDoes tation? One One can can show show by a simple argument that for for arbitrary population population struc structation? argument that tures, x p ( - UU) ) to tures, fitness fitness in a mutation-selection mutation-selection balance balance is reduced reduced from from eexp( to eexp( xp(- U U //( ( 1I - Fsv)). denote the average frequency FST » . To To see this, this, denote the average frequency of of the the deleterious deleterious allele by/7, and the Then Pi ' Then allele by p, and the frequency frequency in the the i th deme deme by p/. -
-
-
Api--
t.l,q i -
(4a) (4a)
sPiqi 4- ~i -'~ m ( f i i - Pi),
Here,/~; among immigrants and m Here, Pi is the the frequency frequency of of the the allele allele among immigrants to to deme deme i, and m is the the migration ffi is the the perturbation perturbation due random drift. migration rate. rate. ?i due to to random drift. Taking Taking the the average average across gives across demes demes gives Aft = i~q-
(4b (4b))
s p q (1 - Fsr) + ~,
m
~r ? is the the fluctuation fluctuation in average average allele allele frequency frequency due due to to drift, drift, which which is on on average average zero. zero. FsT FST is the the standardized standardized variance variance in in allele frequency frequency across across demes. demes. The The flucfluc tuations due due to to drift drift average average out, out, and the effect effect of of migration migration disappears disappears if if we we tuations and the make that migration make only only the the assumption assumption that migration does does not not create create or or destroy destroy alleles. alleles. Taking Taking expectations and and solving solving for equilibrium givesfi givesp = expectations for equilibrium = I~/[s(1 jL/[s( 1 - Fsv)]. FST)]. Since Since Fsv FST is usually usually small, this this remarkably remarkably general general result result implies implies that that subdivision subdivision is unlikely unlikely to increase the to much much increase the mutation mutation load. load. Strictly Strictly speaking, speaking, Eq. Eq. (4b) (4b) applies applies with with FST FST denoting the standardized variance of deleterious alleles, which may differ from denoting the standardized variance of deleterious alleles, which may differ from -
99
The The Evolution Evolution of Metopopulotions Metopopulations
1193 93
that for population structures structures show for neutral neutral markers. markers. Simulations Simulations of of a variety of of population show that when ciently low relative to mutation, mutation, deleterious deleterious when migration migration rates rates are are suffi sufficiently alleles When /-L/m alleles can fix in some some demes, leading leading to high high F FsT's and a high high load. When tz/m ST's and is above above a critical ratio, deleterious deleterious alleles fix everywhere. everywhere. However, this requires requires very Fig. 3). very low low migration migration rates rates ((Fig. By assuming uctuations within assuming that that fl fluctuations within populations populations occur occur on on a much much shorter shorter time scale than if there than changes changes in the overall overall frequency frequency (reasonable (reasonable if there are are very many populations populations and selection of selection is weak), one can can show show that the probability of fixation of deleterious allele FST)(Ne/N)/(exp(4Nes ( 1 fixation of aa deleterious allele is is 2s( 2s(1l -FST)(Ne/N)/(exp(4Nes(1 - FF S TST) )) )- 11), ), where Ne is the effective population, and N is its census size. where Ne effective size of of the the whole whole population, and N Though uctuations in population particular, local Though fl fluctuations population size (and (and in particular, local extinction) extinction) can can reduce reduce the the effective effective size below below the the census census size (see above), above), this this argument argument shows shows that that deleterious deleterious alleles alleles are likely to be fixed in the whole whole species species only if if selection selection is very weak 1 0). Population therefore does weak (say, 4Nes( 4Nes(11 - F FST) < 10). Population subdivision subdivision therefore does ST ) < not not greatly greatly increase increase the the chance chance loss loss of of adaptations. adaptations.
IV. IV. ADAPTATION ADAPTATIONTO TO LOCAL LOCALCONDITIONS CONDITIONS Spatial subdivision subdivision has has potentially important important consequences consequences when when selection selection varies varies from from place place to place. There There has has long long been been controversy controversy over over whether whether geo geographic graphic isolation isolation is required required for for the divergence divergence of of lineages. Darwin Darwin and and Wallace Wallace held that popUlations populations could could adapt adapt to gradually gradually varying conditions conditions across across their their range range and and that this could could lead to speciation. speciation. In contrast, contrast, Wagner Wagner emphasized emphasized the the importance 982, p. importance of of barriers barriers to migration migration in allowing allowing speciation speciation (see Mayr, Mayr, 11982, 562). 1 969) argued 562). More More recently, recently, Ehrlich Ehrlich and and Raven Raven ((1969) argued that that because because genes genes diffuse diffuse slowly through factor impeding impeding through most most species, gene gene flow cannot cannot be be a significant factor divergence. divergence. Population Population genetic theory theory strongly strongly supports supports this view, view, at least if if one one considers adaptations considers adaptations which which can be built up up by individual alleles or or by continuous continuous changes changes in quantitative quantitative traits. We We consider consider the complications complications of of gene gene interactions interactions in the next next section. Whether Whether populations populations can adapt adapt to local conditions conditions depends depends essentially on the the relative relative rates rates of of gene gene flow and selection. selection. The The simplest case is where where an allele allele is favored favored in a population, population, and and has has selective selective advantage advantage s in the the heterozygote. heterozygote. The The allele can be established established despite despite gene gene flow, provided provided that the rate of of immigration immigration of of individuals individuals carrying the alternative alternative allele is lower lower than than the rate rate of of selection selection (m < 93 1 ). One < s; s; Haldane, Haldane, 11931). One can can generalize generalize this to the island model, where where the the allele has has advantage advantage s in a fraction fraction a a of of demes demes and and disadvantage disadvantage - / 3{3s s in the the remainder. than the critical remainder. The The allele can can be be established if if migration migration is lower lower than value {3 ( 1 is favored favored in value merit merit < < (3s/( [~S/([~(1 - a) a) - a), a), or or alternatively, alternatively, if if it it is in at at least least aa /(3) of be fraction fraction aerit acrit = = ((11 - s/m)/( s/m)/(1l + + 11//3) of populations. populations. The The allele allele can can always always be established established if m < < s (heavy (heavy curves curves in Fig. 4a). If habitat, then If genes genes diffuse diffuse at at aa rate rate ss across across aa continuous continuous habitat, then an an allele allele can can
a
1
Nm Nm == 0.05 0.05 Nm Nm == 0.25 0.25 Nm Nm == 1.25 1.25 Nm = Infinite Nm = Infinite
O
O
i
0 0.1
mls m/s |
[
'
'
'
'
i
,
10
,
1
~
I
nit~'
' bl
Nm = 0.25
Nm = 0.05 ~
"t:Z
sf
Nm = 0.25
Nm = 1.25 o0
,
,
,
0.1 0.1
,
I
,
,
,
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FIGURE [:IGIJRI: 4 ~ The The critical critical proportion proportion of of populations, populations, a, c~, required required to to allow allow local local adaptation adaptation in in the the island island model. model. In In aa proportion proportion a a of of populations, populations, an an allele allele is is favored favored by by additive additive selection selection s, while while in in the the {3 = :I + remainder, remainder, it it is is at at aa disadvantage disadvantage - / {3s 3 s ((/3 = 11 in in this this example; example; fitnesses fitnesses are are 11:1 + ss:: 11 + + 22ss or or 2s). (a) (a) Results Results with with no no extinction, extinction, plotted plotted against against the the relative relative rates rates of of migration migration and and I1 :: 11 - ss:: I1 - 2s). selection, selection, mls. m/s. The The heavy heavy curves curves show show the the deterministic deterministic limit, limit, in in which which drift drift is is negligible. negligible. The The lower lower heavy heavy curve curve shows shows the the proportion proportion of of populations populations in in which which allele allele A A is is favored favored that that is is required required for for it it to to invade invade (a (acre, = ((1I - slm)/( s/m)/(1l + + 1/(3)). 1//3)). The The upper upper heavy heavy curve curve shows shows the the corresponding corresponding criterion criterion cei. = for for the the other other allele allele to to invade. invade. Thus, Thus, both both can can be be maintained maintained for for a a between between the the two two curves; curves; polymor polymor). The phism phism and and local local adaptation adaptation is is always always possible possible if if ss > > m m (i.e. (i.e. mls m/s < < I1). The three three pairs pairs of of light light curves curves show m = .25, 0.25, show the the critical critical values values for for N Nm = 11.25, 0.25, and and 0.05: 0.05: as as the the number number of of migrants migrants decreases, decreases, con conditions ditions for for polymorphism polymorphism become become more more restrictive. restrictive. (b) (b) Corresponding Corresponding conditions conditions with with random random ex extinction tinction at at aa rate rate A, A, plotted plotted against against the the relative relative rates rates of of extinction extinction and and selection, selection, Als. A/s. Colonization Colonization involves ). These involves severe severe inbreeding, inbreeding, so so that that colonists colonists are are derived derived from from one one gene gene (cf> (4) = = I1). These are are calculated calculated exactly, exactly, from from the the transition transition matrix. matrix. Migration Migration is is fixed fixed at at m m= = 0.025 0.025 and and selection selection at at ss = 0.05. 0.05. The The I - s/(A heavy heavy curves curves give give the the deterministic deterministic limit limit (a"" (Ogcrit = = ((1 s/(A + + m))/(l m))/(1 + + l/(3)), 1//3)), while while the the three three light light m = 11.25, .25, 0.25, 00, 20, curves curves give give the the limits limits for for N Nm 0.25, and and 0.05 0.05 (2N (2N = = 1100, 20, 4, 4, respectively). respectively). The The shaded shaded m = region region to to the the right, right, which which lies lies between between the the curves curves for for N Nm = 0.05, 0.05, shows shows the the region region in in which which neither neither allele allele can can invade, invade, so so that that the the population population evolves evolves to to fixation fixation for for one one or or the the other. other. -
-
-
-
=
-
=
99
The Metapopulations The Evolution Evolutionof of Metapopulations
1195 95
become established ciently large region, wider established provided that it is favored favored in a suffi sufficiently 973). In if the than than aa characteristic characteristic scale scale Il = = (TIffs o'/x/2-~ (Slatkin, (Slatkin, 11973). In aa linear linear habitat, habitat, if the allele is at an advantage advantage s in some some region and a disadvantage disadvantage - / 3{3s s elsewhere, then it can be established if the region is wider 2tan- I (.J/3)1 ((Nagylaki, Nagylaki, established if wider than than 2tan-~(x/~)l 11975). 975). New of fixation approaching New mutations mutations have a probability probability of approaching 2s 2s if they arise within the appropriate region, but the probability probability falls to zero outside this region (Barton, 11987). 987). A similar condition applies applies to adaptations adaptations based based on quantitative VJ2 Vg , where traits; traits; here, here, the the characteristic characteristic scale scale is is (T-J o'x/Vs/2Vg, where V, Vs is aa measure measure of of the the strength of of stabilizing stabilizing selection, and V Vgg is the additive genetic variance variance (Slatkin, (Slatkin, 11978b). 978b). Since the dispersal is typically much much smaller than dispersal range range (T o-is than the species' species' range, to diverse range, weak weak selection selection can can allow allow aa species species to to adapt adapt to diverse local local conditions. conditions. This can be seen directly. For example, the grass For example, grass Agrostis tenuis tenuis has evolved tolerance to heavy metals MacNair, 11987). 987). metals on mines only a few few meters meters across across ((MacNair, Similarly, narrow clines = ~ 10 km km wide separate races of of Heliconius Heliconius butterflies adapted to different Mallet, 1993). 1 993). Such different mimicry rings ((Mallet, Such examples examples leave leave little doubt doubt that that gene gene flow flow need need not not prevent prevent divergence divergence over over very very short short scales.
A. The Effect Population Turnover Turnover on local Effect of Population Local Adaptation How much much do the local extinctions characteristic of of metapopulation dynamics dynamics impede adaptation? effects: they introduce adaptation? Extinctions have two effects: introduce an additional source of cient; and extinctions increase of random random drift, making selection less effi efficient; increase gene flow, because colonists may be derived from different different habitats. We We can illustrate illustrate these these effects effects using the simple island island model model introduced above, in which fraction must be greater an allele is favored favored in a fraction a a of of populations; this fraction greater than some critical threshold for frequency. Similarly, a for it to increase increase from low frequency. a must be below a second threshold if both conditions threshold if the other allele is to invade; if are some degree are met, a polymorphism can be maintained, maintained, and there there is some degree of of local adaptation (Fig. ( Fig. 4a). Random Random drift reduces under which both alleles reduces the conditions under maintained; if than ~ 1,, then variation is likely to be lost, = 1 can be maintained; if Nm is much less than even if migration is weaker weaker than than selection (m « << s). However, as we argued argued above, ST usually observed above, the the low low values of of F FST observed imply imply fairly fairly high Nm, suggesting suggesting that that local adaptation is not usually much much impeded impeded by drift. Our Our model model includes includes no explicit spatial structure. However, the effect of of random random drift on neutral vari variation in a two-dimensional population population also depends depends on Nm (or equivalently, equivalently, on ' s 11943 Wright 943 neighborhood size), so that Wright's that the arguments arguments may may carry over to this more Malecot, 11948, 948, 11969). 969). more realistic case case ((Mal6cot, We for local We can can now now ask ask how how extinctions, at a rate A, affect the the conditions conditions for adaptation. We We need only consider alleles which are very rare rare in the the migrant pool. ). where where habitat patches im rst the deterministic case (Nm � Consider fi first >> 11). patches are immediately ji, and suppose for the moment moment that selection mediately recolonized recolonized at at frequency frequencyfi, and suppose for the that selection is is stronger stronger than than migration migration (s (s » >> m). In In aa fraction fraction a a of of populations, populations, the the allele allele ji exp(st), (3st). The increases increases as asfi exp(st), while in remainder, remainder, it decreases decreases at a rate rate exp( e x p (- ~st). The probability that the populations will survive for for time t is = ~ exp( e x p (- At); averaging
1196 96
N. H. Barton and Michael Whitlock N.H. Barton and Michael C. C. Whitlock
over migrant pool over this distribution, distribution, the the expected expected contribution contribution to the migrant pool is a Aji/().. (1 -- a )Aji/().. + aAfi/(A - s) + + (1 c0Afi/(A + f3s). /3s). If If this this value value is is greater greater than than the the present present frequency, increase; hence, = frequency, p, fi, then then the the allele will increase; hence, the threshold threshold is a acrit crit = ((1I }..)/( I + ) , then - s/ s/A)/(1 + l/f3). 1//3). If If selection is faster faster than than extinction (s > > }.. A), then the allele is likely to reach reach very very high high frequency frequency before before the the population population goes goes extinct, extinct, and and so invasion invasion is possible possible for arbitrarily small small a. a. The The argument argument is easily extended extended to to include include the the effects effects of of migration, which which reduces reduces the the rate of of increase increase from from low low frequency m)t), leading leading to a m» /( l ++ 1//3). l/f3). Thus, frequency to exp« exp((ss - m)t), O~crit - - ((1l - s/().. s/(A + + m))/(1 Thus, crit = continual continual migration migration and and sporadic sporadic extinction extinction have have the the same same effect, both both tending tending to increase ow. The increase the the effective effective rate of of gene gene fl flow. The allele allele can can always always be be established established if its advantage advantage outweighs outweighs their their combined combined effects effects (s > > ().. (A + + m» m));; otherwise, otherwise, it can can sustained only if if it is favored favored in a suffi sufficiently fraction of of the the metapopbe sustained ciently large fraction metapop ulation ulation (heavy curves curves in Fig. 4b). This This argument argument applies applies only when when populations populations are large enough enough to make make drift drift negligible nd analytic results which in negligible even even during during colonization. colonization. It is hard hard to fi find incorporate corporate the effects effects of of random random drift, drift, because because it is then then likely that the allele will drift to high high frequency frequency in some some populations, populations, even even when when it is rare rare in the the metapop metapopulation as a whole. Analysis of of selection, selection, migration, migration, extinction, and drift drift become become ulation intractably complicated complicated when when alleles become become so common common that that nonlinear nonlinear interac interactions are important. However, However, one one can solve the model model exactly, by calculating the transition that is, the probability transition matrix for for each populationpopulationmthat probability that a deme deme car carrying i copies copies of of the allele will carry carry jj copies copies in the next generation, generation, taking the next taking into into account account migration, migration, selection, extinction, extinction, and and drift. drift. This This shows shows that that extinction can can have consequences when have surprising consequences when the the number number of of migrants migrants is small. As As in Fig. 4b shows a pair pair of of curves for each each value value of of Nm; Nm; these give the critical 4a, Fig. 4b curves for proportions invade. To proportions of of popUlations populations (a) (a) required required to allow allow either either allele to invade. To the under which both alleles left, the region region between between the curves curves shows shows the conditions conditions under which both can invade invade from from low frequency, frequency, allowing allowing an adaptive adaptive polymorphism. polymorphism. As the the rate of becomes less likely (right of of extinction extinction increases, increases, polymorphism polymorphism becomes of Fig. 4b), 4b), as expected expected from from the deterministic deterministic limit. As As the rate rate of of extinction increases increases further, further, the two kinds the curves curves cross, cross, so that when when the the proportions proportions of of the the two kinds of of habitat habitat are equal (a = Nm neither allele equal (a ~ 0.5; shaded shaded region region on right right of of Fig. 4b 4b for for N m = - 0.05), 0.05), neither can for one can invade. invade. This This implies that that the metapopulation metapopulation can can fi fixx for one or or the other other allele, depending depending on starting conditions. conditions.
B. limits Limits to Local Local Adaptation It is striking many striking that that during during drastic drastic changes changes in climate, the distribution distribution of of many species species shifts over over large distances, distances, apparently without much much change change in the range range of environmental conditions which they experience (e.g., Atkinson et at. , 1 987). of environmental conditions experience Atkinson et al., 1987). Species may fail to evolve further under natural selection, because they lack Species may to further under natural selection, because lack the necessary necessary genetic genetic variation, variation, perhaps perhaps reaching reaching some physical limit or or becoming becoming trapped trapped in a coevolutionary coevolutionary race race with competing competing species species (Stenseth (Stenseth and and Maynard Maynard
99
The The Evolution Evolutionof of Metapopulations Metapopulations
1197 97
Smith, 11984). 984). Another ow from populated Smith, Another possibility possibility is is that that gene gene fl flow from the the more more populated areas preventing areas prevents prevents adaptation adaptation at at the the edge edge of of aa species' species' range, range, thereby thereby preventing advance conditions from the ones within the advance into into regions regions with with different different conditions from the ones within the occupied occupied range. range. The arguments arguments so so far far have have assumed assumed that population size size is is independent of The that population independent of genotype receive most most genotype and and habitat. habitat. However, However, since since small small populations populations are are likely likely to to receive immigrants immigrants from from large large populations, populations, their their ability ability to to adapt adapt may may be be substantially substantially 1 996), who limited. An example is Dias et limited. An example is given given by by Dias et al. al. ((1996), who show show that that populations populations of living in island of of the the great great tit tit living in evergreen evergreen habitat habitat on on the the island of Corsica Corsica lay lay their their eggs eggs later in deciduous fledglings later than than do do those those living living in deciduous habitat habitat on on the the mainland, mainland, so so that that fledglings coincide with However, coincide with the the later later profusion profusion of of insects insects emerging emerging on on this this habitat. habitat. However, tits mainland lay lay their tits living living in in the the minority minority evergreen evergreen habitat habitat on on the the mainland their eggs eggs earlier earlier than optimal. There evidence of high rate than appears appears optimal. There is is evidence of aa high rate of of gene gene flow flow into into the the evergreen since strong micro evergreen habitat, habitat, since strong linkage linkage disequilibrium disequilibrium is is found found between between microsatellite combinations characteristic populations in satellite allele allele combinations characteristic of of populations in the the majority majority habitat. habitat. The ow to The potential potential power power of of gene gene fl flow to prevent prevent adaptation adaptation at at the the edge edge of of the the range simple model First, take range can can be be illustrated illustrated by by aa simple model of of aa quantitative quantitative trait. trait. First, take the the pattern population density density as given. The value of trait then pattern of of population as given. The mean mean value of the the trait then changes changes according according to to
Oz 0-20Zz 0log(N) Vg a0log(W) (J2 a2z az a log(N) Oz az � log(W ) . . =. . . ~- (J2 o.2 ~ ~ + + .. + 2 0 xax22 Ox Ox 2 Oz aOtt 2 ax ax 2 az
5) ((5)
The rst term rate (J2 The fi first term represents represents the the diffusion diffusion ooff genes genes from from place place ttoo place place at at aa rate 0 -2 (the while the (the standard standard deviation deviation of of distance distance between between parent parent and and offspring), offspring), while the third third represents tendency of increase mean mean fitness fitness ((Pease Pease et represents the the tendency of natural natural selection selection to to increase et al., al., 11989; 989; Garcia-Ramos 995). The term gives Garcia-Ramos and and Kirkpatrick, Kirkpatrick, 11995). The second second term gives the the effects effects of aa gradient gradient in in population population density, density, which which tends tends to increase the the trait trait mean, of to increase mean, z, z, if if > 0 and z/ax > find the it it is is higher higher where where density density is is higher higher (alog(N)/ax (Olog(N)/Ox > and aOz/Ox > 0). To To find the relative relative importance importance of of regions regions with with different different density, density, define define aa quantity quantity H H which which I and combines combines the the effects effects of of gene gene flow flow and and natural natural selection selection ~ and which which depends depends on on the eld of z(x, t) the whole whole fi field of trait trait means means z(x, t)
=
H[z(x, H[ z (x, t)] =
x- [ V Ix X I N2 N2
(J2 ( az ) 2 ] dx. dx. ax ~xx
Vg log(~W) 10g(W) - 2 2 44
�
(6a) (6a)
One show, using using Eq. always increases: One can can show, Eq. (5), (5), that that H H always increases: aH
Ot = =
at
I
- x
N2
( az ) 2 dx. dx.
at
(6b) (6b)
This is a generalization l 987b) to allow varying density; ~This generalization of Eq. Eq. (2) of Rouhani Rouhani and Barton Barton ((1987b) allow for varying density; a similar expression can allele frequency which is also weighted in proportion similar expression can be found found for allele frequency change, change, which also weighted proportion N2 (Barton (Barton and and Hewitt, to N' Hewitt, 11989). 989).
1198 98
N. H. Borton and Michael Michael C. N.H. Bartonand C. Whitlock Whittock
Thus, H compromise between between the Thus, H tends tends to to aa maximum, maximum, which which is is aa compromise the tendency tendency of of natural gene flow flow to the steepness natural selection selection to to increase increase mean mean fitness fitness and and of of gene to reduce reduce the steepness of in of clines, clines, (az/ax)2. (Oz/Ox) 2. The The important important point point here here is is that that the the effects effects are are weighted weighted in proportion to to the the square of population population density, density, N2 N 2": in in this sense, aa region with proportion square of this sense, region with 10 times OO-fold greater 10 times greater greater density density has has aa l100-fold greater influence influence on on the the effects effects of of gene gene flow flow and and selection. selection. Slatkin 1 973) and 1 975) have Slatkin ((1973) and Nagylaki Nagylaki ((1975) have used used diffusion diffusion equations equations like like Eq. Eq. (5) to study the effects of asymmetric gene flow in swamping adaptations based (5) to study the effects of asymmetric gene flow in swamping adaptations based on using a quan on single single loci. loci. Here, Here, we we illustrate illustrate the the effect effect by by using a simple simple model model of of aa quantitative Assume that titative trait. trait. Assume that the the trait trait has has constant constant genetic genetic variance variance and and is is under under stabilizing cline. The stabilizing selection, selection, with with an an optimum optimumyhich which changes changes linearly linearly along along aa cline. The Vs ) where where zz is is the the mean fitness of population is mean fitness of the the population is then then W = = exp( e x p (- (z (z - f3x)2/2 ~x)Z/2Vs), ' mean is the mean of of the the trait, trait, Vs Vs is is aa measure measure of of the the strength strength of of selection, selection, and and f3 13 is the rate rate of of change change of of the the optimum optimum in in space. space. Genetic Genetic variation variation around around the the optimum optimum reduces reduces mean mean fitness fitness by by L L = = Vg/2Vs V~/2Vs, aa measure measure of of the the load load on on the the population population due due to to ' stabilizing simplicity, we variance. If If stabilizing selection. selection. For For simplicity, we ignore ignore the the environmental environmental variance. population sizes sizes were simple cline, mean population were uniform, uniform, there there would would be be aa simple cline, with with the the mean f3x; Felsenstein, 1 977; Slatkin, tracking indefinite range tracking the the optimum optimum over over an an indefinite range (z (z = =/3x; Felsenstein, 1977" Slatkin, 11978a). 978a). However, w ((i.e., i.e., However, if if the the density density falls falls away away as as aa Gaussian Gaussian with with width width w = No exp( exp(-x2/2w2)), the mean cannot track track the the changing changing optimum optimum as as effec effecN= - x2/2w2)), the mean cannot tively, is adapted to an tively, because because gene gene flow flow from from the the abundant abundant region region which which is adapted to an opti optimum in less less dense mum at at zero zero tends tends to to impede impede adaptation adaptation to to the the different different conditions conditions in dense regions 996). The which maximizes regions (Garcia-Ramos (Garcia-Ramos and and Kirkpatrick, Kirkpatrick, 11996). The solution solution which maximizes H H is zz = = f3/[ /3/[11 + + (2V,/Vg)(a/w)2] (2Vs[Vg)(or/w) 2] = = f3/[ /3/[11 + + L(o'/w)2]. This has a lower reis L(a/w)2]. This has a lower slope, slope, re flecting effect depends flecting aa failure failure to to adapt adapt outside outside the the region region of of high high density. density. The The effect depends on selection (L) (L) and on the the load load due due to to stabilizing stabilizing selection and on on the the distance distance over over which which density density declines, relative declines, relative to to the the dispersal dispersal range range (a/w). (o/w). So given. If So far, far, we we have have taken taken the the population population density density as as given. If density density decreases decreases with mean mean fi fitness, then there there is is aa positive positive feedback, feedback, such such that that populations populations which which with tness, then are less adapted become become smaller, smaller, making making it it harder harder for for them them to to resist resist gene gene are less well well adapted flow, and so so further further reducing reducing their their mean mean fi fitness density ((Kirkpatrick flow, and tness and and their their density Kirkpatrick and 996). Suppose Suppose that mean fitness fitness through Barton, 11996). and Barton, that numbers numbers are are related related to to mean through =0 N Y; Y implies soft while large implies that population N = NoW NoW ~; 3' = 0 implies soft selection, selection, while large y 3' implies that the the population size tness. One perfect adaptation: f3x, size is is very very sensitive sensitive to to mean mean fi fitness. One solution solution is is perfect adaptation" z = =/3x, in which case case numbers constant, the cline is is linear, linear, and first in which numbers are are constant, the cline and gene gene flow flow (the (the first per small peralways stable two terms in in Eq. Eq. (5)) (5)) has has no no effect. effect. This This solution solution is is always stable to to small two terms
turbations, since the However, if if y is turbations, since the second second term term is is proportional proportional to to (z - f3X)2. /3x)2. However, y is within a large enough, enough, there there can can be be another another equilibrium, equilibrium, with with adaptation adaptation only only within a large f3/2)( 1 + limited region. This This second second solution solution is is zz = = wx, ~ox, where where w o~ = = ((/3/2)(1 + limited region. x/(1 - ((/3,.//3) 2) < < / 3f3, , and and the the dimensionless dimensionless parameter parameter/32 = 2 2Vg/~rzy. This sof3Jf3)2) f3 z. = Vg/a2 y. This so -J(l lution is settles to lution is neutrally neutrally stable: stable: the the popUlation population settles to be be adapted adapted to to some some arbitrary arbitrary z, f3 > possibility to but perturbations perturbations can can shift shift it it to to the the left left or or right. right. For For this this possibility to exist, exist,/3 > but /3, where where/32 = 2 2 Vg/a2yVg/orzy~that is, the the change change in in optimum optimum must be rapid rapid ((/3 large), f3c m= that i� must be f3 large), and y large). may not plausible and the the popUlation population sensitive sensitive to to W ((3, large). This This scheme scheme may not be be plausible
99 2
-
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-
o 0
The The Evolution Evolutionof of Metopopulotions Metopopulations
1199 99
-
., -4 ·4
·2 -2
o 0
I
distance distance
22
4
FIGURE FIGURES5 The collapse in adaptation adaptation at the edge of a species' range. range. The graph show show numerical
solutions to a model of population density, N, and the mean of a quantitative trait, z, z, across a one· one-
dimensional habitat. Individuals reproduce reproduce and disperse continuously in time, with with a net growth rate
r( toward an optimum that changes at a rate f3 r(1l - NIK) N/K) - (z (z - zop,)2/2V" Zop,)2/2Vs, and stabilizing selection toward 13 in space line). Assuming that the trait zz is normally distributed with mean space (zop, (Zopt= = f3x; /3x; heavy line). mean z:~ and
= N(z f3x)2/2, = (u2/2)aW/ax2 (o'2/2)OZN/Ox 2 + + rN( rN(1l - NIK) N/K) -- (V/2V,)N (Vg/2Vs)N -- N(~ - fix)2~2, oS/Ot = = (u2/2)a (orz/2)O2z/Ox "k- orzoz/OxO - (z (~ -- f3x)(V flx)(VJ2V~). 'i./ax22 + u2az/axa log(N)/Ox 10g(N)jax /2V,). Scaling, Scaling, the only two parameters are ai/at A (V./rV,) = L/r, B = = 11,, B = 1 .5. Collapse in adaptation is A = = (Vg/rV~) = 2 2L/r, = (f3u/r)j,fiV,. (flo'/r)/2X~s. For this example, A A= = 1.5. possible if B,fi Bx/~ > > A A,, or (f3u/A» (flo'/2x~)>v/-~. shows the equilibrium numbers numbers (bell·shaped (bell-shaped ,j"SL The figure figure shows variance Vg leads leads to equations aN/i)t ON/Ot
curve) and trait mean (light line).
for regulation, for one one trait, trait, since since it it requires requires that that the the strength strength of of density-dependent density-dependent regulation, 'Y, lower than (2 V {32), which which is is the in optimum y, be be lower than (2 V gg//oa2 -2/32), the change change in o p t i m u m over over one one dispersal dispersal range, deviations. However, if the range, measured measured in in genetic genetic standard standard deviations. However, if the optimum optimum for for many the same less re many traits traits changes changes in in the same region, region, this this constraint constraint becomes becomes much m u c h less restrictive, in marginal marginal populations may strictive, suggesting suggesting that that multivariate multivariate adaptations adaptations in populations may collapse collapse in in the the face face of of gene gene flow. flow. This that population population density gene This model model is is simplified simplified by by the the assumption assumption that density and and gene ) . However, However, a model which which follows follows the joint fl ow are directly related flow are directly related (N (N = = NoW NoW Y ~). a model the joint change Fig. 5), 5), with change in in density density and and in in the the trait trait behaves behaves in in a a similar similar way way ((Fig. with adaptation adaptation collapsing collapsing outside outside some some arbitrary arbitrary region region if if the the optimum o p t i m u m changes changes rapidly rapidly enough. enough. There loci. Suppose alleles, There are are analogous analogous results results for for discrete discrete loci. Suppose that that there there are are two two alleles, P P and and Q, Q, with with fitness fitness of of P P given given by by Wp We = = r(l r(1 - N/K) N/K) - rrOp(x), O e ( x ) , and and similarly similarly for Q. The additional death over and for Q. The functions functions Op O e ,, O f~Q represent additional death rates, rates, over and above above the the Q represent density-dependent no migramigra density-dependent term, term, which which differ differ slightly slightly for for the the two two alleles. alleles. With With no tion, a fixed for would equilibrate tion, a population population fixed for Q Q would equilibrate at at N N = = K( K ( 1l - - O 12Qx) for Qx) for 0 (x) < I and would go extinct if 0 (x) > 1 . O is chosen to be large enough f~Q(x) < 1 and would go extinct if l~Q(x) > 1. I~Q is chosen to be large enough Q Q Q � 0). 0). In that declines to large for In the that the the population population declines to zero zero to to the the left left (O (OQQ large for x x << the bulk bulk
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N. H. Barton and Michael N.H. Barton and Michael C. C. Whitlock Whitlock
> 0), has of of the the species species'' range, range, allele allele P P has has disadvantage disadvantage s (D ( ~ oQ > > Dp f~e for for xx > 0), but but has Dp a = S for x < 0). For this case, ) a selective selective advantage advantage S S at at the the edge edge «((Oe - D f~e) ~ S for x < 0). For this case, Q one one can can show show that that if if selection selection is is weak weak (Dp (fie = ~ D flQ), is aa critical value of there is critical value of s, Q )' there above allele P it is is favored above which which allele P cannot cannot be be fixed, fixed, even even though though it favored at at the the edge edge of of the the range would extend could be this critical critical range and and would extend the the range range if if it it could be established. established. Crucially, Crucially, this 2), so must have have a value is small (Scrit value is small (Scrit = ~" S $2), so that that an an allele allele must a very very weak weak disadvantage disadvantage in bulk of if it to be be established selection in in the the bulk of the the range range if it is is to established by by selection in aa sparse sparse and and limited region. limited region. Thus have discussed Thus far, far, we we have discussed adaptations adaptations based based on on one one gene gene or or on on one one quan quantitative this simplest cases, gene flow only if it titative trait. trait. In In this simplest of of cases, gene flow only prevents prevents adaptation adaptation if it exceeds value: the the outcome relative strengths exceeds some some critical critical value: outcome depends depends on on the the relative strengths of of selection flow. Divergence usually occur in this selection and and gene gene flow. Divergence may may usually occur in this way, way, and and new new species the adaptations species may may arise arise as as aa by-product by-product if if the adaptations themselves themselves cause cause reproductive reproductive flies adapted plants isolation. For example, races isolation. For example, races of of Rhagoletis Rhagoletis flies adapted to to different different host host plants mate times, mate on on the the fruit fruit from from which which they they emerged, emerged, and and also also emerge emerge at at different different times, leading Feder et al. 990a,b). In leading to to partial partial isolation isolation ((Feder al.,, 11990a,b). In the the monkey monkey flower flower Mimulus Mimulus guttatus, allele responsible responsible for resistance to guttatus, the the allele for resistance to heavy heavy metals metals interacts interacts with with an another sterility ((MacNair MacNair and 1 989). In In the the next next other gene gene to to cause cause hybrid hybrid sterility and Cumbes, Cumbes, 1989). section, we issues which arise when when certain section, we consider consider the the more more complex complex issues which arise certain combi combicase, it popula nations nations of of genes genes are are required required for for adaptation. adaptation. In In this this case, it may may be be that that populations must be because of tions must be reproductively reproductively isolated isolated before before they they can can adapt-either adapt--either because of some impede some physical physical barrier barrier to to gene gene flow, flow, or or because because of of genetic genetic differences differences that that impede interbreeding. in a interbreeding. Population Population structure structure then then interacts interacts with with selection selection in a more more com complex way, and the the processes processes of of adaptation adaptation and and speciation speciation are are closely plex way, and closely intertwined. intertwined.
V. THE "SHIFTING V. SPECIATION SPECIATIONAND AND THE "SHIFTINGBALANCE" BALANCE" Natural stable Natural selection selection may may cause cause popUlations populations to to evolve evolve toward toward alternative alternative stable states. example, heterozygotes states. For For example, heterozygotes between between different different chromosome chromosome arrangements arrangements may White, may not not pair pair and and segregate segregate properly properly in in meiosis, meiosis, leading leading to to partial partial sterility sterility ((White, 11973). 973). This or the This kind kind of of selection selection against against heterozygotes heterozygotes leads leads to to fixation fixation of of one one or the other other type. type. Thus, Thus, while while species species often often contain contain several several chromosome chromosome arrangements, arrangements, these usually found places, forming races" separated these are are usually found in in different different places, forming "chromosome "chromosome races" separated Hel by alternative equilibria. by narrow narrow clines. clines. Other Other kinds kinds of of selection selection can can lead lead to to alternative equilibria. Helcolour patterns iconius iconius butterflies butterflies are are distasteful distasteful and and have have evolved evolved conspicuous conspicuous colour patterns to advertise their area, there there is is strong to advertise their distastefulness distastefulness to to predators. predators. In In any any one one area, strong selection convergence to common pattern pop selection for for convergence to aa common pattern ("Mullerian ("Mtillerian mimicry"), mimicry"), but but populations 98 1 ). ulations in in different different areas areas have have established established different different patterns patterns (Turner, (Turner, 11981). Where clines which Where these these pattern pattern races races meet, meet, they they are are separated separated by by narrow narrow clines which are are maintained selection against genotypes, and maintained by by selection against heterozygotes, heterozygotes, against against recombined recombined genotypes, and against Mallet, 11993). 993). against rare rare alleles alleles ((Mallet, In In general, general, multiple multiple stable stable states states are are likely, likely, since since different different gene gene combinations combinations may Sewall Wright 1 932) introduced an influmay often often fulfill fulfill the the same same function. function. Sewall Wright ((1932) introduced an influ-
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ential the "adaptive landscape." ential metaphor metaphor for for thinking thinking about about these these multiple multiple states, states, the "adaptive landscape." This is best defi n ed as a graph of mean fi t ness against the state of the population, This is best defined as a graph of mean fitness against the state of the population, which by the which can can be be described described by by allele allele frequencies frequencies or or by the means means of of quantitative quantitative traits traits 1 986). Natural selection tends fitness, and (see Provine, (see Provine, 1986). Natural selection tends to to increase increase mean mean fitness, and so so pop populations evolve toward the the nearest nearest "adaptive "adaptive peak." peak." However, However, this may not not be be the the ulations evolve toward this may global which case will be be impeded, impeded, because because populations global optimum, optimum, in in which case adaptation adaptation will populations cannot cannot evolve evolve toward toward the the global global optimum optimum through through aa sequence sequence of of changes, changes, each each favored by selection. This problem was, of course, a major concern Darwin, favored by selection. This problem was, of course, a major concern for for Darwin, most in his his discussion evolution of the eye Darwin, 11859, 859, Chapter most notably notably in discussion of of the the evolution of the eye ((Darwin, Chapter 6). Multiple stable states states are are also also involved in speciation: hybrids between between popu popuMultiple stable involved in speciation: hybrids lations less fit, conversely, most models of lations at at different different peaks peaks are are less fit, and and conversely, most models of reproductive reproductive 984). isolation lead isolation lead to to multiple multiple equilibria equilibria (Barton (Barton and and Charlesworth, Charlesworth, 11984). Wright 1 932) proposed proposed that Wright ((1932) that species species may may most most efficiently efficiently adapt adapt by by means means of of a this process a "shifting "shifting balance" balance" between between evolutionary evolutionary forces. forces. He He divided divided this process into into three phases. In rst, random uctuations (due, (due, for example, to to sampling sampling drift) drift) three phases. In the the fi first, random fl fluctuations for example, cause local local populations move into into the domain of new cause populations ("demes") ("demes") to to move the domain of attraction attraction of of new adaptive peaks. In phase, selection within populations populations takes to adaptive peaks. In the the second second phase, selection within takes them them to the adaptive peaks peaks compete with each other, by the new new peaks. peaks. Finally, Finally, different different adaptive compete with each other, by aa variety "fitter" peaks peaks tend through the the whole variety of of processes, processes, such such that that "fitter" tend to to spread spread through whole species. species. This This third third phase phase involves involves an an element element of of group group selection. selection. For For example, example, adaptive adaptive peaks peaks may may spread spread if if they they increase increase the the size size of of the the local local population, population, or or the the number not opposed number of of emigrants. emigrants. However, However, selection selection between between adaptive adaptive peaks peaks is is not opposed by selection between by selection between individuals, individuals, and and so so this this component component of of the the "shifting "shifting balance" balance" is models in which group selection is is is more more plausible plausible than than the the more more familiar familiar models in which group selection opposed individual selection selection (e.g., 985; Nunney, N unney, 11985; 985; Wilson, opposed by by individual (e.g., Kimura, Kimura, 11985; Wilson,
1 987). 1987). The rst two is well well The theory theory underlying underlying the the fi first two phases phases of of the the "shifting "shifting balance" balance" is established, established, while while the the third third phase phase has has only only recently recently received received detailed detailed attention. attention. Unfortunately, cance of the shifting Unfortunately, evidence evidence on on the the actual actual signifi significance of the shifting balance balance for for adaptation and and speciation speciation is sparse. We We first summarize the the existing existing understanding adaptation is sparse. first summarize understanding of the specific specific issue issue of of the the shifting shifting balance balance model model and and then then concentrate concentrate on on the of how how local extinctions and generally, variation local extinctions and recolonizations recolonizations (or (or more more generally, variation in in population population structure) structure) affect affect the the process. process.
A. The (Iassical ClassicalShifting Shifting Balance Balance Model The rst requirement The fi first requirement is is that that alternative alternative stable stable states states--roughly speaking, - roughly speaking, adaptive - exist. This is plausible plausible on adaptive peaks peaks--exist. This is on both both theoretical theoretical and and empirical empirical grounds. locus models equilibria, whether fixed or poly grounds. Most Most multi multilocus models admit admit many many equilibria, whether fixed or polymorphic Feldman, 1989). 1 989). Wright 1 935) analyzed what is is perhaps simplest morphic ((Feldman, Wright ((1935) analyzed what perhaps the the simplest case, case, in in which which stabilizing stabilizing selection selection acts acts on on an an additive additive quantitative quantitative trait. trait. Here, Here, many phenotype, and be many combinations combinations of of genes genes lead lead to to the the same same phenotype, and so so each each can can be established established by by selection. selection. If If mutation mutation maintains maintains variation variation around around the the optimal optimal phe pheuilibria increases still further 1 986). More notype, notype, the the number number of of stable stable eequilibria increases still further (Barton, (Barton, 1986). More
q
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Bartonand andMichael MichaelC.C.Whitlock Whitlock N.N.H. H. Borton
generally, if if fifitnesses are randomly randomly assigned assigned to to genotypes, genotypes, there there are are large large numbers numbers generally, tnesses are of local local adaptive adaptive peaks, peaks, which which are are reached reached in in only only aa few few steps steps from from aa randomly randomly of chosen starting starting point point and and may may be be well well below below the the global global optimum optimum ((Kauffman and chosen Kauffman and Levin, 11987). The most most obvious obvious evidence evidence isis that that populations populations which which hybridize hybridize and and 987). The Levin, yet remain remain genetically genetically distinct distinct even even when when living living in in the the same same environment environment have have yet presumably reached reached different different adaptive adaptive peaks peaks ((Harrison, 993). presumably Harrison, 11993). Unfortunately, Unfortunately, while while the the existence existence of of multiple multiple adaptive adaptive peaks peaks is is an an essential essential precondition precondition for for the the shifting shifting balance balance process, process, itit does does not not show show that that the the divergence divergence of populations populations onto onto different different stable stable states states has has occurred occurred in in opposition opposition to to natural natural of selection. Natural Natural selection selection can can cause cause populations populations to to evolve evolve from from some some ancestral ancestral selection. state to to aa variety variety of of fitter fitter states, states, which which may may turn turn out out to to be be incompatible incompatible with with each each state other. For For example, example, Robertsonian Robertsonian fusions fusions between between chromosome chromosome arms arms may may cause cause other. little or or no no meiotic meiotic nondisjunction nondisjunction in in the the heterozygotes heterozygotes with with the the ancestral ancestral unfused unfused little chromosomes. chromosomes. However, However, there there may may be be severe severe sterility sterility in in the the heterozygote heterozygote between between different fusions, fusions, since since many many chromosomes chromosomes may may attempt attempt to to pair pair ((Bickham and different Bickham and Baker, 986; Searle, 986). More Baker, 11986; Searle, 11986). More generally, generally, different different mutations mutations may may arise arise and and be be established by by natural natural selection selection in in different different places, places, and and may may turn turn out out to to be be incom incomestablished patible with with each each other other when when they they meet meet (Bengtsson (Bengtsson and and Christiansen, Christiansen, 11983). It is is patible 983). It plausible plausible that that much much reproductive reproductive isolation isolation evolves evolves in in this this way, way, without without the the need need for any any peak peak shifts shifts in in opposition opposition to to selection selection (Orr, (Orr, 11995). One should should note, howfor 995). One note, how ever, that that while while this this avoids avoids the the first first phase phase of of the the shifting shifting balance, balance, incompatible ever, incompatible gene combinations combinations may may still compete with each other other in the third third phase (below). gene still compete with each in the phase (below). The question of whether different popUlations are often at at different different adaptive adaptive The question of whether different populations are often peaks can be at least peaks can be answered, answered, at least in in principle, principle, by by measuring measuring the the extent extent and and scale scale of "outbreeding between populations reof "outbreeding depression," depression," in in which which crosses crosses between populations lead lead to to re duced There has duced fitness. fitness. There has been been surprisingly surprisingly little little work work along along these these lines, lines. Recent Recent experiments with with plants plants have have suggested suggested significant significant reduction reduction in in hybrid hybrid fitness fitness experiments over 1 994; Burt, Burt, 1995). 1 995). On On the the other other hand, hand, crosses crosses over short short scales scales (Waser ( Waser and and Price, Price, 1994; between between the the much much more more divergent divergent taxa taxa involved involved in in hybrid hybrid zones zones have have given given equivequiv ocal ocal evidence evidence of of reduced reduced hybrid hybrid fitness fitness and and suggest suggest aa more more important important role role for for adaptation 1 995). adaptation to to different different environments environments (Arnold (Arnold and and Hodges, Hodges, 1995). Most Most theoretical theoretical attention attention has has focused focused on on the the probability probability that that random random drift drift will ( 1 94 1 ) will establish establish aa new new adaptive adaptive peak peak in in opposition opposition to to selection. selection. Wright Wright (1941) showed showed that that the the chance chance of of aa new new mutation mutation with with disadvantage disadvantage ss in in the the heterozygote heterozygote being being established established in in aa deme deme of of size size N N is is proportional proportional to to exp(-Ns); exp( Ns); the the diffusion diffusion approximation approximation for for this this probability, probability, which which is is reasonably reasonably accurate accurate even even for for small small is ~s/NTr .JS/N1T eexp( N, is ( Lande, 1979; 1 979; Hedrick, Hedrick, 1981). 1 98 1 ). This This conclusion conclusion extends extends N, x p ( --NNs) s ) (Lande, to to aa wide wide class class of of models models which which can can be be described described by by drift drift and and selection selection across across an an adaptive adaptive landscape" landscape: the the probability probability of of aa peak peak shift shift is is in in general general proportional proportional to W W2N, 2N, where where W W is is the the mean mean fitness fitness of of the the population population in in the the adaptive adaptive valley, valley, to compared compared to to the the original original adaptive adaptive peak peak (Barton (Barton and and Rouhani, Rouhani, 1987). 1 987). Thus, Thus, peak peak shifts shifts must must either eitheroccur occurin in aavery very small small population population or orinvolve involvevery very slight slight reduction reduction in in mean mean fitness fitness ifif they they are are to to occur occur with with reasonable reasonable probability. probability. Migration Migration into into the population popUlation impedes impedes peak peak shifts; shifts; for for example, example, with with selection selection against against heteroheterothe -
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zygotes, - 4Nm zygotes, Nm Nm immigrants immigrants per per generation generation reduce reduce the the probability probability by by aa factor factor 22-4Nm ( Lande, 1979; 1 979; Barton Barton and and Rouhani, Rouhani, 1991). 1991). (Lande, The The chances chances of of aa peak peak shift shift can can be b e greatly greatly increased increased by b y aa severe severe population population bottleneck, as as for for example example during during the the founding founding of of aa new new population. population. During During the the bottleneck, brief period period of of small small population population size, size, selection selection is is negligible negligible relative relative to to random random brief drift, and and the the occurence occurence of of aa peak peak shift shift depends depends mainly mainly on on the the chance chance that that the the drift, population will drift across the "adaptive valley" ( Rouhani and Barton, 1 987a). population will drift across the "adaptive valley" (Rouhani and Barton, 1987a). For an an additive additive quantitative quantitative trait, trait, the the variance variance of of the the population population mean mean is is 2FVg, 2FVg , For where F is the net reduction in heterozygosity and V the initial genetic variance where F is the net reduction in heterozygosity and Vgg the initial genetic variance (Barton and and Charlesworth, Charlesworth, 1984). 1 984). Thus, Thus, with with severe severe inbreeding, inbreeding, and and high high heriheri (Barton tability, aa shift shift of of aa few few phenotypic phenotypic standard standard deviations deviations is is not not unlikely. unlikely. However, However, tability, if substantial substantial reproductive reproductive isolation isolation is is to to arise arise during founder effect, effect, there must if during aa founder there must have been been substantial substantial variation variation in in the the initial initial population, population, and and this this variation variation must must have have been been subject subject to to selection. selection. In In aa variety variety of of models, models, the the expected expected isolation isolation have produced by by aa founder founder event event is is proportional proportional to to the the standing standing genetic genetic load load due due to to produced this variation variation (Barton, (Barton, 1989). 1 989). this Models peak shifts shifts in in a population address address only only the the first first two phases Models of of peak a single single population two phases of the shifting shifting balance balance process. process. A A full full description of the the entire process must must of the description of entire process include the the spread peak through metapopulation. The The simplest simplest include spread of of aa new new adaptive adaptive peak through aa metapopulation. case is is the island model, model, where Wright ' s (1937) ( 1 937) formula formula for equilibrium case the island where Wright's for the the equilibrium between mutation, and complete analysis. In between migration, migration, selection, selection, mutation, and drift drift allows allows aa complete analysis. In a populations of a metapopulation metapopulation consisting consisting of of aa large large number number of of populations of size size N, allele allele frequency is is distributed distributed as as ljJ ~(p) p4Nmff+aNl,-lCNm~+4Nv-1W2N where/7 is the frequency +4Np.- l q"Nmq + 4Nv- 1 W 2N, where p is the ( p ) = p4Nmji allele JL, v rates from Q to allele frequency frequency in in the the migrant migrant pool, pool, and and/x, 1.,the the mutation mutation rates from Q to P P and and vice versa. versa. (There for the vice (There is is aa similar similar formula formula for the distribution distribution of of aa quantitative quantitative trait). trait). This This distribution distribution itself itself determines determines the the composition composition of of the the migrant migrant pool pool (p (/~ = = f6pljJ p ), giving giving an solved numerically numerically (Barton flopd/(p) an equation equation which which can can be be solved (Barton and and (p ) ddp), Rouhani, 993). For Rouhani, 11993). For small small numbers numbers of of migrants, migrants, populations populations shift shift independently independently of likely toward tter peak of each each other. other. Shifts Shifts are are more more likely toward the the fi fitter peak than than away away from from it, it, and and so Because shifts so the the stochastic stochastic equilibrium equilibrium is is biased biased toward toward the the fitter fitter peak. peak. Because shifts are are more more likely likely to to be be to to whichever whichever state state is is commoner commoner in in the the whole whole population, population, there there is is aa positive positive feedback feedback which which increases increases the the bias bias as as the the number number of of migrants migrants increases increases ((left left of of Fig. Fig. 6). However, However, when when the the number number of of migrants migrants is is greater greater than than some some critical ), aa rare critical value value (Nm (Nm > Nmcrit Nmcrit = -~ 11), rare adaptive adaptive peak peak cannot cannot spread spread in in the the face face of of migration migration from from populations populations at at the the commoner commoner peak peak even even if if it it confers confers greater greater fitness. fitness. There There are are then then two two stable stable states states for for the the whole whole metapopulation, metapopulation, and and the the global global optimum optimum cannot cannot be be reached reached (right (right of of Fig. Fig. 6). Adaptation Adaptation is is thus thus most most efficient efficient when when the the number number of of migrants migrants is is just just below below the the critical critical value, value, since since the the bias tter peak bias in in favor favor of of the the fifitter peak is is then then greatest. greatest. This This bias bias can can be be large, large, even even when when the the difference difference in in fitness fitness between between the the two two peaks peaks is is small. small. If If one one adaptive adaptive peak peak also also increases increases the the population population size size or or the the number number of of emigrants, emigrants, then then group group selection selection assists assists its its spread. spread. However, However, this this is is aa weak weak effect, effect, of of second second order order in in selection selection ((Rouhani Rouhani and 993). and Barton, Barton, 11993). Analysis Analysis of of aa population population structured structured in in two two dimensions dimensions is is more more difficult, difficult, but but -~
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Bartonand and Michael MichaelC.C. Whitlock Whitlock N.N.H. H. Barton
leads leads to to qualitatively qualitatively similar similar conclusions. conclusions. A A new new adaptive adaptive peak peak can can be be established established by by chance chance provided provided that that the the number number of of migrants migrants (or (or equivalently, equivalently, the the neighbor neighborhood hood size) size) is is small; small; there there is is no no requirement requirement for for strict strict geographic geographic isolation. isolation. The The probability probability of of aa shift shift is is given given by by the the chance chance that that the the new new adaptive adaptive peak peak is is estab established lished in in an an area area large large enough enough that that its its advantage advantage over over the the old old peak peak outweighs outweighs the the swamping effect effect of of gene gene flow, flow, allowing allowing it it to to spread spread through through the the whole whole population. population. swamping In In aa continuous continuous habitat habitat with with density density p p and and dispersal dispersal rate rate a2 O "2 this this probability probability is is proportional ex), where proportional to to exp( exp(- C C Nb/ Nb/ce), where 00 < < ex cr < < 11 is is aa dimensionless dimensionless measure measure of of the the asymmetry asymmetry between between the the peaks, peaks, and and Nb is is Wright's Wright's neighborhood neighborhood size, size, 4 7Tpa2 7rpo 2 (or (or 4 7TNm 7rNm in in aa stepping-stone stepping-stone model model).). This This applies applies for for both both quantitative quantitative traits traits under Rouhani and 987b) and under disruptive disruptive selection selection ((Rouhani and Barton, Barton, 11987b) and selection selection against against heterozygotes 9 9 1 ) . Just heterozygotes (Barton (Barton and and Rouhani, Rouhani, 11991). Just as as in in the the island island model, model, there there can can be tter peak; nite be aa very very strong strong bias bias in in favor favor of of even even aa slightly slightly fi fitter peak; indeed, indeed, in in an an infi infinite two-dimensional xed. However, two-dimensional habitat, habitat, it it is is impossible impossible for for an an inferior inferior peak peak to to be be fi fixed. However, the the chance chance of of aa shift shift depends depends only only weakly weakly on on the the strength strength of of selection: selection: aa strongly strongly selected shift shift is is less less likely likely to to occur occur by by chance chance over over aa given given area, area, but but need need only only selected spread over over aa smaller smaller area area to to overcome overcome gene gene fl flow from outside. outside. spread ow from Wright tter peaks Wright believed believed that that fi fitter peaks would would spread spread because because the the populations populations in in which they they are are established established would would send send out out more more migrants, migrants, and and hence hence would would inwhich in evitably evitably pull pull neighboring neighboring populations populations into into the the same same state. state. This This kind kind of of determi deterministic spread spread occurs occurs in in the the models discussed above, above, where where a a new peak sweeps sweeps nistic models discussed new peak through aa continuous continuous habitat and Barton, Barton and through habitat (Rouhani ( Rouhani and Barton, 1987b; 1 987b; Barton and Rouhani, Rouhani, 1991). and Rouhani, 1 99 1 ). In In the the island island models models (Barton (Barton and Rouhani, 1993; 1 993; Rouhani Rouhani and and Barton, Barton, 1993), shifts are are stochastic, the evolution 1 993), shifts stochastic, but but the evolution of of the the whole whole metapopulation metapopulation is is also also deterministic. Again, Again, there an advantage deterministic. there is is an advantage to to those those populations populations which which send send out out more spread is driven by more migrants. migrants. However, However, in in both both cases cases the the spread is primarily primarily driven by selection selection -
1
N Nss==I l
__-::::::::: : == : === or=0.1 u=O . ! u=O ~=0
Pp
,
,
,
0.0 0. l 0.2 0.3 FIGURE 66 The The overall overall mean mean allele allele frequency frequency ((p) «(p) =/5), = p), as as aa function function of of the the number number of of migrants migrants FIGU~I: (Nm), 0.0 1 , Ns Ns = 1; I ; calculated calculated from from Eq. Eq. (23a) (23a) of of Barton Barton (Nm), for for selection selection against against heterozygotes; heterozygotes; N/x NIL == 0.01, and Rouhani Rouhani (1993). ( 1 993). Fitnesses Fitnesses of of (QQ, (QQ, PQ, PQ, PP) PP) are are 1I :: 1I - ss ++ ors: as : 1! ++ 2ors. 2as. The The light light curve curve gives gives and the symmetric symmetric case, case, where where the the critical critical number number of of migrants migrants is is Nmcrit Nm,,;, == 0.088. 0.088. The The heavy heavy curve curve is is for for the asymmetry cr a == 0.1 0. 1;; the the critical critical number numberof ofmigrants migrants isis then then Nm,:ri Nm""t == 0.237. 0.237. Even Eventhis this slight slightdifference difference asymmetry in fitness fitness between between the the two two homozygotes homozygotes leads leads to to aa strong strong bias bias in in favor favor of ofthe the fitter fitterpeak peak ififNm Nm is isjust just in less less than thanNmcrit Nm,,;, (upper (upper heavy heavy curve). curve). =
-
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between between individuals, individuals, rather rather than than by by excess excess emigration. emigration. Unless Unless population population density density and the latter has negligible and migration migration rate rate depend depend very very strongly strongly on on genotype, genotype, the latter has negligible 990; Barton, 992). On phase, effect effect (see (see Crow Crow et at., al., 11990; Barton, 11992). On this this view view of of the the third third phase, shifts shifts are are likely likely to to occur occur only only in in regions regions with with low low Nm Nm and and yet yet must must presumably presumably spread species, into Nm. This spread through through the the whole whole species, into regions regions with with large large Nm. This requires requires either either that that Nm Nm varies varies through through time time or or that that it it varies varies gradually gradually across across the the species range, species range, such not swamp such that that asymmetric asymmetric gene gene flow flow from from the the more more abundant abundant regions regions does does not swamp peak peak shifts shifts that that occur occur at at the the margins. margins. The The variations variations in in population population structure structure which which are are central central to to the the idea idea of of aa metapopulation metapopulation facilitate facilitate the the shifting shifting balance balance process. process. However, However, they they also also intro introduce element which which reduces tter peak. duce aa random random element reduces the the bias bias in in favor favor of of the the fi fitter peak. The The shifting evolution of of reproductive shifting balance balance may may be be important important in in the the evolution reproductive isolation, isolation, though though it it is is hard hard to to judge judge whether whether populations populations reached reached different different adaptive adaptive peaks peaks through opposition to selection alone. through random random drift drift acting acting in in opposition to selection, selection, or or through through selection alone. It It seems seems less less likely likely that that the the shifting shifting balance balance contributes contributes significantly significantly to to adaptation: adaptation: species peaks, and species would would need need to to be be divided divided into into very very many many different different adaptive adaptive peaks, and peaks sets of genes would would need need to to be be able peaks involving involving different different sets of interacting interacting genes able to to spread spread independently each other. other. independently of of each
B. Extinction/Recolonization Shifting Balance BI Extinction/Recolonizationand the Shifting Balance The The theory theory for for aa subdivided subdivided population population with with constant constant population population size size and and migration proportional to migration rate rate shows shows that that the the rate rate of of peak peak shifts shifts is is not not directly directly proportional to the the variance to the allele frequencies frequencies across variance of of fluctuations fluctuations due due to to drift, drift, or or to the variance variance in in allele across populations. Thus, Thus, taking as one might populations. taking the the average average variance variance across across populations, populations, as one might when defining "effective population size," us the the chance that a pop when defining "effective population size," does does not not tell tell us chance that a population will demonstrate clearly ulation will will shift shift from from one one peak peak to to another. another. We We will demonstrate this this more more clearly with aa specific example. with specific example. Consider when Consider the the probability probability of of shifting shifting from from one one adaptive adaptive peak peak to to another, another, when the the adaptive adaptive landscape landscape is is determined determined by by the the phenotype phenotype of of one one particular particular trait. trait. Assume that population starts Assume that there there are are two two adaptive adaptive peaks. peaks. The The population starts at at the the lower lower peak peak and and must must cross cross the the region region of of reduced reduced mean mean fitness fitness (the (the "adaptive "adaptive valley"). valley"). The to another been given by The probability probability of of transition transition from from one one peak peak to another has has been given by 1 993, Eq. 3), for populations of given size size and immi Barton Barton and and Rouhani, Rouhani, ((1993, Eq. 113), for populations of aa given and immigration given fitness fitness function. examine the probability of gration rate, rate, and and for for aa given function. Let Let us us examine the probability of transition Imagine a meta transition when when all all populations populations are are initially initially at at the the same same peak. peak. Imagine a metapopulation with variable local population constant number number (Nm) population with variable local population sizes, sizes, but but with with aa constant (Nm) of constant of migrants migrants coming coming into into each each population population in in each each generation. generation. With With aa constant number class of populations will will be number of of migrants, migrants, the the FST FsT of of any any given given class of populations be approxi approximately will assume that the the mately the the same; same; if if the the populations populations are are sufficiently sufficiently old, old, we we will assume that genetic variance within demes constant. However, genetic variance within demes is is approximately approximately constant. However, the the proba probability (see Fig. Fig. 7). The bility of of transition transition to to new new peaks peaks is is not not at at all all constant constant (see The smaller smaller populations populations have have aa much much higher higher probability probability of of transition transition to to new new peaks. peaks. The The difficulty difficulty with with this this simple simple analysis analysis is is that that even even though though the the probability probability of of
206 206
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FIGURE FIGURE 77 The probability probability of transitions transitions between between adaptive adaptive peaks peaks as a function function of population population size. size. Smaller larger populations. Smaller populations populations are much much more more likely likely to undergo undergo peak peak shifts shifts than than are larger populations. These These probabilities peak and have have a probabilities are given given for a metapopulation metapopulation in which which all populations populations start start at one peak constant constant number number of migrants migrants coming coming into into each each deme deme each each generation. generation. The probabilities probabilities of transition transition are given 1 993, Eq. 113a) 3a) based function given given in Barton Barton and Rouhani Rouhani ((1993, based on the fitness fitness function given in that paper. paper. 0.8, I~ n = 1, The parameter values used sense of Barton Rouhani, 11993) 993) are ss = parameter values used here here (in the sense Barton and Rouhani, = 0.8, = 1, a = 0.3, and Nm = 2. a ==0 0, , vv=0.3, andNm=2. forward populations, the forward transition transition from from the the old old peak peak to to new new is is higher higher for for smaller smaller populations, the contribution of smaller populations also smaller, contribution of those those smaller populations to to the the gene gene pool pool is is also smaller, and and therefore uence that have on of other other demes demes is is therefore the the infl influence that they they will will have on the the transitions transitions of much backward transition pop much smaller. smaller. Furthermore, Furthermore, the the backward transition probability probability of of smaller smaller populations also higher, these new new shifts ulations is is also higher, so so these shifts are are less less stable. stable. In directly. In this this kind kind of of model, model, we we can can examine examine the the probability probability of of peak peak shifts shifts directly. We We know know (from (from the the arguments arguments cited cited above) above) that that the the probability probability of of a a successful successful peak with W being fitness of peak shift shift is is proportional proportional to to W 2N, 2u, with being the the fitness of the the population population in in the populations is the adaptive adaptive valley. valley. The The mean mean probability probability of of peak peak shifts shifts across across populations is frequency of populations of This mean mean therefore therefore J f o/;W ~iW 2N" 2Hi, where where 0/; tpi is is the the frequency of populations of size size N; Ni.. This probability to small For example, probability of of aa shift shift is is thus thus extremely sensitive sensitive to small N N values. values. For example, in 0 and 1 90, the the in a a metapopulation metapopulation with with half half its its popUlations populations of of size size 110 and half half of of size size 190, arithmetic mean size is 1 00 and the harmonic mean is 19. However, the constant arithmetic mean size is 100 and the harmonic mean is 19. However, the constant population size size which is about 1 3 if if W = population which has has the the same same overall overall rate rate of of shifting shifting is about 13 = 0.8. The The probability probability of of peak peak shifts shifts is is much much more more influenced by small population 0.8. influenced by small population size is refl ected even mean. size than than is reflected even in in the the harmonic harmonic mean. How new adaptive peaks in in the How does does local local extinction extinction affect affect the the spread spread of of new adaptive peaks the shifting balance? 1 979) showed new chro shifting balance? In In an an elegant elegant analysis, analysis, Lande Lande ((1979) showed that that a a new chroif populations populations go recolonized by mosome arrangement arrangement can can spread spread if go extinct extinct and and are are recolonized by mosome colonists from from a which is is fixed fixed for the new colonists a single single deme, deme, which for the new arrangement. arrangement. If If extinction extinction and respect to the chance and recolonization recolonization are are random random with with respect to genotype, genotype, then then the chance that that a a new be fixed fixed through whole metapopulation metapopulation is is new underdominant underdominant mutation mutation will will be through the the whole equal to it is is fixed fixed within single popUlation. is because because ((by by equal to the the chance chance that that it within aa single population. This This is analogy all the populations in analogy with with the the neutral neutral theory theory of of molecular molecular evolution) evolution) all the populations in a a species must trace back to population, and the rate species must trace back to one one ancestral ancestral population, and the rate of of evolution evolution of of the species must must equal the rate the whole whole species equal the rate of of change change of of that that one one population. population.
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The The Evolution Evolutionof of Metapopulotions Metapopulafions
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In migration between between populations, In reality, reality, new new adaptive adaptive peaks peaks may may spread spread by by migration populations, and be nonrandom. 1 985) and the the process process of of extinction/recolonization extinction/recolonization may may be nonrandom. Lande Lande ((1985) extended extended his his analysis analysis to to include include these these effects effects and and concluded concluded that that an an adaptive adaptive peak peak is likely to between populations populations is likely to gain gain aa greater greater advantage advantage from from its its stochastic stochastic spread spread between than colonization or which it than from from any any increase increase in in colonization or decrease decrease in in extinction extinction which it causes. causes. This relative rates rates of migration and This conclusion conclusion clearly clearly depends depends on on the the relative of migration and extinction: extinction: if an adaptive if almost almost all all spread spread were were by by extinction extinction and and recolonization, recolonization, then then an adaptive peak peak which decreased decreased extinction which extinction could could gain gain aa considerable considerable advantage. advantage. The coin, though, though, is that the the smaller smaller populations populations which which are The other other side side of of the the coin, is that are more likely to be able to drift through an adaptive valley are also subject to more likely to be able to drift through an adaptive valley are also subject to demographic stresses not experienced by larger populations. Therefore, larger demographic stresses not experienced by larger populations. Therefore, larger populations con populations usually usually have have aa higher higher probability probability of of survival, survival, and and make make aa larger larger contribution tribution to to the the migrant migrant pool. pool. This This counteracts counteracts whatever whatever increased increased fitness fitness asso associated small population population may found. A ciated with with aa higher higher peak peak aa small may have have found. A particular particular ex example - sink metapopulations, metapopulations, where the smaller smaller sink ample of of this this is is in in source source-sink where the sink populations likely populations are are more more likely likely to to experience experience genetic genetic drift, drift, but but are are also also more more likely to poor habitat quality, and they are to go go extinct extinct by by demographic demographic stochasticity stochasticity or or poor habitat quality, and they are also also continually continually swamped swamped by by immigrants immigrants from from the the source source popUlation. population. Therefore, Therefore, the size alone, the distribution distribution of of population population size alone, without without considering considering correlated correlated demo demographic cient to graphic parameters, parameters, is is insuffi insufficient to predict predict the the probability probability of of evolution evolution on on aa complex landscape. complex landscape. At the the same same time, time, temporal temporal variance variance in in migration migration rates rates can can increase increase the the prob probAt ability of culties with with the the shifting balance model model is is ability of peak peak shifts. shifts. One One of of the the diffi difficulties shifting balance that probability of rst phase, phase, where to drift that the the probability of the the fi first where drift drift allows allows aa population population to drift to to the is decreased by migration; migration; but prob the domain domain of of attraction attraction of of aa new new peak, peak, is decreased by but the the probability of phase, where new peak is exported to other populations, ability of the the third third phase, where aa new peak shift shift is exported to other populations, migration may may be be increased by by higher higher migration migration rates. rates. Temporally Temporally fluctuating fluctuating migration rates allow for to occur occur successfully; occur during during rates may may allow for both both phases phases to successfully; phase phase one one may may occur periods of of low migration and and phase phase three three later later when when migration migration rates rates are are higher higher periods low migration ((Moore Moore and 994). and Tonsor, Tonsor, 11994). Similar Similar considerations considerations apply apply in in continuous continuous habitats. habitats. The The hybrid hybrid zones zones which which separate by local local barriers barriers to separate different different adaptive adaptive peaks peaks are are easily easily trapped trapped by to gene gene flow flow (Barton, 979). Thus, (Barton, 11979). Thus, aa fitter fitter peak peak may may be be able able to to spread spread through through the the range range of of the the species species only only if if population population structure structure fluctuates fluctuates enough enough for for the the fitter fitter peak peak to to escape barriers. This chance into into the the outcome, outcome, escape local local barriers. This introduces introduces aa large large element element of of chance since population that since an an adaptive adaptive peak peak which which happens happens to to be be in in aa population that expands expands over over aa large spurious advantage -a kind spatial hitch-hikir~g. hitch-hiking. For For large area area will will gain gain aa spurious advantage--a kind of of spatial example, alpine grasshopper grasshopper Podisma pedesris two chromochromo example, the the alpine pedesris is is divided divided into into two some races, which which are zone. One is consistently consistently some races, are separated separated by by aa narrow narrow hybrid hybrid zone. One race race is more (Jackson, 11992), 992), suggesting suggesting aa fitness fitness advantage; more abundant abundant than than the the other other (Jackson, advantage; however, because the on the the main ridge however, it it cannot cannot spread spread because the hybrid hybrid zone zone is is trapped trapped on main ridge of Alpes Martimes. distribution is likely to more of the the Alpes Martimes. The The present present distribution is likely to be be determined determined more by last glaciation, rather than than by by historical historical patterns patterns of of recolonization recolonization after after the the last glaciation, rather by the the relative merits merits of of the the two two karyotypes. karyotypes. relative
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N. H. Borton and Michael Whitlock N.H. Barton and Michael C. C. Whitlock
Most many characters. characters. This is Most hybrid hybrid zones zones involve involve concordant concordant changes changes in in many This is not an an ascertainment ascertainment bias, because contacts contacts identified criterion (e.g., (e.g., plum plumnot bias, because identified in in one one criterion age age or or chromosome chromosome type) type) usually usually show show extensive extensive divergence divergence in in other other genetic genetic systems B arton and 985). This raises another culty for systems as as well well ((Barton and Hewitt, Hewitt, 11985). This raises another diffi difficulty for the the shifting local populations tends shifting balance balance process. process. The The expansion expansion and and contraction contraction of of local populations tends to complex hybrid zones; once once together, together, to bring bring unrelated unrelated differences differences together together in in complex hybrid zones; both inkage disequilibria common influence influence of changing population both llinkage disequilibria and and the the common of changing population structure therefore hard structure tends tends to to keep keep them them together. together. It It is is therefore hard to to see see how how aa new new peak peak could spread its own favorable effect effect on tness, rather could spread as as aa result result of of its own favorable on fi fitness, rather than than as as aa result of its fortuitous association with other peak shifts. The problem is essen result of its fortuitous association with other peak shifts. The problem is essentially process. Though tially that that the the shifting shifting balance balance is is an an asexual asexual process. Though Wright Wright saw saw this this as as an allows gene to be together, it it also an advantage, advantage, in in that that it it allows gene combinations combinations to be kept kept together, also makes it it hard for the the fi fittest gene combination to succeed. succeed. makes hard for ttest gene combination to
C. C. Maintenance of Genetic Genetic Variation in a Metapopulation Population Population structure structure is is potentially potentially important important in in the the maintenance maintenance of of genetic genetic variation. variation. Spatial Spatial heterogeneity heterogeneity in in selection, selection, coupled coupled with with limited limited migration, migration, can can substantially With substantially change change allele allele frequencies frequencies from from one one site site to to another another (see (see above). above). With some sites, genetic is maintained some migration migration among among sites, genetic variation variation is maintained both both locally locally and and globally. globally. Increased population even Increased genetic genetic variation variation can can be be maintained maintained in in aa meta metapopulation even with without multiple adaptive peaks, for out spatial spatial variation variation in in selection. selection. If If there there are are multiple adaptive peaks, for example example due due to to stabilising stabilising selection selection on on aa polygenic polygenic trait, trait, then then local local populations populations can can shift shift from from the the domain domain of of alternative alternative peaks, peaks, allowing allowing genetic genetic variation variation in in allele allele fre frequencies if not in phenotypic increase. Migration among these quencies (even (even if not in phenotypic states) states) to to increase. Migration among these popUlations genotypes, whereby variation populations introduces introduces locally locally unusual unusual genotypes, whereby the the genetic genetic variation within populations populations can within can be be greater greater than than it it would would be be without without spatial spatial population population 992; Barton Rouhani, 1993). 1 993). The structure structure (see (see Goldstein Goldstein and and Holsinger, Holsinger, 11992; Barton and and Rouhani, The following is possible. following analysis analysis shows shows how how this this is possible. Alleles Alleles at at loci loci under under pure pure stabilizing stabilizing selection selection are are essentially essentially under under aa one onelocus selection selection function If stabilizing selection locus function with with heterozygote heterozygote disadvantage. disadvantage. If stabilizing selection on is strong substantially affect fitness, then then the mean of on aa character character is strong enough enough to to substantially affect fitness, the mean of that that character character in in aa population population will will be be close close to to the the most most fi fitt type. type. This This implies implies that that a locus that, say, increases value will a substitution substitution at at one one locus that, say, increases the the character character value will be be com compensated We can pensated by by aa change change at at some some other other locus. locus. We can therefore therefore examine examine changes changes at at single loci, using loci single loci, using the the strength strength of of selection selection on on the the character character and and the the number number of of loci which which affect affect that that character character to to predict predict the the effective effective strength strength of of selection selection against against heterozygotes loci. ((These These assumptions by simu heterozygotes at at one one of of the the loci. assumptions have have been been tested tested by simulations lations of of selection selection on on quantitative quantitative traits traits in in subdivided subdivided populations.) populations.) Figure Figure 8 shows shows some some of of the the results. results. When When selection selection is is strong strong relative relative to to some some critical to adaptive peaks such that the critical migration migration rate, rate, the the populations populations each each go go to adaptive peaks such that the population is frequency metapopulation of given allele is 0.5 and frequency in in the the metapopulation of any any given allele is and each each population is nearly xed for As migration increases, the nearly fi fixed for one one or or the the other other of of the the alleles. alleles. As migration rate rate increases, the
The The Evolution Evolutionof of Metopopulotions Metapopulations
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FIGURE88 The maintenance maintenance of genetic genetic variation in a subdivided with stabilizing selecFIGURE subdivided population with stabilizing selec mutation is approximated approximated by a model model of a single single locus locus with with heterozygote heterozygote disadvantage. disadvantage. In tion and mutation examples, each each local local population is composed composed of 10 with the strength strength of these examples, 10 diploid individuals with stabilizing selection selection ((Vs/VE) equal to 115. frequencies in the pool of migrants migrants will will be on 5 . (a) The allele frequencies stabilizing V,/V,) equal
average 0.5 0.5 until until a critical critical value value of Nm, Nm, after which which a bifurcation bifurcation occurs, occurs, when when the allele allele frequencies frequencies average become either greater greater than than or less less than than a half. half. (b) The total total variance variance in the metapopulation is defined defined these allele frequencies, frequencies, and is shown shown by the upper curve. 0.25 for Nm Nm by these curve. It is at the maximum maximum of 0.25
below the critical critical value and then then decreases. decreases. The lower curve curve shows shows the genetic variance variance within within below This reaches a maximum maximum at the critical migration rate and then then declines with with increasing populations. This increasing migration.
allele frequency frequency in in the the metapopulation goes toward toward either either 0 0 or or 1I [with allele metapopulation goes [with equal equal probability for for these these additively additively acting acting genes genes (see (see Fig 8a)].] . Thus Thus variation probability Fig 8a) variation in in the the metapopulation is is at at a a m a x i m u m anywhere anywhere below below the the critical critical migration rate (see (see metapopulation maximum migration rate Fig 8b).. Variance Variance within within populations populations is is more more complex. complex. With With low low migration migration rate, rate, Fig 8b) local are nearly for one or the other; as as migration local populations populations are nearly fixed fixed for one allele allele or the other; migration rate rate increases, alleles come and increase increases, more more alleles come into into each each population population and increase local local variation. variation. Above the critical the alleles come into a popuAbove the critical migration migration rate, rate, however, however, the alleles which which come into a popu lation a higher of being already there, lation have have a higher probability probability of being identical identical in in effect effect to to those those already there, and the decreases again 8b). A and the local local genetic genetic variation variation decreases again (see (see Fig. Fig. 8b). A shifting shifting balance balance type for traits more type of of process process for traits under under stabilizing stabilizing selection selection can can maintain maintain much much more genetic alone, if u m b e r of genetic variation variation than than can can mutation mutation alone, if the the nnumber of migrants migrants among among poppop ulations selection across across populations ulations is is small small enough. enough. Variation Variation in in selection populations or or through through time for large Nm, time may may cause cause populations populations to to shift shift to to different different adaptive adaptive peaks peaks even even for large Nm, and and hence hence may may maintain maintain variation variation in in the the same same kind kind of of way. way.
VI. DISCUSSION DISCUSSION As As can can be be seen seen from from many many of of the the preceding preceding sections, sections, spatial spatial population population strucstruc ture a n y different ture is is potentially potentially important important to to m many different evolutionary evolutionary processes. processes. UnfortuUnfortu nately, nately, different different population population structures structures which which have have similar similar properties properties from from the the point point of of view view of of one one process process can can have have very very different different properties properties for for other other processes. processes. The obvious The obvious ways ways of of measuring measuring population population structure, structure, such such as as the the effective effective sizes sizes
2210 10
N. H . Barton N.H. Barton and and Michael MichaelCC.. Whitlock Whitlock
of populations, the populations, or of local local populations, the genetic genetic variance variance among among populations, or even even some some stan standardized ST are previous section, dardized measure measure like like F FST are thus thus inadequate. inadequate. As As shown shown in in the the previous section, the the probability probability of of peak peak shifts shifts depends depends greatly greatly on on the the demographic demographic properties properties of of a species, even metapopu a species, even when when one one holds holds FST FST constant. constant. The The effective effective size size of of aa metapopulation sizes (as well as lation depends depends on on both both FST FST and and the the distribution distribution of of population population sizes (as well as on on the tness as the distribution distribution of of population population fi fitness as aa function function of of population population size, size, etc.). etc.). The The probability xation of alleles is completely described ST or probability of of fi fixation of favorable favorable alleles is not not completely described by by F FST or 993). by Barton, 11993). by the the effective effective size size ((Barton, Different Different evolutionary evolutionary processes processes must must be be considered considered separately separately in in order order to to get get the the right right answers. answers. The The distribution distribution of of neutral neutral alleles alleles in in aa metapopulation metapopulation de depends pends on on random random drift drift and and migration, migration, whereas whereas local local adaptation adaptation depends depends primarily primarily ' s view on on migration migration and and selection. selection. Wright Wright's view of of evolution evolution involves involves aa shifting shifting balance balance between between all all three three processes. processes. It It is is critically critically important important that that studies studies of of genetic genetic meta metapopulation structure ST or population structure go go beyond beyond the the usual usual measurement measurement of of F FST or some some such such simple simple measure measure of of the the differentiation differentiation of of neutral neutral markers markers caused caused by by sampling sampling drift. drift. It It is is essential us a knowledge essential that that empirical empirical studies studies are are performed performed which which give give us a broader broader knowledge of of the the demographic demographic and and selective selective forces forces at at work work in in determining determining the the distribution distribution of of genotypes genotypes across across populations. populations. Moreover, Moreover, as as is is clear clear from from the the section section on on peak peak shifts, levels, of shifts, we we must must learn learn more, more, at at both both the the theoretical theoretical and and the the empirical empirical levels, of what what determines determines the the overall overall distribution distribution of of genotypes genotypes across across populations populations and and not not merely merely the the mean mean and and variance. variance. Such Such studies studies are are difficult difficult but but not not impossible, impossible, and and they are are badly badly needed needed for for aa full full picture picture of of the the causes and consequences of spatial they causes and consequences of spatial population population structure. structure.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS We thank the Biotechnology and Biological Sciences Research Council and the Darwin Darwin Trust of of Edinburgh, Edinburgh, whose funds supported this work.
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METAPOPULATIONPROCESSES PROCESSES METAPOPULATION
The processes in metapopulation dynamics dynamics are The principal principal processes in metapopulation are migration, and and colonization, colonization, that that is, is, establishment establishment extinction, extinction, migration, of of new new local local populations. populations. The The key key questions questions are are how how these these propro cesses dynamics and cesses jointly jointly affect affect the the dynamics and evolution evolution of of local local poppop ulations and and the the entire entire metapopulation. metapopulation. The The five five chapters in this this ulations chapters in section extend the of metapopulation section extend the theoretical theoretical treatment treatment of metapopulation ecolecol ogy, and evolution evolution in ogy, genetics, genetics, and in the the previous previous section section by by aa more more detailed examination detailed examination of of particular particular processes, processes, such such as as local local poppop ulation variance in ulation growth, growth, temporal temporal variance in growth, growth, and and the the impact impact of of immigration local dynamics. dynamics. immigration on on local Much of of the interest in in metapopulations metapopulations stems Much the ecological ecological interest stems from that small from the the observation observation that small local local populations populations have have aa high high risk of of extinction, extinction, and and hence the long-term persistence of of aa risk hence the long-term persistence metapopulation consisting consisting of of extinction-prone extinction-prone small small populapopula metapopulation tions tions cannot cannot possibly possibly be be understood understood at at aa smaller smaller scale scale than than the the metapopulation. Foley Foley replaces replaces the the verbal verbal model model of"small-popof "small-pop metapopulation. ulations-have-a-high-probability-of-extinction" with with aa series series of of ulations-have-a-high-probability-of-extinction" mathematical models models of of stochastic stochastic population population growth, which mathematical growth, which predict the the expected expected time time to to population population extinction extinction under under various various predict scenarios, such such as as demographic demographic stochasticity, stochasticity, environmental environmental scenarios,
stochasticity, combinations. Though stochasticity, catastrophies, catastrophies, and and their their combinations. Though nu numerical simulations simulations may may produce produce more more accurate accurate results results for for merical well-studied well-studied populations, populations, the the insight insight provided provided by by general general math mathematical models is ematical models is invaluable. invaluable. Foley Foley also also makes makes an an effort effort to to relate discussing the relate the the models models to to real real populations populations by by discussing the often often hard parameter estimation. Foley identifies hard problems problems of of parameter estimation. Foley identifies aa num number issues that ber of of issues that may may be be important important for for extinction extinction but but are are not not covered covered by by the the standard standard models. models. These These additional additional factors factors in include clude interspecific interspecific interactions, interactions, regional regional stochasticity, stochasticity, the the form form of One additional of density density dependence, dependence, and and so so on. on. One additional problem problem is is the the "noise" "noise" due due to to slowly slowly changing changing environmental environmental conditions. conditions. Arguably, Arguably, many many if if not not most most extinctions extinctions are are due due to to "novel" "novel" events, events, which which could could not not have have been been easily easily anticipated anticipated from from short short records records of of past past population population growth. growth. The calamitous process The calamitous process of of extinction extinction is is countered, countered, in in meta metapopulations, by occoz term populations, by what what Ims Ims and and Y Yoccoz term the the transfer transfer pro processes: emigration, migration, migration, and colonization. Though cesses: emigration, and colonization. Though the the simple models models of metapopulations and field studies studies deal simple of metapopulations and many many field deal with with only only colonization colonization of of empty empty patches, patches, it it is is well-understood well-understood that existing populations populations may that migration migration among among existing may have have significant significant consequences for local dynamics dynamics and metapopulation dy consequences for both both local and metapopulation dynamics (most chapters namics in in the the broad broad sense sense (most chapters in in Sections Sections III III and and IV). actual empirical rates IV). Unfortunately, Unfortunately, actual empirical measurement measurement of of the the rates of hampered by by of emigration, emigration, immigration, immigration, and and colonization colonization are are hampered various regardless of whether one uses indirect various difficulties, difficulties, regardless of whether one uses indirect or or direct dilem direct observations observations or or experiments. experiments. One One of of the the greatest greatest dilemmas mas is is spatially spatially and and temporally temporally varying varying rates rates of of migration. migration. Given level described Given all all the the variability variability at at the the individual individual level described by by Ims Ims and is a and Yoccoz, Yoccoz, it it is a small small wonder wonder that that any any metapopulation metapopulation mod models els show show any any promise promise at at all all for for predictive predictive purposes. purposes. However, However, it it may may tum turn out out that that the the details details of of migratory migratory behavior behavior matter matter little instance, the little for for predicting, predicting, for for instance, the effect effect of of distance distance on on col colonization, onization, aa population-level population-level phenomenon. phenomenon. It It is is questionable questionable whether an an individual-based individual-based model model would would generally generally be be more more whether useful pop useful than than aa population-based population-based model model for for predicting predicting meta metapopulation ulation dynamics, dynamics, though though aa mechanistic mechanistic understanding understanding of of say say migration individual-level study. lesson migration clearly clearly requires requires an an individual-level study. The The lesson is is not not to to funnel funnel into into one one approach approach or or another, another, but but to to realize realize that that different different approaches approaches supply supply complementary complementary information, information, and and the biologists are the clever clever biologists are the the ones ones who who succeed succeed in in forging forging aa synthesis most synthesis most helpful helpful for for aa particular particular task. task. Olivieri selective forces Olivieri and and Gouyon Gouyon recognize recognize that that strong strong selective forces may may operate operate in in metapopulations metapopulations on on migration migration and and other other life lifehistory history traits. traits. Migration Migration has has typically typically certain certain costs, costs, in in the the form form
of mortality mortality during during the the movement movement through through hostile hostile environments environments of and due due to to possible possible relocation relocation into into less less suitable suitable localities localities than than and that of of the the natal natal population. popUlation. Thus, Thus, genes genes promoting promoting migration migration that are are generally generally selected selected against against at at the the level level of of local local population. population. However, However, this this is is only only aa part part of of the the picture, picture, as as at at the the metapopmetapop ulation level migration migration is is selected selected for, for, which which is is most most conspicconspic ulation level uous when when migrants succeed in in establishing establishing new new populations. populations. uous migrants succeed Olivieri and and Gouyon Gouyon term the action action between between these these two two antagantag Olivieri term the onistic forces forces of of selection selection the the "metapopulation "metapopulation effect." effect." This This disdis onistic tinction is between individual tinction is often often seen seen as as aa conflict conflict between individual and and group group selection, but but as as Olivieri Olivieri and and Gouyon Gouyon remark, remark, there there is is no no concon selection, flict between between two two different different types types of of selection, selection, rather rather aa hierarchy hierarchy flict of selection selection components components operating operating at at different different levels levels of of the the biobio of logical there is logical organization. organization. Such Such issues issues apart, apart, there is aa substantial substantial theoretical theoretical and and empirical empirical literature literature on on factors factors affecting affecting and and scesce narios narios of of migration, migration, which which is is reviewed reviewed in in this this chapter. chapter. One One conclusion of potential conclusion of potential importance importance to to conservation conservation of of species species in fragmented fragmented landscapes is that the evolutionarily evolutionarily stable stable mimi in landscapes is that the gration rate is is generally less than would maximize maximize gration rate generally less than the the rate rate that that would the size of of the metapopulation (fraction (fraction of of suitable suitable habitat habitat ococ the size the metapopulation cupied at at equilibrium). into account account that species have have cupied equilibrium). Taking Taking into that species hardly had had time time to to adapt to rapidly rapidly changing changing landscapes landscapes and hardly adapt to and that species species may may be be even even in in an an ecological ecological nonequilibrium nonequilibrium in in that increasingly fragmented fragmented landscapes landscapes (chapters (chapters by by Hanski by increasingly Hanski and and by Nee, May, and and Hassell), Nee, May, Hassell), there there appears appears to to be be substantial substantial scope scope for managed reintroductions. reintroductions. for managed One One of of the the more more frequently frequently used used but but often often misunderstood misunderstood concepts the metapopulation literature is is the rescue effect. effect. concepts in in the metapopulation literature the rescue As by Brown Brown and As originally originally defined defined by and Kodric-Brown, Kodric-Brown, the the rescue rescue effect refers to reduced probability of population extinction due due effect refers to reduced probability of population extinction to immigration. In In mechanistic terms, the the rescue rescue effect effect occurs occurs to immigration. mechanistic terms, because immigration up population which decreases decreases because immigration props props up population size, size, which the risk of extinction (Foley). was the risk of immediate immediate extinction (Foley). The The rescue rescue effect effect was originally envisioned envisioned to to occur in mainland mainland-island originally occur in - island metapopumetapopu lations, but but a may also occur in metapopulations lations, a similar similar effect effect may also occur in metapopulations without a mainland. In this without a mainland. In this case, case, existing existing populations populations provide provide a However, the the complication complication a kind kind of of mutual mutual aid aid to to each each other. other. However, here is is that that individuals that moved moved from from population population A A to to pop pophere individuals that ulation ulation B B reduce reduce the the size size of of population population A A and and hence hence potentially potentially increase increase the the risk risk of of extinction extinction of of the the source source population population (A) (A) while while decreasing decreasing the the risk risk of of extinction extinction in in the the receiving receiving popu population (B). (B). In In practice, practice, the the effect effect of of population population size size on on extinc extinclation tion is is nonlinear, nonlinear, either either because because of of some some deterministic deterministic Allee tion Allee effect, practically practically forcing forcing the the extinction extinction of of small small populations, populations, effect,
or consequences of most severe or because because the the consequences of stochasticity stochasticity are are most severe in in small (Foley). Therefore small populations populations (Foley). Therefore migration migration from from large large to to small metapopu small populations populations may may create create aa rescue rescue effect effect at at the the metapopulation lation level. level. Stacey Stacey and and Taper Taper explore explore in in their their chapter chapter the the ecol ecology butterflies to ogy of of the the rescue rescue effect, effect, in in aa range range of of taxa taxa from from butterflies to small small mammals mammals to to amphibians. amphibians. What What they they find find is is that that migration migration among local populations populations is among local is often often so so extensive extensive that that it it is is likely likely to to affect affect local local dynamics. dynamics. This This means, means, in in terms terms of of theory, theory, that that the models described described by the kinds kinds of of structured structured metapopulation metapopulation models by Gyllenberg, Gyllenberg, Hanski, Hanski, and and Hastings Hastings in in the the previous previous section section should should be be appropriate appropriate and, and, in in terms terms of of dynamics, dynamics, that that complex complex metapopulation metapopulation dynamics dynamics with with alternative alternative stable stable states states may may be be expected occur. expected to to occur. The nal chapter The fi final chapter by by Frank Frank deals deals with with host-parasite host-parasite sys systems tems in in aa broad broad sense sense and and their their evolutionary evolutionary dynamics dynamics in in frag fragmented mented environments. environments. If If there there are are only only small small numbers numbers of of inter interacting acting host host and and parasite parasite genotypes genotypes (or (or species), species), stable stable local local dynamics dynamics may may be be expected. expected. In In contrast, contrast, with with large large potential potential numbers types, inevitable extinctions due numbers of of types, inevitable extinctions due to to small small population population size possibilities for size (of (of some some types) types) open open up up possibilities for resistant resistant hosts hosts and and virulent virulent parasites, parasites, whose whose matching matching parasites parasites and and hosts hosts went went ex extinct, tinct, to to dominate dominate the the local local patch. patch. A A colonization colonization by by the the missing missing type type leads leads to to another another major major perturbation, perturbation, as as the the invader invader now now has a strong selective advantage over the local types. Frank has a strong selective advantage over the local types. Frank relates relates the the theory theory to to aa range range of of observations observations on on plant-pathogen plant-pathogen and cytoplasmic male and bacteria-phage bacteria-phage systems, systems, cytoplasmic male sterility sterility in in plants plants (a (a conflict conflict between between cytoplasmic cytoplasmic and and nuclear nuclear genes), genes), dis disease (allelic variation his ease resistance resistance in in vertebrates vertebrates (allelic variation in in the the major major histocompatibility complex), and tocompatibility complex), and genetic genetic variance variance in in plant plant resist resistance ance to to herbivores. herbivores. Under Under his his hypothesis, hypothesis, which which remains remains aa challenge to spatial scale, scale, coloniza challenge to test test on on aa sufficiently sufficiently large largespatial colonization-extinction tion-extinction dynamics dynamics not not only only maintain maintain the the ever-changing ever-changing composition of local communities, composition of local communities, but but the the turnover turnover rates rates of of allelic types co allelic types are are proportional proportional to to the the number number of of potentially potentially coexisting types. existing types.
10
Extinction Models for local Populations Populations Local Patrick Foley
INTRODUOION I. INTRODUCTION The of metapopulation local extinction The two two underlying underlying processes processes of metapopulation dynamics, dynamics, local extinction and colonization, colonization, are are subject subject to to chance chance (Levins, (Levins, 11970; Hanski and and Gilpin, and 970; Hanski Gilpin, 11991). 99 1 ). Life metapopulation is even when Life in in aa metapopulation is haphazard, haphazard, and and even when deterministic deterministic influences influences are at at work, work, they they may may seem seem stochastic stochastic to to an an ecologist ecologist with with limited are limited information information (Sugihara and and May, May, 11990). chapter reviews reviews stochastic stochastic models of extinction extinction (Sugihara 990). This This chapter models of within is an within aa local local population. population. The The main main thread thread of of the the chapter chapter is an analytic analytic model model of of environmental environmental stochasticity stochasticity in in which which populations populations fluctuate fluctuate between between aa ceiling ceiling and and extinction. diffusion analysis analysis of extinction. A A diffusion of environmental environmental and and demographic demographic stochasticity stochasticity is complemented complemented by by analytic analytic and and numerical numerical investigations investigations into into the robustness of of is the robustness the analysis. Simulations works the analysis. Simulations in in discrete discrete time time show show that that the the diffusion diffusion analysis analysis works ((Foley, Foley, 11994; 994; Hanski 996a). The robustness investigations Hanski et et al., al., 11996a). The robustness investigations of of this this chapter chapter show show that that the the model model can can be be extended extended to to deal deal with with many many biological biological details. details. Al Although though the the model model was was developed developed to to predict predict extinction extinction rates, rates, it it speaks speaks to to the the ques question tion of of colonization colonization rates rates also, also, since since emigration emigration is is often often aa function function of of the the popu population sizes whose fluctuations are analyzed by the same model. lation sizes whose fluctuations are analyzed by the same model. Any Any mathematical mathematical model model is is bound bound to to be be aa simplification simplification of of reality. reality. This This is is aa weakness weakness (and (and virtue! virtue!)) shared shared with with verbal verbal models models and and computer computer simulations. simulations. The The beauty beauty of of aa verbal verbal model model is is that that it it sticks sticks in in your your head head and and you you can can make make use use of of Metapopuialion Metapopulation Biology Biology Copyright 997 by Copyright © 9 11997 by Academic Academic Press, Inc. Inc. All All rights rights of of reproduction reproduction in in any any fonn form reserved. reserved.
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Patrick Foley Foley Patrick
at will. will. However, However, verbal verbal models models of of extinctions extinctions come come down down to to this: this: small small poppop itit at ulations have have higher higher extinction extinction rates rates than than large large ones. ones. The The mathematical mathematical models models ulations presented in in this this chapter chapter do do better. better. They They give give expected expected times times to to extinction extinction and and presented population size size thresholds thresholds for for heightened heightened vulnerability. vulnerability. They They reveal reveal the the paramparam population eters we we need need to to measure measure and and the the ones ones we we can can (at (at smaller smaller risk) risk) neglect. neglect. They They eters allow allow comparisons comparisons between between species species and and between between landscapes. landscapes. A A computer computer simusimu lation can can make make more accurate predictions predictions since since it it can can avoid many simplifying simplifying lation more accurate avoid many assumptions, but but only only if if the the biological biological parameters parameters and and processes processes are known in in assumptions, are known detail. named three generality, realism, detail. Levins Levins (1966) ( 1 966) named three features features of of aa good good model: model: generality, realism, and models of of population typically inaccurate, inaccurate, and accuracy. accuracy. Verbal Verbal models population dynamics dynamics are are typically computer simulations simulations too too specific, specific, and and mathematical mathematical models models unrealistic. unrealistic. Useful Useful computer mathematical models models in in ecology ecology are are robust robust enough enough that that slightly slightly unreal unreal assumpassump mathematical tions produce produce only only slightly slightly inaccurate inaccurate predictions predictions (This (This virtue virtue is is sometimes sometimes called called tions "the structural structural stability stability of of the the model"). model"). The models we we examine are fairly fairly robust. robust. "the The models examine are In view view of of the the recent wave of of discussion individual-based models models In recent wave discussion about about individual-based (Lomnicki, 1988; 1 988; DeAngelis DeAngelis and Gross, 1992; 1 992; Judson, JUdson, 1994), 1 994), the simple analytic analytic (Lomnicki, and Gross, the simple approach may appear not aggreaggre approach of of this this chapter chapter may appear anachronistic. anachronistic. Populations Populations are are not gations of identical atoms; atoms; sex, sex, age, age, genotype, genotype, and history affect ecological dydy gations of identical and history affect ecological namics, and local interactions interactions surely surely influence influence competition competition and predation outout namics, and local and predation comes. When substantial substantial resources resources are computer simulations based on comes. When are available, available, computer simulations based on individual interactions interactions are appropriate. For the Northern Northern individual are particularly particularly appropriate. For example, example, the Spotted Owl Owl (Strix caurina) inspired inspired aa spatially spatially explicit explicit territoryterritory Spotted (Strix occidentalis occidentalis caurina) at. , 1993) 1 993) and and aa spatially spatially explicit explicit indiindi based simulation in Pascal Pascal (McKelvey ( McKelvey et based simulation in et al., et al., al. , vidual-based simulation simulation in in the the object-oriented object-oriented language Smalltalk (Foley ( Foley et vidual-based language Smalltalk 1 993), each each project costing over over 2 person-years of effort. However, However, such such models models 1993), project costing 2 person-years of effort. are are expensive expensive to to build, build, hungry hungry for for parameter parameter values, values, particular particular to to the the case case at at hand, hand, not compelling, and form. Some these not logically logically compelling, and hard hard to to summarize summarize in in useful useful form. Some of of these problems will be problems will be overcome overcome with with conceptual conceptual advances, advances, but but at at present present the the (argua (arguably) illuminating analysis of Northern Northern Spotted Spotted Owl dynamics is is probably probably bly) most most illuminating analysis of Owl dynamics Lande' l 988a) simple simple demographic analysis. For Lande'ss ((1988a) demographic and and metapopulation metapopulation analysis. For insight, insight, logic, is hard to beat Maynard Smith, 974), and logic, and and generality generality it it is hard to beat mathematics mathematics ((Maynard Smith, 11974), and mathematical for the more comcom mathematical models models form form aa secure secure foundation foundation for the construction construction of of more plex plex analyses. analyses. Most metapopulations, island Most important important for for the the theory theory of of metapopulations, island biogeography, biogeography, and and landscape local extinction models allow theory (and computer simula landscape ecology, ecology, local extinction models allow theory (and computer simulation) at at the the higher higher level level to to proceed proceed more more quickly quickly and and gracefully. gracefully. The The fi first models tion) rst models of MacArthur and of island island biogeography biogeography used used demographic demographic stochasticity stochasticity ((MacArthur and Wilson, Wilson, 11967) 967) perhaps perhaps because because it it takes takes the the form form of of the the well-studied well-studied birth birth and and death death pro processes cesses and and it it clearly clearly affects affects small small discrete discrete populations. populations. Many Many recent recent metapopu metapopulation models use 987; Hastings, Hastings, 11990; 990; lation models use catastrophic catastrophic stochasticity stochasticity (Ewens (Ewens et at. al.,, 11987; Mangel 994), perhaps Mangel and and Tier, Tier, 11994), perhaps because because the the time time scales scales of of exponential exponential population population growth growth and and catastrophes catastrophes are are different different enough enough to to permit permit analytic analytic sim simplifications. II have plifications. have focused focused on on environmental environmental stochasticity stochasticity because because it it is is persistpersist-
1100 local LocalExtinction ExtinctionModels Models
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ently ently important important for for small small and and large large populations, populations, it it subsumes subsumes catastrophic catastrophic sto stochasticity it can empirically measured. chasticity (at (at least least in in principle), principle), and and it can be be empirically measured. Most biology, simple models allow Most important important for for conservation conservation biology, simple mathematical mathematical models allow a -benefit a ready ready comparison comparison of of conservation conservation strategies. strategies. Optimization Optimization and and cost cost-benefit analyses Burgman ef analyses depend depend on on such such comparisons. comparisons. Programs Programs such such as as RAMAS RAMAS ((Burgman et ai. 993) and 992) are including age al.,, 11993) and VORTEX VORTEX (Lacy (Lacy and and Kreeger, Kreeger, 11992) are good good at at including age structure structure and and other other biological biological details, details, but but they they need need extensive extensive runs runs to to cover cover much much parameter but at at parameter space. space. They They can can be be made made to to do do the the job job of of comparing comparing strategies, strategies, but some in time, some cost cost in time, generality, generality, and and conceptual conceptual clarity. clarity. Most populations, local Most important important for for an an understanding understanding of of natural natural populations, local extinction extinction models models provide provide explanations explanations about about the the persistent persistent problems problems of of ecology ecology such such as as population - area curves. population fluctuations, fluctuations, density density dependence, dependence, and and species species-area curves. These These ex explanations planations are are testable testable against against the the observed observed patterns patterns of of biogeography biogeography and and popu population rich lation time time series series data. data. This This chapter chapter makes makes some some modest modest advances advances into into this this rich and and varied varied territory. territory.
II. LOCAL EXTINGION DUE LOCALPOPULATIONS POPULATIONSARE AREVULNERABLE VULNERABLETO TO EXTINCTION DUETO TO DEMOGRAPHIC DEMOGRAPHIC AND AND ENVIRONMENTAL ENVIRONMENTALSTOCHASTICITY STOCHASTICITY A population is is extinct extinct when size of zero and it is A population when it it reaches reaches the the size of zero and it is doomed doomed to to prompt extinction when females reaches reaches zero. prompt extinction when the the pooled pooled reproductive reproductive value value of of all all females zero. A single single habitat habitat patch patch in in aa fragmented fragmented landscape landscape has has limited limited resources A resources which which set set a possible size a limit limit to to the the possible size of of the the local local population. population. The The population population size size N N (mea (measured this sured in in number number of of females) females) thus thus lives lives on on the the interval interval that that includes includes zero zero and and this population ceiling. Sooner will go population ceiling. Sooner or or later later any any local local population population will go extinct extinct due due to to the the stochastic processes and variation of the stochastic nature nature of of birth birth and and death death processes and to to the the temporal temporal variation of the environment. Each population is ultimately doomed; doomed; the challenge is predict environment. Each population is ultimately the challenge is to to predict when understand why. why. when and and understand It is is always always possible to model model population growth as as It possible to population growth N(f N(t + + 11)) = = R(N, R(N, f) t) N(t), N(t),
((1) 1)
where 990), iiss aa where R(N, R(N, f), t), the the "fundamental "fundamental net net reproductive reproductive rate" rate" (Begon (Begon ef et al. al.,, 11990), function density dependence) dependence) and permitting sto function of of both both N N (thus (thus providing providing density and ft ((permitting stochasticity). chasticity). R R is is also also aa function function of of the the age age structure structure and and genetic genetic structure structure of of the the important than N and population, these effects population, but but these effects are are usually usually less less important than the the effects effects of of N and random effects R(N, f) random effects which which we we attribute attribute to to f.t. The The probability probability distribution distribution of of R(N, t) becomes population. It is the job of becomes the the critical critical feature feature directing directing the the fate fate of of the the population. It is the job of field ecology job of of field ecology to to describe describe typical typical probability probability distributions distributions for for R R and and the the job theory analyze the is to to understand theory to to analyze the consequences. consequences. The The equivalent equivalent challenge challenge is understand the distribution of f), the the realized realized per rate. the distribution of r(N, r(N, f) t) = = log log R(N, R(N, t), per capita capita reproductive reproductive rate. In will often R(N, t) f) to to R(t) R(f) or R, and the same In the the following, following, II will often shorten shorten R(N, or R, and II do do the same to to r(N, N(t). The logarithm will be used used exclusively. r(N, f) t) and and N(t). The natural natural logarithm will be exclusively.
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Deterministic R and time. Two Deterministic models models assume assume that that R and rr are are constant constant over over time. Two par particular ticular examples examples are are the the constant constant growth growth model, model,
r(N, r(N, t) = = rd,
(2)
for for all all N N and and t, except except at at the the ceiling ceiling value value K, and and the the Ricker Ricker model, model,
r(N, (1 r(N, t) = = rd rd(1 - N/K), N/K),
(3) (3)
which logistic model model ((Burgman Burgman et 1 993). which has has properties properties similar similar to to those those of of the the logistic et al. al.,, 1993). Both population will will go Both models models give give simple simple extinction extinction predictions: predictions" if if rd rd < < 0, the the population go extinct. extinct. For For model model 11 the the expected expected time time to to extinction extinction T Tee is is given given by by T Tee = =
log log (N(O» (N(0))
rd ?'d
.
9
(4) (4)
These for local popu These or or similar similar deterministic deterministic models models may may often often be be appropriate appropriate for local populations, especially especially if Pulliam, 11988) 988) or main lations, if their their patches patches are are sinks sinks (rd < < 0, Pulliam, or the the mainland Wilson, 11967). 967). If If every has negative negative rd un land (rd > > 0, MacArthur MacArthur and and Wilson, every patch patch has r~,, unal., 11996a), 996a), we less, immigration Hanski et less, immigration can can compensate compensate ((Hanski et al., we should should switch switch our our study study to to aa new new metapopulation. metapopulation. Stochastic extinction have populations since Stochastic models models of of extinction have been been applied applied to to local local populations since MacArthur 1 963, 11967) 967) developed island biogeog MacArthur and and Wilson Wilson ((1963, developed their their theory theory of of island biogeography. raphy. MacArthur MacArthur and and Wilson Wilson employed employed demographic demographic stochasticity stochasticity in in their their anal analysis, assuming ysis, assuming constant constant per per capita capita birth birth and and death death rates, rates, aa population population ceiling ceiling K, and population population change change to to be be aa Poisson Poisson process. process. Demographic Demographic stochasticity stochasticity is is and strongest strongest in in tiny tiny populations populations and and negligible negligible compared compared to to environmental environmental stochas stochasticity 1 987a,b; Lande, ticity when when population population sizes sizes reach reach the the hundreds hundreds (Goodman, (Goodman, 1987a,b; Lande, 11993). 993). Demographic it Demographic stochasticity stochasticity is is analogous analogous to to random random genetic genetic drift drift in in that that it depends on the intrinsic uncertainty associated with an individual' s reproduction depends on the intrinsic uncertainty associated with an individual's reproduction and and mortality, mortality, essentially essentially aa sampling sampling problem problem for for mother mother nature. nature. Large Large popula populations tions average average out out birth birth and and death death across across individuals individuals and and the the sampling sampling error error (the (the variance variance of of r(t» r(t)) is is inversely inversely proportional proportional to to N. N. In aa broad broad sense, stochasticity refers refers to to any any randomness randomness imIn sense, environmental environmental stochasticity im posed by r(t). It become conventional posed by the the environment environment on on r(t). It has has become conventional in in the the conservation conservation biology literature distinguish between biology literature to to distinguish between catastrophes catastrophes and and environmental environmental sto stochasticity 98 1 ). Catastrophe chasticity in in the the narrow narrow sense sense (Shaffer, (Shaffer, 11981). Catastrophe models models typically typically as assume by demographic ex sume aa constant constant positive positive rr (modified (modified perhaps perhaps by demographic stochasticity) stochasticity)except habitat destruction), cept when when disaster disaster hits hits (disease, (disease, new new competitor, competitor, temporary temporary habitat destruction), when Hanson and 1 98 1 ; Ewens when rr is is negative negative and and often often large large ((Hanson and Tuckwell, Tuckwell, 1981; Ewens et et al., al., 11987; 987; Mangel 993b; Lande, 993). Mangel and and Tier, Tier, 11993b; Lande, 11993). Environmental Environmental stochasticity stochasticity in in the the narrow narrow sense sense can can plausibly plausibly be be modeled modeled by by assuming assuming that that r(t) r(t) follows follows aa gaussian gaussian distribution distribution with with mean mean rd r~ and and variance variance v" Vr, i.e., i.e.,
r(t) --~ N(rd, Vr),
(5) (5)
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1100 locol LocalExtinction ExtinctionModels Models
which Foley, 1994). 1 994). Efforts Efforts to which provides provides for for good, good, bad, bad, and and indifferent indifferent years years ((Foley, to model and analyze environmental stochasticity have been extensive model and analyze environmental stochasticity have been extensive (Levins, (Levins, 11969b; 969b; Richter-Dyn 972; May, 973a; Ludwig, 1 974, 1976; 1 976; Turelli, Richter-Dyn and and Goel, Goel, 11972; May, 11973a; Ludwig, 1974, Turelli, 11977; 977; Leigh, 1 98 1 ; Nisbet and Gurney, 1 982; Lande and Orzack, 988; Dennis Dennis Leigh, 1981; Nisbet and Gurney, 1982; Lande and Orzack, 11988; et 99 1 ). The models or models et al., 11991). The early early efforts efforts were were based based on on stochastic stochastic logistic logistic models or models with models, but quickly with mixed mixed environmental/demographic environmental/demographic stochasticity stochasticity models, but these these quickly run run into into intractable intractable analysis. analysis. Lande Lande and and Orzack Orzack and and Dennis Dennis et et al. al. assumed assumed no no density 1 993), Foley 1 994), and density dependence dependence whatever. whatever. Recently Recently Lande Lande ((1993), Foley ((1994), and Mid Middleton 1 996) have dleton et et al. al. ((1996) have analyzed analyzed the the simple simple ceiling ceiling model model reviewed reviewed in in this this chapter and obtained succinct, useful results. chapter and obtained succinct, useful results. To To compare compare the the three three modes modes of of stochasticity, stochasticity, consider consider aa model model for for the the prob probability density density of ability of r(N, t), t),
r(N,
r(N {L N(rd, Y(N,, tt))
..~ ~ N(rd, Vr vr + + vJN) v~/N) log(11 - 86)) 10g(
�
-
with with probability probability ((11 - A) A) ' with with probability probability A A '
(6)
(6)
where proportion of where A A gives gives the the probability probability of of aa catastrophic catastrophic year, year, 8 6 gives gives the the proportion of the the population Hanson and 1 98 1 ), and VI reprep population destroyed destroyed in in aa catastrophe catastrophe ((Hanson and Tuckwell, Tuckwell, 1981), and v~ demographic stochasticity in aa single single female. female. The resents variance in resents the the variance in r due due to to demographic stochasticity in The notation was first l 987b, p. p. 2 1 9); Caughley 1 994) notation "V "v~" first employed employed by by Goodman Goodman ((1987b, 219); Caughley ((1994) I " was calls VI should be equal calls this this value value Vd~. Under Under the the usual usual assumptions assumptions v~ should be equal to to the the sum sum of is the of the the mean mean per per capita capita birth birth (b) and and death death (d) rates, rates, while while rd is the difference difference bNotice also also that is the the expected value of not ln(E In(E R). This This - d. Notice that rd is expected value of ln In R, E(ln E(ln R), not distinction caused alarm (see Lande, 993, on distinction has has caused alarm and and confusion confusion in in the the literature literature (see Lande, 11993, on Goodman, 987b, or 977, p. 163, on 975). Goodman, 11987b, or Turelli, Turelli, 11977, p. 163, on Feldman Feldman and and Roughgarden, Roughgarden, 11975). This is clarifi ed further ( 1 989, p. p. This distinction distinction between between growth growth rates rates is clarified further by by Caswell Caswell (1989, 2 1 3). 213). Equation real populations populations have Equation (6) (6) can can only only be be an an approximation, approximation, since since real have dis discrete 1 ) and crete sizes, sizes, while while Eqs. Eqs. ((1) and (6) (6) assume assume aa continuous continuous state state space space for for N. This This approximation interpre approximation will will do do little little damage damage as as long long as as we we make make common common sense sense interpretations of its consequences. must use use N = 1 1 as lower boundary tations of its consequences. For For example, example, we we must as aa lower boundary for usual diffusions for an an extant extant population, population, not not zero. zero. The The boundary boundary behavior behavior of of the the usual diffusions is less problematic 1 985, or is peculiar peculiar at at N = = 0 and and less problematic for for N = = 11 (see (see Gardiner, Gardiner, 1985, or Karlin Karlin and 98 1 , for and Taylor, Taylor, 11981, for discussions discussions of of boundary boundary behavior). behavior). Furthermore, Furthermore, when when aa population sense. The population drops drops below below one one female, female, it it is is not not viable viable in in any any biological biological sense. The upper ecting boundary." upper boundary boundary K is is aa "refl "reflecting boundary." A A good good deal deal of of subtle subtle ecological ecological thinking real thinking has has been been done done on on the the boundary boundary at at infinity, infinity, but but it it hardly hardly concerns concerns real populations. populations. It using a It is is easy easy to to simulate simulate popUlation population growth growth using using Eq.(6) Eq.(6) using a computer computer lan language below guage or or aa spreadsheet spreadsheet program. program. A A population population goes goes extinct extinct when when N goes goes below results can When N goes goes above above K, it it is is reflected reflected below below K. No No analytic analytic results can be be hoped hoped 11.. When for case, especially especially if N. If If A 8) and VI are small for in in the the general general case, if r d depends depends on on N. A (or (or 6) and ]21 are small enough, model is is an enough, then then the the pure pure environmental environmental stochasticity stochasticity model an adequate adequate approx approximation the study imation of of the the population population dynamics, dynamics, providing providing useful useful tools tools for for the study of of struc structured tured metapopulation metapopulation models. models.
Vdl•
r
rd
rd
N
N
N
rd
N.
N=
N
220
Patrick Patrick Foley Foley
III. ENVIRONMENTAL ENVIRONMENTALSTOCHASTICITY STOCHASTICITY A. Environmental Ceiling Model Environmental Stochasticity Stochasticityin a Ceiling The The most most practical practical approach approach to to modeling modeling environmental environmental stochasticity stochasticity is is to to try (Arnold, 1974; 1 974; Turelli, 977; Roughgarden, 979; try aa diffusion diffusion approximation approximation (Arnold, Turelli, 11977; Roughgarden, 11979; Gardiner, 985). The that Gardiner, 11985). The main main disadvantages disadvantages of of diffusion diffusion approximations approximations are are that they time intervals they model model aa discrete discrete state state space space and and discrete discrete time intervals with with aa continuous continuous state niceties state space space and and continuous continuous time time and and that that there there are are subtle subtle and and debatable debatable niceties to that analytic to the the pure pure mathematics mathematics of of diffusions. diffusions. The The greatest greatest advantages advantages are are that analytic results usually easier results are are usually easier to to obtain obtain and and that that the the literature literature on on applied applied diffusion diffusion theory 1 905) and theory is is large, large, embracing embracing Einstein's Einstein's model model of of Brownian Brownian motion motion ((1905) and re recent results results in in the the neutral and selectionist selectionist theories of molecular molecular evolution evolution (Kicent neutral and theories of ( Ki mura, 983; Gillespie, Gillespie, 11991). 99 1 ). Some following analysis be found mura, 11983; Some of of the the following analysis can can be found in in et al. 1 996a). more Foley, 11994; 994; Hanski Hanski et more extended extended form form in in earlier earlier papers papers ((Foley, al.,, 1996a). The natural logarithm of N The simplest simplest approach approach is is to to model model the the change change in in the the natural logarithm of N denoted stochastic differential equation is is denoted by by n. The The appropriate appropriate stochastic differential equation (7) (7)
dn = rddt + ~ r d W ( t ) .
The The drift drift term term rrdd and and the the diffusion diffusion term term Vr Vr we we have have met. met. Wet) W(t) is is the the standard standard Wiener or white noise noise process, N(O, dt). is very Wiener or white process, dW --- N(0, dt). If If Vr Vr is very small, small, the the solution solution to to Eq. Eq. (7) (7) approaches approaches the the deterministic deterministic result result �
(8) (8)
N t - - N o erdt.
A this exponential plateau of A population population ceiling ceiling at at N N = = K cuts cuts off off this exponential growth growth at at the the plateau of the carrying carrying capacity. capacity. In In the the more more familiar familiar logistic logistic growth growth model, size the model, population population size can model it As a technical can exceed exceed the the carrying carrying capacity; capacity; in in the the ceiling ceiling model it cannot. cannot. As a technical no distinction distinction point, has a point, since since the the diffusion diffusion has a constant constant diffusion diffusion term term Vr, Vr, there there is is no between Kloeden and 992, p. p. between the the Ito Ito and and Stratonovich Stratonovich interpretations interpretations ((Kloeden and Platen, Platen, 11992, 1157). 57). The = k) is reflecting boundary boundary and is an absorbing The logarithm logarithm of of K ((=k) is aa reflecting and 0 0 is an absorbing boundary boundary for for the the diffusion diffusion of of n. The The fullest fullest knowledge knowledge that that we we can can obtain obtain about about the the future future history history of of our our population, which probability density pen, t). The population, which starts starts off off at at n = = no, is is the the probability density p(n, The density p(n, satisfies the Kolmogorov Kolmogorov forward forward and and backward backward equations equations ((Karlin density pen, t) satisfi es the Karlin and 1 ; Gardiner, 985). The and Taylor, Taylor, 198 1981; Gardiner, 11985). The backward backward equation equation (KBE), (KBE),
ap(n, Op(n, t) Ot at
--- = =
2p(n, t) Vr ap(n, Vr a02p(n, Op(n, t) + - - ~2 + ~ , '
2 2
On2 an
---
On an
(9) (9)
is Foley, is especially especially useful useful ill in obtaining obtaining results results about about sojourn sojourn and and extinction extinction times times ((Foley, 11994; 994; Hanski 996a). Middleton 1 996) work Hanski et et al., al., 11996a). Middleton et et al. al. ((1996) work with with the the forward forward equa equation KFE, also develop related tion ((KFE, also known known as as the the Fokker-Planck Fokker-Planck equation) equation) to to develop related results results for for population population persistence persistence time. time. Functions Functions of of the the probability probability density density also also satisfy satisfy the ed forms. the forward forward and and backward backward equations equations in in appropriately appropriately modifi modified forms. Some Some eses-
1100
Local LocalEXTInction ExtinctionModels Models
221 221
pecially pecially useful useful functions functions can can be be studied studied in in this this way, way, such such as as the the probability probability of of reaching reaching carrying carrying capacity capacity before before going going extinct. extinct. The the following The expected expected time time to to extinction extinction Te(no) satisfies satisfies the following ordinary ordinary dif differential KBE (Gardiner, 985, p.138; p. 1 38 ; Ewens, 979, ferential equation equation which which follows follows from from KBE (Gardiner, 11985, Ewens, 11979, p. 20), p. 1120), - 1 1
= =
-vrd2Te(no) 2 dn 2
vr d 2Te (no ) dTe (no ) dTe(no) + - ~ + r rd --d ~ . · 2 2 dno dno
-
( ( 1 0 )0)
reflecting boundary Since k is Since is aa reflecting boundary and and 0 0 is is an an absorbing absorbing boundary boundary for for n, the the bound boundary ary conditions conditions are are
Te (O) = 0 Te(O) dTe (k) dT~(k) dno
((11) 1 1) = = 0O. .
Equations (9) and 1 0) can solved in several ways, ways, but the sojourn Equations (9) and ((10) can be be solved in several but the sojourn time time ap approach gives the most information population state proach gives the most information about about the the population state before before absorption absorption (extinction) (extinction) and and has has been been popular popular with with population population geneticists. geneticists. The The sojourn sojourn time, time,
ten; t(n; no no),), is is the the expected expected number number of of generations generations spent spent at at size size n before before the the popupopu ten; no )dn is time spent lation lation goes goes extinct, extinct, or or to to be be more more exact, exact, t(n; no)dn is the the expected expected time spent in in Maruyama, 11977; 977; Ewens, 979, pp. 1 20- 1 23). an size dn an interval interval of of size dn around around n ((Maruyama, Ewens, 11979, pp.120-123). Te for all Te is is the the sum sum of of the the sojourn sojourn times times for all n up up to to k,
Te (no) = fo k t(n, no) dn,
( 2) (12)
where where
t(n, no) = V r ~2( n ) fon tfl(x) dx
2_ no -I ( - (x ;, )) . = J 2
fjJ(x) ten, no t(n, no)) = = vv~6(n)do O(x) dx dx rfjJ(n) 0 rd
fjJ(x) qt(x) = exp e x p ( - 2 ~2~ rod Y dy
for for 0 0 < < n < < no
((13) 1 3)
for for no < < n < < k
((14) 1 4) ((15) 1 5)
.
=
If is easy If rd r~ = 0, 0, Te is easy to to obtain: obtain:
fjJ(x) qJ(x) = = 11
=
ten; t(n; no no)) = 2n/vr 2n/Vr
((16) 1 6) for for 0 0 < < n < < no
ten; t(n; no) = 2nolv 2no/Vrr for for no n 0 <
((17) 1 7)
2 n o ( k - no)
((18) 1 8)
Te(no) =
Vr
"2"
.
222 222
PatrickFoley Foley Patrick
If rd is is not not zero, zero, defi define r J vVrr to to get get If ne s = rd/ =
"'( ~(x) = ee --2sx x) = 2sx
((19) 1 9)
2sn 11 ee --2sn S(n) = = --Sen) 22ss
11 t(n; no) = = -(e --(e 2 2s sn n - 11) ten; no) )
rd Fd
-
(the (the scale scale measure measure which which we we discuss discuss later) later)
(20) (20)
for 0 0 < < n < < no for
(2 1) (21)
ee 22ssnn
2sno) for tten; ( n ; nno) o) = = ~(1 (1 - e e --2s'o) for no no<
rd Fd
Te(n~
1 = ~(e2sk(1 2srd
-
e -2sn~
-
(22) (22)
2sno).
The assumption assumption of of most most metapopulation metapopulation models models is is that that for for aa given given habitat habitat The patch, the extinction extinction rate rate is is constant. This assumption does not not quite quite hold for patch, the constant. This assumption does hold for environmental or or demographic demographic stochasticity, stochasticity, since since Te Te(no) a function function of of the environmental (no) is is a the population size Thus extinction is not not aa Poisson Midextinction is Poisson process process ((Foley, Foley, 1994; 1 994; Mid population size no. Thus dleton eett ai., al., 11996), though for for practical practical purposes purposes it it comes comes close. close. Since Since 996), though dleton k2 2 e2sk e 2sk - 2sk 2sk k 11 Te (k) = Te(k) = -, Vr 2s2k z Vr
(23) (23)
the extinction extinction rate rate e(k) follows the e(k) follows
e(k) I /Tee (K) (K) e(k) = = 1/T V Vrr
22s s 22kk22
kk2 e2sk e 2sk -
2
11 -- 2sk 2sk
(24) (24)
was noticed noticed by by Goodman Goodman When sk is extinction rate rate approaches 2, aas s was is small, small, the the extinction approaches Vr/k vJk2, When sk (1987b). approaches the ( 1 987b). As As sk sk becomes becomes large, large, e(k) e(k) approaches the form form
2
2r e(k) -~ �e2rZe-zsk 2sk e(k) = Vr Vr K-2s 2rdS K -2s,
(25) (25)
as was was pointed pointed out by Lande Lande (1993). ( 1 993). Hanski Hanski (1992a, ( 1992a, 1994b) 1 994b) assumed assumed the form as out by the form of of Eq. Eq. (25) (25) in in aa series series of of papers papers on on incidence incidence functions functions in in metapopulations metapopulations (Han( Han ski, ski, 1992a, 1 992a, 1994b), 1 994b), and and similar similar functions functions characterize characterize species-area species- area curves curves of of the the island island biogeography biogeography literature literature (MacArthur ( MacArthur and and Wilson, Wilson, 1967; 1 967; MacArthur, MacArthur, 1972; 1 972;
1100
Local LocalExtincfion ExtinctionModels Models
223 223
Brown 983). In Brown and and Gibson, Gibson, 11983). In this this literature literature K is is implicitly implicitly or or explicitly explicitly assumed assumed to be proportional to A, the area of the habitat patch or the island. to be proportional to the area of the habitat patch or the island.
B. Values Values of 'drd,, VrVr,, K, p, p, and V1 v~ Estimated Estimated from TIme Time Series Data There There are are many many unsolved unsolved problems problems in in the the estimation estimation of of extinction extinction parameters. parameters. value of over a The estimating rd is The simplest simplest approach approach to to estimating is to to take take the the mean mean value of r(t) r(t) over a time population sizes sizes (Dennis 99 1 ). The technical scruple is that time series series of of population (Dennis et et al., al., 11991). The technical scruple is that ret) is not independent identically random varia r(t) is not drawn drawn from from aa set set of of independent identically distributed distributed random variar values bound bles. The biological problem is that near the carrying capacity bles. The biological problem is that near the carrying capacity r values are are bound the other estimating rr dd by by regressing regressing rr on to to be be underestimates underestimates of of rd rd.' On On the other hand, hand, estimating on N ((Roughgarden, Roughgarden, 11979, 979, p. 306; Dennis Dennis and 994; J. J. E. E. Foley, Foley, P. Foley, Foley, and p. 306; and Taper, Taper, 11994; and seems often true with with territorial territorial S. Torres, Torres, unpublished) unpublished) will will overestimate overestimate rd if, if, as as seems often true animals, density dependence to K. Foley Foley ((1994) 1 994) animals, density dependence is is not not linear linear but but hits hits hardest hardest close close to discusses estimation of discusses the the estimation of rd rd,, Vp Vr, K, and and p p (the (the serial serial autocorrelation autocorrelation of of environ environmental simple, robust mental effects) effects) and and uses uses simple, robust but but nonoptimal nonoptimal parameter parameter estimators estimators on on time time series series including including checkerspot checkerspot butterflies, butterflies, grizzly grizzly bears, bears, mountain mountain lions, lions, and and wolves. More welcome. wolves. More rigorous rigorous estimation estimation procedures procedures would would be be welcome. Table Table II surveys surveys representative representative values values of of the the extinction extinction parameters. parameters. The The rd rdoO estimator is uses the mean estimator is the the arithmetic arithmetic mean mean of of the the rr values. values. The The rdl estimator estimator uses the mean rr from quartiles of population sizes sizes (the is to avoid rr values from the the bottom bottom three three quartiles of population (the idea idea is to avoid values is the in rr over time near ceiling). The near the the ceiling). The Vr Vr estimator estimator is the variance variance in over the the course course of of the the time series. series. This This estimate estimate seems seems rather rather robust robust based based on on simulations, simulations, but but it it may may be be biased by olda ((1978) 1 978) also also reports reports an biased by error error in in census census taking. taking. W Wolda an abundance abundance of of Vr Vr values 1 8 to temperate and insects. The The values (ranging (ranging from from 0.0 0.018 to 0.642) 0.642) for for temperate and tropical tropical insects. estimate is just the serial autocorrelation between consecutive rr values values (Chat estimate for for p p is just the serial autocorrelation between consecutive (Chatfield, 989). K is the maximum maximum value the time but other field, 11989). is taken taken as as the value in in the time series series of of N, but other estimators itself varies varies from estimators may may be be preferable, preferable, especially especially if if K itself from year year to to year year or or if if outbreak populations temporarily Foley, 11994). 994). outbreak populations temporarily shoot shoot above above the the normal normal ceiling ceiling ((Foley, The The extinction extinction predictions predictions are are most most sensitive sensitive to to inaccuracy inaccuracy in in estimating estimating rd r d,, 1 990) intensive intensive regrettably trickiest to Consider Menges regrettably the the parameter parameter trickiest to estimate. estimate. Consider Menges'' ((1990) study 16 furbished populations. His His first study of of 16 furbished lousewort lousewort (Pedicularis (Pedicularis furbishiae) furbishiae) populations. first year 0.64, re year of of data data on on A A (our (our R) is is distinctly distinctly discouraging discouraging (R = = 0.68, 0.68, 0.76, 0.76, 0.64, respectively) Menges, 1990, 1 990, Table spectively) for for all all three three of of the the populations populations studied studied ((Menges, Table 5). 5). The The next because these these three next 22 years years of of R values values look look better, better, partly partly because three populations populations im im.27, 11.03, .03, 0.98) 0.98) and now has prove prove their their growth growth rates rates (R = = 11.27, and partly partly because because he he now has . 1 8) . Over data data for for l133 mostly mostly superior superior populations populations (they (they show show aa mean mean R of of 11.18). Over all all values gives estimate, 33 years, years, an an arithmetic arithmetic mean mean of of yearly yearly mean mean ret) r(t) values gives aa negative negative rd estimate, perhaps rst year. weighted perhaps because because he he studied studied worse worse populations populations in in the the fi first year. An An un unweighted look much lousewort, since since Menges Menges had mean mean of of all all rr values values would would look much better better for for the the lousewort, had more from each population could considered a more data data for for better better years! years! If If data data from each population could be be considered a realization of identical random variable, much confidence limits limits could could realization of an an identical random variable, much tighter tighter confidence patches may be be put put around around an an rd rd,, but but some some habitat habitat patches may be be sinks. sinks.
N
TABLE II Estimated EstimatedValues Valuesofof Extinction ExtinctionParameters Parameters TABLE TT
/dO rdO
/dl rdl
v, I~r
P P
Euphydryas editha, editha, JRC JRC Euphydryas Euphydryas editha, editha, JRH JRH Euphydryas Moth sp. sp. I1,, Rothamsted Rothamsted Moth
26 26 26 26 113 3
0.002 0.002 .052 �- 00.052 0.048 0.048
0.307 0.307 0.126 0. 1 26 0.094 0.094
11.46 .46 0.84 0.84 0.564 0.564
0.24 �- 0.24 .32 �- 00.32
Moth sp. sp. 2, 2, Rothamsted Rothamsted Moth
I11 I
0.083 0.083
0.768 0.768
Moth sp. sp. 3, 3, Rothamsted Rothamsted Moth
110 0
0.154 0. 1 54
Moth sp. sp. 4, 4, Rothamsted Rothamsted Moth
110 0
Moth sp. sp. 5, 5, Rothamsted Rothamsted Moth
Species Species
Metapeira orb orb spider, spider, Metapeira Bull Cay, Cay, Bahamas Bahamas Bull Metapeira orb orb spider, spider, Metapeira Longest Cay Cay Longest Metapeira orb orb spider, spider, Metapeira Dichotomous Cay Cay Dichotomous Metapeira orb orb spider, spider, Metapeira Cay 405 405 Cay Furbished lousewort lousewort Furbished Grizzly bear, bear, Yellowstone Yellowstone Grizzly Wolf, Alaska Alaska Wolf, Blackbird Blackbird Wheatear Great tit Dutch Great Great tit Marley, UK Great Great tit Dean, UK owl Tawny owl
K K
kk
/v,r So So = --- rdO rdo/V
8.89 8.89 7.6 7.6
0.00 1 16 0.00116 �-0.0619 0.061 9 0.085 11 0.08511
0.2 1 027 0.21027 0. 15 0.15 00.16667 . 1 6667
11.61 .6 1
0.05 1 55 0.05155
0.47702 0.47702
0.509 0.509
2.59 2.59
0.05946 0.05946
0. 1 9653 0.19653
00.101 .101
0.093 0.093
0.3511 0.35
0.28775 0.28775
0.26496 0.26496
99
00.117 .1 1 7
0.369 0.369
0.462 0.462
0.25325 0.25325
0.7987 0.7987
44
0.545 0.545
0.971 0.971
234.5 234.5
5.46 5.46
0.561 28 0.56128
44
0.508 0.508
3.025 3.025
1172 72
55.15 .15
0. 1 6793 0.16793
Schoener 1 995) Schoener and and Spiller Spiller ((1995)
44
0.659 0.659
2.103 2. 103
125.5 125.5
4.83 4.83
0.3 1 336 0.31336
Schoener 1 995) Schoener and and Spiller Spiller ((1995)
44 44 28 28 I11I 116 6 116 6 116 6 116 6 116 6 12 12
- 0.588 � - 0.062 � 0.003 0.003 0.03 0.03 00.17 .17 0.08 0.025 0.109 0. 1 09 0.007 0.007 0.053 0.053
3.207 3.207 0.128 0. 1 28 0.011I O.oI 0.18 0. 18 0.174 0. 1 74 0.487 0.487 0.265 0.216 0.2 16 0.345 0.004 0.004
112 2
2.48 2.48
57.97 57.97 2592 2592
4.06 4.06 7.86 7.86
� 0. 1 833 -0.1833 � - 0.4844 0.4844 0.28 1 82 0.28182 0. 1 6667 0.16667 0.97701 0.97701 00.16427 . 1 6427 0.09434 0.09434 0.50463 0.50463 0.02029 0.02029 113.25 3.25
Schoener 1 995) Schoener and and Spiller Spiller ((1995) Menges 1 990)" Menges ((1990)" 99 1 ) Dennis Dennis et et al. al. ((11991) Young 1 944) Young and and Goldman Goldman ((1944) Diamond 1984) Diamond ((1984) Diamond 1 984) Diamond ((1984) Perrins 1 965) Perrins ((1965) Perrins 1 965); Lack 1 966) Perrins ((1965); Lack ((1966) Perrins 1 965) Perrins ((1965) Southern 1 970) Southern ((1970)
0.018 O.o I8 0.175 0. 1 75
0.189 0. 1 89 0.211 0.2 11 0.2511 0.25 0.007 0.007
space and time for 16 ""vr v, Averaged over space 16 populations
- 0.46 � - 00. . 1188 �
7259 7259 11998 998
1 1
1145.5 45.5 85.63 75.19 75. 19 32.14 32. 14
4.98 4.98 4.45 4.45 4.32 4.32 3.47 3.47
SI S1 = - - rd r d l,/ /v, I) r
11.62727 .62727 0.97222 0.97222
0.7 1 321 0.71321 0.97685 0.97685 0.72754 0.72754 11.75 .75
Source Source
Harrison 1 99 1 )) Harrison et et al. al. ((1991 Harrison 99 I )) Harrison eett al. al. ((11991 I.I. Hanski Hanski (personal (personal communication) communication) I.I. Hanski Hanski (personal (personal communication) communication) I.I. Hanski Hanski (personal (personal communication) communication) I.I. Hanski Hanski (personal (personal communication) communication) I.I. Hanski Hanski (personal (personal communication) communication) Schoener 1 995) Schoener and and Spiller Spiller ((1995)
1100
Local LocalExtinction Extinction Models Models
225 225
Or 1988a) survey Owl demography. Or consider consider Lande' Lande'ss ((1988a) survey of of Northern Northern Spotted Spotted Owl demography. 0.96 :±: 0.03 (or equivalently rd is neither His estimate His estimate A h = = 0.96 ___ 0.03 (or equivalently r~ = = - 0 0.04) . 0 4 ) is neither signifi significantly .0 (rd 0.0), which Pacific North cantly different different from from 11.0 (rd = - 0.0), which timber timber interests interests in in the the Pacific Northwest of America would would like like to nor from obtained by by west of North North America to believe, believe, nor from rd rd = = - 0 0.01 . 0 1 obtained 1 984). Alvarez-Buylla 1 99 1 ) longterm Forsman et longterm census census data data ((Forsman et aI. al.,, 1984). Alvarez-Buylla and and Slatkin Slatkin ((1991) provide provide aa recent recent review review of of methods methods to to set set confidence confidence limits limits around around growth growth rate rate estimates obtained from structure models, these models estimates obtained from typical typical ageage-structure models, but but these models assume assume no no density density dependence. dependence. In a metapopulations distinguishing sinks sinks In a metapopulations the the rd rd problem problem becomes becomes one one of of distinguishing from complexities of from sources, sources, aa problem problem made made more more difficult difficult by by the the complexities of migration migration (Watkinson 995). In vexing (Watkinson and and Sutherland, Sutherland, 11995). In general, general, the the uncertainty uncertainty of of rd rd is is aa vexing problem problem for for deterministic deterministic analysis, analysis, which which depends depends entirely entirely upon upon rd rd,, and and only only slightly less problematic slightly less problematic for for the the environmental environmental stochasticity stochasticity analysis, analysis, in in which which the the random random fluctuations fluctuations measured measured by by Vr Vr can can swamp swamp the the effect effect of of rd rd.' Two assume that Two possible possible ways ways to to estimate estimate Vv~I suggest suggest themselves; themselves; either either assume that 1 987b) or regress Yarer) l/N. The method = b + + d d (Goodman, (Goodman, 1987b) or regress Var(r) on on 1/N. The second second method VvlI = demands lot of to my rst method demands aa lot of data data and and has has not not been been done done to my knowledge. knowledge. The The fi first method assumes a process with assumes a standard standard birth-death birth-death process with independence independence across across time time and and among Feller, 11971). 97 1 ). The -death process process theory theory assumes, among individuals individuals ((Feller, The birth birth-death assumes, for for example, does not example, that that competition competition does not operate operate and and that that individuals individuals do do not not compen compensate sate for for demographically demographically bad bad periods periods (b is is the the instantaneous instantaneous birth birth rate, rate, d d the the corresponding Recall that corresponding death death rate). rate). Recall that aa demographic demographic accident accident is is not not necessarily necessarily a cases, a a failure failure to to accumulate accumulate energy energy reserves reserves or or to to survive survive hardship. hardship. In In many many cases, a female may do female who who fails fails to to reproduce reproduce early early in in the the season season may do so so later later or or save save the the extra next year. compensation exists, extra resources resources for for next year. We We know know that that such such individual individual compensation exists, especially especially in in perennial perennial plants plants and and large large vertebrates; vertebrates; but but we we need need to to discover discover em empirically pirically how how far far it it will will go go to to diminish diminish demographic demographic stochasticity. stochasticity. Competitors Competitors may may also also take take opportunistic opportunistic advantage advantage of of the the demographic demographic accidents accidents of of their their un unVI ranges ranges from rd l to fortunate holds, then fortunate neighbors. neighbors. If If no no such such compensation compensation holds, then v~ from IIrdl to several rd l . Thus to environenviron several times times IIr~l. Thus Y T= = VV~/Vr, the relative relative effect effect of of demographic demographic to I /V" the mental single female, range in to about 1. mental stochasticity stochasticity for for aa single female, will will range in practice practice from from 0 0 to about 1. A A local local population's population's extinction extinction rate rate due due to to environmental environmental stochasticity stochasticity de depends rough pends on on all all these these parameters parameters and and the the initial initial population population size size No No,, but but aa rough zero and prediction prediction can can be be obtained obtained by by assuming assuming that that rr dd and and p p are are close close to to zero and that that No is is close 1 8) implies implies that that Te = ((log log K)2/Vr K)2!vr .. Extinction close to to K. Then Then Eq. Eq. ((18) Te = Extinction param param' s ranging eters give To 30 1 0 years years for for the eters for for populations populations of of Table Table 11 give Te'S ranging from from 3010 the tawny tawny owl near years for the Metapeira Metapeira orb spider of in the Bahamas. owl near Oxford Oxford to to 2 2 years for the orb spider of Cay Cay 405 405 in the Bahamas. The value is is 54 years for Marley, England. England. So we see The median median value 54 years for the the great great tit tit in in Marley, So we see that that range from to 0.5 populations per year. typical typical local local extinction extinction rates, rates, e, range from 0.0003 0.0003 to 0.5 populations per year. Healthy populations which can raise Healthy populations will will usually usually have have positive positive rd r~ values values which can raise persistence times greatly. much greater one, T(K) pro persistence times greatly. When When sk sk is is much greater than than one, T(K) becomes becomes proportional shown in of Table have somewhat portional to to K2s K 2s as as shown in Eq. Eq. (25). (25). The The populations populations of Table 11 have somewhat dynamics influenced influenced substantially substantially by intermediate intermediate sk sk values, values, with with dynamics by both both chance chance (vr) and positive population popUlation growth. (Vr) and the the upward upward pressure pressure (rd (r~)) of of aa positive growth. -
-
226 226
Patrick PatrickFoley Foley
C. Environmental Stochasticify Explain Species-Area C. Environmental StochasticityMay Explain Species-Area Curves Curves MacArthur 1 967, p. the speciesarea curves for MacArthur and and Wilson Wilson ((1967, p. 8) examined examined the species-area curves for several number of several archipelagoes. archipelagoes. The The relationship relationship between between the the number of species species S S on on an an island often fits fits the island and and island island area area A often the equation equation S = = cA", cA z, S
(26) (26)
where region under is where c depends depends on on the the taxon taxon and and the the region under study, study, and and zz is is aa value value that that is consistently 1 5 for consistently close close to to 0.3 0.3 for for islands, islands, but but only only 0. 0.15 for continental continental habitat habitat patches patches that MacArthur and Wilson, 11967; 967; Diamond Diamond and May, that are are not not very very isolated isolated ((MacArthur and Wilson, and May, 11976). 976). The island depends on colonization The number number of of species species on on an an island depends on colonization rates rates and and community which affects affects extinction extinction rates, rates, but much of the community community structure, structure, which but much of the community influence effects on influence can can be be subsumed subsumed under under its its effects on rd rd,, K, and and Vr• Vr. Consider no ec Consider first first the the "disconnected "disconnected community" community" in in which which species species have have no ecological then ological interactions. interactions. If If the the species species colonizing colonizing an an island island are are independent, independent, then community depending on constant colonization community assembly assembly is is aa simple simple process process depending on constant colonization (m) and and extinction extinction (e) rates. rates. With With aa continent continent to to assure assure colonization, colonization, the the probability probability of species incidence incidence on Hanski, 11992a). 992a). of aa particular particular species on an an island island is is ((Hanski, p = =
P
m . m ++ e e. m
--
(27) (27)
If is if If m is is small small compared compared to to e, that that is if the the islands islands are are depauperate depauperate compared compared to to the the continent, continent, then then we we have have p p
~ mTe mTe,'
=
(28)
where should be to A. Referring Referring to to Eq. Eq. (25), where Te depends depends on on K which which should be proportional proportional to (25), we we then then obtain obtain approximately approximately p = 2rdsmK 2s.
(29) (29)
persisting species Since Since S, S, in in this this disconnected disconnected community, community, is is just just the the sum sum of of all all persisting species number of from which is for S from the the continent continent which is aa reservoir reservoir for Sma species, the the expected expected number of maxx species, species es species satisfi satisfies Smax Smax
ESS = = L ~ Pi Pi.E
ii=1 =i
(30) (30)
If for each the island, If we we assume assume that that K and and s and and m are are similar similar for each species species on on the island, we we obtain obtain approximately approximately ES ~ 2rdSmSmax K2s,
(3 1) (31)
which area curve 1 ) with with which has has the the form form of of the the standard standard speciesspecies-area curve (4. (4.1)
z ~ 2s = 2 r~. Vr
(32) (32)
10 1 0 Local Locol Extinction Extinction Models Models
227 227
This result result suggests suggests that that if if environmental environmental stochasticity stochasticity sets sets extinction extinction rates rates and and This 0. 1 5 on on if communities communities are are disconnected, disconnected, then then typical typical ss values values are are close close to to ss == 0.15 if islands. islands. Consider next next the the "zero-sum "zero-sum community" community" of of competitors competitors who who share share aa comcom Consider capacity Doubling the species number on an island might mon carrying capacity Ktota~. Doubling the species number on an island might carrying mon halve the the effective effective carrying carrying capacity capacity for for each each species species on on average average in in this this guild. guild. halve Then Then
Ktotal'
p=c
p ~- c and on on aa typical typical island island and
(Ktotal) 2S
S ' ,
SI + 2s = cSmaxK2s
(34) (34)
S 1 + 2s ~ cSmaxg2S S ~
(33) (33)
(cSmax) 1/1 +2sg2s/l+2s,
K
(35) (35)
again us the form of Eq. (26) if K is is proportional proportional to to area. area. Given Given the two again giving giving us the form of Eq. (26) if the two extremes the disconnected the zero-sum we get get extremes of of the disconnected and and the zero-sum community, community, we 2s
2 2 ss
--
1I + + 2 s 2s ' '
(36) (36)
or equivalently or equivalently
zZ zZ - ::5 --< s s ::5 _< .. 2 2 ( 1 - z) z) 2 2(1 -
(37) (37)
which 1 5 ::5 which means means that that typically typically 0. 0.15 -< ss ::5 -< 0.22 0.22 on on islands. islands. The The power power form form of of the the speciesarea curve calculated ss values values are species-area curve is is consistent consistent with with the the theory, theory, and and the the calculated are plausible as plausible as can can be be seen seen from from Table Table I. I. On On archipelagoes archipelagoes in in "relaxation" "relaxation" with with no no migration migration among among islands, islands, the the in incidence cidence patterns patterns are are entirely entirely decided decided by by extinction extinction dynamics. dynamics. For For example, example, ' s ((1978) Brown 1 978) data Brown's data on on montane montane mammals mammals of of the the Great Great Basin Basin Ranges Ranges suggest suggest similar similar values values as as the the MacArthur MacArthur and and Wilson Wilson table. table. Montane Montane birds birds show show aa lower lower z than than do do mammals, mammals, and and Brown Brown reasonably reasonably argues argues that that this this is is due due to to the the added added infl uence of influence of colonization. colonization. It It is is also also possible possible that that his his birds birds may may have have lower lower ss values, values, or or less less community community connectivity. connectivity. The area relation The above above ideas ideas touch touch the the surface surface of of the the literature literature on on speciesspecies-area relation' s ((1975b, ships. 1 975b, 11984), 984), the ships. For For other other insights insights see see Diamond Diamond's the Gilpin Gilpin and and Diamond Diamond ' s ((1992a, ((1976, 1 976, 11981), 98 1 ), and 1 992a, 11994b; 994b; this and Hanski Hanski's this volume) volume) series series of of papers papers on on in incidence cidence functions. functions. The The form form of of Eq. Eq. (26) (26) is is traditionally traditionally explained explained in in terms terms of of ' s ((1962) Preston 1 962) canonical Mac Preston's canonical lognormal lognormal distribution distribution for for species species abundances abundances ((MacArthur, 972; May, 975), but Arthur, 11972; May, 11975), but the the canonical canonical distribution distribution still still needs needs explanation. explanation. There There may may well well be be aa deep deep connection connection between between the the theory theory of of this this section section and and ' s observations. Preston Preston's observations.
PatrickFoley Foley Patrick
228 228
IV. MODElS MODELSWITH WITH DEMOGRAPHIC DEMOGRAPHICAND AND ENVIRONMENTAL ENVIRONMENTALSTOCHASTICITY STOCHASTICITY IV. A. Adding AddingDemographic Demographicto to Environmental EnvironmentalStochasticify Stochasticity A. The environmental environmental stochasticity stochasticity models models developed developed in in the the previous previous sections sections The are mathematically mathematically tractable tractable because because the the diffusion diffusion function function Vr Vr is is constant. constant. Small Small are and large large populations populations are are all all under under the the same same hand hand of of nature. nature. To To add add demographic demographic and stochasticity to to the the model model means means uglier (at least least more more involved) involved) mathematics mathematics stochasticity uglier (at (Leigh, 11981). The diffusion diffusion coeffi coefficient is now now aa function function of of popUlation population size, and (Leigh, 98 1 ) . The cient is size, and ' s ((1977) I employ 1 977) recommendation. employ the the Ito Ito interpretation interpretation on on Turelli Turelli's recommendation. Following Following (6), we we must must replace replace Vr Vr by by Vr Vr + + Vv~/N. This leads leads to to I /N. This O(x) = exp ( _ 2f0 x
ra
V r "31"
Vl/N dn
)
1 == exp (( _ 2 r~Vr Lx)(x 1 + (Vl/Vr) e-e-"n dn )) \ == exp (( - 2 ;,r~Vr [x + log(1 + yyee-x)-X ) - log(1 + y)]])! (1 y)2s )2S' (U + y y)2s' (38) where y = V1/Vr, SS = The integral was evaluated using Gradshteyn and Ryzhik (1980, p. 92). Note that if y is zero, ~(x) reverts to the earlier form in exp
-2 �
exp
-2
1
dn
O I + (vJvr)
Vr
- 10g( 1 + y) ]
[x + 10g( 1 +
+ ,)/)2s (1 + (ex +
(38)
VI /V" = rd/vr where y F d / V r.• The integral was evaluated using Gradshteyn and Ryzhik ( 1980, p. 92). Note that if y is zero, !/I(x) reverts to the earlier form in Eq. 19). The integrate in in closed closed form. Eq. ((19). The new new !/I(x) ~(x) is is apparently apparently hard hard to to integrate form. Approxi Approximations are required required for for further exploration. Karlin and Taylor mations are further exploration. Karlin and Taylor (1981, ( 1 98 1 , p. p. 194) 1 94) discuss the use scale function discuss the use of of the the scale function =
= 1n
Sen) = f0 n !/I(x) S(n) ~(x) dx dx
(39) (39)
and the and the speed speed function function
=
men) = m(n)
1
(40) (40)
------
(Vrr (v
+ vl/exp(n))tp(n) vJexp(n» !/I(n) "[-
to obtain obtain more more insight insight into into the the diffusion diffusion process. process. (Their (Their formulas formulas are are more more general general to than ours). ours). As As already already suggested, suggested, Eq. Eq. (39) (39) may may not not be be integrated integrated in in closed closed form. form. than From From Eq. Eq. (24), (24), we we see see
= e-2sx ( ye-X)-2' 2sT e-2sx ( + -l+y
qKx) = e-2"x( 11 ++ -'Ye~:~'~-:s 1I ++7 Y J
!/J(x)
=
e -2sx
11 @ ~
2sy X
1 + Y
x "31+
sy(2sy 1) s y ( 2 s y -- 1)
(( 1l ++3 , )y)22
2))
Xx2
(4 1 ) (41)
''
10 1 0 Local local Extinction Extinction Models Models
229 229
')' 1I [ I ( I) ] " ( s n +sn~ )+] Sen) 1I -2se -2sn ++ -2 I1 + 3"s- [ ~-2- e-- 2es-2sn 2s')';/. ]e 1 ++2 3 ' [ 1 - ( 1 + ( = 2s(1 (42) (42) 2:(1 +:�3')) [ I - I + 1I �+ ~3"jl s')') e -2sn ] " Notice that that as as nn gets gets large, large, S(n) Sen) approaches approaches the the value value Notice 2')' 1I ++ 23' (43) (43) S((0) == 2s(1 S(~) 2s0 ++ 3')')') and that when when nn is is very very small small and that (44) S(n) (44) Sen) ~ n. In fact fact the the critical critical value value of of n n turns turns out out to to be be In 2')' 1I ++ 23' (45) (45) S(oo) = 2s(1 nc == S(oo) nc + 2s(1 + ')'3')) For below n Sen) follows follows Eq. (44), and for n n somewhat somewhat above above it, it, For nn somewhat somewhat below nc, Eq. (44), and for O ' S(n) Sen) S(n) ~ S(oo). S(~). This This plateau-like plateau-like shape shape of of Sen) S(n) is is common common to to any any model model (demo (demographic graphic and/or and/or environmental) environmental) when when rd > > O. 0. This This makes makes the the critical critical value value nc (and (and Nc) ofof some some interest. interest. an equation which begins The The sojourn sojourn time time near near k starting starting from from k is is given given by by an equation which begins as as aa rewritten rewritten version version of of Eq. Eq. (13), ten t(n;; kk)) = = 2m(n)S(n) n e2sn 22(1 ne2S, ( 1 + ')'e-n)2s (46 for 0 < < n < < nc (46)) for Vr Vr (1 + ')')3')2s2s 2sn I + 2')'0 ee2s'l 23'(1 + ')'e -n)2s - 1 for for nc < < n < < k, (47) 2s + Vr S (1 +-Jr-')')3')2s+1 Vr which which is is continuous continuous and and piecewise piecewise differentiable. differentiable. Since Since almost almost all all of of the the sojourn sojourn time time is is concentrated concentrated above above nc if if kk is is larger larger than than nc, we we get get 2s - (enc + ')') 3')2s2s (1 + 2')' 23')) _ ((eekk ++ ')')3')2s_ Te k) Te ((k) ~ fn k ten, t(n; kk)) dn 2Srd + 3')2s+1' ')' (1 2srd + )2 I ' (48) fknc . which which may may be be improved improved by by adding adding in in the the integral integral of of the the sojourn sojourn time time below below nco nc. If If rd is is zero, zero, aa simpler simpler calculation calculation than than Eqs. Eqs. (38)-(48) (38)-(48) gives gives k2 (_ 1_) . (49) (49) Te (k) = 1 + Vr 1 y')' This This result result shows shows the the most most exaggerated exaggerated influence influence of of demographic demographic stochasticity stochasticity on k) since on Te Te((k) since when when s is is large large populations populations hover hover around around K. K. When When s is is close close to to
which we we get get by by expanding expanding the the second second product product around around xx == 0. O. Thus Thus which -
S(n) ~=
e -2sn
3"
')'S
•
=
=
-I + 3"e-n) 2s-I
=
+ 3'e-n) 2s-1
S
I
(enc _+_
_
s+
c
Te(k)
=
-
Vr
230 230 TABLE TABLE IIII
Patrick Patrick Foley Foley Effects and y Size Effects of of ss and 3' on on N" N,, the the Critical Critical Population Population Size 'Y
s s
0 0
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
I1
1.2 1.2
1.4 1.4
0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0. 0.11 0. 15 0.15 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1
22026 22026 4 1 60 4160 1265 1265 5518 18 259 259 148 148 28.0 28.0 12.2 12.2 5.3 5.3 3.5 3.5 2.7 2.7 2.3 2.3 2.0 2.0 11.9 .9 11.7 .7 1.6 1.6
1116619 166 1 9 116684 6684 4 1 60 4160 1468 1468 653 653 341 341 48.9 48.9 118.5 8.5 7.0 7.0 4.3 4.3 3.2 3.2 2.6 2.6 2.3 2.3 2.1 2.1 11.9 .9 11.8 .8
3835 18 383518 44994 44994 9737 9737 3089 3089 11265 265 6 19 619 72.7 72.7 24.9 24.9 8.5 8.5 5.0 5.0 3.6 3.6 2.9 2.9 2.5 2.5 2.2 2.2 2.0 2.0 11.9 .9
936589 936589 94687 94687 118424 8424 5398 5398 2077 2077 968 968 97.8 97.8 331.1 1.1 9.9 9.9 5.6 5.6 4.0 4.0 .1 33.1 2.7 2.7 2.4 2.4 2. 2.11 2.0 2.0
18756 10 1875610 1168896 68896 30256 30256 8331 8331 3055 3055 1370 1370 123 123 37.0 37.0 111.1 1.1 6. 6.11 4.2 4.2 3.3 3.3 2.8 2.8 2.5 2.5 2.2 2.2 2.1 2.1
326901 32690177 268337 268337 44994 44994 111790 1 790 4 1 60 4160 1808 1808 148 148 42.5 42.5 112.2 2.2 6.5 6.5 4.5 4.5 3.5 3.5 2.9 2.9 2.6 2.6 2.3 2.3 2.1 2.1
55150197 1 50197 391910 391910 62253 62253 115664 5664 5355 5355 2269 2269 173 173 47.6 47.6 113.1 3.1 6.9 6.9 4.7 4.7 3.6 3.6 3.0 3.0 2.6 2.6 2.4 2.4 2.2 2.2
7521930 7521930 537371 537371 8 1 595 81595 119847 9847 66 10 6610 2743 2743 196 196 52.4 52.4 14.0 14.0 7.2 7.2 4.9 4.9 3.7 3.7 3 .1 3.1 2.7 2.7 2.4 2.4 2.2 2.2
zero, for demographic zero, population population sizes sizes more more frequently frequently fall fall in in the the danger danger zone zone for demographic stochasticity. provides insight insight into into the the parameter stochasticity. Still Still Eq. Eq. (49) provides the effect effect of of the parameter y. y. If If y uctuate into y is is 11 and and s is is close close to to zero zero (so (so that that population population sizes sizes fl fluctuate into the the demographic demographic risk zone), extinction extinction rates risk zone), rates will will be be doubled. doubled. Defi ne a colonization as Define a successful successful colonization as one one that that produces produces aa population population that that reaches carrying capacity it goes goes extinct. this suc reaches carrying capacity before before it extinct. Then Then the the probability probability of of this success is cess is
s
S(n) S(n) P (n) - Pk(n) S(k) ,' k
-
(50) (50)
S(k)
where, probability of where, to to be be precise, precise, P Pk(n) is the the probability of reaching reaching k before before 0, starting starting at at n k (n) is (Ewens, 979, p. p. 1119; 1 9; Karlin Taylor, 11981, 98 1 , p. p. 1195). 95). Equations Equations (43) (Ewens, 11979, Karlin and and Taylor, (43) and and (44) (44) then insights. First, then lead lead to to two two major major insights. First, populations populations greater greater in in size size than than Nc Nc are are fairly fairly sure success, and sure of of success, and second, second, populations populations below below Nc Nc can can expect expect success success in in propor proportion log N. tion to to log N. Pure Pure environmental environmental stochasticity stochasticity implies implies y 7 = 0, so so Eq. Eq. (45) becomes becomes
nncc
= -. 2s 1
-
2s
=
(51) (5 1)
Table Nc should Table IIII gives gives critical critical values values of of Nc Nc for for ranges ranges of of typical typical ss and and y y values. values. Nc should
1100
local Extinction Models LocalExtinction Models
231 231
be be interpreted interpreted for for aa particular particular population population in in this this way: way:
Pk(no) ~" no k
if No < K < Nc
(52) (52)
Pk(no)~n~
if N o < N c < K
(53) (53)
if Nc < No < K.
(54) (54)
nc
Pk(no) ~ 1
A A nice nice feature feature of of the the critical critical population population size size concept concept is is that that it it does does not not depend depend on populations. on K, but but only only on on ss and and ,}" y, facilitating facilitating comparisons comparisons among among populations.
B. Demographic DemographicStochasticity StochasticityAlone Alone Models best analyzed with dis Models involving involving only only demographic demographic stochasticity stochasticity are are best analyzed with discrete 1 967), Nisbet Nisbet and crete models models as as done done by by MacArthur MacArthur and and Wilson Wilson ((1967), and Gurney Gurney ((1982), 1 982), Talent 1 990), and 199 1 ) . Real Talent ((1990), and Renshaw Renshaw ((1991). Real populations populations surely surely undergo undergo some stochasticity models models are are some environmental environmental stochasticity, stochasticity, so so pure pure demographic demographic stochasticity mainly sake of of completeness mainly of of theoretical theoretical interest. interest. Still Still for for the the sake completeness and and comparison, comparison, we Assume that that vr we employ employ our our usual usual diffusion diffusion methods methods for for the the analysis. analysis. Assume Vr = = 0 0 and and define define Sd Sd = rd/vr r d / V l . • Then Then =
(55) (55)
ff/(X) = e -2sd(e~-l)
S(n) = e2s~(E1 ( - 2So en) --
=
S(n) ~- n S(n)
) -(
E1 (-- 2So))
(56)) (56
for for 0 0 < < n < < nc nc
(57)) (57
for for nncc < < n < < k
(58 ) (58)
for 0 0 < < n < < nnoo for
(59) (59)
for N Noo < < n < for
(60)) (60
if /2sd < if l1/2Sd < k.
(61 (6,1))
S(n) ~ e2s~E1 (2Sa) = In ~-ln
t(n; no) =
In(2» ln(2)) 11 + + 2sd 2So /
2 --ene2sd(en-1)S(n)
V1
t(n; no) =
2
-- ene 2sd(e"-l)S(nO) Vl
t(n; k) ~
2
- - e n e 2 s d e " E 1 (2So) V1
t(n; k)
-- ~
1
2rd
ene2sd(e " -
1)
232
Patrick Foley Foley Patrick
The critical critical nc is is here here The
( --)
n c ~ e2s~E1 (2Sd)
= In In
( l n ( 2 )In(2» ) 11 + + 2sd 2Sd
(62) (62)
and the the probability probability of of success success is is given given by by and
P e kk((n) n)
= S(n)/S(k) -- S ( n ) / S ( k )
4Sd(n--Sd n2)
for0
(63) (63)
as as long long as as 11 < < 2sddk. 2sddk. Equations Equations (56) (56) and and (58) use use exponential exponential integral integral functions, functions, special functions functions of of mathematical mathematical physics physics (Olver, (Olver, 11974, p. 40). The The sojourn sojourn time time special 974, p. of Eq. Eq. (6 (61) can be be integrated integrated to to give give of 1 ) can e 2s(ek-1)
Te(k) ~
1
--
if 1/2Sd < k.
4rdSd
(64)
ROBUSTNESSOF THE MODEL MODEL V. ROBUSTNESS OF THE A. Aufocorrelafed AutocorrelatedEnvironmenfal EnvironmentalSfochasficify Stochasticity A. Diffusion models of environmental environmental stochasticity stochasticity traditionally Diffusion models of traditionally assume assume that that year to year year fl fluctuations in the the environment environment are are not autocorrelated. In year to uctuations in not autocorrelated. In the the real real world, assumption fails. fails. Foley a corrected corrected value value of of v" Vr, the the world, this this assumption Foley ((1994) 1 994) derived derived a effective effective environmental environmental variance variance Vre vre ,' that that satisfies satisfies the the equation equation
-
l1 ++ p p vre = = Vr Vr -Vre 1 - p ,, 1
p
(65) (65)
is the effects between between consecutive consecutive years. years. where where p p is the autocorrelation autocorrelation of of environmental environmental effects An adjustment has An equivalent equivalent adjustment has been been developed developed but but rarely rarely used used in in population population gege netics Gillespie and and Guess, Guess, 1978). Turelli (1977, 1 99 1 ; Gillespie 1 978). Turelli ( 1 977, p. p. 148) 1 48) exex netics (Gillespie, (Gillespie, 1991; amined the processes with more general general autocorrelation autocorrelation amined the convergence convergence of of stochastic stochastic processes with more structure to structure to white white noise. noise. Equation that positive autocorrelation leads Equation (65) (65) shows shows that positive autocorrelation leads to to exaggerated exaggerated swings swings ( because one one bad bad year year is is likely likely to to be be followed followed by by another) another) and and that that negative negative (because autocorrelation autocorrelation dampens dampens swings swings in in r. r. Simulations Simulations confirm confirm the the utility utility of of Eq. Eq. (65) (65) (Foley, ( Foley, 1994). 1 994). Formulas Formulas for for extinction extinction times, times, extinction extinction probabilities probabilities and and so so forth forth by using using Vre Vre in in place place of of Vr. Vr• can be be adjusting adjusting for for nonzero nonzero pp by can Many values, more more negneg Many population population size size time time series series show show slightly slightly negative negative pp values, ative accounted for ative than than can can be be accounted for by by the the bias bias of of the the standard standard estimator. estimator. Some Some of of the negativity negativity may may be be due due to to the the inevitable inevitable bounciness bounciness near near the the ceiling ceiling and and near near the extinction for data extinction for data sets sets that that have have been been selected selected to to analyze. analyze. Some Some of of the the variance variance in apparent rr may attributable to ( t ) and may be be attributable to error error in in estimating estimating NN(t) and subsequent subsequent corcorin apparent
1100
Local LocalExtinction ExtinctionModels Models
233 233
rections. structure disequilibrium but not in most rections. Some Some may may be be due due to to ageage-structure disequilibrium ((but not in most insects !). This insects!). This remains remains aa statistical statistical puzzle puzzle and and perhaps perhaps aa biological biological one one also. also.
B. Logistic Logistic Density Density Dependence Dependence The ceiling, a The theory theory presented presented in in this this chapter chapter assumes assumes that that K is is aa ceiling, a barrier barrier above which a above which a population population cannot cannot remain remain for for long long to to wander. wander. The The ceiling ceiling as assumption sumption is is about about as as realistic realistic as as the the logistic logistic assumption, assumption, and and it it is is easier easier to to do do the the math. eld observations math. A A substantial substantial body body of of verbal verbal theory, theory, simulations simulations and and fi field observations about vagueness" (Strong, 986), "spreading risk" (den Boer, 11968, 968, about "density "density vagueness" (Strong, 11986), "spreading the the risk" (den Boer, 11981), 98 1 ), and 954), mainly and "density "density independence" independence" (Andrewartha (Andrewartha and and Birch, Birch, 11954), mainly by by insect population biologists, has driven a rebellion against the pervasive density insect population biologists, has driven a rebellion against the pervasive density dependence ecologists have dependence of of the the logistic logistic model. model. Vertebrate Vertebrate ecologists have usually usually supported supported the model, despite despite the their subjects. the logistic logistic model, the territorial territorial nature nature of of many many of of their subjects. Terri Territoriality may density dependence near the the toriality may be be expected expected to to lead lead to to strong strong density dependence only only near population However, vertebrates wild population population flucfluc population ceiling. ceiling. However, vertebrates rarely rarely show show the the wild tuations way. tuations of of invertebrates invertebrates and and thus thus appear appear to to be be regulated regulated in in aa more more consistent consistent way. Nonetheless, If so, Nonetheless, most most of of this this regulation regulation may may be be at at or or very very near near the the ceiling. ceiling. If so, there inverte there may may be be less less difference difference than than there there appears appears between between vertebrate vertebrate and and invertebrate density dependence. brate population population regulation regulation with with respect respect to to density dependence. The The classical classical logistic logistic model model of of Verhulst, Verhulst,
dN-rN(1
_N),
(66) (66)
shows shows awkward awkward behavior behavior in in the the vicinity vicinity of of K. Populations Populations above above K are are supposed supposed to retreat retreat toward toward K with with the the same same force force with with which which they they approach approach K from from below. below. to It is is possible possible to to generalize generalize the the logistic logistic ((Hassell al.,, 1976), but there there is 1 976), but is no no It Hassell et at. consensus in real consensus about about the the prevalent prevalent form form of of density density dependence dependence in real populations populations ((Royama, Royama, 1992, 1 992, discusses possibilities at it is discusses the the possibilities at length), length), and and it is difficult difficult to to detect detect density 987; Vickery Nudds, density dependence dependence from from time time series series (Pollard (Pollard et at. al.,, 11987; Vickery and and Nudds, S. Torres, logistic model 11991; 99 1 ; J. E. E. Foley, Foley, P. P. Foley, Foley, and and S. Torres, unpublished). unpublished). The The logistic model peculiarity becomes becomes es becomes becomes markedly markedly peculiar peculiar if if r < < 0 and and N > K. This This peculiarity especially pecially significant significant in in the the analysis analysis of of environmental environmental stochasticity stochasticity because because r(t) will will above K. often often fall fall below below zero, zero, even even if if N is is above For intelligible comparison, the model For these these reasons reasons and and for for the the sake sake of of an an intelligible comparison, the model of of pervasive pervasive density density dependence dependence examined examined here here includes includes these these features: features: K remains remains - k ) . In by rd( words our our model becomes a a ceiling, ceiling, and and rd r d is is replaced replaced by r~(1l - eenn-x). In other other words model becomes a a truncated, truncated, stochastic stochastic Ricker Ricker equation equation
N>
N
dn = ro (1
- e "-k)
dt + ~ r dW(t).
(67) (67)
(7) which We do usual diffusion diffusion Compare Compare this this with with Eqs. Eqs. (3) and and (7) which it it generalizes. generalizes. We do the the usual
234 234
Patrick Foley
analysis to analysis to get get ~(X)
- - e - 2 S X e 2s(ex-k-e-k)
(68)
2sn -- e e-2sn 11 -
2sn -- e e -2sn k e -2sk -2sk 11 ke 2 + < SSen) (n) < < + -- ((e2s~-e-*) _ 11). e S( 1 - e -k) ). --- < 22 2s 2s 22s s
(69) (69)
Unfortunately closed form. A good good approxi Unfortunately SSen) ( n ) appears appears difficult difficult to to obtain obtain in in closed form. A approximation is the the scale function for mation and and certainly certainly aa lower lower bound bound is scale function for the the case case of of no no density density dependence. by using using the of !fJ(k) the dependence. The The upper upper bound bound is is obtained obtained by the value value of qJ(k) and and the concavity of the takes its its lower concavity of the exponential exponential function. function. The The sojourn sojourn function function takes lower bound bound and and aa good good approximation approximation in in the the form form t(n;k) t(n;k)
1 - -2sn 2 s n e -2s ~ - e 22sn e ' - kLe -e-k k) ) -1 - e e -2sn -e e - 2 s ((en-
=
� Vr
2s 2S
1
= - - (e 2s" -
1)e
-2s(e'-k-e-k).
(70) (70)
Fd
This This is is also also hard hard to to integrate integrate in in closed closed form, form, but but numerical numerical approximations approximations are are easy. ratio of model to easy. Remarkably, Remarkably, the the ratio of Te(k) Te(k) with with the the ceiling ceiling Ricker Ricker model to the the Te(k) Te(k) without rather simply numerical without the the logistic logistic term term depends depends mainly mainly on on s and and rather simply so. so. As As aa numerical rule rule of of thumb, thumb,
Te(k) Te(k) Ricker Ricker Te(k) Te(k)
=
11 - 0 .0.7s 7s
for f o r ss<< 11..
(7 1) (71)
This result result is is so so consistent, consistent, it it should should be be analytically analytically obtainable. obtainable. Simulations Simulations reveal reveal This that stochastic Ricker without a ceiling at that the the stochastic Ricker model model without a ceiling at K K leads leads to to aa higher higher Te than than in Eq. (71) since sojourn sojourn times above n add add to the integral, integral, but but as as shown 1 ) since times above to the shown above, above, in Eq. (7 this this model model has has some some built built in in biological biological unreality. unreality. values, extinction rates conclusion of The conclusion of this this is is analysis analysis is is that that for for small small s values, extinction rates The do logistic model do not not differ differ much much between between the the ceiling ceiling model model and and the the logistic model of of density density dependence. dependence.
C. C. Other Other Adjustments Adjustments to the Model Allee et aI., 949, p. p. Allee effects effects due due to to social social aggregation aggregation requirements requirements (Allee (Allee et al., 11949, culties to in Dennis 1 99 1 ) and 393) pose pose few few diffi difficulties to the the model. model. As As discussed discussed in Dennis et et al. al. ((1991) and 1 994), the lower population boundary changes changes from = n a , the the Foley ((1994), Foley the lower population boundary from n = = 0 to to n = na, critical log population that permits This is is equiv critical minimal minimal log population size size that permits social social aggregation. aggregation. This equiva . Note may substantially alent alent to to lowering lowering k by by the the amount amount n na. Note that that this this adjustment adjustment may substantially diminish diminish demographic demographic stochasticity stochasticity if if nnaa is is large large enough. enough. All All the the demographic demographic stochasticity may persist on stochasticity may occur occur in in populations populations too too small small to to persist on deterministic deterministic grounds. grounds. The 1 975) analyzed The ceiling ceiling K K need need not not be be constant. constant. Roughgarden Roughgarden ((1975) analyzed aa logistic logistic model model with with fluctuating fluctuating K K and and found found that that populations populations would would fluctuate fluctuate with with aa time time lag represents a lag due due to to the the effects effects of of recent recent density density dependence. dependence. If If K K represents a ceiling, ceiling, this this
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Locol Extinction Models LocalExtinction Models
235 235
lag to those lag should should be be smaller, smaller, and and if if the the fluctuations fluctuations are are small small compared compared to those of of r, r, ' s model the using an the results results of of this this chapter chapter's model should should still still hold hold approximately approximately using an ef effective all of of the the environmental stochasticity is is fective K close close to to the the arithmetic arithmetic mean. mean. If If all environmental stochasticity ignoring attributable attributable to to K rather rather than than r, r, the the most most straightforward straightforward extinction extinction model model ((ignoring looking for demographic examine the distribution of demographic stochasticity) stochasticity) is is to to examine the distribution of K, looking for the the below one. one. This we know the full probability probability that that K drops drops below This approach approach assumes assumes that that we know the full In a probability probability distribution distribution of of K rather rather precisely, precisely, aa doubtful doubtful proposition. proposition. In a sense, sense, this would imply knowledge of the rate of catastrophes. this would imply knowledge of the rate of catastrophes.
VI. AND Vl. CATASTROPHES CATASTROPHES AND GENETIC GENETICIMPOVERISHMENT IMPOVERISHMENT Catastrophes Catastrophes are are extraordinary, extraordinary, almost almost unpredictable unpredictable population population threats. threats. Typ Typpopulation mortality mortality rate Hanson ical models assume assume a ical models a catastrophe catastrophe rate rate A A and and aa population rate {) 6 ((Hanson may follow some random and 98 1 ; Lande, 1 993). Of and Tuckwell, Tuckwell, 11981; Lande, 1993). Of course course {) 6 may follow some random distri distribution; it Mangel and 993b, 11994). 994). Extraordinary, Extraordinary, bution; it need need not not be be constant constant ((Mangel and Tier, Tier, 11993b, unpredictable events events are study, especially unpredictable are hard hard to to study, especially when when they they include include agents agents as as diverse predators or diverse as as disease disease outbreaks, outbreaks, hurricanes, hurricanes, superinvading superinvading organisms organisms ((predators or competitors), competitors), asteroids, asteroids, widespread widespread fires, fires, and and acid acid rain. rain. For For aa popUlation population to to become become extinct extinct due due to to catastrophe, catastrophe, one one of of three three scenarios scenarios seems I ) aa clean seems necessary: necessary: ((1) clean sweep sweep catastrophe, catastrophe, (2) aa catastrophe catastrophe sequence, sequence, or or (3) (3) catastrophes catastrophes mixed mixed with with lesser lesser problems. problems. In In the the clean clean sweep sweep scenario, scenario, every everyone dies; i.e., is one. is high so one dies; i.e., {) 6 is one. In In the the catastrophe catastrophe sequence sequence scenario, scenario, A h is high enough enough so that close sequence, sequence, before that two two or or more more catastrophes catastrophes can can occur occur in in close before aa population population has plausibly, in the mixed mixed model, model, catastro has time time to to get get back back to to K. Perhaps Perhaps most most plausibly, in the catastrophes bring population down down low low enough enough for phes bring the the population for normal normal environmental environmental stochas stochasticity nish the clean ticity or or demographic demographic stochasticity stochasticity to to fi finish the job. job. The The analysis analysis of of the the clean sweep easy; the is exactly sweep scenario scenario is is easy; the extinction extinction rate rate e is exactly A. h. Local Local dynamics dynamics do do not not matter. matter. The The mixed mixed catastrophe catastrophe scenario scenario does does not not lend lend itself itself to to simple simple analytic analytic results, although 1 993a) show construct generating results, although Mangel Mangel and and Tier Tier ((1993a) show how how to to construct generating matrices concept of matrices to to obtain obtain Te Te.. The The catastrophe catastrophe sequence sequence scenario scenario stretches stretches the the concept of catastrophe common occurrence; occurrence; nonetheless catastrophe into into aa common nonetheless theorists theorists frequently frequently resort resort to to it Hanson and Tuckwell, 11978, 978, 198 1 ; Lande, 993). In the relative it ((Hanson and Tuckwell, 1981; Lande, 11993). In comparing comparing the relative importance in the importance of of environmental environmental stochasticity stochasticity ((in the narrow narrow sense) sense) and and catastrophes, catastrophes, Lande 1 993) concludes Lande ((1993) concludes that that it it depends depends upon upon the the parameter parameter values. values. Unfortunately, Unfortunately, little little is is known known about about real real catastrophe catastrophe parameter parameter values. values. To To get get aa quantitative quantitative handle handle on on catastrophes, catastrophes, let let us us consider consider disease disease out outbreaks. disease is is a population. Most breaks. Mortality Mortality due due to to disease a normal normal part part of of the the life life of of aa population. Most variation will then variation in in disease-based disease-based mortality mortality will then fall fall under under the the category category of of environ environmental stochasticity (although (although disease shows predictably predictably periodic periodic dynam mental stochasticity disease often often shows dynamics). ics). However, However, the the primary primary outbreak outbreak of of aa new new viral viral or or bacterial bacterial strain strain can can of�n often have the relationship the disease have severe severe effects. effects. After After such such an an outbreak, outbreak, the relationship between between the disease and usually becomes less virulent and the the host host usually becomes less virulent for for aa variety variety of of reasons reasons including including host host immunity, 99 1 ; immunity, host host evolution evolution and and disease disease agent agent evolution evolution (Anderson (Anderson and and May, May, 11991;
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Patrick Patrick Foley Foley
TABLE III III
Candidates Candidates for for Catastrophes Catastrophes Caused Caused by by Disease Disease in in Natural Natural Populations Populations
Species
Disease Disease
Grey fox Wolf, Wolf, Superior Superior NF NF Wolf Wolf pups Rabbit, Australia Lion, Lion, Serengeti Domestic dogs, Africa Domestic Ringed seal, Lake Baikal Cheetah, Cheetah, captive colony Ungulates, Africa
Canine distemper Parvovirus Parvovirus Myxomatosis Canine distemper Canine distemper Phocid distemper FIP Rinderpest
{j #
0.78 High 0.75 0.99 0.25 0.35 0.2 0.42 0.9
A A
1 ?9 0.44 ?9 ?9 0.2 ?9 ?9 ?9
Source Source
Davidson et 1 992) et al. al. ((1992) Mech and Goyal ((1993) 1 993) 1 994) Johnson et et al. al. ((1994) Fenner and Myers ((1978) 1 978) Morell ((1994) 1994) Alexander and 1 994) and Appel ((1994) Grachev et 1989) et al. ((1989) 1 985) O'Brien et al. ((1985) McCallum and Dobson ((1995) 1 995)
Ewald, 994). It that behaves behaves like Ewald, 11994). It is is the the primary primary outbreak outbreak that like the the catastrophe catastrophe of of theory. virulent. Table theory. Such Such outbreaks outbreaks can can be be rare, rare, unpredictable, unpredictable, and and highly highly virulent. Table III III gives in natural gives examples examples of of high high mortality mortality induced induced by by disease disease in natural populations. populations. Some Some barely le; most they have have either barely fit fit the the catastrophe catastrophe profi profile; most do do not, not, because because they either high high A A or or made in in the low low O. 6. Little Little progress progress has has been been made the literature literature so so far far to to estimate estimate A A and and 0, 6, although threats to Mc although the the increasing increasing interest interest in in disease disease threats to conservation conservation efforts efforts ((McCallum 995) should A particular may only Callum and and Dobson, Dobson, 11995) should change change that. that. A particular disease disease may only sweep catastrophically we sweep catastrophically through through aa population population (or (or metapopulation) metapopulation) once, once, so so that that we kind of need need to to estimate estimate the the pooled pooled A A for for all all potential potential diseases. diseases. This This kind of data data is is available only only for for humans humans and and domesticated domesticated animals, animals, but but even even the the most most substantial substantial available 1 99 1 ) con efforts ecological effects disease (Anderson efforts to to study study the the ecological effects of of disease (Anderson and and May, May, 1991) concentrate centrate on on specifi specificc diseases. diseases. Disease-caused Disease-caused catastrophes catastrophes interact interact in in aa complicated, complicated, poorly poorly understood understood way place a way with with the the genetic genetic structure structure of of host host populations. populations. It It remains remains difficult difficult to to place a value genetic variability value. Survivorship value on on genetic variability although although we we know know it it has has value. Survivorship and and 1 988), but fertility fertility may may be be depressed depressed by by inbreeding inbreeding (Ralls (Ralls et et at. al.,, 1988), but populations populations can can 1 98 1 , p. p. 229). Furthermore Furthermore evolve Falconer, 1981, evolve to to compensate compensate for for the the problems problems ((Falconer, within the host, the genetic variability the immune immune system within the individual individual host, the somatic somatic genetic variability of of the system defenses evolves in Franklin ((1980) 1 980) defenses evolves in aa fashion fashion heterodox heterodox to to population population genetics. genetics. Franklin derived inbreeding de derived magic magic numbers numbers for for critical critical popUlation population sizes sizes vulnerable vulnerable to to inbreeding depression and based on pression and genetic genetic impoverishment. impoverishment. However, However, the the latter latter was was based on aa doubtful doubtful 1 992), and application 's early quantitative genetics genetics theory Foley, 1992), application of of Lande Lande's early quantitative theory ((Foley, and the the former selection against former often often fails fails due due to to the the purging purging effects effects of of natural natural selection against deleterious deleterious recessives. the convoluted relationship recessives. At At present present we we have have no no magic magic numbers, numbers, and and the convoluted relationship between variability and between the the loss loss of of genetic genetic variability and catastrophic catastrophic (or (or other) other) extinction extinction re remains mysterious. For biology, the the wisest wisest strategy is to to conserve mains mysterious. For conservation conservation biology, strategy is conserve founder (Lacy, 11989), 989), but founder genome genome equivalents equivalents (Lacy, but we we have have no no real real idea idea how how valuable valuable an an extra extra genome genome equivalent equivalent is is for for aa natural natural population. population. Novel living and evolving agents Novel parasites, parasites, predators, predators, competitors, competitors, and and other other living and evolving agents
1100
local LocalExtinction ExtinctionModels Models
237 237
of will surely which is of catastrophe catastrophe will surely test test aa population's population's ability ability to to evolve, evolve, which is approx approximately proportional 958). Therefore, with the the imately proportional to to genetic genetic variability variability (Fisher, (Fisher, 11958). Therefore, with possible of abiotic abiotic catastrophes, possible exception exception of catastrophes, the the genetic genetic structure structure of of aa population population is how to we is hard hard to to ignore. ignore. Unfortunately Unfortunately we we do do not not yet yet know know how to use use it. it. Nor Nor do do we have have established established methods methods for for estimating estimating the the parameters parameters A A and and 8, 6, so so the the application application of of catastrophic catastrophic extinction extinction theory theory to to real real populations populations remains remains problematic. problematic.
VII. IMPLICATIONS VII. IMPLICATIONSFOR FOR METAPOPUlATiON METAPOPULATIONDYNAMICS DYNAMICS A. Structured StructuredMetopopulotions Metapopulations Levins 1970) assumed would usually be either at carrying Levins ((1970) assumed aa local local population population would usually be either at carrying capacity extinct. This This simplifies capacity or or extinct. simplifies the the math math and and approximates approximates reality reality if if successful successful new approach carrying stay there. there. Structured new propagules propagules rapidly rapidly approach carrying capacity capacity and and stay Structured metapopulation models models take population sizes sizes within patch, metapopulation take explicit explicit account account of of population within aa patch, allowing intermediate values values ((Hastings Hastings and and Wolin, 989; Gyllenberg allowing them them intermediate Wolin, 11989; Gyllenberg and and Populations subject to environenviron Hanski, 1992; 1 992; Gyllenberg Hanski, Gyllenberg et et at., al., this this volume). volume). Populations subject to mental will fluctuate positive. We We mental stochasticity stochasticity will fluctuate near near the the carrying carrying capacity capacity if if rd rd is is positive. can times to popu can use use the the sojourn sojourn times to obtain obtain the the probability probability distribution distribution of of extant extant population sizes, the lation sizes, the mean mean and and the the variance variance of of N. N. 's Defi ne !p(n; no) as the population Definefp(n; as the the probability probability density density of of n conditional conditional on on the population's o . Similarly ne the the cumulative cumulative distri persistence given a persistence given a starting starting value value of of nno. Similarly defi define distriof the the sojourn bution bution function function F Fp(n; Then from from the the definition definition of sojourn time, time, p(n; no). Then
ten, no) t(n, ~ , ! p(n; no) fp(n; no) = = -r~(no) . Te(no)
(72) (72)
This density density allows allows us us to to calculate calculate the the expectation expectation and and variance variance of of n, En, En, and and Vn. V n. This The The two two most most interesting interesting starting starting points points no are are the the carrying carrying capacity capacity k (since (since propagule size the establish populations populations above above nc are are effectively effectively at at k) and and the the propagule size at at the establisho . Foley 1 994) and Hanski et at., ((1996a) 1 996a) ment ment of of aa new new population, population, aa very very small small nno. Foley ((1994) and Hanski et al., derive n; no), En and V n for the environmental stochasticity model, and these derive ! fp(n; En and Vn for the environmental stochasticity model, and these / results results are are given given in in Appendix Appendix 3. 3. Appendix Appendix 33 shows shows that that if if s is is close close to to zero, zero, aa population population will will spend spend much much of of its well below condition the density on its time time well below the the carrying carrying capacity capacity whether whether we we condition the density on aa probability distribution is not not starting near k. The starting point point of of near near zero zero or or near The probability distribution of of n is gaussian, is N normal, as would occur occur in boundaries in in gaussian, nor nor is N log log normal, as would in the the absence absence of of boundaries logistic Lewontin and Cohen, 11969; 969; Levins, logistic or or density-independent density-independent models models ((Lewontin and Cohen, Levins, 11969b; 969b; Ludwig, 974; Nisbet 982; Lande 988; Tul Ludwig, 11974; Nisbet and and Gurney, Gumey, 11982; Lande and and Orzack, Orzack, 11988; Tuljapurkar, 990). In is japurkar, 11990). In fact fact if if s is is zero, zero, the the conditional conditional probability probability density density of of n is uniform, result expected Feller, uniform, aa result expected from from the the standard standard theory theory of of random random walks walks ((Feller, 11971). 97 1 ). The from a The conditional conditional probability probability density density of of N N starting starting from a small small propagule propagule most of its time closer to the extinction extinction threshold is is then then 1/(N 1/(N In In K K);); i.e., i.e., N N spends spends most of its time closer to the threshold
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Patrick Foley PatrickFoley
than to the ceiling. This resembles resembles the pattern of of herbivorous insects capable capable of of outbreaks ((Ito, outbreaks Ito, 11980). 980). One metapopulation models is to One way way to use the formulas of of Appendix Appendix 3 in metapopulation calculate calculate a more more realistic colonization rate based based on the mean population size size of of extant Hanski et 996a). A practical application of extant populations populations ((Hanski et al., al., 11996a). of the formulas could include parameters Trdd and Vr from include the the rough estimation of of the the extinction parameters and Vr the the empirical distribution of of n. This This would would require either either a long long time time series series or observations from several uncorrelated uncorrelated habitat patches patches with estimatable estimatable carrying carrying capacities. Most interestingly, the sojourn sojourn times for population sizes permit a careful analysis of of the erosion of of genetic variability due to random genetic genetic drift. Such an investigation has been undertaken 1 99 1 ), Hedrick undertaken with simulations by Gilpin ((1991), and Gilpin (this volume), and Ewens et 1 987) for et al. al. ((1987) for catastrophe catastrophe driven systems. Analytic Analytic results for environmental and demographic demographic stochasticity become become pos possible, given the the sojourn times of of this chapter. chapter. Population Population size affects affects molecular molecular clock rates and genetic distances. distances. While strictly neutral theory is independent Kimura, 11983), 983), under under independent of of population population size ((Kimura, nearly neutral models small populations evolve at faster rates, and and much much of of mo molecular 972; Foley, 1987). 1 987). Genetic lecular evolution is apparently nearly nearly neutral neutral (Ohta, 11972; Genetic distances distances between between local populations will thus depend depend on population densities densities as well as on migration, so explicit models of of some use in of local dynamics can be of metapopulation genetics fluctu metapopulation genetics analysis analysis for species species with substantial population fluctuations.
B. Regional Regional Stochasticity: Stochasticity: Correlated Correlated Environmental Environmental Fluctuations Fluctuations Demographic mi Demographic stochasticity is not spatially correlated (barring (barring substantial substantial migration between between populations), but environmental stochasticity often often is, and catas catastrophes due to weather, weather, invaders, invaders, and and disease disease outbreaks outbreaks are are likely to be. Most metapopulation theory assumes patch wise independence patchwise independence of of extinction rates, and and modification of this theory is vulnerable vulnerable to the charge charge of of unreality. How much modification of the theory is needed needed remains remains an open question although simulations have explored 989; Gilpin, 11990; 990; Akqakaya Ak9akaya and some of of the possibilities (Harrison (Harrison and Quinn, 11989; Ginzbug, 99 1 ; Lahaye et et al., 994). As Levins ((1969a) 1 969a) argues from his diffusion Ginzbug, 11991; al., 11994). analysis, correlated extinctions lead to lower average average N values, and as the simu simulations show, shorter shorter metapopulation persistence persistence times. In the extreme extreme case, local populations undergo persistence undergo uncorrelated uncorrelated extinctions, so that that metapopulation persistence M ' depends time, T TM, depends only on the analog to demographic stochasticity that Hanski ((1992a) 1 992a) calls immigration-extinction 978; immigration-extinction stochasticity (Gurney (Gumey and Nisbet, 11978; Nisbet and Gurney, 11982). 982). At the other extreme, realized realized extinction rates are so M = perfectly perfectly correlated that T TM = Te for local populations. What What about intermediate cases, cases, and what are the patterns of of correlation in actual landscapes? landscapes? The California Spotted Owl populations of southern California appear The California of appear to
N
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Locol Extinction Models LocalExtinction Models
239 239
fonn with substantial form aa metapopulation metapopulation of of upland upland forested forested habitat habitat patches patches with substantial regional regional stochasticity ((Hanski, Hanski, 11992a) 992a) due stochasticity due to to drought. drought. Pairwise Pairwise correlation correlation estimates estimates for for rainfall in three three habitat 1 1 to and the the previous previous rainfall in habitat patches patches ranges ranges from from 0.8 0.811 to 0.896, 0.896, and ' s rainfall year 52% of this region La year's rainfall explains explains 52% of the the variation variation in in owl owl fecundity fecundity in in this region ((Lahaye et ai. al.,, 11994). of environmental environmental correlation correlation will will sharply haye 994). This This level level of sharply lower lower the the expected persistence time for Is in Den Boer Boer ((1981), 1 98 1 ), Pollard expected persistence time for spotted spotted ow owls in the the region. region. Den Pollard ((1991), 199 1 ), C. c. D. 1 99 1 ), Taylor 1 986), and Hanski and Woiwod ((1993) 1 993) D. Thomas Thomas ((1991), Taylor ((1986), and Hanski and Woiwod have have documented documented numerous numerous examples examples of of spatially spatially correlated correlated population population fluctua fluctuations Regional tions in in insects, insects, much much of of it it due due to to correlated correlated environmental environmental fluctuations. fluctuations. Regional stochasticity is stochasticity is aa very very real real feature feature of of metapopulations, metapopulations, population population data data and and cli climatological (e.g., Lamb, 985), and matological data data are are piling piling up up (e.g., Lamb, 11985), and it it cries cries out out for for analysis. analysis. Levins 969 analysis 1 969a) to Levins did did not not extend extend his his 11969 analysis ((1969a) to metapopulation metapopulation extinction extinction times, 1 978) Nisbet 1 982) analyzed times, and and Gurney Gurney and and Nisbet Nisbet ((1978) Nisbet and and Gurney Gurney ((1982) analyzed the the extreme extreme case case of of immigration-extinction immigration-extinction stochasticity stochasticity without without regional regional stochas stochasticity. Regional questions. How ticity. Regional stochasticity stochasticity presents presents several several entangled entangled questions. How do do cor correlated ex related ret) r(t) values values between between two two patches patches translate translate into into correlated correlated parametric parametric ex)? How How does translate tinction tinction rates, rates, e(t e(t)? does distance-based distance-based environmental environmental correlation correlation translate into global environmental correlation of metapopulation? How How does into the the global environmental correlation of the the metapopulation? does aa shared realized extinction extinction shared parametric parametric extinction extinction rate rate translate translate into into aa correlated correlated realized rate? rate? ' s ((1969a) Consider 1 969a) model, with H H defined the Consider aa slight slight extension extension of of Levin Levin's model, with defined as as the the number number patches in in a number of of suitable suitable habitat habitat patches a metapopulation, metapopulation, Q(t) as as the number of of the patches patches occupied occupied at at time time t, e(t) as as the the extinction extinction rate rate at at time time t, and and met) m(t) as as the colonization time t. Then colonization rate rate at at time Then
dQ
d_._Q_Qd= = t
m
(
)
Q(t» Q(t) \ met) - e(t) e(t) Q(t) Q(t) m(t) Q(t) (11 - -| H H /
may, may, if if met) m(t) is is constant constant and and ferential ferential equation equation
(73) (73)
e(t) N(E, Ve), Ve), be modeled by dif --- N(E, be modeled by the the stochastic stochastic dif-
(t) - h) dt dq(t) = m( m(1l - eq e q(t)-h) d t -- E E + dW(t), + X~e Fe dW(t), d q(t ) = -
(74) (74)
(t ) = is similar identical where where q q(t) = log log Q(t) and and h = = log log H. This This equation equation is similar (but (but not not identical in in this this chapter, it can be extended in fonn) form) to to those those analyzed analyzed already already in chapter, and and it can be extended similarly similarly to VdQ , to include include immigration-extinction immigration-extinction stochasticity stochasticity by by changing changing Ve to to Ve Ve + + VJQ, - death process process stochasticity experienced by by one where where VI V~ represents represents the the birth birth-death stochasticity experienced one population. The metapopulation is extinct when below 0 has a ceiling population. The metapopulation is extinct when q q drops drops below 0 and and has a ceiling at at q q= = h. Note Note that that random random movements movements during during colonization, colonization, and and the the independence independence of populations prevent local analysis, of local local populations prevent some some of of the the simplifications simplifications of of the the local analysis, such such as downplaying of density dependence as the the downplaying of logistic logistic density dependence and and demographic demographic stochasticity. stochasticity. II am with efforts efforts to to am presently presently engaged engaged in in the the analysis analysis of of these these equations equations together together with answer the other questions Foley, 11996). 996). answer the other questions of of this this section section ((Foley, Stochastic eld theory theory (Vanmarcke, 983) and Stochastic approaches approaches such such as as random random fi field (Vanmarcke, 11983) and
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Patrick PatrickFoley Foley
point 980; Thompson, 988) and point process process theory theory (Cox (Cox and and Isham, Isham, 11980; Thompson, 11988) and statistical statistical techniques 993) for with spatial techniques (Cressie, (Cressie, 11993) for dealing dealing with spatial correlational correlational patterns patterns are are be becoming accessible accessible to to the the ecological ecological research research community. community. Thus, Thus, much of the the prob probcoming much of lem of regional stochasticity stochasticity should should soon yield to to analysis field work. work. lem of regional soon yield analysis and and field
C. Scales for Growth, Growth, Extinction, C. Scales Extinction, and Colonization Colonization The The population population size size assumption assumption of of Levins, Levins, either either N N = = 0 0 or or N N = = K, is is also also somewhat Harmlessly, K should be somewhat weakened weakened by by an an explicit explicit stochastic stochastic analysis. analysis. Harmlessly, should be replaced CN). A new population can expect to reach this level in a time replaced by by E Ep(N). A new population can expect to reach this level in a time r T.(n o) given Ts(no) given by by lo...: (N) 0) = :; E -"pc.;.. __ -"o Ts(no) -~ _ logg_ Ep.__(N) - nno Tin
rrdd
(75) (75)
about 10 to 00 generations I. Meanwhile about 10 to 1100 generations given given the the parameter parameter values values of of Table Table I. Meanwhile expected 00 to to the the expected extinction extinction times times for for the the same same populations populations are are on on the the order order of of 1100 astronomical. going extinct astronomical. However, However, many many populations populations are are going extinct at at rates rates comparable comparable potential overlap is threat to would go to the the rate rate they they would go from from No No to to K. This This potential overlap in in scale scale is threatening to the Levins metapopulation model. If a metapopulation is to persist, high ening to the Levins metapopulation model. If a metapopulation is to persist, high extinction rates by high colonization rates. rates. If colo extinction rates need need to to be be compensated compensated for for by high colonization If colonization rates high enough, enough, then populations may begin to to show nization rates are are high then neighboring neighboring populations may begin show coupled dynamics, very complex, complex, potentially potentially chaotic coupled dynamics, and and very chaotic and and destabilizing destabilizing dy dynamics isolated, spatially namics may may ensue ensue as as suggested suggested by by the the literature literature on on weakly weakly isolated, spatially explicit, Hastings, 11990; 990; Hastings 1 994; Bas explicit, deterministic deterministic models models ((Hastings, Hastings and and Higgins, Higgins, 1994; Bascompte 994, 1995). 1 995). compte and and Sole, So16, 11994, There There are are other other effects effects of of the the local local dynamics dynamics which which are are not not inevitably inevitably harmful harmful to relationship between to the the Levins Levins metapopulation metapopulation model. model. Consider Consider the the functional functional relationship between local emigration. Emigration likely to to increase increase as pop local population population density density and and emigration. Emigration is is likely as population densities increase. This holds true for butterflies (Ehrlich, 1 984) and ver ulation densities increase. This holds true for butterflies (Ehrlich, 1984) and vertebrates, between density need not be tebrates, but but the the relationship relationship between density and and emigration emigration rate rate need not be linear Hansson, 1991 species emigrate densities linear ((Hansson, 1991).). Many Many species emigrate disproportionately disproportionately at at densities close Wynne-Edwards, 1962), 1 962), so this close to to carrying carrying capacity capacity ((Wynne-Edwards, so that that aa knowledge knowledge of of this become valuable. valuable. It relationship relationship and and the the probability probability density density fp(n) become It is is doubtful doubtful that that modifications to Levins model will gracefully modifications to the the colonization colonization assumptions assumptions of of the the Levins model will gracefully accomodate accomodate all all natural natural dispersal/density dispersal/density relationships. relationships. Complex Complex dynamics dynamics may may result result as as suggested suggested in in the the previous previous paragraph. paragraph.
D. Metacommunity Dynamics Dynamics The The single-species single-species local local population population dynamics dynamics explored explored in in this this chapter chapter have have inevitable implications implications for for community community interactions interactions at at the the local local and and the the landscape landscape inevitable level. Populations pred level. Populations that that fluctuate fluctuate have have erratic erratic interactions interactions with with competitors, competitors, predators, 1 954, ators, prey, prey, and and mutualists. mutualists. As As has has been been argued argued by by Andrewartha Andrewartha and and Birch Birch ((1954,
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241 241
p. 20), Wiens ((1977), Strong ((1986), and others, others, random random populations populations lead lead to to less p. 20), Wiens 1 977), Strong 1 986), and less deterministic makes competition harder to to deterministic and and more more diffuse diffuse competition. competition. This This makes competition harder detect in in the the field, field, encouraging encouraging irresolvable irresolvable controversy. controversy. However, However, stochasticity stochasticity detect has real real consequences consequences for for communities. In fluctuating fluctuating environments, environments, some rehas communities. In some re sources often unutilized. This may diminish diminish competition competition and sources often go go unutilized. This may and provide provide oppor opportunities for invasion and and evolution evolution of high-dispersal competitors. competitors. Or com tunities for invasion of new new high-dispersal Or competition can become occasionally transient bursts bursts of petition can become occasionally more more intense, intense, leading leading to to transient of natural selection for character displacement (Grant and Grant, 1 989). Fluctuating natural selection for character displacement (Grant and Grant, 1989). Fluctuating predator-prey communities communities can lead to predator-prey can lead to transient transient refugia refugia for for prey. prey. Population Population fluctuations may may erode mutual isms, since since they predictable reciproca fluctuations erode mutualisms, they depend depend on on predictable reciprocation. on these consequences of stochasticity is is vast vast (recent tion. The The literature literature on these consequences of local local stochasticity (recent reviews include include Chesson Chesson and Huntly, 11989; 989; Chesson, Chesson, 11990; 990; Pimm, Pimm, 11991). 99 1 ). HowHow reviews and Huntly, ever, do not ever, we we do not know know how how much much community community disruption disruption arises arises from from particular particular forms of stochasticity and forms of local local stochasticity and extinction. extinction. May 1 973a), Chesson 1 98 1 , 11990), 990), Caswell 1 993), and May ((1973a), Chesson ((1981, Caswell and and Cohen Cohen ((1993), and others others have models of in flux, flux, but but most depends have analyzed analyzed models of communities communities in most empirical empirical work work depends on comparative, macroecological approaches approaches (Brown, (Brown, 1995; but see on comparative, macroecological 1 995; but see Bengtsson Bengtsson and 1 995, for recent, fairly tight lab and Milbrink, Milbrink, 1995, for aa recent, fairly tight lab study study with with green green algae algae and and two two Daphnia species). Hassell 1 978), Hastings 1 977, 11978, 978, 1990), 1 990), Kareiva 1 989), Daphnia species). Hassell ((1978), Hastings ((1977, Kareiva ((1989), Taylor 1 990, 11991), 99 1 ), and Taylor ((1990, and others others have have paid paid special special attention attention to to the the enhancement enhancement of stability by of predator-prey predator-prey stability by spatial spatial and and temporal temporal heterogeneity heterogeneity although although the the distinction level processes processes and processes has distinction between between metapopulation metapopulation level and local local processes has not not always been Taylor, 11990, 990, 11991). 99 1 ). The always been empirically empirically established established ((Taylor, The analysis analysis of of any any ecological the local ecological interaction interaction can can be be improved improved only only by by an an understanding understanding of of the local extinction extinction dynamics dynamics of of individual individual species, species, but but aa species species embedded embedded in in the the com community, munity, tightly tightly coupled coupled to to other other species, species, is is likely likely to to have have distinctive distinctive dynamics dynamics that for that are are poorly poorly modeled modeled by by aa single-species single-species stochastic stochastic process. process. If If most most of of VVrr for a lynx lynx population population can be attributed attributed to to fluctuations fluctuations in in snowshoe snowshoe hares hares (Royama, (Royama, a can be 11992), 992), then then aa two-dimensional two-dimensional stochastic stochastic analysis analysis is is called called for. for. The The SIR SIR models models of local disease the finite of epidemiology epidemiology lead lead to to local disease agent agent extinctions, extinctions, partly partly due due to to the finite size populations, partly size of of host host populations, partly due due to to the the predator-prey-like predator-prey-like deterministic deterministic dy dynamics May, 1991; 1 99 1 ; Mollison, Mollison, 1995). 1 995). Much Much of namics (Anderson (Anderson and and May, of environmental environmental "sto "stochasticity" is is presumably presumably attributable attributable to to the the deterministic, deterministic, perhaps perhaps chaotic, chaotic, dy dychasticity" namics of of the the community community in in which which aa species species is is embedded embedded (Sugihara (Sugihara and and May, namics May, 11990). 990). It clear whether It is is not not yet yet clear whether the the "predictablility" "predictablility" of of deterministic deterministic dynamics dynamics for the 'unpredictability' for diffuse diffuse interactions interactions differs differs from from the 'unpredictability' of of stochastic stochastic dynamics dynamics when extinction. when it it comes comes to to extinction. The The other other major major metapopulation metapopulation process, process, immigration, immigration, may may also also depend depend on on the MacArthur and and Wilson, 1 967) or the presence presence of of competing competing species species ((MacArthur Wilson, 1967) or it it may may not not (Simberloff, 978b). Community single-species stochasticity, (Simberloff, 11978b). Community interactions, interactions, single-species stochasticity, and and colonization colonization interplay interplay sometimes sometimes tightly, tightly, sometimes sometimes diffusely diffusely in in metapopulations, metapopulations, and likewise. However, and the the appropriate appropriate analytic analytic tools tools vary vary likewise. However, these these problems problems take take us well beyond beyond this this chapter. us well chapter.
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Patrick Foley Foley Patrick
CONClUSION VIII. CONCLUSION Local extinction extinction can can be be plausibly plausibly modeled modeled in in many many cases cases by by aa population population Local undergoing environmental environmental stochasticity, stochasticity, with with aa ceiling ceiling set set by by resource resource caps caps and and undergoing lower threshold threshold set set by by the the minimum minimum number number needed needed to to successfully successfully reproduce. reproduce. aa lower The environmental simple analytic The environmental stochasticity stochasticity model model presented presented here here generates generates simple analytic predictions of of extinction extinction rates rates and and population population densities. densities. These These can can be be used used in in predictions metapopulation models models to to improve improve our our analysis of the the fundamental fundamental processes of metapopulation analysis of processes of extinction and and colonization. colonization. For For extinction extinction dynamics, dynamics, demographic demographic stochasticity stochasticity extinction may than has may play play aa smaller smaller role role than has sometimes sometimes been been assumed, assumed, but but catastrophes catastrophes rere main hard hard to to study study empirically. empirically. It It is is easy easy to to estimate estimate the the parameters parameters of of the the enen main vironmental stochasticity stochasticity mode, mode, but but surprisingly surprisingly little little work work has has been been done done estiesti vironmental fundamental parameter parameter of of demographic demographic stochasticity, stochasticity, or or 6{) and and A, A, mating v~, V I ' the mating the fundamental the catastrophe parameters. parameters. We We still still know know little effect of of genetic genetic imim the catastrophe little about about the the effect poverishment on on the population. poverishment the viability viability of of aa population. The model has has many many largely unexplored impliimpli The environmental environmental stochasticity stochasticity model largely unexplored cations for metapopulation dynamics. Emigration rates rates are by local cations for metapopulation dynamics. Emigration are influenced influenced by local fluctuations, rates become populations, the time scales scales fluctuations, extinction extinction rates become correlated correlated across across populations, the time of local growth Most of of these discussed, of local growth and and extinction extinction may may overlap. overlap. Most these have have been been discussed, but not not mathematically mathematically analyzed. Hopefully this this chapter chapter will will provide provide tools tools with with but analyzed. Hopefully which to to solve problems of of metapopulation metapopulation dynamics, dynamics, which solve some some of of the the outstanding outstanding problems especially those those related related to to colonization colonization and and regional especially regional stochasticity. stochasticity. The processes that create environmental stochasticity are fairly well well known known The processes that create environmental stochasticity are fairly in aa general way, but but the of the environmental in general way, the systematic systematic quantitative quantitative partitioning partitioning of the environmental variance Vr Vr into into its its components components for for a range of of representative representative species species would would help help variance a range to disentangle disentangle the the classic problems of of popUlation population regulation, competition inten intento classic problems regulation, competition sity, density density vagueness, vagueness, regional sity, regional stochasticity, stochasticity, and and metapopulation metapopulation persistence. persistence. Population fluctuations fluctuations are are probably probably better better understood understood in in light light of of the the variance variance in in Population r 978) than log N (e.g., 99 1 ). To r (e.g. (e.g.,, Wolda, Wolda, 11978) than in in the the variance variance of of N or or log (e.g., Pimm, Pimm, 11991). To understand understand the the local local extinction extinction patterns patterns of of tightly tightly coupled, coupled, nondiffuse nondiffuse ecological ecological interactions disease) will will require two interactions (e.g., (e.g., mountain mountain lion lion and and deer, deer, infectious infectious disease) require twodimensional stochastic models to to less-isoless-iso dimensional stochastic analysis. analysis. The The transition transition from from Levins Levins models lated patch patch models models needs needs to to be be examined examined more more carefully, carefully, since since rescue rescue effects effects lated become become increasingly increasingly important important in in local local persistence persistence (Brown (Brown and and Kodric-Brown, Kodric-Brown, Hanski and and Gyllenberg, Gyllenberg, 11993) and populations populations become more synchronous synchronous 11977; 977; Hanski 993) and become more in in their their fluctuations. fluctuations. So So there there is is plenty plenty of of work work to to do. do.
N
N
ACKNOWLEDGMENTS ACKNOWLEDGMENTS Dr. DVM, UC been a persistent source of discussion Dr. Janet Janet Foley Foley ((DVM, UC Davis) Davis) has has been persistent source discussion about about disease. disease. Ilkka Ilkka Hanski Hanski has kept kept me me metathinking. metathinking. The The work work of Levins, Levins, MacArthur, MacArthur, and Wilson Wilson continues continues to inspire. inspire.
1100 locol Extinction Models LocalExtinction Models
243 243
APPENDIX 11 Symbols Symbols Used Used in This This Paper Symbol Symbol
Explanation Explanation
N N R (N, t) R(N,
Population Population size size measured measured in in females; females; n = = 10ge(N) loge(N ) Realized Realized fundamental fundamental net net reproductive reproductive rate: rate:
K ?'d p
Vr
Vre
(N, t); N(t + + 11)) = = N(t) R R(N, t); rr = = 10ge(R) loge(R)
in females) 10ge(K ) Carrying Carrying capacity capacity ((in females) considered considered as as aa ceiling; ceiling; k k = = loge(K) The cient or expected The drift drift coeffi coefficient or deterministic deterministic component component of of r, the the expected value value of of rr Serial autocorrelation between consecutive Serial autocorrelation in in environmental environmental effects effects between consecutive years years Variance Variance in in rr due due to to environmental environmental stochasticity stochasticity Effective Effective environmental environmental variance variance taking taking p p into into account: account: VYre re =
1l ++p P vVrl-r p I
-_
s
Sd
A o
P
Variance demographic stochasticity Variance in in rr for for one one female female due due to to demographic stochasticity : Vr + The cient, the The diffusion diffusion coeffi coefficient, the variance variance in in rr'Vr + vv~/N l /N I /Vr : relative importance importance of of demographic demographic and and environmental environmental sto stoVv~/Vr'relative chasticity chasticity rd/ to environmental rd/Vr'Strength of population population growth growth (rd (rd)) relative relative to environmental Vr : strength of stochasticity stochasticity rd/v rd/v~'strength of population population growth growth (rd (rd)) relative relative to to demographic demographic sto stoI : strength of chasticity chasticity Rate close to Rate at at which which catastrophes catastrophes occur, occur, aa number number close to zero zero Mean fraction fraction of of individuals individuals killed in aa catastrophe, catastrophe, aa number number close close Mean killed in to one one to
-
"'(x) qJ(x)
eexp( x p ( - 2 I(xx � r_ddy), an an unnamed, unnamed, but useful function but useful function J3oo vv
S en) S(n)
Scale measure Scale measure of of the the diffusion: diffusion:
m(n) m en) ten; t(n; no) Te(n) P Pk(n) k (n) p en, t) p(n, NC
!afon ~(x) dx --v(n)~(n)
"'(x) dx
11 v(n) "'(n) Sojourn ten; no) = Sojourn time time near near n: n:t(n;no) = 2m(n)S(n) 2m(n)S(n) for for n < < nno o , , t == 2m(n)S(no) for n > no 2m(n)S(no) for n > no k ten, Expected time time ttoo extinction extinction starting starting at at n: Te(no) Te(no) = t(n, no)dn Expected Speed measure Speed measure of of the the diffusion: diffusion:
!afo
Probability going extinct: Probability of of reaching reaching k before before going extinct: S(n)/S(k) S(n)/S(k) Probability et): En value and Probability density density of of n n(t): En and and Vn Vn are are the the expected expected value and variance, respectively, respectively, of of n variance, The value for popUlation success (nc = e (Nc)): Pk(nc) Pk (nJ = The critical critical value for population success (nc = 10g loge(Nc)): ~
Pk(k) P k ( k)
Patrick Foley Foley Patrick
244 244
APPENDIX 22 Results of Stochastic Stochastic Extinction Extinction Theory with a Population Population Ceiling Ceiling at KK Main Results Stochasticity No Stochasticity no
Te(no)
if rd < 0
rd
otherwise Te(no) = ~
Environmental Stochasticity Stochasticity (Exact ( Exact Results Results Given Given Diffusion/Continuity Limitations) limitations) Environmental
1
if rd = 0
(e 2s~ (1 - e -2'n~ - 2sno)
Te(no)
2Srd
k) Te(k) Te(
11 (e 2sk
nc
2s
-
2Srd 2srd (ez'k
Vr
k2
11 - 2sk) 2sk)
--
ifra = 0
Vr
1 Demographic Stochasticity O) Demographic Stochasticity(More Approximate, for fdrd ~> O) no ek - e "~ + 1 Te(no)
V1
e2s(e k-l)
Te(k)
1
__
ek(k - 1) + 1
if 1 < 2sdk
4rdS d
e2SdEl(2Sd) ~ In
11 c
1 +
121
= 00
iif f Frd d =
if rd = = 0 0 if
ln(2)) 2Sd
Environmental (Also Approximate, Environmental and Demographic Demographic Stochasticity Stochasticity(Also Approximate, for T (k) Te(k) e
nc
nc
( e kk ++ y T)2S--(enc (e )2S - (e n,
4-T)es +
2sra 2srd
11 ++2 72y 2s(11 + + y) 7) 2s(
Environmental Environmental Autocorrelation Autocorrelation Vre Vre
1l ++ p 11 - p
P
v -~ Vrr _
P
)k2 y)2s (1 +22T y) + y) , y ) 22 s+ s+l1 ((11 +
- (--) . k2 ( 11 ) V'r--~ 11 + + Y v
rd >
rd ~ O) O)
ifrd = 00 If rd =
1100 Locol LocalExtincTIon ExtinctionModels Models
245 245
Rule Thumb: Effect Dependence on Environmental Environmental Stochasticity Rule of Thumb: Effect of Logistic LogisticDensity Density Dependence Stochasticity with logistic Te(k) Te(k) with logistic density density dependence dependence ----' '--'--'-�---''---....!---density dependence Te(k) Te(k) without without logistic logistic density dependence
11 --0 . 70s.7 s for f o r 00<< s <s < 11
=
APPENDIX APPENDIX 33 Conditional Predicted by Models Models with Environmental Conditional Population Population Densities Densities Predicted Environmental Stochasticity Stochasticity
In General General n
Fp(n; no) =
fp(n" no) = t(n; no) Te(no)
fok
Ep(n) =
nfp(n; no) dn
fp(N; No) = fv(l~
log(No)) N
Vp(n) =
fo
fO
fp(X; no) dx
(n - Ep(n))2fp(n; no) dn
using formula using the the change change of of variable variable formula
Starting Propagule Size) Starting from Small Small no no ((Propagule h fp(n" k) p (n '- k)
=
Ep(n) = = Ep(n)
2 s e 22s(n - k) s(n-k) 2se -2skk 11 - ee -2s -
sk 2ske 2zsk 2skk + -k- 11 2ske ee 2s sk 22s(e s ( e 22sk -- 1) 1)
Vp(n)
=
E Ep(N) = p (N) -
2
k2 1122
hp (N)
Il ++
0 if rd = 0 ifr~ =
� [ �ski] _k2[[ ]
=
=
2 s N 2 s - 1s - 1 2sN2 f fp(N) -~ K 22s p (N) = K s - 11
F Fp(N) p (N)
11 fp (n ; k) = k
7k
if sk sk < < 11 if
(S3k)2 ] 1 + + (sk)2
if if sk sk < < 11
11 kN
if i f rrdd = = 0 0
3
= -
N z s2s - 11 N K2s s m 1 1 K2
2s(KZs+ _ 11)) s + 11 2s(K2 (2s + + 11)(K - 11)) )(K22ss -
=
K K [ 1 + sk] k [1 + sk]
k
if if sk sk < < I1
246 24,6
Patrick Patrick Foley Foley
(N ) VpVp(N) K2s+! 2s(2s 1)(K 2)(K 2s+ 1 _ 1)2 K2s+2 - 1)1 ) -- 44s2(2s )2(K2S - 1)( s 2(2s + 2)( 2s(2s ++ 11)2(K2s 1)2 S 2 )2(K2 (2s + 2)(2s + 1)2(K2s 1)2 (2s + 2)(2s + 1 - 1) K2[112k++ ssk]k] (( 1 2(1 + + Sk) sk)'~) 2k k } ifif sk << 1 --
2s+2
_
+
_
--
--'-
1 + + ---'--
=
~..--~ K 2 [
Starting from from k (or above above n() Starting n~) 2s(ee 2sn - 1) fp (n; k)k) = ee 2s2sk2s( 2sk - 11 k - 2sk -
_
fp(n;
1)
for for 0 0 < < n < < k
n k 2n
fp(n; k) = ~ -
k(k( k - l/ 1 /2s) 2 s ) -- 1/2s 1 / 2 s -- sk sk2 ------'Ep (n) = e22ssk--'-e2sk-'---'2 s k -- 1 - 2sk Ep(n)
=
-
---
-
if r~ = 0
k [ �3 Sk18 J ifif sskk << 4 l2s ++ l/2S 2) - l/2s 2 - 2sP/3 - (E(Ep(n)) (n) == e2sk (k2 - kk/2s VpVp(n) p(n))2 e2sk 2sk 2sk- 1 sk k2 [�18 + �J 135 ifif sk << 4 e 2sk - -
=
~k
e2sk(k
2 --
+
1 / 2 S 2) - - 1 / 2 S 2 - - 2 s k 3 / 3
e 2sk _ --
_
=
+
Studying Studying Tronsfer Transfer Processes Processesin Metopopulotions Metapopulations Emigration, Emigration, Migration, Migration, and Colonization Colonization Rolf A. Ims
Nigel Nigel G. G. Yoccoz Yoccoz
I. INTRODUGION INTRODUCTION The transfer of individuals is a process in in metapopulations. The transfer of individuals across across space space is a key key process metapopulations. In sets of ex In fact, fact, metapopulations metapopulations are are often often seen seen as as sets of local local populations populations the the very very existence exchange) of istence of of which which is is dependent dependent on on mutual mutual transfer transfer (or (or exchange) of individuals individuals ((Hanski Hanski and volume). Components transfer rates and Simberloff, Simberloff, this this volume). Components of of transfer rates such such as as em emigration, migration migration (dispersal), and colonization colonization become become critical critical variables. variables. igration, (dispersal), and These These variables variables must must be be understood understood and and analyzed analyzed in in quantitative quantitative terms terms for for aa better understanding understanding of of metapopulation metapopulation dynamics. dynamics. Unfortunately, Unfortunately, there there are are often often better great with obtaining data on great difficulties difficulties associated associated with obtaining data on the the transfer transfer process process in in meta metapopulations. habitat patches, patches, populations. Events Events such such as as the the emigration emigration of of individuals individuals from from habitat the individuals are the colonization colonization of of empty empty patches, patches, and and the the behavior behavior of of migrating migrating individuals are difficult this reason, transfer rate be inferred inferred with with difficult to to observe. observe. For For this reason, the the transfer rate must must often often be indirect validity. indirect approaches approaches based based on on rather rather stringent stringent assumptions assumptions of of uncertain uncertain validity. In we will will draw together the the various to the In this this chapter chapter we draw together various empirical empirical approaches approaches to the study of emigration, migration, and colonization processes in a metapopulation study of emigration, migration, and colonization processes in a metapopulation context. we stress limitations and context. In In particular, particular, we stress the the limitations and advantages advantages of of different different ap approaches assess critically critically the the lines lines of inferences that reliably be proaches and and assess of inferences that can can reliably be drawn drawn from them. of our our conclusions conclusions may seem overly our knowledge from them. Some Some of may seem overly pessimistic: pessimistic: our knowledge is still very what can learned from is still very sparse, sparse, and and there there are are severe severe limitations limitations on on what can be be learned from Metapopularion Biology Metapopulation Biology Copyright © 997 by Academic Press, reproduction in any fOIm 9 11997 Press, Inc. All rights of of reproduction form reserved.
247 247
Rolf Nigel G. Rolf A. A. Ims Ires and and Nigel G. Yoccoz Yoccoz
248 248
the are convinced knowing these the different different approaches. approaches. Nonetheless, Nonetheless, we we are convinced that that knowing these lim limitations designs and itations allow allow us us to to devise devise better better experimental experimental and and observational observational designs and methods methods of of analysis. analysis.
II. TRANSFER OF INDIVIDUALS IN METAPOPULATIONS: TRANSFEROF INDIVIDUALSIN METAPOPULATIONS:DEFINING DEFININGTHE THE COMPONENTS COMPONENTS We er as which an is We will will define define transf transfer as the the process process by by which an individual individual organism organism is brought is brought (actively (actively or or passively) passively) from from one one site site to to another. another. Often Often the the transfer transfer is associated cycle phase, phase, e.g., associated with with aa change change in in life life cycle e.g., migration migration (natal (natal dispersal) dispersal) precedes is rather rather similar precedes reproduction reproduction in in mammals mammals and and birds. birds. Our Our definition definition is similar to to the organization and the one one used used by by students students of of social social organization and demography, demography, in in particular particular of of 989), for primate primate populations populations (e.g., (e.g., Clutton-Brock, Clutton-Brock, 11989), for which which the the transfer transfer concept concept relates to the exchange of individuals between social groups. relates to the exchange of individuals between social groups. Generally viewed as Generally transfer transfer of of individuals individuals can can be be viewed as aa three-stage three-stage process process ((Fig. Fig. I1). ). It which the leaves its It is is initiated initiated by by an an emigration stage stage in in which the individual individual leaves its home is the home site. site. The The emigration emigration event event is is followed followed by by aa migration migration stage stage which which is the displacement individual away home displacement process process (movement) (movement) that that brings brings the the individual away from from its its home site. We which is site. We will will here here use use migration migration as as synonymous synonymous to to dispersal which is aa term term often often favored in favored by by population population ecologists ecologists to to describe describe spatial spatial displacement displacement processes processes in settings other than metapopulations metapopulations (see (see Wiens, Wiens, this this volume). volume). The The migration migration settings other than phase individual to home site where it phase may may or or may may not not bring bring the the individual to aa new new home site where it can can settle. settle. We will term We will term this this settlement settlement stage stage immigration regardless regardless of of whether whether settlement settlement takes takes place place at at aa site already inhabited inhabited by by conspecifics conspecifics or or at a site which which is empty. Immigration of a new Immigration in in an an empty empty patch patch followed followed by by successful successful establishment establishment of a new
Transfe Transferr
Emigration Emigration
Migration Migration
Immigration Immigration Colonization ColoniT.ation
Social Pressure Pressure Social
Movement Movement cues cues
Conspecific attraction Conspecitic attraction
Genetic Polymorphism Polymorphism Genetic
Mortality Mortality
Interspecific facilitation Interspecitic facilitation
characteristics Boarder characteristics
FIGURE FIGURE 11
Social Social fence
The stages causal mechamecha stages of the transfer transfer process process in metapopulations metapopulations and some some of the causal nisms stage. nisms that that may may be involved involvedat at each each stage.
1111
Studying Studying Transfer Transfer Processes Processesin Metapopulafions Metapopulations
249 249
population Alternatively the to a population is is colonization. colonization. Alternatively the migration migration stage stage may may not not lead lead to a settlement, metapopulation context context the scopes of settlement, if if it it is is terminated terminated by by death. death. In In aa metapopulation the scopes of these Below these three three general general stages stages of of the the transfer transfer process process are are somewhat somewhat restricted. restricted. Below we we address address the the emigration, emigration, migration, migration, and and colonization/immigration colonization/immigration as as they they per pertain settings of the tain to to the the particular particular settings of metapopulations. metapopulations. We We will will be be very very brief brief on on the ultimate and each stage ultimate and proximate proximate mechanisms mechanisms that that may may be be involved involved in in each stage as as such such considerations considerations have have been been dealt dealt with with extensively extensively in in several several recent recent reviews reviews (for (for reviews reviews on on factors factors that that trigger trigger emigration emigration and and dispersal, dispersal, see, see, e.g., e.g., Chepko-Sade Chepko-Sade and Halpin, 11987; 987; Johnson Gaines, 11990; 990; Hansson, 99 1 ; Stenseth and Halpin, Johnson and and Gaines, Hansson, 11991; Stenseth and and Lid Lidicker, 992a; Olivieri Olivieri and Gouyon, this volume, the icker, 11992a; and Gouyon, this volume, the migration/dispersal migration/dispersal process process itself by, e.g., Ims, 1995; 1 995; and itself is is dealt dealt with with in in detail detail by, e.g., Ims, and regarding regarding colonization/im colonization/immigration, see, see, e.g., 967; Bazzaz, 986; Ebenhard, migration, e.g., MacArthur MacArthur and and Wilson, Wilson, 11967; Bazzaz, 11986; Ebenhard, 11991). 99 1 ).
A. Emigration Emigration ned as is Emigration is is defi defined as leaving leaving aa local local population population whose whose spatial spatial extent extent is determined is determined by by the the extent extent of of the the respective respective habitat habitat patch. patch. The The suitable suitable habitat habitat is typically (or bath, typically restricted restricted to to discrete discrete patches patches imbedded imbedded in in aa matrix matrix (or bath, or or nonha nonhabitat: 976; Czaran 992), an not suited for long-term bitat: Levin, Levin, 11976; Czaran and and Bartha, Bartha, 11992), an area area not suited for long-term survival survival and and reproduction. reproduction. Emigration Emigration events events therefore therefore requires requires crossing crossing aa bound boundary recognized as ary between between habitat habitat and and nonhabitat. nonhabitat. This This boundary boundary may may be be recognized as aa distinct distinct physical Holland et 1 99 1 ). The physical patch patch edge edge or or it it may may be be more more like like aa gradient gradient ((Holland et al., 1991). The sharpness patch, ex sharpness of of the the border border zone zone may may affect affect the the emigration emigration rate rate from from the the patch, expecially pecially if if behavioral behavioral mechanisms mechanisms are are involved involved in in the the emigration emigration process process (Stamps (Stamps et al., 11987; Durelli et et al. al.,, 11990). There may, some situation situation even even et 987; Durelli 990). There may, however, however, be be some in in aa metapopulation metapopulation setting setting for for which which this this habitat-nonhabitat habitat-nonhabitat dichotomy dichotomy does does not leave a local patch not apply apply to to the the emigration emigration stage. stage. Sometimes Sometimes organisms organisms leave a local patch when when it exploitation by by the the local population it has has been been depleted depleted for for resources resources either either by by exploitation local population or itself ((Whitlock, Whitlock, 1992b, 1 992b, Roff, or by by the the independent independent dynamics dynamics of of the the patch patch itself Roff, 11994b). 994b). In In this the patch any more this case case the patch does does not not any more constitute constitute habitat habitat for for reproduc reproduction and and survival. tion survival.
B. Migration The involves displacements/movements The migration migration stage stage involves displacements/movements in in nonhabitat nonhabitat (ma (matrix), often distances. The trix), often over over long long distances. The critical critical points points here here are are the the hazards hazards associated associated with well as movement trajectories. Both mortality mortality and with migration, migration, as as well as the the specific specific movement trajectories. Both and movement patterns may inside habhab movement patterns may differ differ substantially substantially from from what what may may be be typical typical inside itat patches ((Ims, lms, 11995). 995). itat patches As models ((MacArthur MacArthur and Wilson, As aa heritage heritage from from the the original original island-type island-type models and Wilson, 11967; 967; Levins, 969a, 11970), 970), present even in Levins, 11969a, present metapopulation metapopulation studies studies even in terrestrial terrestrial environments tend terrestrial environments tend to to treat treat the the matrix matrix as as if if it it was was water. water. However, However, terrestrial matrices are homogeneous as implicitly matrices are typically typically not not as as featureless featureless and and homogeneous as is is often often implicitly
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assumed may contain physical and assumed (Wiens, (Wiens, this this volume). volume). Terrestrial Terrestrial matrices matrices may contain physical and biological in space space and time. Ma biological features features which which are are distributed distributed heterogeneously heterogeneously in and time. Matrix migration. This This applies to trix features features will will to to varying varying extents extents impede impede or or enhance enhance migration. applies to all all organisms, organisms, whether whether they they migrate migrate by by active active movements movements (most (most animals) animals) or or by by passive -matrix heterogeneity passive displacements displacements (wind (wind dispersal dispersal of of seeds). seeds). Migration Migration-matrix heterogeneity interactions tend to make the density distribution of dispersing individuals much interactions tend to make the density distribution of dispersing individuals aa much more solely by patch specific specific emiemi more complex complex function function than than if if it it was was determined determined solely by patch gration global mean migration dis gration rates rates together together with with the the global mean and and variance variance of of the the migration distances. tances.
C. C. The Settlement Settlement Stage: Stage: Immigration and Colonization Colonization Migrating Migrating organisms organisms in in metapopulations metapopulations may may or or may may not not encounter encounter aa new new habitat dis habitat patch patch depending depending on on the the spatial spatial distribution distribution of of patches patches (interpatch (interpatch distances) specific migration tances) in in relation relation to to species species specific migration capacity capacity and and its its interaction interaction with with matrix to be less matrix properties. properties. Spatially Spatially isolated isolated patches patches will will often often tend tend to be encountered encountered less often close to patches. For this reason reason patch often than than patches patches that that are are located located close to other other patches. For this patchspecific isolation indices, functions neighboring patches patches specific isolation indices, functions of of distances distances to to occupied occupied neighboring weighted used to weighted by by the the sizes sizes of of these these populations, populations, are are used to describe describe encounter encounter rates rates in spatially explicit models ((Hanski, Hanski, 11994a, 994a, this Note, in spatially explicit metapopulation metapopulation models this volume). volume). Note, however, that always decrease with increasing increasing patch however, that such such encounter encounter rates rates may may not not always decrease with patch isolation. example, if uses distant cues to isolation. For For example, if the the migrating migrating individual individual uses distant cues to orientate, orientate, isolated Bell, 1991). 1 99 1 ). isolated patches patches may may attract attract aa disproportionate disproportionate number number of of migrants migrants ((Bell, Properties also be physical features in the the Properties of of the the matrix matrix may may also be decisive. decisive. Certain Certain physical features in matrix matrix may may act act as as corridors corridors and and thus thus channel channel migrating migrating individuals individuals to to certain certain patches expense of 1 995). patches on on the the expense of others others (Ims, (Ims, 1995). For For passively passively dispersing dispersing organisms, organisms, for for example example wind-dispersed wind-dispersed seeds, seeds, the the stochastic components of encounter rates with the the majority majority stochastic components of patch patch encounter rates tend tend to to be be large large with of all. The probability of of a of dispersers dispersers not not reaching reaching any any new new habitat habitat at at all. The probability a disperser disperser reaching selective pressure reaching aa new new patch patch may may be be an an important important selective pressure shaping shaping life life history history traits structure. Organisms traits of of organisms organisms with with aa metapopulation metapopulation structure. Organisms that that disperse disperse pas passively large sively and and thus thus have have low low patch patch encounter encounter probabilities probabilities typically typically produce produce large numbers of expense of numbers of offspring offspring at at the the expense of the the quality quality of of each each offspring. offspring. The The opposite opposite is to orient is true true for for actively actively migrating migrating animals animals that that are are able able to orient toward toward patches patches from from some case the the encounter may be ciently high some distance. distance. In In this this case encounter rate rate may be suffi sufficiently high to to permit permit the the individual individual to to sample sample several several patches patches before before settlement settlement and and thus thus to to increase increase the likelihood of habitat selection selection involves the likelihood of successful successful establishment. establishment. Patch Patch or or habitat involves decisions patch characteristics) characteristics) decisions based based on on the the assessment assessment of of environmental environmental cues cues ((patch that that reflect reflect the the quality quality of of the the patch. patch. Apart Apart from from patch patch characteristics characteristics such such as as patch patch size and also depend the patch is already size and quality, quality, the the settlement settlement may may also depend on on whether whether the patch is already occupied cs or Immigration into occupied by by conspecifi conspecifics or not. not. Immigration into already already established established populations populations may (by conspecific may be be facilitated facilitated (by conspecific attraction attraction or or facilitation; facilitation; Smith Smith and and Peacock, Peacock, 11990; 990; Wood 987) or 1 982) Wood and and del del Moral, Moral, 11987) or prohibited prohibited (social (social fences; fences; Hestbeck, Hestbeck, 1982) by cs may by the the presence presence of of conspecifics. conspecifics. Heterospecifi Heterospecifics may similarly similarly facilitate facilitate or or impede impede
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immigration Danielson and Gaines, 11987; 987; Tilman, 993). However, immigration ((Danielson and Gaines, Tilman, 11993). However, colonization colonization of patches (no must by be possible possible in in a of empty empty patches (no conspecifics conspecifics present) present) must by definition definition be a metapopulation context. metapopulation context. Colonization Colonization and and immigration immigration in in metapopulations metapopulations are are not not conceptually conceptually very very different processes under settings. General theories of of habitat different from from the the same same processes under other other settings. General theories habitat selection such Rosenzweig ((1981) 1 98 1 ) may applicable to to metapopulations metapopulations stud selection such as as Rosenzweig may be be applicable studies ((Danielson, Danielson, 1992; 1 992; Morris, 992). However, However, generally would expect expect that ies Morris, 11992). generally one one would that the stochastic stochastic components habitat selection selection are the components of of habitat are greater greater in in metapopulations metapopulations than than otherwise the more get information information otherwise given given the more restricted restricted opportunities opportunities for for organisms organisms to to get on habitats being being unpredictably scattered in time and on the the availability availability of of habitats unpredictably scattered in time and space. space.
III. INDIREG, GLOBAL APPROACHES INDIRECT,GLOBAL APPROACHESTO TO PARAMETER PARAMETERESTIMATION ESTIMATION O Off the the three three transfer transfer stages stages defined defined above, above, colonization colonization has has been been ooff principal principal interest interest in in both both theoretical theoretical and and empirical empirical investigations investigations of of metapopulation metapopulation dynam dynamics. The historical; colonization ics. The reason reason for for this this is is partly partly historical; colonization is is the the only only stage stage that that appears Levin, 11969a, 969a, 11970). 970). ColCol appears explicitly explicitly in in Levins' Levins' metapopulation metapopulation model model ((Levin, onization onization rate rate has has also also remained remained the the key key transfer transfer parameter parameter in in the the most most recent recent extensions of 's model model ((Hastings Hastings and 994). extensions of Levins Levins's and Harrison, Harrison, 11994). Although Pokki, Although colonization colonization rate rate may may occasionally occasionally be be measured measured directly directly ((Pokki, 11981; 98 1 ; Solbreck, 99 1 ; Whitlock, Whitlock, 11992b; 992b; Valone Brown, 1995), 1 995), in in most most cases cases Solbreck, 11991; Valone and and Brown, it to estimate by direct colonization rates rates it is is an an impossible impossible task task to estimate by direct observation observation how how colonization change instance as change for for instance as aa function function of of interpatch interpatch distances. distances. For For both both practical practical and and conceptual most diffi cult of the three conceptual reasons, reasons, colonization colonization is is the the most difficult of the three transfer transfer stages stages to study. The culty hinges to study. The practical practical diffi difficulty hinges on on the the fact fact that that colonization colonization events events are are typically rare rare and and thus thus require require studies studies of of exceptionally exceptionally large spatial and temporal typically large spatial and temporal extents ((Wiens, Wiens, 11989a). 989a). In contrast, emigration emigration and migration are more local extents In contrast, and migration are more local and and continuous hinge on the fact fact that continuous processes. processes. The The conceptual conceptual difficulties difficulties hinge on the that coloni colonization the two two preceding zation is is the the most most complex complex transfer transfer stage, stage, being being aa function function of of the preceding stages (emigration (emigration and and migration) migration) of of the the transfer transfer process. process. stages Due Due to to the the great great difficulties difficulties of of obtaining obtaining direct direct estimates estimates of of colonization colonization rates, rates, most most empirical empirical metapopulation metapopulation studies studies have have employed employed indirect indirect approaches. approaches. The The simplest the colonization rate simplest approach approach to to getting getting an an indirect indirect global global estimate estimate of of the colonization rate for to solve the model known ex for aa Levins-type Levins-type metapopulation metapopulation would would be be to solve the model for for aa known extinction Unfortunately, tinction rate, rate, and and assume assume that that the the metapopulation metapopulation is is at at equilibrium. equilibrium. Unfortunately, direct to obtain direct estimates estimates of of extinction extinction rates rates are are at at least least as as hard hard to obtain as as estimates estimates of of colonization colonization rate. rate. Estimates Estimates of of extinction extinction rates rates require require long-term long-term data data sets, sets, which which are methods make are available available mainly mainly for for islands. islands. Even Even then, then, available available methods make homogeneity homogeneity assumptions assumptions (temporal (temporal and and spatial) spatial) which which need need to to be be tested tested before before these these methods methods can 994). A realistic empirical empirical approach can be be used used (Clark (Clark and and Rosenzweig, Rosenzweig, 11994). A more more realistic approach may two other other processes to study, may be be to to obtain obtain estimates estimates of of the the two processes which which are are easier easier to study, the the emigration emigration rate rate and and the the distribution distribution of of migration migration distances. distances. Such Such estimates estimates would migrating individuals individuals would give give aa measure measure of of the the expected expected density density distribution distribution of of migrating
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in expected encounter migrants with with empty in the the matrix matrix and and hence hence the the expected encounter rate rate of of migrants empty and and occupied patches. incidence function, function, describing occupied patches. Combined Combined with with the the incidence describing the the spatial spatial distribution of extinct extinct and and extant extant patches patches in in the the landscape, expected encounter encounter distribution of landscape, the the expected rates colonization (and extinction) rates rates by numerical rates may may be be used used to to derive derive colonization (and extinction) by numerical techniques. 1 994) have used such techniques. Recently Recently Hanski Hanski et et ai. al. ((1994) have used such an an approach approach to to develop develop spatially spatially explicit explicit metapopulation metapopulation models models (allowing (allowing for for spatially spatially variable variable popu population lation sizes sizes and and interpopulation interpopulation distances distances in in steady-state steady-state metapopulations) metapopulations) and and estimate estimate patch-specifi patch-specificc colonization colonization probabilities. probabilities. Compared Compared to to the the earlier, earlier, more more general 99 1 , 11994a; 994a; Hanski 993; Hastings general models models (see (see Hanski, Hanski, 11991, Hanski and and Gyllenberg, Gyllenberg, 11993; Hastings and 994), these and Harrison, Harrison, 11994), these new new spatially spatially explicit explicit models models represent represent aa significant significant advance specific meta advance for for predicting predicting the the equilibrium equilibrium and and transient transient dynamics dynamics of of specific metapopulations. still go populations. These These approaches approaches still go under under the the heading heading "global "global approaches" approaches" as as the estimates represent the parameter parameter estimates represent averages averages at at the the metapopulation metapopulation level; level; the the trans transfer spatially explicit sense that fer parameters parameters are are not not spatially explicit in in the the sense that per per capita capita rates rates might might vary sizes, matrix vary according according to to patch patch sizes, matrix conditions, conditions, etc. etc. This This assumption assumption about about ho homogeneous mogeneous processes processes may may limit limit the the applicability applicability of of this this approach approach as as predictive predictive tools tools in in real real management management situations. situations. Our Our greatest greatest concern concern in in this this context context relate relate to to how how the the estimates estimates are are obtained. obtained. First First of of all, all, this this approach approach requires requires aa relatively relatively large large number number of of parameters parameters to to be be estimated, most likely example, in model by estimated, most likely from from the the same same data data set. set. For For example, in the the model by 1 994) there parameters of related to to Hanski Hanski et et al. al. ((1994) there were were eight eight parameters of which which five five were were related I , carrying patch patch (population) (population) specifi specificc dynamics dynamics (0' (a --~, carrying capacity; capacity; r, r, density-indepen density-independent dent growth growth rate; rate; e, 0, aa parameter parameter measuring measuring the the strength strength of of density density dependence; dependence; and finally, finally, Ssdd and and Se Se,, two tWO parameters parameters measuring, measuring, respectively, respectively, demographic demographic and and and environmental stochasticities stochasticities in extinction rates). environmental in extinction rates). Three Three parameters parameters describe describe the the two dispersal (c, constant two transfer transfer stages stages emigration emigration and and dispersal constant emigration emigration rate; rate; T, z, aa parameter describing the p." constant parameter describing the migration migration distances; distances; and and/.~, constant mortality mortality rate rate dur during migration). migration). While While aa few few of of the the parameters were obtained by direct direct estimation estimation ing parameters were obtained by based cally designed -recapture studies i.e., emigration rate based on on specifi specifically designed capture capture-recapture studies ((i.e., emigration rate and others were curve fitting five population population and dispersal dispersal distances), distances), others were obtained obtained by by curve fitting (the (the five specifi specificc parameters) parameters) or or (subjective) (subjective) adjustment adjustment procedures procedures (mortality (mortality of of mi migrants). in grants). This This approach approach to to deriving deriving estimates estimates of of demographic demographic parameters parameters stands stands in contrast the art contrast to to the the state state of of the art of of recent recent demographic demographic research research which which emphasizes emphasizes ((1) 1 ) rigorous rigorous designs designs for for sampling sampling statistical statistical populations populations (e.g., (e.g., Burnham Burnham et et al., al., 11987; 987; Skalski 992), (2) Skalski and and Robson, Robson, 11992), (2) analyses analyses based based on on the the formulation formulation of of al alternative their ternative statistical statistical models models for for estimating estimating single single demographic demographic parameters parameters and and their et ai., statistical intervals intervals (under (under the the most most appropriate appropriate statistical statistical model) model) (Lebreton (Lebreton et al., statistical 11992, 992, 11993), 993), and the outcome and (3) (3) tests tests of of specifi specificc hypotheses hypotheses deducing deducing the outcome of of inter interactions between mechanisms mechanisms and actions between and processes. processes. For For example, example, in in the the case case of of the the three three transfer 1 994) it not clear transfer parameters parameters of of Hanski Hanski et et al. al. ((1994) it is is not clear whether whether the the data data collected collected for for estimation estimation purposes purposes was was aa representative representative sample sample for for the the whole whole metapopula metapopulation. was assumed, assumed, but but not not tion. Further, Further, spatial spatial constancy constancy of of the the parameter parameter estimates estimates was tested. Finally, no nor could could they they have have been tested. Finally, no statistical statistical intervals intervals were were provided, provided, nor been as as the the parameters parameters were were not not obtained obtained independently independently of of each each other. other.
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However, models such However, we we will will not not dispute dispute the the fact fact that that models such as as those those of of Hanski Hanski 1 994) may ((1994a) 1 994a) and and Hanski Hanski et al. ((1994) may be be useful useful for for predicting predicting the the dynamics dynamics of of the the specific specific metapopulation metapopulation for for which which they they were were developed. developed. In In fact, fact, sensitivity sensitivity analyses analyses showed showed that that the the predictions predictions did did not not depend depend very very much much on on the the choice choice of of specific specific values values of of the the parameters parameters determining determining transfer transfer rates. rates. However, However, this this may may imply imply that that the the dynamics dynamics of of these these butterfl butterflyy metapopulations metapopulations are are mostly mostly affected affected by by other other processes. processes. The The dynamics dynamics of of other other metapopulations metapopulations may may be be more more transfer transfer rate such cases power of is likely rate dependent. dependent. In In such cases the the predictive predictive power of models models is likely to to be be much much more more dependent dependent on on precise precise and and unbiased unbiased estimates estimates of of emigration, emigration, migration, migration, and and immigration/colonization immigration/colonization rates, rates, and and knowledge knowledge about about how how these these rates rates are are shaped shaped by various various ecological ecological factors. factors. Clearly Clearly there there is is aa need need for for assessing the reliability by assessing the reliability of of empirical empirical metapopulation metapopulation models, models, in in particular particular through through forecasting, forecasting, i.e., i.e., pre pre1 995). This not been diction diction under under aa different different set set of of conditions conditions (Conroy (Conroy et al. al.,, 1995). This has has not been done done to to our our knowledge, knowledge, but but would would probably probably provide provide knowledge knowledge of of the the most most crit critical ical parameters. parameters. Another obtaining indirect, indirect, global rates in Another method method of of obtaining global estimates estimates of of transfer transfer rates in metapopulations use population genetic methods methods to to estimate flow. The metapopulations is is to to use population genetic estimate gene gene flow. The estimation flow between between populations, pioneered by estimation of of gene gene flow populations, as as pioneered by Wright Wright (reviewed (reviewed in 969), has studies. In use in Wright, Wright, 11969), has been been the the subject subject of of numerous numerous studies. In particular, particular, the the use of ST statistics methods have have been been extensive extensive of Wright' Wright'ss F Fs~statistics and and spatial spatial autocorrelation autocorrelation methods (see 989; Epperson, 1 993). Such methods aim (see Slatkin Slatkin and and Barton, Barton, 11989; Epperson, 1993). Such methods aim at at estimating estimating the population size size of the product product of of Ne Ne,, the the effective effective population of each each local local population, population, and and m, the immigration rate. could therefore indirect measure measure the immigration rate. These These methods methods could therefore provide provide an an indirect of components of metapopulation dynamics, of one one of of the the essential essential components of metapopulation dynamics, immigration, immigration, and themselves either and colonization colonization rates. rates. However, However, such such indirect indirect methods methods are are by by themselves either unsuitable unsuitable for for metapopulations metapopulations as as they they assume assume spatially spatially homogeneous homogeneous popula populations 993) or is at tions (Epperson, (Epperson, 11993) or they they assume assume that that the the metapopulation metapopulation is at aa genetic genetic and and demographic equilibrium (Olivieri 1 99 1 ; Slatkin, 1 994, 1995). 1 995). demographic equilibrium (Olivieri et al., 1991; Slatkin, 1994, When indirect methods based When direct direct methods methods of of estimating estimating transfer transfer rates rates and and indirect methods based on they have been found on gene gene flow flow have have been been used used for for the the same same species, species, they have often often been found to give rather different different answers; indirect methods greater exto give rather answers; indirect methods typically typically indicating indicating greater ex tent 987). For For example, Slatkin (1987) ( 1 987) tent of of dispersal dispersal than than direct direct methods methods (Slatkin, (Slatkin, 11987). example, Slatkin described Euphy described very very limited limited dispersal dispersal distances distances for for the the checkerspot checkerspot butterfl butterflyy Euphydryas editha, using but extensive gene flow flow using using allele using direct direct observations, observations, but extensive gene allele frequencies. Similar results frequencies. Similar results have have been been found found for, for, e.g., e.g., the the Colombian Colombian ground ground columbian us ((Dobson, Dobson, 11994), 994), banner-tailed kangaroo rats squirrel, squirrel, Spermophilus Spermophilus columbianus banner-tailed kangaroo rats Dipodomys 1 99 1 ), and Dipodomys spectabilis spectabilis (Waser (Waser and and Elliott, Elliott, 1991), and grey-crowned grey-crowned babbler babbler Pomatostomus Edwards, 11993), 993), all which direct direct methods methods Pomatostomus temporalis temporalis ((Edwards, all species species for for which indicated limited dispersal. within a indicated limited dispersal. Moreover, Moreover, genetic genetic differentiation differentiation within a metapop metapopulation will will be origin (e.g., or different patches) ulation be dependent dependent on on the the origin (e.g., from from the the same same or different patches) and virgin or and nature nature (e.g., (e.g., virgin or inseminated inseminated females) females) of of the the dispersers, dispersers, and and therefore therefore the without supplementary the latter latter characteristics characteristics cannot cannot be be inferred inferred without supplementary observations observations ((McCauley, McCauley, 11991). 99 1 ). Slatkin 1 994) suggested Slatkin ((1994) suggested that that the the results results of of direct direct and and indirect indirect species is methods methods should should be be combined combined to to determine determine whether whether the the species is at at genetic/degenetic/de-
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mographic mographic equilibrium equilibrium and, and, more more generally, generally, the the consistency consistency of of the the different different apap proaches. proaches. We conclude conclude that that the the existing existing indirect indirect methods methods of of obtaining obtaining metapopulation metapopulation We level estimates estimates of of demographic demographic or or genetic genetic parameters parameters will will be be themselves themselves provide provide level little opportunity opportunity for for predicting predicting present present transfer transfer rates rates between between populations. popUlations. HowHow little ever, the the incidence incidence function function approach approach (Hanski, ( Hanski, 1992a, 1 992a, 1994a, 1 994a, this this volume) volume) and and ever, et al., af., 1994) 1 994) may may propro the genetic genetic methods methods (Hastings ( Hastings and and Harrison, Harrison, 1994; 1 994; Morin Morin et the vide useful useful information information on on migration migration on on longer longer time time and and larger larger spatial spatial scales scales than than vide direct methods methods could. could. The The fact fact that that metapopulation metapopulation dynamics dynamics may may change change dradra direct matically on on aa long long time time scale scale (Solbreck, (Solbreck, 1991) 1 99 1 ) shows shows that that this this is is of of great great emem matically pirical interest. pirical interest.
IV. DIRECT, D1REG, PATCH-SPECIFIC PATCH·SPECIFIC APPROACHES APPROACHES The rich textures textures of of real landscapes is is playing significant role re The rich real landscapes playing aa significant role for for the the resulting metapopulation dynamics (Wiens et et al., af., 1993). 1 993). Habitat Habitat characteristics characteristics sulting metapopulation dynamics (Wiens such quality, spatial distribution of of patches, of such as as patch patch area area and and quality, spatial distribution patches, and and presence presence of dispersal corridors or or barriers barriers (see (see Ims, Ims, 1995, 1 995, for for aa review) review) may influence local local dispersal corridors may influence population dynamics as as well as exchanges exchanges between between patches. In the way that that population dynamics well as patches. In the same same way demographic have focused focused on the sensitivity the demographic analyses analyses of of single single populations populations have on the sensitivity of of the population growth rate different demographic parameters (e.g., population growth rate to to the the different demographic parameters (e.g., Eberhardt Eberhardt et 994), metapopulation should use use similar et af., al., 11994), metapopulation studies studies should similar sensitivity sensitivity analyses analyses to to identify (Alvarez-Buylla, 1994). 1 994). The identify the the critical critical parameters parameters and and assumptions assumptions (Alvarez-Buylla, The great great difficulties dynamics of single populations difficulties involved involved in in the the analysis analysis of of the the dynamics of single populations will will surely surely propagate propagate at at the the metapopulation metapopulation level. level. We We do do not not think think the the indirect indirect ap approaches proaches outlined outlined above above could could be be developed developed much much further further without without focusing focusing more more 995). on (Conroy et on the the specific specific mechanisms mechanisms at at the the population population level level (Conroy et af., al., 11995). In In addition addition to to providing providing parameter parameter estimates estimates for for predictive predictive models, models, patch patchspecific specific studies studies may may provide provide tests tests about about common common assumptions assumptions made made in in metapop metapopulation ulation models. models. For For instance, instance, patch patch specific specific studies studies may may tell tell whether whether local local pop populations are demographically self-sustained or not, i.e., whether they are ulations are demographically self-sustained or not, i.e., whether they are sources sources or or sinks sinks in in aa deterministic deterministic sense, sense, and and whether whether exchange exchange rates rates between between patches patches are are sufficiently sufficiently low low to to justify justify the the common common assumption assumption of of independent independent dynamics dynamics of of local but see this volume). local populations populations ((but see Gyllenberg Gyllenberg et et af., al., this volume). Below we we review review some some of of the the most most recent recent methods methods that that can can be be used used to to obtain obtain Below patch-specifi patch-specificc estimates estimates of of emigration emigration and and immigration/colonization immigration/colonization rate rate as as well well as as methods methods of of studying studying the the spread spread of of individuals individuals (migration/dispersal) (migration/dispersal) in in the the ma matrix. trix. We We suggest suggest that that the the choice choice of of methods methods must must be be guided guided by by explicit explicit hypotheses hypotheses about about which which ecological ecological mechanisms mechanisms underlie underlie these these rates. rates. Moreover, Moreover, we we emphasize emphasize that that the the methods methods must must be be sound sound with with respect respect to to general general statistical statistical principles principles for for sampling sampling and and modeling. modeling. We We focus focus mainly mainly on on observational observational methods methods since since itit ap appears pears to to us us that that large-scale large-scale metapopulation metapopulation processes processes are are not not often often amenable amenable to to replicated, replicated, manipulative manipulative experiments. experiments. Experimental Experimental model model system system (EMS) (EMS) studies studies
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((Ims Ims and 989; Wiens 993), which for certain and Stenseth, Stenseth, 11989; Wiens et et at., al., 11993), which are are feasible feasible for certain "scaled-down" "scaled-down" systems, systems, may may be be important important as as aa general general method method of of exploring exploring ec ecological ( lms et 1 993). ological mechanisms mechanisms that that may may operate operate under under various various conditions conditions (Ims et at., al., 1993). EMS Yoc EMS studies studies may may also also provide provide tools tools for for validating validating new new statistical statistical methods methods ((Yoccoz, 1 994). For a summary of our survey, see Table I. coz, 1994). For a summary of our survey, see Table I.
A. Studies Studies of Migration In In plants, plants, migration migration in in metapopulations metapopulations is is generally generally restricted restricted to to seed seed dis dispersal, will usually persal, since since vegetative vegetative reproduction reproduction will usually not not allow allow for for colonizing colonizing distant distant patches. numerous, and relative importance importance patches. The The agents agents for for seed seed dispersal dispersal are are numerous, and their their relative varies between and within species Hughes et et at., 994). varies enormously enormously both both between and within species (e.g., (e.g., Hughes al., 11994). However, However, we we may may distinguish distinguish between between passive passive physical physical dispersal, dispersal, such such as as wind wind dispersal dispersal and and migration migration by by animals, animals, since since they they require require different different research research ap approaches. Some animals, also paspas proaches. Some animals, small small insects insects in in particular, particular, and and fungi fungi disperse disperse also sively (Shaw, (Shaw, 11995). 995). sively Wind been Wind dispersal dispersal is is aa physical physical process, process, and and the the seed seed shadow shadow has has therefore therefore been analyzed analyzed recently recently using using mechanistic mechanistic (physical) (physical) models models (e.g., (e.g., Greene Greene and and Johnson, Johnson, 1989a,b; 989; Andersen, 99 1 ). Such Such models models have have the the ad 1989a,b; Okubo Okubo and and Levin, Levin, 11989; Andersen, 11991). advantage c , even if more vantage of of not not being being sitesite- or or species-specifi species-specific, even if more empirical empirical approaches, approaches, using 1 987) are using for for example example physical physical models models of of seeds seeds (Augspurger (Augspurger and and Franson, Franson, 1987) are needed. the dispersal usually rather rather needed. While While the the average average or or mode mode of of the dispersal distribution distribution are are usually well such models Okubo and 1 989), the the tail tail of well explained explained by by such models (e.g., (e.g., Okubo and Levin, Levin, 1989), of long longdistance well known, known, partly because of sampling problems, problems, partly distance dispersal dispersal is is not not well partly because of sampling partly because of lumping for distances ((Portnoy Portnoy and Willson, 1993). 1 993). Most Most because of data data lumping for long long distances and Willson, seeds short distances distances (a giving only only small small sample sizes for for seeds disperse disperse short (a few few meters) meters) giving sample sizes long distances McEvoy and 987). However, it is is the the long tail of the long distances ((McEvoy and Cox, Cox, 11987). However, it long tail of the distribution which which is is of relevance to metapopulation dynamics. distribution of relevance to metapopulation dynamics. Local Local factors, factors, such such as patterns, may the distri as landscape landscape features features influencing influencing wind wind patterns, may greatly greatly modify modify the distribution expected expected in in more homogeneous environments McEvoy and and Cox, Cox, 11987). 987). bution more homogeneous environments ((McEvoy We are not aware for long We are not aware of of studies studies quantifying quantifying the the role role of of landscape landscape features features for long distance most studies distance seed seed dispersal, dispersal, as as most studies have have been been focusing focusing on on the the colonization colonization part transfer process process (e.g., Wood and 1 987; Augspurger part of of the the transfer (e.g., Wood and del del Moral, Moral, 1987; Augspurger and and Kitajima, 992; Tilman, 993). Kitajima, 11992; Tilman, 11993). Animal migration involving involving active movements is Animal migration active movements is typically typically more more complex complex than modes of dispersal. Still rather simple models of than passive passive modes of dispersal. Still rather simple mechanistic mechanistic models of ver vertebrate migration reasonably good migration tebrate migration may may show show reasonably good fit fit to to empirical empirical data data on on migration distances 1 987; Miller Miller and 989; Caley, 1 99 1 ; however, however, distances [e.g., [e.g., Buechner, Buechner, 1987; and Carroll, Carroll, 11989; Caley, 1991; Porter and 1 993) have this may from a bias]. Porter and Dooley Dooley ((1993) have shown shown that that this may result result from a sampling sampling bias]. These mechanistic models inadequate in setting These mechanistic models are, are, however, however, inadequate in aa metapopulation metapopulation setting as only parameter monotonically changchang as the the only parameter explicitly explicitly considered considered is is aa constant constant or or monotonically ing probability probability of stopping (settling dying) some distance from home site site ing of stopping (settling or or dying) some distance from the the home in homogeneous homogeneous landscape landscape ((Waser, Waser, 1985; 1 985; Buechner, 1 987; Caley, 99 1 ; Miller Miller in Buechner, 1987; Caley, 11991; and 989). and Carroll, Carroll, 11989).
TABLE TABLE II
Methodological MethodologicalApproaches Approachesto to the the Study Study of of Transfer Transfer inin Metapopulations Metapopulations
Approach
Spatial Spatial scale scale
Advantages
Inconvenients Inconvenients
References References
Incidence function
Metapopulation
Metapopulation Metapopulation
Experimental model model system system Experimental
Metapopulation: Metapopulation: Small Small absolute absolute spatial spatial scale scale
Large Large parameter parameter uncertainty poorly Mechanisms poorly known Large Large parameter parameter uncertainty uncertainty Ecological mechanisms mechanisms Ecological unknown unknown Artificial Artificial environments environments
Hanski ((1992a, 1992a, 11994a) 994a)
Gene Gene flow flow
Only Only presence/absence presence/absence data data and Large spatial and temporal scales scales temporal Large Large spatial spatial and and temporal scales scales temporal
Capture-recapture studies studies Capture-recapture
Metapopulation Metapopulation and and patch patch
Brownie et et al. al. ((1992); 1 992); Brownie Nichols et et al. al. ((1992) Nichols 1 992)
Radiotelemetry Radiotelemetry
Metapopulation Metapopulation and and patch patch
Require Require large large sample sample sizes sizes Components of of transfer transfer Components not separated separated not Small Small samples samples
Experimental Experimental release release in in matrix matrix Experimental Experimental release release in in patch patch
Metapopulation Metapopulation Patch Patch
Origin Origin of of individuals individuals Origin Origin of of individuals individuals
Artificial Artificial empty empty patches patches
Metapopulation and and patch patch Metapopulation
Harrison ((1989) Harrison 1989) Augspurger and and Kitajima Kitajima Augspurger Danielson and and ((1992); 1 992); Danielson Gaines ((1987) Gaines 1 987) Schoener ((1974); Schoener 1 974); Whitlock ((1992b) Whitlock 1 992b)
Enclosed Enclosed populations populations
Patch Patch
Access Access to to the the mechanisms influencing influencing the the components components of of transfer transfer Estimation Estimation of of patch patchspecific demography, demography, specific population sizes sizes and and population exchange rates rates exchange Allow Allow study study at at the the individual individual level level Aim Aim at at dispersal dispersal stage stage Aim Aim at at settlement settlement stage stage
Aim Aim at at colonization; colonization; control of of patch patch size size control and and isolation isolation Aim at at emigration/ emigration/ Aim colonization colonization
Artificial Artificial nature nature of of patches patches Small Small scale scale Fence artifacts artifacts Fence
Slatkin ((1994) Slatkin 1 994)
Huffaker ((1958); Forney Huffaker 1 958); Forney and Gilpin Gilpin ((1989) and 1 989)
White and and Garrott Garrott ((1990) White 1 990)
Johnson and and Gaines Gaines Johnson Valone ((1985, 1 985, 11987); 987); Valone and Brown Brown ((1995) and 1 995)
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Mechanistic Mechanistic computer computer simulation simulation models models provide provide aa more more realistic realistic approach approach to This approach involves computer to the the study study of of migration migration in in metapopulations. metapopulations. This approach involves computer simulations explicit mosaics consisting of simulations of of movement movement patterns patterns in in spatially spatially explicit mosaics consisting of lat lattices on the movement tices of of suitable suitable and and nonsuitable nonsuitable habitat. habitat. The The focus focus here here is is on the movement process factors. In simplest mod process itself itself and and its its interaction interaction with with environmental environmental factors. In the the simplest models the based on random els the algorithms algorithms are are usually usually based on simple simple or or first-order first-order correlated correlated random walks (e.g., 989; Wiens Wiens and 989; Johnson Johnson et 992a,b). walks (e.g., Gardner Gardner et et al., al., 11989; and Milne, Milne, 11989; et al., al., 11992a,b). The The cell cell size size of of the the lattice lattice corresponds corresponds to to the the grain grain size size (the (the smallest smallest spatial spatial scale scale at Wiens, 11989a1Kotliar 989a; Kotliar and at which which an an organism organism recognizes recognizes spatial spatial heterogeneity; heterogeneity; Wiens, and Wiens, 990). Spatial Wiens, 11990). Spatial structures structures at at larger larger scales scales emerge emerge as as clusters clusters of of cells. cells. The The clusters not be networks; Gardner 989; clusters may may or or may may not be connected connected (percolating (percolating networks; Gardner et et al., al., 119891 992a), depending that has Johnson Johnson et et al., al., 11992a), depending on on the the fraction fraction of of the the cells cells that has been been des designated the ignated as as suitable suitable area. area. Such Such spatial spatial mosaics mosaics can can be be modeled modeled to to resemble resemble the 992b) or spatial of real habitats (e.g., Johnson et spatial structures structures of real matrix matrix habitats (e.g., Johnson et at., al., 11992b) or may may be be constructed given proportion suitable and nonsuitable cells constructed by by assigning assigning aa given proportion of of suitable and nonsuitable cells at at random 989). random (Gardner (Gardner et et at., al., 11989). The individual movements may The computer computer simulation simulation approach approach to to modeling modeling individual movements may serve two two related related purposes. First, it serve as serve purposes. First, it may may serve as aa null null model model against against which which real real movement patterns of individuals individuals may may be be compared compared (e.g., (e.g., Turchin, Turchin, 11986). The movement patterns of 986). The second check whether whether a is a reasonable approximation second purpose purpose is is to to check a random random walk walk is a reasonable approximation of could be spatially explicit of real real movement movement behaviors behaviors and and hence hence could be assumed assumed in in spatially explicit at., 11994). 994). metapopulation Hanski, 1994a, 1 994a, Hanski metapopulation models models ((Hanski, Hanski et et al., Insects Insects and and other other ground ground living living invertebrates invertebrates moving moving slowly slowly on on the the ground ground are studies because because they easy are particularly particularly amenable amenable for for movement movement pathway pathway studies they are are easy to whole pathway the moving moving individuals together with with to observe observe such such that that the the whole pathway of of the individuals together relevant lengths (for discretized pathways) relevant parameters parameters such such as as speed speed or or step step lengths (for discretized pathways) and and tortousity quantified. Such measurements may to some tortousity can can be be measured measured and and quantified. Such measurements may even even to some extent be be obtained obtained by of radiotelemetry radiotelemetry (Andreassen (Andreassen et et al., al., 1993, 1 993, 1996a, 1 996a, extent by means means of 11996b) 996b) or 1 987; Oksanen, 1993) for or snow snow tracking tracking (Tegelstrom (Tegelstr6m and and Hansson, Hansson, 1987; Oksanen, 1993) for organisms organisms that that are are hard hard to to observe observe directly. directly. In In particular, particular, radiotelemetric radiotelemetric studies studies provide indicate the provide aa way way to to measure measure long-distance long-distance dispersal, dispersal, which which may may indicate the scale scale of more than of specifi specificc metapopulations. metapopulations. Radiotelemetry Radiotelemetry also also provides provides more than any any other other techniques mortality techniques opportunities opportunities to to obtain obtain direct direct estimates estimates of of rates rates and and causes causes of of mortality in the the matrix matrix (see below). in (see below). Wiens and and colleagues colleagues have have compared compared observed observed movement movement patterns patterns of of ground ground Wiens beetles and mechanistic simulation beetles and ants ants against against the the expectations expectations from from mechanistic simulation models models at., 1992b1Crist 1 992b; Crist et 992). Real Real pathways pathways (Wiens and Milne, 11989; 989; Johnson (Wiens and Milne, Johnson et et al., et at. al.,, 11992). usually differ expectations based walks usually differ from from the the expectations based on on first first order order correlated correlated random random walks models. also the (turning angles angles and models. This This is is also the case case when when the the input input parameters parameters (turning and step step 1992) and the lattice lengths) (Crist (Crist et lengths) et al., al., 1992) and the the spatial spatial structure structure of of the lattice (Johnson (Johnson et et al., al., 11992b) 992b) are empirical data. real movements movements have have a higher are based based on on empirical data. Generally, Generally, real a higher displacement less complex movements (Crist (Crist et displacement rate rate and and are are less complex than than simulated simulated movements et al., al., 11992). 992). This patterns have This is is likely likely to to be be caused caused by by the the fact fact that that most most movement movement patterns have a directional bias Kennedy, 119511 95 1 ; Johnson, Johnson, 11969). 969). a directional bias (e.g., (e.g., Kennedy,
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Not Not surprisingly, surprisingly, displacement displacement rates rates of of ground-living ground-living insects insects are are dependent dependent on the the spatial spatial structure structure of of the the vegetation vegetation as as well particular species species studied studied on well as as the the particular 992). Furthermore, Furthermore, the ((Wiens Wiens and 989; Crist and Milne, Milne, 11989; Crist et et ai. al.,, 11992). the insects insects apparently apparently respond to scale-dependent changes changes in (Johnson et respond to scale-dependent in the the vegetation vegetation structure structure (Johnson et ai. al.,, 11992a). 992a). However, intriguing and potentially important these However, the the most most intriguing and potentially important result result from from these studies is studies is that that the the structure structure of of the the pathways pathways (measured (measured by by their their fractal fractal dimension) dimension) appears appears to to be be similar similar for for different different species, species, in in different different vegetation vegetation types types and and over over the range of spatial scales being studied (Crist et at., 1 992). This gives hope for the range of spatial scales being studied (Crist et al., 1992). This gives hope for establishing that are establishing certain certain generalizations generalizations about about movement movement patterns patterns that are valid valid across across species, species, spatial, spatial, and and temporal temporal scales. scales. At At the the same same time, time, the the widespread widespread capacity capacity for for spatial spatial memory memory and and orientation orientation in 969; Danthanarayana, 986; Bell, Bell, 1991) 1 99 1 ) brings brings into in many many animals animals (Johnson, (Johnson, 11969; Danthanarayana, 11986; into question null model model approach. approach. Including more biologically plau question the the value value of of this this null Including more biologically plausible decisions, for sible assumptions assumptions about about movement movement decisions, for example example elements elements of of optimal optimal search Bell, 11991), 99 1 ), can models more more useful useful for for search theory theory (see (see Bell, can make make the the simulation simulation models improving our understanding understanding of patterns in in relation relation to to spatial improving our of dispersal dispersal patterns spatial structures. structures. Useful lines have have been made ((Kareiva Kareiva and Useful developments developments along along such such lines been recently recently made and Odell, 11987; 987; Turchin, 987, 1991; 1 99 1 ; Odendaal 1 988; Crist 992; MorMor Odell, Turchin, 11987, Odendaal et et ai., al., 1988; Crist et et at., al., 11992; ris, 11993; 993; Vail, Vail, 11993). 993). ris, Migration Migration may may be be risky, risky, especially especially in in metapopulation metapopulation settings settings characterized characterized by exposure times times to to unfavorable by long-distance long-distance migration migration and and long long exposure unfavorable environmental environmental conditions biological enemies. migration is is likely likely to to be conditions and and biological enemies. Mortality Mortality during during migration be aa more modifier of distribution for more significant significant modifier of the the migration migration distance distance distribution for species species in in metapopulations ltering effect" metapopulations than than otherwise. otherwise. To To understand understand and and predict predict the the "fi "filtering effect" of the the migration migration stage stage ((Lomolino, both on on demographic demographic and and genetic genetic proof Lomolino, 11993) 993) both pro cesses is important the importance importance cesses in in metapopulations, metapopulations, it it is important to to identify identify and and quantify quantify the of agents. While While the wind-dispersed seed may of the the different different mortality mortality agents. the fate fate of of aa wind-dispersed seed may largely events, whether displaced to to a largely depend depend on on chance chance events, whether it it happens happens to to be be displaced a suitable suitable patch more nonrandomly patch or or not, not, mortality mortality factors factors are are likely likely to to act act more nonrandomly among among actively actively migrating animals. animals. Unfortunately, Unfortunately, very very little little is is actually actually known known about about the the hazards migrating hazards involved dispersal in (Johnson and 1 990; Stenseth Stenseth and involved in in animal animal dispersal in general general (Johnson and Gaines, Gaines, 1990; and Lidicker, 992a). Lidicker, 11992a). Although mortality Although the the methodological methodological difficulties difficulties of of studying studying the the causes causes of of mortality during animals, recent recent advances in radioteleradiotele during migration migration are are formidable formidable for for most most animals, advances in metric Kenward, 11987; 987; White 990) bear promises for for metric techniques techniques ((Kenward, White and and Garrott, Garrott, 11990) bear promises species transmitters. Studies by Small species large large enough enough to to carry carry transmitters. Studies such such as as those those by Small et et ai. al. ((1993), 1 993), Larsen Boutin ((1994), 1 994), Steen 1 994), and Larsen and and Boutin Steen ((1994), and van van Vuren Vuren and and Armitage Armitage reasons for for more more optimistic optimistic expectations expectations for for the than those those ((1994) 1 994) give give reasons the future future than conveyed 1 990) in in their their general conveyed by by Johnson Johnson and and Gaines Gaines ((1990) general review review of of dispersal dispersal studies on given a migrating studies on birds birds and and mammals. mammals. In In particular, particular, given a large large sample sample of of migrating radiotagged radiotagged individuals, individuals, even even rough rough mortality mortality rate rate estimates estimates may may be be obtained obtained (unbiased, still large statistical intervals size limitations). (unbiased, but but with with still large statistical intervals due due to to sample sample size limitations). Hazard cally tailored of radiotagged radiotagged animals Hazard rate rate models models specifi specifically tailored to to data data on on the the fates fates of animals
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((Pollock Pollock et al., 989) provide cation of data with al., 11989) provide possibilities possibilities for for stratifi stratification of the the data with respect respect to (sex, age, etc.) as well as allowing one to characteristics characteristics of of the the dispersing dispersing animals animals (sex, age, etc.) as well as allowing one to to include include relevant relevant environmental environmental covariates covariates such such as as matrix matrix characteristics. characteristics. Clearly, the eco Clearly, such such analyses analyses could could contribute contribute greatly greatly to to our our understanding understanding of of the ecological logical processes processes determining determining transfer transfer rates rates in in metapopulations. metapopulations. The The sample sample size size requirements requirements may may still still be be beyond beyond the the logistic logistic constraints constraints of of most most research research projects, projects, and individuals for for most and precise precise estimates estimates of of mortality mortality rates rates of of migrating migrating individuals most species species may may only only be be obtained obtained from from extensive, extensive, adequately adequately designed designed and and analyzed analyzed capture capturerecapture studies (see (see below). recapture studies below).
B. Estimation Estimation of Transfer Transfer Rates Rates Using Using Capture-Recapture Capture-Recapture Methods Methods Capturing, has been been used Capturing, marking, marking, and and recapturing recapturing individuals individuals has used for for aa long long time Ford, time for for estimating estimating survival survival rates rates and and population population sizes sizes (e.g. (e.g.,, Fisher Fisher and and Ford, 11947) 947) as well as (Jackson, 11939). 939). However, is mainly mainly since since the the as well as migration migration rates rates (Jackson, However, it it is 1964), Jolly 1965), and development development of of the the stochastic stochastic models models of of Cormack Cormack ((1964), Jolly ((1965), and Seber Seber flexibility and (see Lebreton ((1965) 1 965) that that the the flexibility and usefulness usefulness of of the the models models become become clear clear (see Lebreton 992). Most Most models the analysis and and North, North, 11992). models focused focused on on the analysis of of single single populations, populations, or or the comparison of different populations (e.g., Burnham et al., 1 987; Lebreton the comparison of different populations (e.g., Burnham et al., 1987; Lebreton et et al., 992), with al., 11992), with fewer fewer developments developments regarding regarding the the estimation estimation of of exchanges exchanges be between 973). In rates are tween populations populations (e.g., (e.g., Amason, Amason, 11973). In the the former former models, models, survival survival rates are local survival rates rates in in the the sense sense that that they that an an individual local survival they measure measure the the probability probability that individual released will be the same + I1.. These esti released at at time time t will be alive alive in in the same population population at at time time t + These estimates Similarly, the mates confound confound mortality mortality and and emigration emigration rates. rates. Similarly, the recruitment recruitment rates rates will often will often confound confound immigration immigration and and reproduction reproduction in situ. situ. We We will will below below describe describe recent recent methods methods which which have have been been proposed proposed to to disentangle disentangle these these processes. processes. We We will focus focus on on methods methods which which do do not not assume assume any any spatial spatial or temporal homogeneity homogeneity will or temporal in the the migration migration processes processes (e.g., (e.g., Matsuda Matsuda and and Akamine, Akamine, 11994), as we we are are indeed indeed 994), as in interested interested in in the the spatial spatial and and temporal temporal variability variability of of these these processes. processes.
1. Estimation Estimation of of Immigration Immigration and and Reproduction Reproduction in in Situ Situ Using Using
Capture-Recapture
Capture- Recapture Studies Studies of of Single Single Populations Populations
Nichols and 1990) assumed Nichols and Pollock Pollock ((1990) assumed that that the the total total adult adult population population at at time time t+ ), is is made adults that present as + I1,, N(ad, N(ad, t + + 11), made of of three three groups: groups: adults that were were present as adults adults + 11); ); adults that immigrated immigrated between in in the the population population at at time time t, N(aa, N(aa, t + adults that between time time t and N(ai, t + adults which which were were present trappable young and time time t + + I1,, N(ai, + 1); adults present as as trappable young in in the the popUlation at time t, N(ay, that maturation time is to one population at time N(ay, t + + 11)) (assuming (assuming that maturation time is equal equal to one could then using time step). N(ai, time step). N(ai, t + + I1)) could then be be estimated estimated using N(aa, N(aa, tt + + I1)) = = ¢ad,t ~)ad,t Nc(ad, Nc (ad, t), t), N(ay, N(ay, t + + 11)) = = ¢y,t thy,tNc(Yo, Nc(yO, t), t),
1 ), N(aa, t + N(ai, N(ai, t + + I1)) = = Nc(ad, Nc(ad, t + + I1)) - N(ay, N(ay, t + + I1)) - N(aa, + 1),
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where ¢y.t) are young) survival where ¢ad,,( ~bad.,(4~y,,) are the the adult adult ((young) survival rates rates from from time time tt to to tt + 11 estimated estimated using using open open population population models, models, and and Nc Nc denotes denotes population population sizes sizes estimated estimated using using closed closed population population models. models. This This approach approach is is also also useful useful in in documenting documenting whether whether the population population has has aa rate rate of of increase increase which which is is larger larger or or smaller smaller than than one, one, in in the the the absence of of immigration: immigration: we we use use the the ratio ratio [N(aa, [N(aa, tt + + 11)) ++ N(ay, N(ay, tt + + 11)]/ absence )]/ ( Yo, t)/Nc(ad, Nc(ad, Nc(ad, t) t) = - ¢ad,t qSaa,, + + ¢y,t ~by,,Nc Nc(yO, t)/Nc(ad, t). t). The The latter latter ratio ratio represents represents the the repro reproductive rate rate for for the the adults. It may may be be compared compared to to more more direct direct estimation, estimation, using, using, ductive adults. It e.g., data data on on clutch clutch size size and and proportion proportion of of breeders. breeders. e.g., Another recent recent approach approach is is provided provided by by Pradel Pradel ((1996). Instead of of considering considering 1 996). Instead Another capture histories histories forward forward ((i.e., from time time tt to to tt + + 1), we we could could read read it it back backcapture i.e., from ward: "survival" "survival" rates rates would would then then be be the the probability probability that that an an animal animal caught caught at at time time ward: tt + + 11 was was present present in in the the population population at at time time t.t. Such Such probabilities probabilities are are called called se seniority probabilities probabilities by by Pradel Pradel ((1996) and represents represents the the resident resident fraction fraction of of the the niority 1 996) and population. These These probabilities probabilities could could be be estimated estimated using using similar approaches than than population. similar approaches are used for survival rates 1 990; Lebreton et al., 992). As As poppop are used for survival rates (Pollock (Pollock et et al., al., 1990; Lebreton et al., 11992). ulation growth growth rates rates are are related related to to both both survival rates and and seniority seniority probabilities probabilities ulation survival rates Pradel's approach approach could could also also be be used used to estimate population population growth growth rates rates ((inPradel's to estimate in cluding immigration), without without relying relying on on estimation of population population sizes. sizes. The The ro rocluding immigration), estimation of bustness of survival survival rate rate estimation estimation ((Lebreton et al., al., 11992) also apply apply to to the the 992) may may also bustness of Lebreton et estimation of seniority seniority probabilities and therefore therefore to to the the estimation of population estimation of probabilities and estimation of population growth rates. rates. This This approach approach would would then then be be an an alternative alternative to to the the use of closed growth use of closed population models models when when the the latter latter cannot cannot be be used. used. population Both making no no specifi assumptions about Both methods methods present present the the advantage advantage of of making specificc assumptions about the dynamics of population studied; estimating the the rate increase the dynamics of the the population studied; they they aim aim at at estimating rate of of increase of the the population, population, with and without is one of the crucial of with and without immigration, immigration, which which is one of the crucial parameters metapopulation dynamics. statistical models, parameters in in metapopulation dynamics. However, However, as as all all statistical models, they they rely growth rates rates of rely on on some some assumptions, assumptions, in in particular particular about about the the growth of the the young young for for Nichols aI., 1993) 1 993) and and about about the the absence absence of of Nichols and and Pollock's Pollock's approach approach (Yoccoz ( Yoccoz et et al., age-dependency in the Pradel's approach (i.e., can only only be age-dependency in the PradeI ' s approach ( i.e., parameters parameters can be time-detime-de pendent but for the pendent but not not age-dependent; age-dependent; Pradel, Pradel, 1996). 1 996). Nevertheless, Nevertheless, at at least least for the NichNich ols and Pollock's appropriate study study design design could ensure that that these ols and Pollock' s approach, approach, appropriate could ensure these asas sumptions are checked checked (using (using individual growth rates) sumptions are individual growth rates) and and eventually eventually modified modified and Lambin, Lambin, 1997). 1 997). ((Yoccoz Yoccoz and
and Emigration between Populations 2. Estimation of of Immigration and PopUlations Using Using Capture-Recapture Capture - Recapture Studies Statistical for estimating estimating survival survival rates Statistical models models for rates from from capture-recapture capture -recapture data data in in open open populations populations are are rather rather robust robust to to heterogeneity heterogeneity of of capture capture probabilities probabilities or or trap trap response, response, aa common common problem problem in in many many ecological ecological studies. studies. Survival Survival rate rate eses timation is is therefore therefore recommended recommended for for testing testing biological biological hypotheses hypotheses instead instead of of timation population Population size population size size estimates estimates (Lebreton ( Lebreton et et al., al., 1992). 1 992). Population size estimates estimates are are based based on on closed closed population popUlation models models for for which which heterogeneity heterogeneity and and trap trap response response cannot al., 1990; 1 990; Yoccoz Yoccoz et et al., al. , 1993). 1 993). cannot be be simply simply modeled modeled (Pollock ( Pollock et et al.,
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There is is of of course course no no way way that that emigration emigration rates rates could could be be estimated estimated without without There trapping outside outside the the study study area. area. In In contrast, contrast, immigration immigration can can be be directly directly estimated estimated trapping using, for for example, example, two two different different approaches approaches (see (see above). above). One One obvious obvious way way of of using, estimating emigration emigration rates rates has has been been to to assume assume the the equality equality of of immigration immigration and and estimating emigration rates rates ((Waser et al., al., 11994; see also also Alberts Alberts and and Altmann, Altmann, 11995). In the the 994; see 995). In emigration Waser et context of of metapopulation dynamics, the the approach approach of of Waser Waser et et al. al. ((1994) seems context metapopulation dynamics, 1994) seems to be be of of limited limited interest, interest, since since it it is is assuming assuming that that all all populations populations are are at at demo demoto graphic equilibrium. equilibrium. This This assumption is often often not not validated demographic graphic assumption is validated using using demographic data (survival (survival and and reproductive reproductive rates), rates), and and the the accuracy accuracy of of this this approach approach is is there theredata fore difficult difficult to to assess. assess. In In practice, practice, to to estimate estimate emigration emigration rate, rate, it it is is necessary necessary to to fore monitor populations connected connected by by movements movements of of individuals. individuals. While While this this monitor several several populations approach will provide approach will provide us us only only with with estimation estimation of of rates rates of of exchanges exchanges (which (which result result from emigration emigration from from aa given given population population and a n d mortality mortality during during the the dispersal dispersal move movefrom ment), it it is is aa necessary necessary fi first step in in the the direction direction of of direct direct estimation estimation of of emigration emigration ment), rst step rate. rate. Existing models models for for estimating estimating exchange exchange rates rates based on capture capture-recapture Existing based on - recapture or resighting data cases of models (sensu or sightsight-resighting data are are special special cases of multiple multiple strata strata models (sensu Brownie et et al., Nichols et et al., these models models the the strata strata correspond correspond Brownie al., 11992; 992; Nichols al., 11992). 992). In In these to different different populations in the the study. These models an extension extension to populations included included in study. These models provide provide an of usual usual capture capture-recapture models, where where there there is is only only one one survival survival rate: of -recapture models, rate: the the survival rate within the the focal focal population. population. This approach has successfully survival rate within This approach has been been successfully used for for estimating estimating exchange exchange rates rates in geese ((Hestbeck et al., al., 1991), 1 99 1 ), but but as as these these used in geese Hestbeck et authors recognize recognize the the method method has has limitations limitations which which can can probably be overcome overcome authors probably be through appropriate appropriate study study design. design. through The fi first limitation is is due due to to the the large large number number of of parameters the model, model, The rst limitation parameters in in the since we need to all pairwise pairwise exchange since we need to incorporate incorporate all exchange rates rates between between populations. populations. The parameters vary with populations The parameters may may also also vary with time. time. For For example, example, with with five five populations studied rates + studied during during 3 years, years, we we have have five five local local survival survival rates + 2 •X 10 1 0 exchange exchange rates rates per survival parameters. yearly interval, interval, that that is is 50 survival parameters. If If we we add add to to this this the the recapture recapture per yearly parameters, parameters in in the the full full model, model, which which will will obviously obviously parameters, we we end end up up with with 60 parameters require aa large large data data set set to draw any any reliable assumed require to draw reliable inferences. inferences. Moreover, Moreover, it it is is assumed that the the probability probability of survival for for aa given that of survival given individual individual (within (within or or in in another another poppop ulation) was independent of of its its previous previous history history (in ( in particular it has ulation) was independent particular whether whether it has previously between populations previously moved moved or or not). not). Obviously, Obviously, if if transfer transfer between populations is is aa unique unique event in the the individual individual lifetime, lifetime, this this assumption will not not be be satisfied. satisfied. Brownie Brownie et et event in assumption will al. al. (1992) ( 1 992) have have analyzed analyzed models models where where this this assumption assumption was was relaxed relaxed (by ( by including including additional additional parameters), parameters), but but as as they they pointed pointed out, out, it it resulted resulted in in models models with with aa very very large large number number of of parameters, parameters, probably probably too too large large for for most most available available data data sets. sets. Our opinion is that this estimates of Our opinion is that this approach approach is is necessary necessary to to get get reliable reliable estimates of exchange exchange rates. rates. However, However, it it will will probably probably be be possible possible to to implement implement it it mainly mainly for for either either experimental experimental model model systems systems (for (for which which the the number number of of parameters parameters can can be be reduced reduced by by design: design: assuming assuming equal equal exchange exchange rates rates between between identical identical patches) patches) or or species for which species for which it it is is possible possible to to obtain obtain large large samples samples (Hestbeck ( Hestbeck et et al., al., 1991). 1 99 1 ).
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Roll A. Ims Irns and Nigel Nigel G. G. YOCCOI Yoccoz Rolf
V. EXPERIMENTAL EXPERIMENTALAPPROACHES APPROACHESTO TO STUDYING STUDYINGTRANSFER TRANSFERRATES RATES V. METAPOPULATIONS IIN N METAPOPULATIONS Experimental manipUlations manipulations of of popUlations populations have have been been used used to to identify identify some some Experimental of the the causal causal mechanisms mechanisms affecting affecting each each of of the the components components of of transfer transfer between between of populations, emigration, emigration, migration, migration, and and colonization. colonization. In In the the following, following, we we will will populations, review the the experimental experimental approaches approaches that that have have been been used used to to explore explore these these three three review components. components.
A. Emigration Emigration A. Apart from from aa few few studies studies in in which which emigration emigration has has been been studied studied by by direct direct Apart observation of of marked marked individuals individuals leaving leaving habitat habitat patches patches (e.g., (e.g., Lawrence, Lawrence, 11987b; observation 987b; Odendaal et et ai., al., 11989), many experimental experimental studies studies have have relied relied upon upon certain certain de deOdendaal 989), many vices to to trap trap individuals individuals attempting attempting to to leave leave the the experimental experimental popUlations. populations. The The vices use 978a; use of of fences fences (enclosures) (enclosures) has has been been extensive extensive in in vertebrate vertebrate studies studies (Gill, (Gill, 11978a; Johnson 985, 11987; 987; Danielson 987; Bondrup-Nielsen, Johnson and and Gaines, Gaines, 11985, Danielson and and Gaines, Gaines, 11987; Bondrup-Nielsen, 11992), 992), but Hertzberg, 1996). 1 996). Individuals but also also in in aa few few studies studies on on invertebrates invertebrates ((Hertzberg, Individuals attempting to to leave leave the the population population are are forced forced to to move move along along fences fences and and ultimately ultimately attempting enter aa fence fence trap trap (Johnson (Johnson and and Gaines, Gaines, 11985, Bondrup-Nielsen, 11992; Aars enter 985, 11987; 987; Bondrup-Nielsen, 992; Aars et et al., ai., 11995). 995). Emigration Emigration rates rates are are then then measured measured by by the the proportion proportion of of individuals individuals caught in in the the fence fence traps. traps. However, However, with with this this method method it may be problematic to to caught it may be problematic distinguish between between emigration emigration and and short-distance short-distance exploratory exploratory movements movements out outdistinguish side the the habitat habitat patch. patch. This This problem problem can can be be somewhat somewhat eased eased by by increasing increasing the the side spatial of the the experiment, for instance instance by by allowing allowing for for aa quite quite large large distance spatial scale scale of experiment, for distance from border to (e.g., Hertzberg, Hertzberg, 1996) from the the patch patch border to the the fence fence (e.g., 1 996) or or by by using using semipersemiper meable border zones, meable border zones, for for instance instance water water (Ims, ( lms, 1989). 1 989). Experimental studies on the effect on emigration emigration have have examined examined the effect of of factors factors such such Experimental studies as population density (e.g., (e.g., Kareiva, patch size as popUlation density Kareiva, 1985; 1 985; Lawrence, Lawrence, 1987b), 1 987b), patch size (Kareiva, ( Kareiva, 1985; patch shape et al., ai., 1993), 1 993), and and edge edge characteristics characteristics 1 985; Turchin, Turchin, 1986), 1 986), patch shape (Harper ( Harper et (Back, on emigration rate. Not surprisingly, the experiments have shown ( Back, 1988b) 1 988b) on emigration rate. Not surprisingly, the experiments have shown that different species species often differently to to such such factors factors depending depending on on social social that different often respond respond differently structure (Wiens and perception of the patch edges structure (Wiens et et al., ai., 1993) 1 993) and perception of the "hardness" "hardness" of of patch edges (Stamps et ai., 1987; 1 987; Ims, Ims, 1995). 1 995). (Stamps et al.,
B. Migration Migration The The most most common common experimental experimental approach approach to to migration/dispersal migration/dispersal studies studies is is to habitat for for the to release release individuals individuals on on areas areas assumed assumed to to be be matrix matrix habitat the species. species. One One problem approach is problem with with this this approach is that that migration migration may may be be imposed imposed on on individuals individuals not not motivated motivated to to migrate. migrate. The The migration migration stages stages may may be be easy easy to to distinguish distinguish in in some some organisms (e.g., (e.g., seed seed in in plants plants or or certain life stage stage in in insects; insects; Johnson, Johnson, 1969), 1 969), but but organisms certain life there there may may be be much much individual individual variation variation in in the the migration migration tendency tendency in in other other oror ganisms (Swingland (Swingland and and Greenwood, Greenwood, 1983; 1 983; Chepko-Sade Chepko-Sade and and Halpin, Halpin, 1987). 1 987). ganisms
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Combining to define individuals prone prone Combining studies studies on on emigration emigration (e.g., (e.g., using using fences) fences) to define individuals to relevant migration experiments might of getting to emigrate emigrate with with relevant migration experiments might be be aa way way of getting around around the animals. The the problem problem of of studying studying migration migration on on an an inadequate inadequate sample sample of of animals. The most most common in the common factor factor affecting affecting migration migration in the release release studies studies is is the the physical physical structure structure of instance, Crist Crist et 1 992) manipulated of the the matrix matrix habitat. habitat. For For instance, et al. ((1992) manipulated the the structure structure Eloides beetle beetle of vegetation in studies of movement patterns three Eloides of the the vegetation in their their studies of movement patterns of of three species. species. The The possible possible role role of of linear linear habitats habitats as as dispersal dispersal corridors corridors in in highly highly fragmented fragmented landscapes (see (see Simberloff Simberloff et al., 11992; 992; Hobbs, Hobbs, 1992) 1 992) has landscapes has been been addressed addressed by by aa few experimental studies studies on LaPolla and Barrett, 11993; 993; Andreassen Andreassen few experimental on small small rodents rodents ((LaPolla and Barrett, et al., 11996a,b). 996a,b). For 1 996a) manipulated width For example, example, Andreassen Andreassen et et al. ((1996a) manipulated the the width of dispersal corridors root voles of dispersal corridors and and found found that that root voles Microtus Microtus oeconomus oeconomus showed showed highest highest transfer transfer rates rates between between patches patches connected connected with with corridors corridors of of intermediate intermediate width line movement movement in in the width due due to to both both high high emigration emigration rates rates and and straight straight line the corridor. corridor. There experimental studies studies on the effect corridor There is is aa great great need need for for experimental on the effect of of various various corridor designs for benefit of conservation measures Mann and and Plum designs for the the benefit of more more effective effective conservation measures ((Mann Plummer, 993). Inglis 1 992) give guidelines about mer, 11993). Inglis and and Underwood Underwood ((1992) give useful useful guidelines about the the design design of of experiments experiments addressing addressing transfer transfer rates rates in in patch patch systems systems connected connected with with corri corridors. dors.
C. Colonization/Immigration Colonization/Immigration Without why a hab Without additional additional information information it it remains remains often often an an open open question question why a habitat patch patch which which is is seemingly seemingly suitable suitable for for aa species nonetheless empty. There itat species is is nonetheless empty. There are First, the be suitable suitable but but is is empty because the are two two possibilities. possibilities. First, the patch patch may may be empty because the local population population has extinct or because it dispersal range from local has gone gone extinct or because it is is beyond beyond the the dispersal range from any in some any extant extant population. population. Although Although observational observational studies studies may may in some cases cases provide provide useful information information (e.g., (e.g., long-term long-term studies studies may may reveal colonization events; useful reveal colonization events; Valone Valone and Brown, 11995), 995), and the migration and Brown, and incidence incidence functions functions may may indicate indicate the migration ranges ranges ((Peltonen Peltonen and Hanski, 1991), 1 99 1 ), experimental required to and Hanski, experimental studies studies are are often often required to determine determine under possible. under which which conditions conditions colonization/immigration colonization/immigration onto onto aa given given patch patch is is possible. Harrison 1 989) used in which which individually marked Harrison ((1989) used an an experimental experimental approach approach in individually marked checkers pot butterflies were released in the matrix habitat habitat at increasing distance checkerspot butterflies were released in the matrix at increasing distance from an an empty empty habitat patch in order to to estimate estimate colonization colonization probabilities as aa from habitat patch in order probabilities as function the distance distance from the release release point. point. Colonization rates may also be be function of of the from the Colonization rates may also estimated by by introducing introducing artifi cial patches proved to within migration migration estimated artificial patches ((proved to be be suitable) suitable) within range local populations populations (Schoener, 1 974; Whitlock, Whitlock, 11992b). 992b). range from from known known local (Schoener, 1974; Second, for colonization in its its present Second, the the patch patch is is not not in in fact fact suitable suitable for colonization in present state state for most common determining whether whether empty for the the species. species. The The most common approach approach to to determining empty patches suitable or not is is translocation/transplantation experiment (e.g., (e.g., Har patches are are suitable or not translocation/transplantation experiment Harrison, 1 989; Massot Massot et al., 11994). 994). More More information information about which factors factors determine rison, 1989; about which determine patch suitability suitability for for colonization colonization have obtained by varying patch have been been obtained by experiments experiments varying patch size (Schoener Spiller, 1995), 1 995) , pro patch characteristics characteristics such such as as patch/island patch/island size (Schoener and and Spiller, propagule 973; Ebenhard, 987), the presence of pagule size size (Crowell, (Crowell, 11973; Ebenhard, 11987), the presence of predators predators (Schoener (Schoener
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and Spiller, Spiller, 1995), 1 995), and and conspecific conspecific and and heterospecific heterospecific residents residents in in the the patch patch (Dan(Dan and ielson and and Gaines, Gaines, 1987; 1 987; Tilman, Tilman, 1993; 1 993; Valone Valone and and Brown, Brown, 1995). 1 995). All All these these ielson factors have have been been shown shown to to have have effects effects on on immigration immigration rate rate and and colonization colonization factors success, but but the the strength, strength, and and sometimes sometimes even even the the sign, sign, of of the the effect effect may may vary vary success, between between species. species. For For example, example, the the presence presence of of conspecifics conspecifics may may decrease decrease (intra( intra specific specific competition; competition; Danielson Danielson and and Gaines, Gaines, 1987) 1 987) or or increase increase the the success success of of immigrants (conspecific (conspecific attraction; attraction; Stamps, Stamps, 1991, 1 99 1 , or or Allee-effect) Allee-effect) and and the the same same immigrants applies to to the the presence presence of of heterospecifics heterospecifics (negative (negative effects effects due due to to interspecific interspecific applies competion, Valone Valone and and Brown, Brown, 1995; 1 995; Tilman, Tilman, 1993; 1 993; versus versus interspecific interspecific facilifacili competion, tation or or nursing nursing effect, effect, Wood Wood and del Moral, Moral, 1987). 1 987). The The outcome outcome of of transplantransplan tation and del tation experiments experiments may may depend depend on on the the time time scale. scale. It It seems seems important important to to extend extend tation the experiment experiment beyond beyond the the settlement settlement phase phase as as certain certain patch-specific patch-specific factors factors may may the take take aa long long time time to to exert exert their their effects effects (e.g., (e.g., Schoener Schoener and and Spiller, Spiller, 1995). 1 995). Such Such experiments should pay more more attention attention to to characteristics characteristics of of the the translocated translocated inin experiments should pay dividuals as this may have the success the propagules dividuals as this may have profound profound influence influence on on the success of of the propagules ( Bright and Morris, 1994). 1 994). (Bright and Morris,
CONClUSION VI. CONCLUSION In this we have have focused focused on migration, and and immigration/ immigration/ In this chapter chapter we on emigration, emigration, migration, colonization as processes, which which need need to to be be approached empirically by by colonization as separate separate processes, approached empirically different study At the the same same time, time, the the three three transfer transfer stages stages are clearly different study designs. designs. At are clearly linked both both methodologically methodologically and and biologically. biologically. For For example, example, emigration, dis linked emigration, dispersal, and immigration/colonization must must be be translated translated into into transfer transfer rates rates to to be be persal, and immigration/colonization used predictive models models at metapopulation level. used in in predictive at the the metapopulation level. Likewise, Likewise, the the three three transfer transfer processes also linked linked to processes are are also to patch-specific patch-specific demographic demographic parameters parameters (Stacey (Stacey and and Taper, volume). For For example, example, emigration emigration rate rate may may be be aa function of population population Taper, this this volume). function of density 983), and density and and growth growth rate rate (Stenseth, (Stenseth, 11983), and immigration immigration and and colonization colonization suc success may colonists (Back, 1 988a,b; cess may depend depend on on local local densities densities and and number number of of colonists (Back, 1988a,b; Lawrence, 1 987a; Augspurger 1 992; Stenseth 992a). Lawrence, 1987a; Augspurger and and Kitajima, Kitajima, 1992; Stenseth and and Lidicker, Lidicker, 11992a). Methods will Methods used used for for estimating estimating popUlation population growth growth rates rates or or density density dependence dependence will often in particular particular capture often be be the the same same as as the the ones ories for for estimating estimating exchange exchange rates rates ((in capture- recapture methods). The choice of most effi cient study study design design should should be recapture methods). The choice of the the most efficient be guided by parameters as well as guided by the the critical critical parameters as well as how how complementary complementary information information about about these these parameters parameters may may be be obtained. obtained. We We believe believe that that no no single single approach approach will will suffice if we we are are to to advance advance our our understanding understanding of of the the transfer transfer processes. processes. A A com comsuffi ce if bination of of individual-based individual-based approaches approaches to to study study dispersal dispersal patterns, patterns, patch-specifi patch-specificc bination demographic demographic studies studies to to estimate estimate exchange exchange rates rates between between populations, populations, as as well well as as specific specific experiments experiments for for exploring exploring the the causal causal links links between between environmental environmental factors factors and will provide and the the focal focal processes processes will provide aa robust robust framework framework for for empirical empirical transfer transfer rate rate studies. studies. We We have have advocated advocated the the importance importance of of patch-specific patch-specific parameter parameter estimates estimates in in the context context of of proper proper hypothesis testing, study study design, design, and and statistical statistical modeling. modeling. the hypothesis testing,
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Global Global approaches approaches to to obtaining obtaining average average values values over over the the population population of of patches patches may ce to models. In case, on their may or or may may not not suffi suffice to parameterize parameterize specifi specificc models. In any any case, on their own, likely to insight into in metameta own, they they are are not not likely to provide provide new new insight into the the role role of of transfer transfer in populations. estimates are the assumptions un populations. Patch-specific Patch-specific estimates are needed needed to to evaluate evaluate the assumptions underlying unoc derlying the the global global approaches approaches to to parameter parameter estimation. estimation. For For example, example, are are unoccupied cupied patches patches at at all all suitable? suitable? Are Are occupied occupied patches patches sources sources or or sinks sinks in in aa deterministic sense? sense? If so, what the actual values of the rate increase/ deterministic If so, what are are the actual values of the rate of of increase/ decrease in the absence of emigration and which fac decrease in the absence of emigration and immigration? immigration? Furthermore Furthermore which factors growth rates? rates? The last question param tors determine determine patch-specifi patch-specificc growth The last question will will require require parameters that related to eters that are are related to the the demographic demographic processes processes (mortality (mortality and and reproduction) reproduction) rather simple descriptions descriptions of patch-specific rather than than simple of the the habitat habitat characteristics characteristics and and patch-specific densities. For example, both densities. For example, both empirical empirical and and theoretical theoretical studies studies have have shown shown that that density such may may not good measure habitat quality density as as such not be be aa good measure of of habitat quality in in a a demographic demographic sense Home, 11983, 983, Pulliam, 988; Danielson, Danielson, 11992; 992; Kellner Kellner et 1 993). sense (van (van Home, Pulliam, 11988; et al., al., 1993). It asking for for more more than than can It may may be be argued argued that that we we are are asking can be be achieved achieved in in most most field field studied studied and and that that the the "details" "details" we we ask ask for for may may be be redundant redundant and and complicate complicate rather improve the Estimating transfer will remain rather than than improve the matters. matters. Estimating transfer rates rates will remain aa crucial, crucial, but but difficult difficult task task in in metapopulation metapopulation studies, studies, because because of of the the large large sample sample size size required. in model model required. It It is is most most probably probably through through aa combination combination of of direct direct approaches approaches in systems metapopulations that systems and and global global approaches approaches in in natural natural metapopulations that further further progress progress will be made. will be made.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS We thank Hertzberg, D. Hjerrnann, S. S. Mesnager, manu thank K. K. Hertzberg, D. Hjermann, Mesnager, and N. Rioux Rioux for comments comments on the manuscript extensive editorial help. RAI was supported Pro script and Ilkka Ilkka Hanski Hanski for his extensive editorial help. supported by UiO Support Support Programme/ELF NGY by the "Programme "Programme Environnement gramme/ELF Petroleum Petroleum Norge Norge AS and NGY Environnementdu CNRS-Meth CNRS--M6thodes, odes, Modeles, ModUles,Theories." Th6ories."
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Migration Migration within within Metapopulations Metapopulations The The Impact Impact upon upon Local LocalPopulation PopulationDynamics Dynamics Peter B. B. Stacey Stacey
Veronica Veronica A. Johnson Johnson
Mark L. L. Taper Taper
I. INTRODUmON INTRODUCTION The interest in 99 1 , The recent recent interest in metapopulation metapopulation systems systems (e.g., (e.g., Gilpin Gilpin and and Hanski, Hanski, 11991, and references references therein, therein, Hanski Hanski and and Simberloff, Simberloff, this this volume) volume) stems stems from from the and the fact fact that not distributed in space, that members members of of aa species species usually usually are are not distributed continuously continuously in space, but but are are often often clumped clumped together together as as aa result result of of variation variation in in the the geophysical geophysical and and ecolog ecological characteristics characteristics of of the the landscape. landscape. The The concentration concentration of of individuals individuals within within aa ical particular constitutes the the local local population, population, and it is this level level of particular area area constitutes and it is at at this of organi organization most behavioral, behavioral, genetic and ecological interactions occur. occur. In tum, zation that that most genetic and ecological interactions In turn, local un local populations populations are are generally generally separated separated from from one one another another by by more more or or less less unsuitable habitats, where low or Traditionally, suitable habitats, where densities densities of of the the species species are are low or zero. zero. Traditionally, most community level level studies have focused focused on most population population and and community studies have on the the analysis analysis of of individuals individuals living living within within aa specifi specificc area, area, with with the the assumption assumption that that events events within within the local population population were both representative and generally the local were both representative and generally sufficient sufficient to to under understand most important important phenomena. phenomena. However, all species species have have evolved stand most However, almost almost all evolved mechanisms individuals to unsuitable habitats mechanisms that that enable enable individuals to cross cross unsuitable habitats at at some some stage stage of of their cycle, and of a their life life cycle, and thus thus most most local local populations populations of a species species are are potentially potentially con connected Many studies nected to to other other populations populations through through dispersal dispersal and and migration. migration. Many studies have have shown limited amount migration can have a shown that that even even a a very very limited amount of of migration can have a profound profound effect effect upon population. For example, on two mimiupon the the recipient recipient population. For example, on aa genetic genetic level, level, one one or or two
Merapopiliarion Metapopulation Biology Biology Copyright 1997 by Academic Press, Inc. All rights reproduction in any fonn reserved. Copyright © 9 1997 Academic Press, rights of of reproduction form reserved.
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grants popUlations to grants per per generation generation will will cause cause two two otherwise otherwise isolated isolated populations to behave behave as as if between them Wright, 1951, 1 95 1 , 11978). 978). On if mating mating between them is is panmictic panmictic ((Wright, On aa demographic demographic level, simulation models models of populations have sur level, recent recent simulation of multiple multiple populations have shown shown that that aa surprisingly small number between three five prisingly small number of of immigrants immigrants per per year year (often (often between three and and five adults: 992) will will allow individual populations popUlations to to persist adults: e.g., e.g., Stacey Stacey and and Taper, Taper, 11992) allow individual persist in stochastic in stochastic environments environments where where they they would would otherwise otherwise quickly quickly go go extinct extinct (e.g., (e.g., Fahrig and and Merriam, Merriam, 11985; Gilpin, 11987; Hanski, 1991; Beier, 1993). These reFahrig 985; Gilpin, 987; Hanski, 1 99 1 ; Beier, 1 993). These re sults indicate collection of popUlations that together through through sults indicate that that aa collection of populations that is is connected connected together migration higher level level of organization, the the metapopulation, migration can can function function at at aa higher of organization, metapopulation, and and that these systems emergent phenomena phenomena that that these systems can can exhibit exhibit important important emergent that can can be be under understood stood only only by by considering considering the the entire entire system system of of popUlations populations as as aa whole. whole. A central issue in the analysis of metapopulations is the A central issue in the analysis of metapopulations is the frequency frequency of of migra migration, or demographic connectivity, among component populations. For most tion, or demographic connectivity, among component populations. For most anal analyses, popu yses, the the actual actual number number of of migrants migrants that that successfully successfully move move between between two two populations lations per per breeding breeding season season or or generation generation is is the the most most important important measure measure of of the the level of will be level of connectivity connectivity between between them. them. Migration Migration frequency frequency will be aa continuous continuous variable variable and and may may range range from from zero, zero, where where populations populations are are completely completely isolated isolated from from one one another, another, to to aa value value that that may may be be nearly nearly equal equal to to the the number number of of individuals individuals in in each each unit, unit, in in which which case case the the two two units units function function as as aa single single popUlation. population. This This emphasis emphasis on on connectivity connectivity allows allows us us to to differentiate differentiate aa metapopulation metapopulation from from other other population population complexes. complexes. Specifically, Specifically, we we consider consider aa metapopulation metapopulation to to be be aa system system of isolated populations which there is suf of geographically geographically or or ecologically ecologically isolated populations within which suf-
ficant impact ficient migration migration among among populations populations to have a signi significant impact on either the demography population. As in demography or genetic structure structure of of each component component population. As aa result, result, in a metapopulation, metapopulation, the the dynamics of each each component population cannot cannot be be fully fully a dynamics of component population understood without to the whole. understood without reference reference to to other other populations populations and and to the system system as as aa whole. Thus, populations may Thus, not not all all population population complexes complexes are are metapopulations. metapopulations. Two Two populations may exist close close together is regular individuals exist together in in space, space, but but unless unless there there is regular movement movement of of individuals between in one will not population, between them, them, the the events events in one population population will not affect affect the the other other population, and will not metapopulation. Similarly, and they they will not constitute constitute aa metapopulation. Similarly, if if migration migration is is so so fre frequent among members two "populations" ran quent that that matings matings among members of of the the two "populations" are are essentially essentially random ((i.e., i.e., the individual mating with any other individual dom the probability probability of of an an individual mating with any other individual in in aa », then then the system actually sexually species is is nearly nearly 1/2(NI l /2(N1 + sexually reproducing reproducing species + N N2)), the system actually 2 functions genetically and single population popUlation that functions both both genetically and demographically demographically as as aa single that oc occupies patches. cupies different different habitat habitat patches. In specific direction timing of In addition addition to to connectivity, connectivity, the the specific direction and and timing of migration migration among popUlations are among component component populations are important, important, since since it it can can occur occur either either before before or or after be either unidirectional or after local local population population extinctions, extinctions, and and it it may may be either unidirectional or omnidi omnidirectional. Levins, rectional. For For example, example, in in many many early early models models of of metapopulations metapopulations (e.g., (e.g., Levins, 11970; 970; see review by Hanski, 1991) 1 99 1 ) migration from a see review by Hanski, migration from a source source population population to to aa target target population occurs only goes extinct. population occurs only after after the the target target popUlation population goes extinct. In In this this situation, situation, the each has nite probability probability of ex the metapopulation metapopulation consists consists of of units units where where each has aa fi finite of extinction, pee), p(e), and and (re)colonization, (re)colonization, p(c). Currently Currently extant extant populations populations serve serve as as tinction, sources of reestablish extinct extinct popsources of migrants migrants that that can, can, with with aa certain certain probability, probability, reestablish pop-
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ulations, and and thus thus the the overall overall system system can can persist persist longer longer than than the the same same number number of of ulations, populations that that are are isolated isolated from from each each other. other. The The nontrivial nontrivial equilibrium equilibrium solution solution populations p* = = 1 1 -- e/c. e/c. In In this this for the the fraction fraction of of extant extant populations populations in in this this model model is is P* for conceptualization, migration is important only as it impacts the frequency of conceptualization, migration is important only as it impacts the frequency of recolonization, and and the the primary primary dynamic dynamic of of the the metapopulation metapopulation system system is is the the recolonization, extinction/recolonization event. event. The The identities, identities, locations, locations, and and temporal temporal histories histories extinction/recolonization of individual individual populations populations are are not not relevant, relevant, and and the the models models are are therefore therefore "unstruc"unstruc of tured" in in both both time time or or space. space. In In the the field, field, the the existence existence of of periodic periodic population population tured" extinction and recolonization events is the primary indicator that this type of of extinction and recolonization events is the primary indicator that this type 1 980; Menges, 1 990). metapopulation system is present (e.g., Smith, metapopulation system is present (e.g., Smith, 1980; Menges, 1990). A second second group group of of early early analyses analyses examined examined individuals individuals that that occurred occurred in in A 1 974; Chesson, 1 98 1 ; Crowley, Crowley, 1981). 1 98 1 ). Most Most of of these these patchy habitats habitats (e.g., (e.g., Roff, Roff, 1974; patchy Chesson, 1981; studies involve involve situations situations where where migration migration rates rates (connectivity) (connectivity) among among patches patches are studies are high enough enough that that the the probability probability of of individuals individuals mating mating among among patches patches is is not not very very high different from from that that of of mating mating within within patches. patches. In In such such situations, situations, the the system system is is best best different considered as as a single population, population, with with spatial spatial variation variation in in the the distribution distribution of of considered a single individuals. individuals. More recently, recently, attention has also also been given to to situations situations where where connectivity connectivity More attention has been given prior to, as as well well as as after, an is limited, limited, but but migration migration among among populations populations occurs occurs prior is after, an extinction event (e.g., Pulliam, 1 988; Harrison, Harrison, 1991; 1 99 1 ; Hastings, Hastings, 1991; Schoener, Schoener, extinction event (e.g., Pulliam, 1988; 11991; 99 1 ; Stacey 992; GyUenberg Hanski, 1992; 1 992; Hanski Hanski and Stacey and and Taper, Taper, 11992; Gyllenberg and and Hanski, and Gyl Gyl993; see 973; Brown Kodric-Brown, lenberg, lenberg, 11993; see also also Boorman Boorman and and Levitt, Levitt, 11973; Brown and and Kodric-Brown, 11977). 977). The The source source population population can can potentially potentially affect affect the the dynamics dynamics of of the the target target popUlation time and cient new population at at any any time and may may provide provide suffi sufficient new immigrants immigrants to to "rescue" "rescue" local local populations populations before before they they go go extinct. extinct. These These types types of of metapopulations metapopulations are are more more difficult which populations difficult to to detect detect than than those those in in which populations "wink "wink off off and and on," on," because because local extinctions may local extinctions may rarely rarely occur occur under under normal normal conditions. conditions. Individual Individual elements elements in time, and in the the system system can can in in fact fact appear appear quite quite stable stable through through time, and unless unless migration migration among assessed (e.g., among the the populations populations is is disrupted disrupted or or can can be be directly directly assessed (e.g., Stacey Stacey and and Taper, 992), the Taper, 11992), the importance importance of of metapopulation metapopulation exchange exchange may may not not be be apparent. apparent. It It is is also also more more difficult difficult to to construct construct models models of of this this type type of of metapopulation, metapopulation, because because the the system system is is structured structured in in time time and and space space and and the the dynamics dynamics of of each each population population cannot cannot be be described described in in simple simple linear linear fashion fashion using using extinction extinction and and recolonization recolonization probabilities. probabilities. Migration Migration can can occur occur at at any any time, time, and and its its frequency frequency may may depend depend upon upon the the size size and and locations locations of of both both the the source source and and the the target target popUlations. populations. Because Because of of the the resulting resulting complexity, complexity, these these systems systems are are best best analyzed analyzed using using either either rather rather complex 992; Hanski complex analytic analytic models models (e.g., (e.g., Gyllenberg Gyllenberg and and Hanski, Hanski, 11992; Hanski and and Gyl Gyllenberg, 993; Gyllenberg lenberg, 11993; Gyllenberg et al. al.,, this this volume) volume) or or simulation simulation approaches approaches (see (see be below). low). One One special special case case in in which which migration migration can can occur occur before before extinction extinction is is the the source - sink metapopulation, source-sink metapopulation, which which exhibits exhibits similarities similarities to to the the situations situations de de967; Diamond scribed scribed by by island island biogeography biogeography (e.g., (e.g., MacArthur MacArthur and and Wilson, Wilson, 11967; Diamond and and May, 976). In May, 11976). In these these systems, systems, one one or or more more source source populations, populations, typically typically large large in in size size or or that that occupy occupy prime prime habitats, habitats, regularly regularly produce produce an an excess excess of of individuals individuals that that
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disperse (sinks). The recipient target target disperse to to smaller smaller populations populations in in less less optimal optimal habitat habitat (sinks). The recipient populations zero populations may may have have population population growth growth rates rates that that are are consistently consistently less less than than zero 1 ); however, they are prevented from going (or the net reproductive rate, R o , is < (or the net reproductive rate, Ro, is < 1); however, they are prevented from going extinct immigrants from productive populations extinct by by the the constant constant input input of of immigrants from more more productive populations (where ). This of system can apply to populations living on (where Ro Ro > > 11). This type type of system can apply both both to populations living on islands islands that that are are isolated isolated from from aa mainland mainland and and surrounded surrounded by by unsuitable unsuitable water water 988; Schoener, habitats habitats (e.g., (e.g., Pimm Pimm et et al. al. 11988; Schoener, 1991) or or to to patches patches of of varying varying size size and Brown and Kodric-Brown, 11977; 977; and quality quality in in aa fragmented fragmented landscape landscape (e.g. (e.g.,, Brown and Kodric-Brown, Pulliam, 988; Hanski 993; Valone Brown, 1995). 1 995). The Pulliam, 11988; Hanski and and Gyllenberg, Gyllenberg, 11993; Valone and and Brown, The central characteristic these systems is that remains central characteristic of of these systems is that the the direction direction of of migration migration remains consistent through excess of in consistent through time; time; some some populations populations consistently consistently produce produce an an excess of individuals, while dividuals, while other other populations populations are are consistently consistently rescued rescued from from extinction extinction by by immigration. immigration. In In this this paper, paper, we we focus focus on on the the more more general general "rescue "rescue effect" effect" metapopulation metapopulation system, in extinctions, but but where system, in which which migration migration can can prevent prevent local local population population extinctions, where there both the rate and the direction migration. there may may be be stochastic stochastic variation variation in in both the rate and the direction of of migration. Thus, identities of populations can can change Thus, the the identities of the the "source" "source" and and "sink" "sink" populations change unpre unpre99 1 ; Stacey 992). The shifting of dictably dictably through through time time (Harrison, (Harrison, 11991; Stacey and and Taper, Taper, 11992). The shifting of sources and likely to occur in in situations in which which local local population population sources and sinks sinks is is most most likely to occur situations in growth stochastic. In particular year, growth rates rates are are both both highly highly variable variable and and stochastic. In any any particular year, re reproduction and survivorship within within one populations may may be be high, high, and production and survivorship one or or more more populations and thus populations may may produce surplus of individuals and as sources. thus those those populations produce aa surplus of individuals and act act as sources. In following year reverse itself, In aa following year the the situation situation may may reverse itself, and and these these populations populations may may suffer suffer declines declines to to the the point point of of accepting accepting immigrants immigrants and and becoming becoming sinks. sinks. This This system sink models in that habitat occupied by the system differs differs from from sourcesource-sink models in that no no one one habitat occupied by the species better or more productive species is is predictably predictably better or more productive than than any any other other occupied occupied area. area. Metapopulations metapopulation models) models) that involve sto Metapopulations (and (and structured structured metapopulation that involve stochastic variation direction constitute constitute the chastic variation in in migration migration frequency frequency and and direction the most most inclu inclusive sive case, case, and and the the extinction/recolonization extinction/recolonization and and source/sink source/sink systems systems can can be be in incorporated within within them them as as special special situations. situations. Thus, Thus, within within aa particular complex corporated particular complex of un of populations, populations, the the dynamics dynamics of of very very small small and and isolated isolated units units may may be be best best understood an extinction/recolonization populations derstood using using an extinction/recolonization approach, approach, whereas whereas larger larger populations that closer together may interact di that are are closer together may interact through through migration migration that that changes changes in in both both direction frequency through Harrison and volume; Thomas rection and and frequency through time time ((Harrison and Taylor, Taylor, this this volume; Thomas and Hanski, this conceptualization is will depend and Hanski, this volume). volume). Which Which conceptualization is most most useful useful will depend on on the in many the particular particular characteristics characteristics of of the the populations populations involved, involved, and and in many cases cases the the most appropriate simply be question of scale, depending most appropriate model model may may simply be aa question of scale, depending upon upon both both the their tendency tendency to move the dispersal dispersal abilities abilities of of the the individual individual organisms organisms and and their to move prior prior to to reproduction reproduction (see (see below). below). One important characteristics metapopulation systems is that the One of of the the important characteristics of of all all metapopulation systems is that the overall metapopulation metapopulation can much more than the the component populations, overall can be be much more stable stable than component populations, because Kodric-Brown, 11977) 977) inin because migration migration can can buffer buffer or or "rescue" "rescue" (Brown (Brown and and Kodric-Brown, dividual populations their local dividual populations from from negative negative stochastic stochastic and and genetic genetic events events in in their local 98 1 ; Foley, 994). This be environments environments (see (see also also Leigh, Leigh, 11981; Foley, 11994). This characteristic characteristic may may be
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particularly relevant relevant to to conservation, conservation, since since human human activities activities often often fragment fragment once once particularly continuous continuous habitats, habitats, creating creating artificial artificial metapopulations, metapopulations, or or disrupt disrupt the the ability ability of of individuals to to disperse disperse to to different different areas areas within within natural natural metapopulations metapopulations (e.g., (e.g., individuals et al., al. , 1995; 1 995; Gibbs, Gibbs, 1993; 1 993; McKelvey McKelvey et et al., al., 1993; 1 993; Ouborg, Ouborg, 1993). 1 993). In In Robinson et Robinson fact, there there is is evidence evidence (e.g., (e.g., Harrison, Harrison, 1991, 1 99 1 , and and below) below) to to suggest suggest that that metapometapo fact, pulations that that include include stochastic stochastic variation variation in in migration migration and and rescue rescue effects effects may may be be pulations common in many many species species that fragmented environments. environments. Unfortunately, Unfortunately, common in that occupy occupy fragmented because these these systems systems can can be be difficult difficult to to recognize recognize in in nature because of of the the lack lack because nature because of regular regular extinction/recolonization extinction/recolonization events, events, many many studies studies of of metapopulations metapopulations inin of volving strong strong rescue rescue effects effects have have depended depended upon upon indirect indirect methods methods and and utilize utilize volving either either an an analysis analysis of of genetic genetic population population structure structure to to estimate estimate previous previous levels levels of of migration and gene gene flow flow (Olivieri (Olivieri et et al., al., 1990; 1 990; Gilpin, Gilpin, 1991, 1 99 1 , pp. pp. 17-38; 1 7 -38; Hastings Hastings migration and 1 994; Stacey Stacey and and Johnson, Johnson, 1996) 1 996) or or mathematical mathematical models models to to gengen and Harrison, Harrison, 1994; and erate specific predictions predictions that that can can be be tested tested in in the the field. field. Below, Below, we we use use aa simulation simulation erate specific model variation in model that that includes includes stochastic stochastic variation in migration migration to to illustrate illustrate how how migration migration can affect both both local local population population dynamics dynamics and and the the overall overall persistence persistence of of the the metameta can affect population. We then examine some of the empirical evidence for metapopulation population. We then examine some of the empirical evidence for metapopulation systems that may include include strong strong rescue rescue effects effects in in several several different different taxonomic taxonomic systems that may groups for further groups and and offer offer suggestions suggestions for further research research that that would would be be useful useful in in detecting detecting and understanding understanding these these types types of of systems. systems. and
THE EFFEG OF STOCHASTIC VARIATION IN IN MIGRATION MIGRATION RATE RATE ON ON II. THE EFFECTOF STOCHASTICVARIATION METAPOPULATION DYNAMICS DYNAMICS METAPOPULATION Theoretically, Theoretically, there there is is considerable considerable a priori priori reason reason to to expect expect that that many many pop populations will be to ulations will be connected connected together together by by levels levels of of migration migration that that are are sufficient sufficient to local population prior to, local extinction affect local population dynamics dynamics both both prior to, as as well well as as after, after, local extinction affect events. For Smith and Peacock, events. For example, example, as as aa result result of of conspecific conspecific attraction attraction (e.g., (e.g., Smith and Peacock, 99 1 ), dispersing individuals will will often often join join an Ray et et al., al., 11991), dispersing individuals an extant extant population population 11990; 990; Ray when wait to to immigrate immigrate until until the when there there is is available available breeding breeding space, space, rather rather than than wait the target population is extinct, as assumed under under the the most most simple target population is extinct, as assumed simple extinction/recoloni extinction/recolonization models. Metapopulation systems that involve significant migration migration before before zation models. Metapopulation systems that involve significant extinction (as opposed only after be difficult to detect detect in the extinction (as opposed to to only after extinction) extinction) can can be difficult to in the field, simply because individual individual popUlations will appear to remain stable field, simply because populations will appear to remain relatively relatively stable through 1 992) examined through time. time. For For example, example, Stacey Stacey and and Taper Taper ((1992) examined the the popUlation population (Melanerpes dynamics of an isolated population of acorn woodpeckers dynamics of an isolated population of acorn woodpeckers (Melanerpes formici formicivorus) in in Water Water Canyon, Canyon, located located in in the the mountains mountains of of central central New New Mexico, which varus) Mexico, which had had been been extant extant in in the the area area for for at at least least 50 years. years. Detailed Detailed data data on on annual annual repro reproductive ductive success success and and survival survival of of juveniles juveniles and and adults adults in in the the population population had had been been collected collected from from individually individually marked marked birds birds as as part part of of aa long-term long-term study study of of coop coop979; Stacey 987, 11991). 99 1 ). erative erative breeding breeding in in this this species species (Stacey, (Stacey, 11979; Stacey and and Ligon, Ligon, 11987, Each demographic demographic variable variable exhibited exhibited large large annual annual variation, presumably as as aa Each variation, presumably result environment, including including rainfall result of of stochastic stochastic events events in in the the local local environment, rainfall and and acorn acorn
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Peter B.B. Stacey Stacey etet al. al. Peter
production. production. Using Using data data from from the the field field study study in in aa simple simple simulation simulation model model of of single single population 1 992) found population growth growth through through time, time, Stacey Stacey and and Taper Taper ((1992) found that that if if this this pop population ulation was was considered considered to to be be completely completely closed, closed, without without immigration, immigration, demo demographic stochasticity stochasticity would would lead lead to to rapid rapid extinction extinction (usually (usually < < 20 20 years, years, depend dependgraphic ing on the particular assumptions about density dependence in growth rates ing on the particular assumptions about density dependence in growth rates in in the the model; see Fig. l A). Extinction always occurred because, simply by chance, model; see Fig. 1A). Extinction always occurred because, simply by chance, there there would would be be aa series series of of bad bad years years for for either either reproduction reproduction or or survival survival that that would would drive drive population population numbers numbers so so low low that that they they could could not not recover. recover. However, However, allowing allowing mi migration from from other other populations populations rapidly rapidly increased increased local local persistence persistence times, times, and and even even gration four four immigrants immigrants per per year year (8% of of population population at at carrying carrying capacity) capacity) enabled enabled the the B). The local local population population to to continue continue for for > > 250 years years (Fig. (Fig. l1B). The number number of of immigrants immigrants required required by by the the model model to to obtain obtain long-term long-term persistence persistence was was similar similar to to the the migration migration rate actually actually observed observed in in the the study study population. population. rate 1 992) suggests The The study study of of Stacey Stacey and and Taper Taper ((1992) suggests that that the the Water Water Canyon Canyon pop population ulation of of acorn acorn woodpeckers woodpeckers is is part part of of aa larger larger metapopulation, metapopulation, whose whose existence existence and cient data and importance importance had had not not been been suspected suspected until until there there were were suffi sufficient data to to conduct conduct detailed detailed analyses analyses of of local local population population dynamics dynamics and and particularly particularly the the impact impact of of stochastic stochastic variation variation in in growth growth rates rates on on persistence persistence times. times. The The local local population population by by itself was not not likely to continue continue for for very very long, but it it was was presumably presumably rescued rescued itself was likely to long, but from extinction extinction by by regular regular migration migration from from other other populations. populations. As As aa result, result, it it ap apfrom peared relatively stable through through time time even even though though its dynamics were were affected peared relatively stable its dynamics affected in in large measure by migration from other other populations. populations. Thus, Thus, while while the pop large measure by migration from the single single population model model predicted rapid extinction, it within a metapop metapopulation predicted relatively relatively rapid extinction, placing placing it within a ulation with among-population migration predicted ulation complex complex with among-population migration predicted continued continued persistpersist ence. ence. This study study illustrates illustrates how how migration migration within within aa metapopulation metapopulation can can buffer buffer inin This dividual populations populations from from stochastic stochastic variation in growth rates that might otherother dividual variation in growth rates that might lead to to extinction. extinction. To To explore explore this this phenomenon phenomenon further, further, we we developed developed aa wise wise lead more complex, complex, spatially spatially and and temporally temporally structured structured model model that that allows allows for migra more for migration in in variable variable frequency frequency and and direction direction (depending (depending on on local local growth rates at at any any tion growth rates particular point point in in time) time) and and for for immigration immigration either either before before or or after after target target popupopu particular lations lations go go extinct. extinct. The The details details of of this this model model will will be be discussed discussed elsewhere elsewhere (Taper ( Taper and Stacey, Stacey, 1997); 1 997); here, here, we we describe describe some some of of the the results results from from the the model model to to illusillus and trate some of the the important important consequences consequences of of migration migration within metapopulations trate some of within metapopulations for individual for individual population population dynamics. dynamics. Because Because of of the the potential potential complexities complexities of of inin dividual population population size size trajectories trajectories within within this this type type of of stochastic stochastic system, system, we we dividual focus on on the the question question of of how how long long all all populations populations within within aa metapopulation metapopulation persist persist focus under under different different conditions conditions and and with with different different initial initial population popUlation sizes. sizes. The The model model is designed designed to to apply apply primarily primarily to to vertebrate vertebrate populations populations with with distinct distinct breeding breeding is periods; other other simulation simulation models models that that incorporate incorporate different different assumptions assumptions are are availavail periods; able (e.g., (e.g., RAMAS; RAMAS; see see Akgakaya Ak�akaya and and Ferson, Ferson, 1992; 1 992; Wu Wu et et al., al., 1993). 1 993). able In this this model, model, aa predetermined predetermined number number of of local local populations populations are are established established In and and allowed allowed to to grow grow or or decline decline based based upon upon an an annual annual growth growth rate rate parameter parameter (r) (r) that has has both both aa mean mean and and aa variance. variance. Dispersal Dispersal within within the the system system is is density density dede that pendent; populations (K ) populations that that grow grow above above aa preset preset maximum maximum carrying carrying capacity capacity (K) pendent;
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FIGURE !1 (A) (A) An An example example of of the the annual annual changes changes in in the the size size of of aa closed closed acorn acorn woodpecker woodpecker poppop FIGURE ulation from from aa typical typical computer computer simulation, simulation, using using demographic demographic parameters parameters taken taken from fromdata datacollected collected ulation over overaa 10-year l O-year field field study study in in Water WaterCanyon, Canyon, New New Mexico. Mexico. Simulations Simulations were were started started atatthe themaximum maximum (K = 52) 52) and and continued continued until until the the population populationdeclined declined to to zero. zero. Because Because the the observed observed populationsize size (K population annual reproductive reproductive and and survival survival rates rates were were highly highly variable, variable, most most simulated simulatedpopulations populations went wentextinct extinct annual less than than 20 20 years. years. (B) (B) Increase Increase inin simulated simulated persistence persistence times times of ofthe the same same population population when when itit isis inin less open open and andthere there isis migration migration into intothe thepopulation population with withdifferent different annual annual frequencies frequencies (from (from Stacey Stacey and and Taper, Taper, 1992). 1992). =
274 274
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produce produce emigrants emigrants that that leave leave their their home home populations populations and and encounter encounter new new popu populations new population population is is cur lations with with aa transition transition probability probability t, where where t < < 11.. If If the the new currently below its population is rently below its carrying carrying capacity, capacity, it it accepts accepts an an immigrant; immigrant; if if the the population is at at K, the the immigrant to another For each the immigrant moves moves again again to another population. population. For each transition, transition, the emigrant specified probability dying before reaches a emigrant has has aa specified probability m m of of dying before it it reaches a new new population. population. The becomes a popu The process process is is repeated repeated until until the the individual individual becomes a member member of of aa new new population size of is calculated calculated after lation or or dies. dies. The The size of each each population population is after all all births, births, deaths, deaths, and and dispersal dispersal movements movements have have occurred, occurred, and and then then the the process process repeated repeated for for another another reproductive "year." Key parameters of the model that can be varied for reproductive "year." Key parameters of the model that can be varied for each each analysis populations in system, their initial analysis include include the the number number of of component component populations in the the system, their initial sizes sizes and and carrying carrying capacities, capacities, the the mean mean and and variance variance in in growth growth rates rates of of each each population, and population, and the the correlation correlation among among the the growth growth rates rates of of all all populations populations in in the the system. Metapopulation connectivity likelihood that system. Metapopulation connectivity is is varied varied by by specifying specifying the the likelihood that aa dispersing will encounter encounter another it dies. dispersing individual individual will another population population before before it dies. If If connec connectivity tivity equals equals zero, zero, all all populations populations are are isolated isolated from from each each other other and and the the system system does by inde does not not function function as as aa metapopulation. metapopulation. Temporal Temporal structure structure is is included included by independently simulation "year." pendently tracking tracking the the size size of of each each population population through through each each simulation "year." Spatial Spatial structure structure is is incorporated incorporated into into the the model model by by specifying specifying the the connectivity connectivity of of each pair pair of matrix such such that pairs that that are are close each of populations populations independently independently in in aa matrix that pairs close together low together have have high high connectivity connectivity values, values, while while those those that that are are far far apart apart have have low values. However, values. However, in in the the results results described described here here we we considered considered the the most most simple simple case case where where individuals individuals have have equal equal probabilities probabilities of of encountering encountering any any other other pop population ulation within within the the system system after after migration. migration. Finally, Finally, the the type type of of density density dependence dependence in be most most typical popula in migration migration frequency frequency modeled modeled here here may may be typical of of vertebrate vertebrate populations, whereas invertebrates may density tions, whereas many many insects insects and and other other invertebrates may exhibit exhibit more more density independent result, this may independent reproduction reproduction and, and, therefore, therefore, migration. migration. As As aa result, this model model may be less applicable applicable to taxa. be less to those those taxa. We dynamics of metapopulations where We are are able able to to compare compare the the dynamics of metapopulations where immigration immigration occurs occurs only only after after extinction extinction with with continuous continuous migration migration by by changing changing the the conditions conditions under populations. When under which which immigrants immigrants enter enter new new populations. When modeling modeling generalized generalized res rescue cue effect effect dynamics, dynamics, immigrants immigrants enter enter aa target target population population if if the the population population size size (n) is is 2: -> 0 and and < < K. When When modeling modeling strict strict extinction extinction-recolonization dynamics, - recolonization dynamics, immigration immigration to to aa population population occurs occurs only only if if n n = = 0 for for that that population. population. Local Local pop populations whenever population ulations are are declared declared extinct, extinct, at at least least temporarily, temporarily, whenever population size size de declines below sexually reproducing the entire clines below two two (for (for sexually reproducing organisms); organisms); extinction extinction of of the entire system system occurs occurs when when all all local local populations populations are are extinct extinct simultaneously. simultaneously. Simulations Simulations values, and were were repeated repeated 200 times times for for each each set set of of parameter parameter values, and either either the the median median or or the the geometric geometric mean mean time time to to extinction extinction for for the the entire entire metapopulation metapopulation was was de determined. termined. Most buffering ef Most metapopulation metapopulation models models have have focused focused on on determining determining the the buffering effect have in fect that that migration migration can can have in either either preventing preventing extinction extinction and/or and/or allowing allowing for for reestablishment gone extinct. threat of reestablishment of of aa population population after after it it has has gone extinct. The The threat of extinction extinction is likely to small, fragmented Gilpin, 11987; 987; GoodGood is likely to be be higher higher in in small, fragmented populations populations (e.g., (e.g., Gilpin, at., 11993) 993) and Foman, 987a, Pimm man, 11987a, Pimm et et al., and where where growth growth rates rates vary vary stochastically stochastically ((Fo-
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ley, 11994, 994, this illustrates the environmental and ley, this volume). volume). Figure Figure 2 illustrates the effects effects of of environmental and demographic our model, without the demographic variation variation in in our model, both both with with and and without the buffering buffering effect effect of of migration. migration. These These graphs graphs illustrate illustrate several several points. points. First, First, even even when when the the growth growth (r = 0.00 1 ), all populations eventually go extinct rate is slightly positive rate is slightly positive = 0.001), all populations eventually go extinct as as aa result of random fl u ctuations (if there were no variance in growth rates, all pop result of random fluctuations (if there were no variance in growth rates, all populations would continue Second, when when there ulations would continue to to grow grow indefinitely, indefinitely, albeit albeit slowly). slowly). Second, there is no migration in the system and all populations are isolated from each other is no migration in the system and all populations are isolated from each other (i.e., individuals concentrated in one two (i.e., there there is is no no metapopulation metapopulation structure), structure), individuals concentrated in one or or two large populations will will persist persist longer longer than the same same number number of individuals divided large populations than the of individuals divided up more, but populations. However, However, when migration is up into into more, but smaller, smaller, populations. when migration is allowed, allowed, an an intermediate number number of containing an intermediate number intermediate of popUlations, populations, each each containing an intermediate number of of individuals, last populations or many small individuals, last longer longer than than either either aa few few large large populations or many small ones. ones. This results from This is is an an emergent emergent property property of of the the metapopulation metapopulation structure structure that that results from the the buffering effects migration. Because vari buffering effects of of among-population among-population migration. Because of of the the stochastic stochastic variation in growth growth rates, rates, when populations are may be be inin ation in when some some populations are declining, declining, others others may creasing to produce immigrants that growth in creasing at at aa rate rate sufficient sufficient to produce immigrants that can can supplement supplement growth in the the declining declining populations. populations. Thus, Thus, aa connected connected group group of of populations populations can can actually actually have have aa greater greater probability probability of of persistence persistence than than aa few few large, large, but but isolated, isolated, popula populations. important, these these results illustrate how local popula tions. Most Most important, results illustrate how the the dynamics dynamics of of local populations by metapopulation tions can can be be strongly strongly affected affected by metapopulation exchange exchange and and that that the the continued continued stability of each reference to to the the system stability of each population population cannot cannot be be understood understood without without reference system as whole. as aa whole. Figure 3 illustrates the the timing of migration Figure 3 illustrates the effect effect of of the timing of migration and and compares compares the the persistence of in which which migration occurs only only after persistence of metapopulations metapopulations in migration occurs after extinction extinction (n = 0), with time as = 0), with the the situation situation where where immigration immigration can can occur occur at at any any time as long long as as n< All other model are constant. In In both < K. All other parameters parameters in in the the model are held held constant. both situations, situations, an longer than than an intermediate intermediate number number of of populations populations of of intermediate intermediate size size persist persist longer either large populations both individual either aa few few large populations or or many many small small ones. ones. However, However, both individual populations last longer if mimi populations and and the the metapopulation metapopulation system system as as aa whole whole always always last longer if gration will be before, gration can can occur occur before before extinction, extinction, because because populations populations will be rescued rescued before, as well as events. The as well as after, after, extinction extinction events. The pronounced pronounced "advantage" "advantage" of of the the strong strong rescue effects metapopulation system system over over the - recolonization model rescue effects metapopulation the extinction extinction-recolonization model suggests species in evolved highly highly suggests that that many many species in fragmented fragmented environments environments may may have have evolved efficient Without such efficient mechanisms mechanisms for for migration migration across across unsuitable unsuitable habitats. habitats. Without such mechanisms, species may in such mechanisms, many many species may not not be be able able to to persist persist in such landscapes. landscapes. For For example, one species of in the example, one species of particular particular interest interest and and political political importance importance in the United United States owl (Strix the Mexican States is is the the spotted spotted owl (Strix occidentalis). occidentalis). Two Two of of the the subspecies, subspecies, the Mexican (S. o. lucida) owls, occur occur in in natunatu lucida) and and the the California California (S. o. occidentalis) occidentalis) spotted spotted owls, rally mountain ranges American southwest rally fragmented fragmented habitats habitats in in the the mountain ranges in in the the American southwest and and southern structure with southern California, California, and and aa functional functional metapopulation metapopulation structure with frequent frequent mi migration for the gration among among the the populations populations may may be be necessary necessary for the continued continued persistence persistence of of the species species in in this this region region (Lahaye (Lahaye et et al., 1994; Stacey, 11994). contrast, the the the 1 994; Stacey, 994). In In contrast, 988a; McKelvey McKelvey et 993), whose Northern subspecies (S. o. caurina; Northern subspecies caurina; Lande, Lande, 11988a; et al., 11993), whose habitats forests in habitats were were once once primarily primarily contiguous contiguous forests in the the Pacific Pacific northwest, northwest, may may not not
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Results from from a computer simulation model that illustrate how migration among among popu populations in a metapopulation can "buffer" those populations against the effects effects of stochastic variation in annual growth rates and decrease decrease the probability of extinction. Each graph graph gives the mean time to extinction in 200 200 computer simulation runs for systems containing the same total number of individ individuals divided into different numbers of patch patch sites or populations, each with a size equal equal to 120 120 divided number of patches (e.g., one population population of 120 120 individuals, two populations of 60 60 individuals by the number each, three three populations populations of 40 40 individuals each, etc.). Shown here here is the effect of different combinations combinations of demographic demographic and environmental environmental variation variation on metapopulation persistence times (A, C, and E).
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Comparison of of persistence persistence times times for for two two different different types types of of metapopulation metapopulation systems. systems. A Comparison rescue effect effect model model where where migration migration can occur occur both both before before and and after after a population population might might go go extinct rescue (triangles). The The more more restrictive restrictive condition of the the extinction/recolonization extinction/recolonization model model where where migration migration (or (or (triangles). condition of recolonization) occurs occurs only after after population population extinction extinction (circles). (circles). Results Results of of simulations simulations are are given given as recolonization) parameters except except the timing of migration migration events were for each each comparison. comparison. timing of were the same same for in Fig. 2; all parameters each case, case, metapopulations metapopulations that include include migration migration before before extinction extinction persist persist longer longer than than those sys sysIn each that do do not include include the rescue rescue effect. tems that
have the ability to shift a metapopulation metapopulation structure structure with have the ability to shift into into a with efficient efficient migration migration once artificially fragmented and the once their their habitat habitat has has been been artificially fragmented by by human human activities, activities, and the survival subspecies may survival of of this this subspecies may be be much much more more problematic. problematic. The effect of The buffering buffering effect of the the metapopulation metapopulation depends depends on on the the ability ability of of some some populations populations .to to act act as as sources sources of of migrants migrants at at the the same same time time that that other other populations populations are declining ready to are declining and and ready to accept accept those those migrants. migrants. Because Because of of this, this, migration migration will will have effect on on individual have little little effect individual persistence persistence times times if if all all populations populations are are increasing increasing or also Harrison 1 989; Gilpin, Gilpin, 1990; 1 990; Burgman Burgman or declining declining together together (see (see also Harrison and and Quinn, Quinn, 1989;
FIGURE FIGURE 2 (Continued) (Continued) Systems Systems in in which which there there is is no no migration migration among among populations populations (between-population (between-population connectivity, connectivity, meamea sured function as sured as as transition transition survival survival probability probability -= 0), 0), and and therefore therefore they they do do not not function as metapopulations metapopulations (B, (B, D, D, and and F). F). All All of of the the same same parameter parameter values values except except that that migration migration among among populations populations is is possible, possible, and the connectivity or survival probability dispersing individuals 0.2 (20% (20% of of all all dispersing individuals survive survive and the connectivity or transition transition survival probability is is 0.2 to immigrate immigrate into into another Systems that that include include migration migration and and function function as as metapopulations metapopulations to another population). population). Systems persist effect (see persist longer longer in in all all cases cases than than do do isolated isolated populations populations as as aa result result of of the the rescue rescue effect (see text text for for details). details).
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et al., 993). Thus, al., 11993). Thus, the the rescue rescue effect effect is is likely likely to to be be most most important important when when growth growth rates rates of of local local populations populations in in the the system system are are either either uncorrelated uncorrelated or or negatively negatively cor correlated. related. The The impact impact of of among-population among-population correlations correlations in in growth growth rates rates on on the the per persistence sistence of of our our model model metapopulation metapopulation is is shown shown in in Fig. Fig. 4. 4. These These results results suggest suggest that metapopulations with strong rescue effects may be most common in situations that metapopulations with strong rescue effects may be most common in situations where where there there is is little little correlation correlation in in the the environments environments of of different different habitat habitat fragments fragments and therefore therefore in in the the annual annual growth growth rates rates of of the the populations populations that that occupy occupy each each of of and those those fragments. fragments. Unfortunately, Unfortunately, very very few few empirical empirical studies studies have have examined examined the the degree degree of of among among population population correlations correlations in in growth growth rates rates because because it it requires requires de detailed tailed and and long-term long-term demographic demographic data data gathered gathered over over an an area area large large enough enough to to include include aa number number of of separate separate populations. populations. Studies Studies of of this this nature nature are are just just beginning beginning 994; Dennis 996; Martin (Lahaye (Lahaye et e t aal., / . , 11994; Dennis et al., al., 11996; Martin et aI., al., in in press; press; Stacey Stacey and and Martin, 11997). Martin, 997). The results results of of both both the the simulation simulation model model presented presented here here and and earlier earlier models models The indicate indicate that that migration migration within within aa metapopulation metapopulation can can have have aa major major impact impact upon upon local population persistence times local population dynamics dynamics and and increase increase the the persistence times of of both both individual individual populations populations and and the the entire entire metapopulation metapopulation system system under under conditions conditions that that may may be be relatively relatively common common in in certain certain taxa taxa and and types types of of environments. environments. Directly Directly testing testing the the predictions however, will will be cult in cases because predictions of of these these models, models, however, be diffi difficult in many many cases because observing observing migration migration directly directly in in the the field field requires requires identifying identifying and and following following aa large large number unpredictable and number of of individuals individuals whose whose migration migration paths paths may may be be unpredictable and for for whom whom 11000 000
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FIGURE 4 4 Effect Effect of of different different degrees degrees of of among-population among-population correlations correlations (rho) (rho) in in population population growth growth FIGURE 2; rates on on mean mean persistence persistence times times of of the the metapopulation. metapopulation. Results Results of of simulations simulations are are given given as as in in Fig. Fig. 2; rates all all other other parameters parameters in in the the model model were were held held constant constant for for each each simulation. simulation. Persistence Persistence times times increase increase as the the correlation correlation in in growth growth rates rates becomes becomes more more negative. negative. Functionally, Functionally, this this represents represents aa situation situation as where some some populations populations in in the the system system are are growing growing large large enough enough to to produce produce emigrants emigrants at at the the same same where time that that other other populations populations are are declining declining to to the the point point where where they they need need immigrants immigrants to to avoid avoid extinction. extinction. time
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mortality high. As result, much the mortality rates rates may may be be high. As aa result, much of of the the current current evidence evidence for for the importance of migration in local population dynamics, and for the existence importance of migration in local population dynamics, and for the existence of of metapopulations follow metapopulations with with strong strong rescue rescue effects, effects, is is necessarily necessarily indirect. indirect. In In the the following to illustrate ing section, section, we we review review selected selected studies studies to illustrate both both where where these these systems systems are are likely is needed to detect them. likely to to occur occur and and the the evidence evidence that that is needed to detect them.
III. EMPIRICAL EMPIRICALEVIDENCE EVIDENCE The The importance importance of of migration migration in in population population dynamics dynamics is is indicated indicated not not only only by studies; there evidence for the role rescue by theoretical theoretical studies; there is is growing growing empirical empirical evidence for the role of of rescue effects systems. We exhaustive effects in in natural natural metapopulation metapopulation systems. We have have not not attempted attempted an an exhaustive review review of of all all possible possible natural natural metapopulations; metapopulations; rather, rather, as as examples, examples, we we focus focus on on several several taxonomic taxonomic groups groups in in which which metapopulations metapopulations are are likely likely to to occur occur and and where where a a sufficient sufficient number number of of studies studies have have been been published published to to explore explore the the nature nature of of mi migration gration in in these these systems. systems. These These groups groups include include butterflies, butterflies, small small mammals, mammals, and and amphibians. patterns and amphibians. Each Each group group illustrates illustrates somewhat somewhat different different patterns and suggests suggests pro productive research. ductive directions directions for for future future research. First, it it is is important needed to First, important to to establish establish the the type type of of evidence evidence needed to demonstrate demonstrate the with strong strong rescue the existence existence of of aa metapopulation metapopulation with rescue effects effects mediated mediated by by migra migration. ((1) 1 ) As all metapopulation systems, the species must must inhabit distinct habitat tion. As in in all metapopulation systems, the species inhabit distinct habitat patches consistent patches surrounded surrounded by by areas areas of of unsuitable unsuitable habitat. habitat. (2) There There must must be be consistent movement movement between between already already occupied occupied habitat habitat patches patches such such that that the the migration migration rates rates are the simpler simpler extinction/recoloni are relatively relatively high high compared compared to to predictions predictions from from the extinction/recolonization zation systems systems where where migration migration occurs occurs only only after after extinction. extinction. (3) Specific Specific demo demographic be distinguishable graphic and and genetic genetic effects effects of of migration migration should should be distinguishable at at the the local local population level. For population level. For example, example, observed observed demographic demographic parameters parameters such such as as popu population lation size size and and density, density, age age structure, structure, sex sex ratio, ratio, and and population population growth growth rate rate should should be be different different from from those those predicted predicted for for an an isolated isolated population. population. Similarly, Similarly, genetic genetic measures levels, and measures such such as as heterozygosity, heterozygosity, inbreeding inbreeding levels, and genetic genetic differentiation differentiation should population (e.g., should be be different different from from those those expected expected in in an an isolated isolated population (e.g., Hastings Hastings 994). Genetic to distinguish and and Harrison, Harrison, 11994). Genetic parameters parameters also also may may be be used used to distinguish among among 1 99 1 , pp. pp. the systems. For example, Gilpin Gilpin ((1991, the different different types types of of metapopulation metapopulation systems. For example, 117-38) 7 38) noted be a the extinction/re noted that that low low heterozygosity heterozygosity should should be a feature feature of of the extinction/recolonization repeated founder that occur colonization metapopulations metapopulations as as aa result result of of the the repeated founder effects effects that occur with recolonization of migrants. In with the the recolonization of areas areas by by small small numbers numbers of of migrants. In contrast, contrast, het heterozygosity migration as erozygosity should should be be higher higher in in metapopulations metapopulations with with frequent frequent migration as aa result of high gene component populations, populations, and system should should result of high gene flow flow among among component and the the system act like a large unit unit (Stacey Johnson, 1996). 1 996). (4) act genetically genetically more more like a single single large (Stacey and and Johnson, Finally, be greater Finally, if if metapopulation metapopulation exchange exchange is is important, important, there there should should be greater persist persistence well as ence or or stability stability of of the the local local population population as as well as of of the the overall overall metapopulation, metapopulation, reflected reflected in in terms terms of of decreased decreased probability probability of of extinction extinction and and dampened dampened popUlation population size in the the same size fluctuations fluctuations relative relative to to an an isolated isolated population population in same environment. environment. -
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A. Butterflies Butterflies In one one of of the the fi first detailed studies of the the population population dynamics dynamics of of butterflies, butterflies, In rst detailed studies of Baker 1 969) analyzed patches and Baker ((1969) analyzed movements movements of of adults adults out out of of natal natal patches and concluded concluded that species should exhibit low then, many many researchers researchers that most most species should exhibit low migration migration rates. rates. Since Since then, have assumed that limited migration migration abilities have assumed that butterflies butterflies have have extremely extremely limited abilities and and tend tend to exist in relatively isolated populations ((Warren, Warren, 1987a,b; 1 987a,b; Baguette Neve, to exist in relatively isolated populations Baguette and and Nbve, 11994). 994). Thus, Thus, when when metapopulation metapopulation theory theory emerged, emerged, butterflies butterflies seemed seemed to to be be prime prime examples of examples of extinction-recolonization extinction-recolonization metapopulations metapopulations in in which which small, small, isolated isolated populations could could decline decline to recolo populations to extinction, extinction, and and habitat habitat patches patches could could then then be be recolonized by events. Yet evidence suggest nized by rare rare long-distance long-distance migration migration events. Yet several several lines lines of of evidence suggest that that migration migration among among populations populations may may be be much much more more common common in in some some taxa taxa of of butterflies butterflies than than was was originally originally thought thought (see (see also also Thomas Thomas and and Hanski, Hanski, this this volume), volume), and and long-distance long-distance migration migration may may increase increase the the persistence persistence of of local local populations populations through through both both demographic demographic and and genetic genetic effects. effects. 1. Checkerspot Checkerspot and and Fritillary Fritillary Butterflies Butterflies
Most in North Most checkerspot checkerspot butterflies butterflies in in the the genus genus Euphydryas in North America America ap appear which occur pear to to specialize specialize on on one one or or aa few few species species of of plants plants which occur in in patchy patchy distri distributions. For butions. For example, example, Euphydryas gillettii, gillettii, aa northern northern Rocky Rocky Mountain Mountain endemic, endemic, lays which is lays its its eggs eggs only only on on the the black black twinberry twinberry (Lonicera involucrata) which is re restricted 988; Debinski, Debinski, stricted to to wet, wet, sunny, sunny, early early successional successional patches patches (Williams, (Williams, 11988; 11994). 994). A confined to to wet, wet, low A northeastern northeastern species, species, E. phaeton, is is confined low areas areas that that contain its its primary plant, turtlehead contain primary food food plant, turtlehead (Chelone glabra; Brussard Brussard and and Vawter, Vawter, butterflies tend 11975). 975). As As aa result, result, many many Euphydryas butterflies tend to to occupy occupy distinct distinct habitat habitat patches habitat, predisposing predisposing them them toward patches separated separated by by areas areas of of unsuitable unsuitable habitat, toward aa metapopulation structure. metapopulation structure. Unlike extinction -recolonization metapopulations, metapopulations, relatively Unlike traditional traditional extinction-recolonization relatively high high rates populations, resulting resulting in rates of of migration migration have have been been detected detected among among existing existing populations, in clear clear demographic demographic and and genetic genetic effects. effects. In In aa mark-recapture mark-recapture study study of of E. anicia, aa species Mountains, White 1 980) directly directly species restricted restricted to to high high peaks peaks in in the the Rocky Rocky Mountains, White ((1980) observed migration single season, season, approximately approximately observed migration between between two two populations. populations. In In aa single 3% of of the the individuals individuals in in the the source source population population successfully successfully migrated migrated to to aa new new population. high enough to eliminate population. White White remarked remarked that that such such aa rate rate should should be be high enough to eliminate genetic between populations. genetic drift, drift, minimizing minimizing genetic genetic differentiation differentiation between populations. Indeed, Indeed, at at the time been detected the time of of publication, publication, no no significant significant genetic genetic differences differences had had been detected (White, 11980). 980). Other studies have rates from (White, Other studies have inferred inferred relatively relatively high high migration migration rates from genetic Brussard and Vawter ((1975) 1 975) ex genetic data. data. In In an an early early study study of of E. phaeton, Brussard and Vawter examined in three populations. amined protein protein differentiation differentiation at at seven seven allozyme allozyme loci loci in three local local populations. They nd distinct because each They expected expected to to fi find distinct genetic genetic differentiation differentiation because each population population had had no migration a a small small effective effective population population size size (Ne = ~ 20-200) 20-200) and and no migration had had been been de detected tected through through aa preliminary preliminary mark-recapture mark-recapture study. study. Yet Yet only only one one of of the the seven seven protein populations, and protein loci loci showed showed any any statistical statistical difference difference between between populations, and all all pop populations exhibited exhibited high heterozygosity and ulations high heterozygosity and intrapopulation intrapopulation genetic genetic diversity. diversity.
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Brussard 1 975) concluded Brussard and and Vawter Vawter ((1975) concluded that that gene gene flow, flow, and and thus thus migration, migration, must must be 1 983) be occurring occurring frequently frequently enough enough to to cause cause these these genetic genetic effects. effects. Ehrlich Ehrlich ((1983) reviewed reviewed data data from from various various Euphydryas species species and and calculated calculated that that individual individual populations 50 that populations with with effective effective population population sizes sizes (Ne) (Ne) = ~ 3030-50 that do do not not immedi immediately rebound rebound to to higher higher levels levels will will be be unable unable to to maintain maintain themselves themselves in in isolation isolation ately and inevitably inevitably decline decline to to extinction. extinction. Thus, Thus, the the apparent apparent persistence persistence of of the the small small and populations populations studied studied by by Brussard Brussard and and Vawter Vawter suggest suggest that that migration migration in in E. phaeton prevents prevents loss loss of of heterozygosity heterozygosity and and genetic genetic differentiation, differentiation, and and can can apparently apparently rescue popUlations populations from from extinction. extinction. rescue SSimilarly, imilarly, in in aa more more recent recent study study of of genetic genetic differentiation differentiation in in E. gillettii, Debinski 1 994) expected nd high Debinski ((1994) expected to to fi find high levels levels of of differentiation, differentiation, inbreeding, inbreeding, and and homozygosity in local populations of Glacier National Park, because little homozygosity in local populations of Glacier National Park, because little mi migration Holdren and gration had had been been directly directly detected detected in in transplant transplant experiments experiments ((Holdren and Ehrlich, Ehrlich, 11981). 98 1 ). She 9 loci She analyzed analyzed isozyme isozyme electrophoretic electrophoretic differences differences at at 119 loci in in two two local local populations populations and and calculated calculated unexpectedly unexpectedly low low values values of of inbreeding inbreeding and and high high values values 1; H calculated that of of heterozygosity heterozygosity (F (FsT = 0.04 0.041; H = = 0.079, 0.079, 0.084). 0.084). She She calculated that Nm, the the ST = number migrants between number of of migrants between these these populations populations per per generation, generation, must must be be as as high high as Debinski, 11994). 994). Although as 5.80 5.80 in in order order to to cause cause the the observed observed genetic genetic effects effects ((Debinski, Although increased increased population population stability stability or or persistence persistence was was not not directly directly inferred inferred in in this this study, study, Ehrlich 1 983) observed Ehrlich ((1983) observed that that genetic genetic variability variability and and polymorphism polymorphism tend tend to to be be maintained maintained in in Euphydryas species species despite despite large large population population size size fluctuations. fluctuations. One One possible 1 994) results possible explanation explanation suggested suggested by by Debinski Debinski ss ((1994) results is is that that sufficient sufficient migration exists exists to to preserve preserve genetic genetic variability variability and and rescue rescue small small populations migration populations ((EhrEhr lich, 11983), 983), increasing populations as as well well as overall lich, increasing the the persistence persistence of of those those populations as of of the the overall metapopulation. metapopulation. Several studies of of European European butterfly butterfly species species also provide evidence meta Several studies also provide evidence for for metapopulations with with strong rescue effects. The bog bog fritillary fritillary (Proclossiana eunomia) populations strong rescue effects. The is where its plant, is found found in in natural natural wet wet meadows meadows in in Belgium, Belgium, where its only only larval larval food food plant, Polygonum bistorta grows. grows. Thus, this butterfly also occupies occupies distinct distinct habhab Polygonum Thus, this butterfly species species also itat areas of itat patches patches surrounded surrounded by by areas of unsuitable unsuitable habitat. habitat. Using Using mark-recapture mark- recapture techniques, Baguette Baguette and and N~ve Neve (1994) ( 1 994) demonstrated demonstrated that that high high levels levels of of adult adult techniques, migration occur occur between between these these patches. patches. At At least least 4.6% 4.6% of of all all males males and and 10.5% 1 0.5% of of migration all between populations. These researchers suggested that all females females moved moved between populations. These researchers suggested that high high levels movement may lead to genetic variability, levels of of movement may lead to high high levels levels of of genetic variability, explaining explaining why why populations the bog despite habitat populations of of the bog fritillary fritillary have have persisted persisted despite habitat loss loss (Baguette ( Baguette and and N~ve, Neve, 1994). 1 994). In aa similar similar study study in in Britain, Britain, Warren Warren (1987a) ( 1 987a) used used mark-recapture mark- recapture experiexperi In ments to to examine examine mobility mobility of of the the heath heath fritillary fritillary (Mellicta (Mellicta athalia), athalia), an an endangered endangered ments species that et al., al. , 1984). 1984). that exists exists in in small, small, discrete discrete populations populations or or colonies colonies (Warren ( Warren et species He found found that that at at least least 1.7% 1 .7% of of all all males males and and 1.3% 1 .3 % of of all all females females moved moved between between He populations nearly nearly 11 km km distant distant from from each each other. other. No No specific specific demographic demographic or or populations genetic effects effects were were attributed attributed to to this this substantial substantial migration migration rate, rate, but but Warren Warren genetic ( 1 987a,b) noted noted that that these these populations populations were were unlikely unlikely to to be be genetically genetically isolated isolated (1987a,b) from each each other other and and that that local local population popUlation growth growth rates rates appeared appeared to to be be uncorreuncorrefrom
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lated. lated. This This suggests suggests that that the the substantial substantial migration migration observed observed may may help help dampen dampen overall population size fluctuations rescue effect. overall population size fluctuations through through aa rescue effect. Thomas ai., ((1992) 1 992) performed of study four Thomas et al., performed an an entirely entirely different different kind kind of study of of four rare rare British British butterfly butterfly species: species: Piebejus Plebejus argus, argus, Hesperia Hesperia comma, comma, Thymeiicus Thymelicus ac acteon, and and M. athalia. All All of of these these skippers skippers and and fritillaries fritillaries are are restricted restricted to to distinct distinct habitat habitat patches patches due due to to very very specific specific vegetation vegetation requirements requirements for for feeding feeding and and laying laying eggs. Rather than than directly directly assessing migration, these these researchers researchers examined examined pat pateggs. Rather assessing migration, terns context of theory. Using Using terns of of patch patch occupancy occupancy in in the the context of island island biogeography biogeography theory. logistic relation logistic regression, regression, they they analyzed analyzed presence presence and and absence absence of of each each species species in in relation to local to patch patch size size and and isolation isolation distance. distance. They They found found that that the the presence presence of of aa local population was was negatively negatively correlated correlated with with patch patch isolation distance. Because Because iso isopopulation isolation distance. lation migration rate, lation distance distance in in tum turn should should be be negatively negatively correlated correlated with with migration rate, pop populations ulations within within the the local local area area presumably presumably persisted persisted because because they they were were close close enough migration ((Thomas Thomas et ai., enough to to be be in in aa metapopulation metapopulation structure structure linked linked by by migration al., 11992; 992; Thomas Hanski, this Thomas and and Hanski, this volume). volume). Hanski most detailed Hanski and and his his co-workers co-workers have have conducted conducted what what is is perhaps perhaps the the most detailed analysis of analysis of metapopulation metapopulation structure structure for for any any butterfly butterfly species species to to date date with with the the series of in southwest southwest Finland Finland Glanville Glanville fritillary fritillary Meiitaea Melitaea cinxia cinxia on on aa series of islands islands in (e.g., 994, 11995a,b). 995a,b). This This and summarized in in (e.g., Hanski Hanski et ai., al., 11994, and similar similar studies studies are are summarized Thomas (this volume). provides convincing convincing evidence evidence not not Thomas and and Hanski Hanski (this volume). Their Their study study provides only that aa metapopulation metapopulation structure structure exists exists in in M. cinxia, but that there there is only that but also also that is sufficient the dynamics dynamics of sufficient migration migration within within the the system system to to affect affect the of local local populations, populations, including including their their probability probability of of extinction extinction and and subsequent subsequent recolonization. recolonization. All of All of these these studies studies suggest suggest aa pattern pattern of of evidence evidence for for important important rescue rescue effects effects in in butterfly butterfly metapopulations. metapopulations. Many Many species species are are clearly clearly predisposed predisposed toward toward aa meta metapopulation structure structure because because they they occupy occupy distinct habitat patches patches as as aa result result of of population distinct habitat specialized specialized vegetation vegetation requirements requirements during during some some stage stage of of their their life life cycle. cycle. Though Though traditionally viewed primarily - recolonization metapopulations, traditionally viewed primarily as as extinction extinction-recolonization metapopulations, many species show many of of these these species show relatively relatively high high rates rates of of migration migration detected detected directly directly through -recapture and by genetic studies. Migration through mark mark-recapture and suggested suggested indirectly indirectly by genetic studies. Migration in butterflies has (suggesting a rescue in butterflies has been been connected connected to to high high genetic genetic variability variability (suggesting a rescue effect low genetic effect rather rather than than extinction/recolonization extinction/recolonization system), system), low genetic differentiation, differentiation, and and greater greater persistence persistence of of local local populations. populations. However, However, further further research research is is needed needed to directly directly quantify quantify the the demographic demographic effects effects of of migration migration and and demonstrate demonstrate that to that high migration rescuing local local populations high migration rates rates in in various various species species are are actually actually rescuing populations from extinction. from extinction.
B. Small B. Small Mammals Mammals Considerable migration in in mammam Considerable research research has has been been directed directed toward toward studying studying migration mals (see Chepko-Sade Halpin, 1987). 1 987). These have rarely linked mimi mals (see Chepko-Sade and and Halpin, These studies studies have rarely linked gration metapopulation systems, systems, perhaps gration patterns patterns to to population population dynamics dynamics and and metapopulation perhaps because local populations has not because the the spatial spatial structure structure of of different different local populations has not been been studied studied in detail. detail. Particular Particular migration migration patterns patterns have have been been connected connected to to observed observed genetic genetic in population population structure structure in in small small mammals, mammals, so so many many popUlations populations may may exhibit exhibit metameta-
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population dynamics dynamics if if it it is is assumed assumed that that genetic genetic structure structure must must be be the the result result of of population spatial population population structure structure ((Lidicker and Patton, Patton, 11987). Yet Amarasekare Amarasekare ((1994) 987). Yet 1994) spatial Lidicker and notes that that small small mammals mammals such such as as the the banner-tailed banner-tailed kangaroo kangaroo rat rat (Dipodomys (Dipodomys notes spectabilis) can can exhibit exhibit some some spatial spatial structure structure without without occupying occupying the the discrete discrete spectabilis) habitat patches patches required required for for metapopulation metapopulation structure. structure. Therefore, Therefore, we we consider consider only only habitat examples examples of of possible possible metapopulations metapopulations with with strong strong rescue rescue effects effects that that satisfy satisfy the the criterion criterion of of discrete discrete populations, populations, either either as as aa result result of of natural natural habitat habitat patchiness, patchiness, social structure, structure, or or where where the the species species habitat habitat has has been been artificially artificially fragmented fragmented into into social distinct patches patches by by human human activities. activities. distinct Pikas - - Natural Natural Habitat Habitat Patchiness Patchiness 1. Pikas-
A. Smith Smith and and his his colleagues' colleagues' research research on on pikas pikas (Ochotona princeps; Smith, Smith, A. and this this volume) volume) provides provides aa direct direct verifi verification of the the rescue rescue effect effect in in aa 11980, 980, and cation of
natural metapopulation. metapopulation. Pika Pika populations inhabit talus talus slopes surrounded by by un unnatural populations inhabit slopes surrounded suitable sagebrush desert desert habitat. habitat. Smith Smith tested tested the the Brown and Kodric-Brown Kodric-Brown suitable sagebrush Brown and original prediction prediction that that immigration immigration rate rate should should increase increase with decreasing ((1977) 1 977) original with decreasing isolation distance, depressing rates rates of of local local population extinction. He censused isolation distance, depressing population extinction. He censused pika populations populations in in 11972 and again again in in 1977 to examine examine population population persistence. persistence. 972 and 1 977 to pika Though extinction extinction and and recolonization recolonization were were common common on on the most isolated isolated habitat Though the most habitat patches, migration was negatively correlated with with isolation isolation distance, distance, apparently apparently patches, migration was negatively correlated resulting in reduced reduced extinction extinction rates rates of of the the less isolated populations. populations. In addition, resulting in less isolated In addition, some patches patches were were too too small small to to support support breeding breeding populations, populations, yet yet they they persisted some persisted presumably because because migration migration rates rates were were high high enough enough to to create create aa demographic demographic presumably rescue effect effect (Smith, (Smith, 11980). In aa detailed detailed study study of of juvenile juvenile migration among the the rescue 980). In migration among same populations, populations, Peacock Peacock ((1995) found that that most most young young pikas tended to to disperse disperse 1 995) found pikas tended same to neighboring to neighboring talus talus patches. patches. However, However, multilocus multilocus DNA DNA fingerprint fingerprint data data demdem onstrated that that within-population band-sharing scores did not differ differ from what what onstrated within-population band-sharing would assorting randomly would be be expected expected if if individuals individuals were were assorting randomly among among all all patches patches in in each among habitat each generation. generation. Migration Migration among habitat fragments fragments appeared appeared to to be be important important in in maintaining genetic variation maintaining genetic variation within within the the metapopulation metapopulation and and preventing preventing the the loss loss of The mean mean heterozygosity the fragof heterozygosity heterozygosity in in component component populations. populations. The heterozygosity in in the frag mented essentially the (H == 0.736) was was essentially the same same as as that that found found in in aa continuous continuous mented system system (H "mainland" population Together, these stud"mainland" population nearby nearby (H = = 0.709; 0.709; Peacock, Peacock, 1995). 1 995). Together, these stud ies of the evidence available available that ies provide provide some some of the best best evidence that apparent apparent migration migration within within mammalian metapopulations can maintain genetic genetic diversity diversity and and rescue mammalian metapopulations can maintain rescue local local poppop ulations ulations from from extinction. extinction. 2. o g s --Social - S o c i a l Systems 2. Voles Voles and and Prairie Prairie D Dogs Systems That That Create Create Spatial Structure Spatial Structure
Stoddart early research research of Stoddart (1970) ( 1 970) reported reported on on early of water water voles voles (Arvicola terrestris) in in Scotland. Scotland. These These microtine microtine rodents rodents aggregate aggregate into into spatially spatially distinct distinct populations populations along along riverbanks, riverbanks, not not because because the the suitable suitable habitat habitat is is organized organized into into discrete discrete patches, patches, but but rather rather because because each each population population represents represents aa discrete discrete social social group group and and few individuals individuals are are ever ever found found outside outside aa social social unit. unit. The The entire entire system system exhibits exhibits few migration patterns similar metapopulations based migration patterns similar to to those those of of rescue rescue effect effect metapopulations based on on
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distinct recapture experiments distinct habitat habitat patches. patches. Stoddart Stoddart conducted conducted markmark-recapture experiments and and that most were confi ned within but found that most movements movements were confined within aa particular particular social social group, group, but found several female female voles voles dispersed dispersed long distances and and joined social groups. groups. several long distances joined new new social Though research is Though further further research is necessary necessary to to determine determine whether whether or or not not such such migration migration could rescue rescue populations populations from provides early evidence that could from extinction, extinction, this this study study provides early evidence that social structure structure can can cause cause spatial spatial population population structure structure and migration patterns patterns con social and migration consistent theoretical rescue rescue effect sistent with with theoretical effect metapopulation metapopulation structure. structure. Studies of the black-tailed prairie dog (Cynomys Studies of the black-tailed prairie dog (Cynomys ludovicianus) ludovicianus)provide provide more more detailed and convincing evidence that metapopulations with strong rescue detailed and convincing evidence that metapopulations with strong rescue effects effects may may be be common common among among social social mammals. mammals. Black-tailed Black-tailed prairie prairie dog dog populations populations or or colonies Populations remain colonies consist consist of of several several family family groups groups called called coteries. coteries. Populations remain spa spatially (King, 11955; 955; tially distinct distinct from from each each other other because because of of this this rigid rigid social social structure structure (King, Smith, 958). In Smith, 11958). In addition, addition, social social groups groups actively actively alter alter the the vegetation vegetation around around col colonies plants onies by by clipping clipping tall tall plant plant species species to to encourage encourage the the growth growth of of favored favored food food plants ((Bonham Bonham and Lerwick, 11976), 976), in effect creating distinct habitat patches across and Lerwick, in effect creating distinct habitat patches across aa 1 988) used used radio relatively homogenous landscape. Garrett relatively homogenous natural natural landscape. Garrett and and Franklin Franklin ((1988) radio telemetry assess migration populations. telemetry to to assess migration of of prairie prairie dogs dogs between between these these discrete discrete populations. They They detected detected consistent consistent migration migration between between colonies colonies by by male male yearlings yearlings and and aa few few adult females began breeding new colonies adult females and and verified verified that that several several individuals individuals began breeding in in new colonies after after migration. migration. Though Though the the direct direct effects effects of of migration migration were were not not monitored, monitored, Gar Garrett rett and and Franklin Franklin stated stated that that their their research research supports supports the the hypothesis hypothesis that that migration migration functions minimize inbreeding. ux of functions to to minimize inbreeding. The The regular regular infl influx of male male immigrants immigrants should should reduce reduce inbreeding inbreeding within within local local populations populations and and elevate elevate levels levels of of heterozygosity, heterozygosity, potentially rescuing rescuing populations populations from from inbreeding inbreeding depression depression and and possible possible extinc extincpotentially tion Hoogland, 11982). 982). Indeed, 1 983) examined tion ((Hoogland, Indeed, Foltz Foltz and and Hoogland Hoogland ((1983) examined four four poly polymorphic heterozygotes (H = morphic loci loci and and found found aa consistent consistent excess excess of of heterozygotes = 0.068) that that they male migration migration between between colonies. This they attributed attributed in in part part to to the the high high rate rate of of male colonies. This suggests metapopulation structure genetic rescue rescue effects in suggests the the existence existence of of aa metapopulation structure and and genetic effects in black-tailed prairie prairie dogs. dogs. However, However, Chesser Chesser ((1983) found high levels of of inbreed inbreed1 983) found high levels black-tailed ing poly ing (F]s (Fis = = 0.3297) and and reduced reduced heterozygosity heterozygosity in in aa separate separate study study of of seven seven polymorphic black-tailed prairie dogs. Further morphic protein protein loci loci in in black-tailed prairie dogs. Further research research is is clearly clearly needed in prairie needed to to determine determine accurately accurately the the patterns patterns of of migration migration and and their their effects effects in prairie dogs and dogs and other other social social mammals. mammals.
3. Rescue Effects in 3. Mice, Chipmunks, Rats, and SquirrelsSquirrels~Rescue
Fragmented Fragmented Habitats Habitats
Forests United States Forests that that were were once once contiguous contiguous in in the the eastern eastern United States and and Canada Canada are and urban are now now highly highly fragmented fragmented as as aa result result of of agricultural agricultural and urban development. development. Farmlands by Farmlands often often now now contain contain distinct distinct woodlots woodlots separated separated from from each each other other by fields, patches fields, buildings, buildings, and and roads, roads, thereby thereby creating creating aa mosaic mosaic of of suitable suitable habitat habitat patches 972; Middleton 98 1 ). for species ((MacArthur, MacArthur, 11972; for many many woodland woodland species Middleton and and Merriam, Merriam, 11981). White-footed persisted in in this White-footed mice mice (Peromyscus (Peromyscus leucopus) leucopus) have have persisted this network network of of woodlot routes ((Middleton Middleton woodlot patches patches and and farm farm buildings buildings connected connected by by migration migration routes and 98 1 ), creating distinct spatial woodand Merriam, Merriam, 11981), creating aa distinct spatial population population structure. structure. If If each each wood-
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lot local population lot patch patch can can be be said said to to contain contain aa distinct distinct local population of of white-footed white-footed mice, mice, then then both both population population persistence persistence and and migration migration between between populations populations have have been been 1 98 1 ) experimentally directly assessed for species. Middleton Middleton and directly assessed for this this species. and Merriam Merriam ((1981) experimentally removed to monitor monitor immigration. immigration. removed all all white-footed white-footed mice mice from from one one farm farm woodlot woodlot to They individuals that the woodlot woodlot and They trapped trapped aa total total of of 54 individuals that immigrated immigrated into into the and detected farm buildings detected additional additional migration migration between between three three other other woodlots woodlots and and farm buildings included by these high levels levels included in in the the study. study. They They also also found found that that populations populations linked linked by these high of linked by of migration migration had had higher higher growth growth rates rates than than populations populations linked by lower lower levels levels of of migration. be as individuals, rapid rapid migration. Since Since winter winter populations populations can can be as small small as as two two individuals, population migration) could population growth growth in in the the spring spring (apparently (apparently facilitated facilitated by by migration) could be be critical 198 1). critical for for the the persistence persistence of of local local populations populations (Middleton (Middleton and and Merriam, Merriam, 1981). Such in the short term term led Such clear clear evidence evidence for for rescue rescue effect effect in the short led Fahrig Fahrig and and Merriam Merriam ((1985) 1 985) to model migration migration and in P. leucopus in in order to to model and population population structure structure in order to examine patterns of predicted that examine patterns of long-term long-term population population persistence. persistence. The The model model predicted that population should be higher in migration population growth growth rates rates should be higher in populations populations connected connected by by migration (as sizes should should conse (as opposed opposed to to isolated isolated populations), populations), that that winter winter population population sizes consequently quently be be higher higher in in these these metapopulations, metapopulations, and and that that persistence persistence of of local local popu populations the rate migration. lations and and the the overall overall metapopulation metapopulation would would depend depend on on the rate of of migration. They used both field data to verify population They then then used both tracking tracking and and trapping trapping field data to verify that that population growth rates rates were were indeed indeed higher higher in in populations linked by migration (Fahrig (Fahrig and and growth populations linked by migration 985). Gottfried 1 979) also also examined examined population persistence of Merriam, Merriam, 11985). Gottfried ((1979) population persistence of P. distance. He populations in leucopus in in relation relation to to patch patch isolation isolation distance. He found found that that populations in woodlots less than 0.5 km persisted, apparently because of high den woodlots isolated isolated by by less than 0.5 km persisted, apparently because of high densities sities and and high high rates rates of of migration. migration. Other species exhibit exhibit rescue Other small small mammal mammal species rescue effect effect metapopulation metapopulation dynamics dynamics in cially fragmented mosaics. Eastern chipmunks (Tamias stria in artifi artificially fragmented agricultural agricultural mosaics. Eastern chipmunks stria(us) numbers migrating tus) breed breed in in farm farm woodlots woodlots and and have have been been trapped trapped in in high high numbers migrating between woodlots and resident chipmunks chipmunks between woodlots and immigrating immigrating into into patches patches where where all all resident 1 985). Migration had Henderson et al., 1985). had been been experimentally experimentally removed removed ((Henderson Migration probably probably facilitates populations that facilitates patch patch persistence, persistence, because because populations that had had declined declined to to two two to to three three nonbreeding usually rebounded become sizable sizable breeding popula nonbreeding individuals individuals usually rebounded to to become breeding populations. They stability of chipmunk tions. They concluded concluded that that patterns patterns of of occupancy occupancy and and the the stability of chipmunk populations woodlots depend rescue effect. populations in in the the entire entire mosaic mosaic of of woodlots depend on on the the rescue effect. Woodlot Woodlot patches populations in patches may may be be too too small small to to support support discrete discrete local local populations in aa metapopulation metapopulation structure; however, if structure; however, if rescue rescue effect effect dynamics dynamics are are occurring occurring at at this this smaller smaller scale, scale, they level. they also also may may be be present present at at the the metapopulation metapopulation level. One nal set One fi final set of of evidence evidence for for rescue rescue effect effect in in small small mammals mammals comes comes from from Before this studies studies of of Columbian Columbian ground ground squirrels squirrels (Spermophilus columbianus). Before this century, century, ground ground squirrel squirrel habitat habitat in in western western North North America America was was virtually virtually contin continuous, but it, producing uous, but agricultural agricultural development development has has since since fragmented fragmented it, producing islands islands of of habitat Weddell, 11991). 99 I ). Boag Murie ((1981) 1 98 1 ) studied habitat ((Weddell, Boag and and Murie studied migration migration of of squirrels squirrels to to and and from from aa single single colony colony in in one one habitat habitat patch patch and and estimated estimated that that 20% of of all all juveniles population each juveniles successfully successfully migrated migrated from from the the population each year. year. A A smaller smaller but but detectable detectable proportion proportion actually actually became became established established breeders breeders in in other other populations. populations.
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Boag relatively small Boag and and Murie Murie also also noted noted that that local local popUlation population fluctuations fluctuations were were relatively small that compared compared to to those those of of other other species species in in the the genus genus Spermophilus, Spermophilus, suggesting suggesting that 1 99 1 ) migration between populations migration between populations may may increase increase population population stability. stability. Weddell Weddell ((1991) used 5%. In used radio radio telemetry telemetry to to estimate estimate aa juvenile juvenile migration migration rate rate of of 115%. In addition, addition, patch patch occupancy occupancy was was negatively negatively correlated correlated with with isolation isolation distance distance such such that that most most occupied mi occupied patches patches were were isolated isolated by by less less than than 11 km. km. Weddell Weddell concluded concluded that that migration gration between between habitat habitat patches patches was was critical critical for for the the continued continued persistence persistence of of local local squirrel squirrel populations. populations. He He further further emphasized emphasized the the importance importance of of migration migration by by observing observing that that connected connected popUlations populations persisted persisted despite despite eradication eradication attempts attempts by by local local farmers. farmers. Though Though he he does does not not use use the the term, term, it it is is clear clear that that Weddell Weddell ((1991) 199 1 ) is ground is describing describing aa rescue rescue effect effect metapopulation metapopulation of of Columbian Columbian ground squirrels. squirrels. The rescue effect The studies studies discussed discussed above above point point out out patterns patterns of of evidence evidence for for rescue effect metapopulations our current metapopulations in in small small mammals mammals and and highlight highlight gaps gaps in in our current knowledge. knowledge. Many Many diverse diverse species species of of small small mammals mammals may may be be predisposed predisposed toward toward metapopu metapopulations because sociality or lations because they they show show spatial spatial population population structure structure as as aa result result of of sociality or habitat habitat fragmentation. fragmentation. High High rates rates of of migration migration have have been been detected detected through through track tracking trapping efforts, in species that exhibit ing and and trapping efforts, but but more more studies studies of of migration migration in species that exhibit natural spatial spatial structure structure are are clearly clearly needed. needed. Migration Migration has has been been linked linked to to some some demographic demographic and and genetic genetic effects, effects, but but it it has has more more frequently frequently been been implicated implicated in in local local population population and and metapopulation metapopulation persistence persistence and and stability stability through through island island bio biogeography migration for geography studies. studies. Such Such studies studies address address the the consequences consequences of of migration for meta metapopulation persistence, result is is achieved, achieved, either either population persistence, but but the the process process by by which which that that result through extinction extinction-recolonization, effect, or or both, both, remains remains to be deter deterthrough -recolonization, rescue rescue effect, to be mined. mined. The The previous previous examples examples also also illustrate illustrate that that the the spatial spatial structure structure and and migration migration patterns several differdiffer patterns of of metapopulations metapopulations with with strong strong rescue rescue effects effects can can arise arise in in several ent ways. way in which ent ways. Natural Natural habitat habitat patchiness patchiness may may be be the the least least common common way in which spatial Social structure spatial structure structure is is created created for for small small mammals. mammals. Social structure also also appears appears to to encourage migration encourage spatial spatial aggregation aggregation and and the the threat threat of of inbreeding inbreeding may may promote promote migration 1 987) found no consistent consistent relationship relationship in social systems. Lidicker and in social systems. Yet Yet Lidicker and Patton Patton ((1987) found no between may be between social social systems systems and and migration migration patterns, patterns, so so such such generalizations generalizations may be too too simplistic. simplistic. The The most most common common way way in in which which spatial spatial structure structure is is created created appears appears 1 99 1 ) suggested suggested that that mammals to to be be habitat habitat fragmentation. fragmentation. Weddell Weddell ((1991) mammals in in histori historically contiguous evolved strategies long-distance migra cally contiguous habitats habitats never never evolved strategies for for very very long-distance migration populations, so unable tion or or persistence persistence in in small small isolated isolated populations, so these these species species may may be be unable to with strong rescue to persist persist in in human-created human-created metapopulations. metapopulations. Metapopulations Metapopulations with strong rescue effects, effects, in in which which local local populations populations are are close close enough enough to to accommodate accommodate short-dis short-distance tance migration, migration, may may be be the the inevitable inevitable outcome outcome of of habitat habitat fragmentation fragmentation for for these these species. unoc species. Similarly, Similarly, black-tailed black-tailed prairie prairie dogs dogs apparently apparently never never disperse disperse to to unoc988), so they could could not not exhibit cupied habitat cupied habitat patches patches (Garrett (Garrett and and Franklin, Franklin, 11988), so they exhibit extinction/recolonization extinction/recolonization metapopulation metapopulation dynamics. dynamics. Conspecific Conspecific attraction attraction could could promote to rescue metapop promote higher higher rates rates of of successful successful migration, migration, leading leading to rescue effect effect metapopulation dynamics. ulation dynamics.
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C. C. Amphibians Amphibians Wetland usually separated Wetland or or pond pond habitats habitats are are naturally naturally patchy, patchy, usually separated by by terrestrial terrestrial habitats unsuitable for including many habitats that that may may be be unsuitable for wetland-associated wetland-associated species, species, including many amphibians. Thus, many species may metapopulation dynamics. amphibians. Thus, many of of these these species may exhibit exhibit metapopulation dynamics. For example, example, Gibbs Gibbs ((1993) assumed aa metapopulation metapopulation mosaic mosaic to to model model the the loss For 1 993) assumed loss of of small small wetland wetland patches patches and and its its effect effect on on wetland wetland animals. animals. Using Using demographic demographic data distances from previously published empirical stud data and and observed observed migration migration distances from previously published empirical studies of of wetland wetland species, species, including including some some amphibians, amphibians, he that species ies he found found that species with with high migration migration rates rates appeared appeared to to be be buffered from local the rescue high buffered from local extinctions extinctions by by the rescue effect. species that the highest persistence were effect. The The species that showed showed the highest probabilities probabilities of of persistence were amphibians. The studies discussed below support 1 993) theoretical amphibians. The studies discussed below support Gibbs' Gibbs' ((1993) theoretical evi evidence metapopulations in dence for for the the occurrence occurrence of of rescue rescue effect effect metapopulations in numerous numerous wetland wetland systems. systems.
1. Red-Spotted Red-Spotted Newts Newts In empirical investigation 1 978a,b) studied In an an early early empirical investigation of of metapopulations, metapopulations, Gill Gill ((1978a,b) studied the viridescens). These the population population dynamics dynamics of of red-spotted red-spotted newts newts (Notophthalmus (Notophthalmus viridescens). These newts States, traveling into sur newts breed breed in in mountain mountain ponds ponds in in the the eastern eastern United United States, traveling into surrounding terrestrial terrestrial habitat habitat only only to to enter enter winter winter hibernacula. Despite this this distinct distinct rounding hibernacula. Despite spatial Gill ((1978a,b) 1 978a,b) found spatial structure, structure, Gill found evidence evidence for for aa relatively relatively high high migration migration rate ponds. Though migration event, event, he rate between between ponds. Though he he directly directly observed observed only only one one migration he inferred site, since since approximately 50% inferred high high migration migration from from ponds ponds off off his his study study site, approximately 50% of recruits. In In addition, of his his breeding breeding population population was was replaced replaced each each year year by by new new recruits. addition, juveniles in distinctive "eft" life stage is juveniles in these these populations populations pass pass through through aa distinctive "eft" life stage that that is absent not exhibit metapopulation spatial absent in in coastal coastal populations populations that that do do not exhibit metapopulation spatial structure. structure. Gill ((1978a) 1 978a) suggested this life may be Gill suggested that that this life stage stage may be particularly particularly adapted adapted for for migra migration effects of migration on tion in in aa metapopulation metapopulation context. context. The The direct direct effects of migration on local local dynam dynamics 1 978b) calculated calculated very very low low inbreeding ics were were not not extensively extensively assessed, assessed, but but Gill Gill ((1978b) inbreeding coeffi cients for well as the overall metapopulation (F = coefficients for each each population population as as well as the overall metapopulation = 0.000 I to 1 ). In 30.0001 to 0.005 0.0051). In addition, addition, no no local local extinctions extinctions were were observed observed during during the the 3year despite often reproductive success, suggesting demographic year study study despite often negligible negligible reproductive success, suggesting demographic rescue populations by large numbers numbers of immigrants. Gill Gill ((1978a) 1 978a) con rescue of of local local populations by large of immigrants. concluded cluded that that the the populations populations he he studied studied were were completely completely dependent dependent on on migration migration for most immigrants mosaic of for persistence, persistence, and and that that most immigrants came came from from aa shifting shifting mosaic of "meta "metapopulation proceeded to population centers" centers" outside outside of of his his study study area. area. He He proceeded to model model this this meta metapopulation, showing showing that colonization rates rates are high that population, that migration migration or or colonization are so so high that extinc extinction time, even tion almost almost never never occurs occurs and and virtually virtually all all patches patches are are occupied occupied all all the the time, even ' s model if would be if reproductive reproductive success success is is low. low. In In current current terminology, terminology, Gill Gill's model would be considered a rescue effect considered a rescue effect metapopulation. metapopulation. 2. Natterjack Natterjack Toads Toads Sinsch ((1992) 1 992) studied similar system natterjack toads toads (Bufo Sinsch studied aa similar system of of natterjack (Bufo calamita) in use of in Rhineland, Rhineland, Germany. Germany. These These toads toads make make more more extensive extensive use of terrestrial terrestrial habhab-
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itats breeding in in ponds, ponds, itats than than the the red-spotted red-spotted newts, newts, but but they they are are still still restricted restricted to to breeding and they they concentrate concentrate their their terrestrial terrestrial activities activities near near these these breeding breeding sites. sites. Through Through and direct telemetry, Sinsch most direct observation, observation, mark-recapture, mark-recapture, and and radio radio telemetry, Sinsch found found that that most adult juveniles migrate between breeding adult female female toads toads and and some some juveniles migrate between breeding ponds ponds within within seasons, 6seasons, while while most most males males never never leave leave their their first first breeding breeding sites. sites. During During the the 6year study, one pond was consistently a source of migrants. Yet this system was year study, one pond was consistently a source of migrants. Yet this system was not aa traditional traditional sourcesource-sink metapopulation because because the the other other breeding breeding sites not sink metapopulation sites shifted between being sources and sinks; therefore, the location of source popu shifted between being sources and sinks; therefore, the location of source populations populations over time. Sinsch 1 992) did lations moved moved through through the the mosaic mosaic of of populations over time. Sinsch ((1992) did not discuss the of frequent not discuss the demographic demographic consequences consequences of frequent migration migration except except to to con conclude in the red-spotted newt, newt, natterjack persist, clude that, that, as as in the red-spotted natterjack toad toad populations populations may may persist, despite failure, because because of despite reproductive reproductive failure, of immigration immigration and and the the rescue rescue effect. effect. To To assess used allozyme elec assess the the genetic genetic consequences consequences of of high high migration migration rates, rates, he he used allozyme electrophoresis to calculate populations. All trophoresis to calculate genetic genetic distances distances between between the the local local populations. All cal cal0.0646), though culated Nei's D = culated genetic genetic distances distances were were low low ((Nei's = 0.0023 0.0023 to to 0.0646), though the the most distant area genetic differentiation. Overall, Sinsch con most distant area showed showed greater greater genetic differentiation. Overall, Sinsch concluded that exhibit metapopulation metapopulation dynamics similar to to those those cluded that natterjack natterjack toads toads exhibit dynamics similar modeled by Gill ((1978a). l 978a). Most Most sites sites are continually occupied, modeled by Gill are continually occupied, and and local local popula populations tions persist persist due due to to immigration immigration and and the the rescue rescue effect. effect.
3. 3. Pool Pool Frogs Frogs
Along the Baltic coast of of Sweden, Sweden, pool frogs (Rana (Rana lessonae) lessonae) occur occur in in natural natural Along the Baltic coast pool frogs metapopulations, only in bodies. Sjogren ( 1 99 1 ) and metapopulations, reproducing reproducing only in distinct distinct water water bodies. Sj6gren (1991) and Sjogren Gulve ((1994) 1 994) studied studied extinction patterns in frog metapopulation, metapopulation, Sj6gren Gulve extinction patterns in one one frog incorporating island island biogeography biogeography theory. theory. Over Over aa 6-year 6-year period, period, he found that incorporating he found that populations greater than went extinct, less isolated isolated poppop populations isolated isolated by by greater than 1I km km went extinct, while while less ulations ulations tended tended to to persist. persist. He He attributed attributed this this persistence persistence to to the the demographic demographic and and genetic since most most dispersing less than genetic consequences consequences of of migration, migration, since dispersing frogs frogs travel travel less than I1 km probably received km and and persisting persisting populations populations probably received 22- 1155 immigrants immigrants per per gener generation, as ation, as estimated estimated from from demographic demographic data data (Sjogren, (Sj6gren, 1991 1991).). Migration Migration apparently apparently had replacing individuals in had the the demographic demographic effect effect of of replacing individuals lost lost to to pike pike predation predation in heavily depredated 994). Sjogren 1 99 1 ) also heavily depredated ponds ponds (Sjogren (Sj6gren Gulve, Gulve, 11994). Sj6gren ((1991) also examined examined the in egg local the proportion proportion of of fertilized fertilized eggs eggs in egg masses masses and and determined determined that that small small local populations populations showed showed no no evidence evidence of of inbreeding inbreeding depression, depression, aa possible possible result result of of genetic rescue rescue effect. effect. Finally, Finally, immigration immigration may may have have also also increased increased population population genetic stability uctuations, which stability by by mitigating mitigating population population size size fl fluctuations, which can can be be large large in in this this species. Sjogren 1 994) concluded species. Sj6gren Gulve Gulve ((1994) concluded that that the the metapopulation metapopulation structure structure of of pool pool frogs Gill ((1978a) 1 978a) frogs resembles resembles the the rescue rescue effect effect metapopulation metapopulation model model proposed proposed by by Gill for for red-spotted red-spotted newts. newts. Several Several researchers researchers have have found found evidence evidence for for rescue rescue effect effect in in amphibian amphibian spe species. Since cies. Since amphibians amphibians are are restricted restricted to to breeding breeding in in water, water, they they are are predisposed predisposed toward toward spatial spatial metapopulation metapopulation structures. structures. Migration Migration has has rarely rarely been been directly directly de detected, new recruits tected, but but high high rates rates of of migration migration have have been been inferred inferred from from arrivals arrivals of of new recruits
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and and instances instances of of population population persistence persistence despite despite reproductive reproductive failure. failure. Genetic Genetic ef effects migration, such low inbreeding inbreeding coeffi cients and lack of inbreeding fects of of migration, such as as low coefficients and lack of inbreeding depression, usually inferred inferred from depression, have have been been measured. measured. Demographic Demographic effects effects are are usually from observed populations has has observed population population persistence; persistence; therefore, therefore, persistence persistence of of local local populations been attributed to to migration migration and and the the rescue rescue effect. effect. Population Population stability stability may may also also been attributed be to dampen be enhanced enhanced by by the the rescue rescue effect, effect, because because migration migration appears appears to dampen local local popUlation uctuations. population size size fl fluctuations.
IV. CONClUSIONS CONCLUSIONS The The studies studies discussed discussed above above suggest suggest that that metapopulations metapopulations with with strong strong rescue rescue 1 99 1 ; effects (see also Harrison, 1991; effects may may be be more more common common than than currently currently supposed supposed (see also Harrison, Harrison research was Harrison and and Taylor, Taylor, this this volume). volume). In In most most cases, cases, the the research was not not designed designed to published before before the to provide provide evidence evidence for for aa rescue rescue effect, effect, and and some some were were published the term term was was even even introduced. introduced. Many Many organisms organisms live live in in naturally naturally fragmented fragmented environments, environments, and will be and areas areas of of suitable suitable habitat habitat often often will be too too small small to to support support viable viable populations populations over highly stochastic over the the long long term, term, particularly particularly in in highly stochastic environments, environments, if if they they are are completely is not not surprising many completely independent independent of of other other populations. populations. Thus, Thus, it it is surprising that that many species have developed species have developed the the ability ability to to disperse disperse over over considerable considerable distances. distances. Both Both our collection of our models models and and those those of of others others clearly clearly show show that that aa collection of popUlations populations connected connected together together through through migration migration can can persist persist longer longer than than the the same same number number of sizes that of populations populations of of identical identical sizes that are are isolated isolated from from one one another. another. Migration Migration and and metapopulation will buffer individual populations populations against metapopulation exchange exchange will buffer individual against "bad" "bad" years, years, and species to and they they can can allow allow species to persist persist in in fragmented fragmented habitats habitats where where they they might might otherwise extinct. The otherwise quickly quickly go go extinct. The ability ability to to disperse disperse successfully successfully on on aa regular regular basis across lter" that determines the basis across unsuitable unsuitable areas areas may may act act as as aa "fi "filter" that determines the presence presence or or absence absence of of particular particular species species in in particular particular areas. areas. As As aa result, result, the the best best way way to to look look for for metapopulations metapopulations with with strong strong rescue rescue effects effects may may be be to to examine examine habitat habitat types, types, rather rather than than particular particular taxa. taxa. Metapopulation Metapopulation systems systems are are likely likely to to be be widespread widespread throughout taxa, but throughout many many different different taxa, but all all should should be be found found within within patchy patchy habitats, habitats, because because patchiness patchiness confers confers the the spatial spatial population population structure structure that that is is aa prerequisite prerequisite of all all metapopulation metapopulation models. models. For For example, example, the the amphibians amphibians discussed discussed above above were were of fi rst identified potential metapopulation first identified as as potential metapopulation systems systems because because they they occupied occupied spa spatially ponds; therefore, is likely tially distinct distinct ponds; therefore, it it is likely that that other other amphibian amphibian species species and and other other wetland-associated will exhibit rescue effect wetland-associated species species will exhibit rescue effect metapopulation metapopulation dynamics. dynamics. Additional Additional habitat habitat types types that that should should be be examined examined include include tide tide pools pools in in the the Pacifi Pacificc northwest, United States, States, and northwest, riparian riparian areas areas throughout throughout the the western western United and mountain mountain tops tops in the 1 996). in the desert desert southwest southwest (Stacey (Stacey and and Johnson, Johnson, 1996). As As is is true true of of many many biological biological phenomena phenomena that that have have aa spatial spatial component, component, aa central in metapopulation scale. For the distinction central issue issue in metapopulation structure structure is is scale. For example, example, the distinction between between one one popUlation population that that occupies occupies aa number number of of different different patches patches and and aa meta metapopulation populations in population system system that that is is composed composed of of individual individual populations in discrete discrete habitats, habitats,
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will into the will not not always always be be clear. clear. One One system system will will necessarily necessarily grade grade into the other other because because spatial there spatial distribution distribution is is aa continuous continuous rather rather than than discrete discrete variable. variable. However, However, there may these systems, the behavior the orga may be be aa real real difference difference in in these systems, based based upon upon the behavior of of the organisms. In In aa patchy patchy environment, environment, individuals individuals should have aa relatively relatively equal equal prob probnisms. should have ability breeding with within the the natal patch and in nearby ability of of breeding with other other individuals individuals both both within natal patch and in nearby patches; this probability probability in be a function of the migration patches; this in turn turn will will be a function of the migration capabilities capabilities of of the species. In populations, the the species. In aa metapopulation metapopulation consisting consisting of of individual individual populations, the indi indiwith an vidual vidual will will have have aa much much lower lower probability probability of of mating mating with an individual individual in in another another population than one. Mating highly discontinuous. population than in in their their natal natal one. Mating probabilities probabilities are are highly discontinuous. In In terms terms of of demographic demographic models, models, this this means means that that the the growth growth of of the the population population that occupies a collection of that occupies a collection of patches patches can can be be described described as as the the sum sum of of birth birth and and deaths system, we deaths within within that that set set of of patches. patches. In In contrast, contrast, in in aa metapopulations metapopulations system, we must must describe describe both both local local population population processes processes and and migration migration frequency frequency among among populations. will often populations. Because Because migration migration will often be be independent independent of of local local demography, demography, metapopulations will will exhibit exhibit emergent (such as metapopulations emergent properties properties (such as continued continued persistence persistence through its parts. through rescue rescue effects) effects) that that are are greater greater than than the the sum sum of of its parts. Similarly, Similarly, the the distinction distinction between between metapopulations metapopulations where where extinction extinction and and sub subsequent is sequent recolonization recolonization is is the the primary primary dynamic dynamic and and systems systems where where migration migration is frequent be primarily primarily a matter frequent enough enough to to prevent prevent most most local local extinctions extinctions may may also also be a matter of spatially and local populations to go go of scale, scale, both both spatially and temporally. temporally. All All local populations are are likely likely to extinct observed for long enough enough period. period. Similarly, extinct eventually, eventually, if if they they are are observed for aa long Similarly, many many 978a; Sj6gren, Sjogren, 11991; 99 1 ; Sinsch, of the the mammal mammal studies discussed above above (e.g., (e.g., Gill, Gill, 11978a; Sinsch, of studies discussed 11992) 992) have have found found that that persistence persistence of of many many small small populations populations is is aa matter matter of of their their degree of of isolation isolation from from other other populations. populations. Adjacent Adjacent populations populations may may last degree last longer longer because they rescue effect with frequent because they are are part part of of aa rescue effect metapopulation metapopulation with frequent migration, migration, where rescued less less frequently frequently and where more more distant distant populations populations are are rescued and exhibit exhibit extinction/ extinction] recolonization dynamics. Although Although these models may may vary vary in in some recolonization dynamics. these models some predictions predictions about local population or processes processes (e.g., about local population characteristics characteristics or (e.g., the the level level of of genetic genetic di diversity within the they are likely to to be versity within the metapopulation), metapopulation), they are not not likely be mutually mutually exclusive exclusive in metapopulations can in nature. nature. Real Real world world metapopulations can and and apparently apparently do do exhibit exhibit more more than than one model all of the above above examples, rescue effects to occur one model dynamic. dynamic. In In all of the examples, rescue effects seemed seemed to occur in isolated populations probably in the the central, central, least least isolated populations while while extinction/recolonization extinction/recolonization probably occurred the peripheral, peripheral, most most isolated isolated populations. occurred among among the populations. The The combination combination of of strong strong rescue rescue effects effects and and extinction/recolonization extinction/recolonization dynamics dynamics may may be be common common in in nature, rates should be negatively negatively correlated isolation nature, because because migration migration rates should be correlated with with isolation distance Brown and 977). Differences in migration may distance ((Brown and Kodric-Brown, Kodric-Brown, 11977). Differences in migration rates rates may be which metapopulation im be the the primary primary factor factor determining determining which metapopulation dynamic dynamic is is most most important, there is no a that dispersing dispersing individuals individuals will portant, as as there is no a priori priori reason reason to to expect expect that will settle settle only only in in habitats habitats that that are are currently currently unoccupied. unoccupied. In In each each case, case, aa central central chal challenge systems will will be lenge in in understanding understanding these these systems be to to determine determine the the frequency frequency and and timing of resulting impact upon local local poppop timing of among-population among-population migration migration and and its its resulting impact upon ulation ulation dynamics. dynamics. Metapopulations popMetapopulations can can be be more more resistant resistant to to extinction extinction than than independent independent pop-
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Migration Migrationwithin within Metopopulotions Metapopulations
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ulations that while ulations because because the the stochastic stochastic nature nature of of variation variation in in growth growth rates rates means means that while some are likely some populations populations are are declining, declining, other other elements elements in in the the system system are likely to to be be increasing 987a). The ability of increasing and and producing producing potential potential immigrants immigrants (Goodman, (Goodman, 11987a). The ability of organisms this effect; organisms to to successfully successfully migrate migrate among among populations populations is is critical critical for for this effect; even persistence even small small increases increases in in system system connectivity connectivity result result in in large large increases increases in in persistence times. We this volume. times. We thus thus repeat repeat the the conclusions conclusions reached reached by by other other authors authors in in this volume. For or For natural natural metapopulations, metapopulations, any any change change in in the the environment environment (either (either natural natural or human to the the decline human caused) caused) that that makes makes migration migration more more difficult difficult may may lead lead to decline and and possible extinction extinction of of both both individual individual populations populations and and the the entire entire metapopulation. metapopulation. possible Management usually focus single largest, Management and and preservation preservation efforts efforts usually focus on on the the single largest, or or most most representative, population or ecosystem. Smaller popUlations or habitats are often representative, population or ecosystem. Smaller populations or habitats are often ignored Even though large populations ignored and and destroyed destroyed by by development. development. Even though large populations may may ap appear to be stable and self-sustaining, they may actually be highly dependent pear to be stable and self-sustaining, they may actually be highly dependent on on other other populations populations in in aa metapopulation metapopulation system system for for their their continued continued persistence persistence (Sta (Sta992). Without cey and and Taper, Taper, 11992). Without immigration, immigration, these these populations populations may may collapse, collapse, even even cey though though their their local local habitat habitat has has been been protected. protected. Migration Migration has has long long been been considered considered to negative effects to be be important important to to prevent prevent the the negative effects of of inbreeding inbreeding in in small small populations; populations; our our analyses, analyses, and and those those of of other other researchers, researchers, emphasize emphasize that that it it may may have have even even more more immediate immediate value value in in terms terms of of demographic demographic rescue. rescue. Recognition Recognition of of the the po potential tential importance importance of of metapopulation metapopulation structure structure in in the the dynamics dynamics of of species species will will make make conservation conservation efforts efforts more more challenging, challenging, but but also also more more likely likely to to succeed succeed in in the the long long run. run.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS Ric Roche helped with the code for the simulation model. We thank thank Elisabeth Ammon, Ammon, Erik Doerr, improved this work. Doerr, Brenda Brenda Johnson, Johnson, and Patricia Patricia Zenone for for comments comments that substantially improved P.B.S. is supported supported by NSF NSF Grant DEB-9302247, V.A.J. by a University of of Nevada Graduate FelP.B.S. DEB-9302247, V.A.J. Fel lowship, and DEB94 I 1 770. and M.L.T. by EPA EPA Grant CR-820086 CR-820086 and NSF Grant DEB9411770.
sdfsdf
13
Evolution of Migration Rate and Other Traits Metapopu/ation Effect The Metapopulation Isabelle Olivieri
Pierre-Henri Gouyon
I.I. INTRODUGION INTRODUCTION Traditionally, have put put aa strong Traditionally, population population geneticists geneticists have strong emphasis emphasis on on the the ev ev952, 11969; 969; olutionary Wright, 11952, olutionary consequences consequences of of subdivision subdivision within within aa population population ((Wright, Malecot, 948, 11969; 969; Maruyama, 970; Kimura 97 1 ; Nei, Mal6cot, 11948, Maruyama, 11970; Kimura and and Maruyama, Maruyama, 11971; Nei, 11987). 987). Population Population structure structure has has been been viewed viewed as as aa way way to to produce produce spatial spatial differ differentiation and and founder founder effects effects through through stochastic stochastic processes processes (e.g., (e.g., Wright' Wright'ss shifting shifting entiation 969; Maruyama, 970; Maruyama 980). More balance balance theory, theory, 11969; Maruyama, 11970; Maruyama and and Kimura, Kimura, 11980). More recently, recently, another another class class of of models, models, hereafter hereafter called called metapopulation metapopulation evolutionary evolutionary models, models, has has focused focused on on the the evolutionary evolutionary consequences consequences of of population population extinctions extinctions 97 1 ). In and Van Valen, and recolonizations recolonizations ((Van Valen, 11971). In these these latter latter models, models, the the emphasis emphasis is is on on the the selective selective pressures pressures created created by by population population turnover turnover within within aa metapopulation. metapopulation. The The focus focus is is on on the the evolution evolution of of particular particular traits traits (migration, (migration, sex-ratio, sex-ratio, life-history life-history traits, traits, etc.) etc.) whose whose genetic genetic determinism determinism is is usually usually unknown. unknown. These These two two classes classes of of models models have have generated generated two two terminologies terminologies (see (see Hanski Hanski and i.e., aa set and Simberloff, Simberloff, this this volume) volume) that that amount amount to to the the same same object object ((i.e., set of of indi individuals viduals subdivided subdivided into into more more or or less less ephemeral ephemeral subsets) subsets):: classical classical population population genetics genetics considered considered aa population population subdivided subdivided into into subpopulations, subpopulations, or or demes, demes, or or neighborhoods, neighborhoods, while while metapopulation metapopulation evolution evolution is is about about aa metapopulation metapopulation made made of of ephemeral ephemeral local local populations. populations. These These two two approaches approaches have have not not yet yet converged, converged, MefOIJ()pulafioll MetapopldationBiology Biology
Copyright 997 by Copyright © 9 11997 by Academic Academic Press. Press, Inc. Inc. All All rights rights 01 el reproduction reproduction in in any any fonn form reserved. reserved.
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Isabelle Olivieri Isabelle Olivieri and Pierre-Henri Pierre-Henri Gouyon Gouyon
though way. Models Models of subdivided pop though some some efforts efforts in in this this direction direction are are under under way. of subdivided populations dynamics (e.g., ulations now now include include the the effects effects of of local local extinctions extinctions and and local local dynamics (e.g., Barton 1 977; Slatkin Wade, Barton and and Whitlock, Whitlock, this this volume; volume; see see also also Slatkin, Slatkin, 1977; Slatkin and and Wade, 11978; 978; Lande, 1 985a; Ohta, 992; Michalakis Olivieri, 1993; 1 993; Nichols Lande, 1985a; Ohta, 11992; Michalakis and and Olivieri, Nichols and and Hewitt, 11994; 994; see 1 993, for these concepts Hewitt, see Hauffe Hauffe and and Searle, Searle, 1993, for an an application application of of these concepts to mice). Metapopulation models have have so to Robertsonian Robertsonian fusion fusion in in mice). Metapopulation evolutionary evolutionary models so far far been the evolution migration rate rate ((Van Van Valen, 97 1 ; been mostly mostly used used to to study study the evolution of of migration Valen, 11971; 980; Levin 984; Olivieri Olivieri and 985; Venable, Comins Comins et et aI., al., 11980; Levin et et al., al., 11984; and Gouyon, Gouyon, 11985; Venable, 11993; 993; but 982, for for a dis but see see Caswell, Caswell, 11982, a study study of of the the influence influence of of demographic demographic disequilibrium this chapter, we will will focus metapopulation equilibrium on on life-history life-history traits). traits). In In this chapter, we focus on on metapopulation evolutionary wish to that the functioning of me evolutionary models. models. We We wish to show show that the demographic demographic functioning of metapopulations properties that influence the tapopulations creates creates intrinsic intrinsic emergent emergent properties that influence the evolution evolution of of major this effect, major biological biological traits traits such such as as migration migration rate. rate. We We suggest suggest that that this effect, which which be taken into account when studying we we call call the the metapopulation m e t a p o p u l a t i o n effect, effect, should should be taken into account when studying the the evolution evolution of of life-history life-history traits traits and and genetic genetic systems systems in in species species which which are are com composed metapopulation. posed of of transient transient populations, populations, where where the the unit unit of of evolution evolution is is the the metapopulation. Migration it Migration plays plays aa central central role role in in metapopulation metapopulation dynamics dynamics and and evolution: evolution: it contributes contributes to to metapopulation metapopulation spatial spatial structure, structure, local local dynamics, dynamics, and and metapopu metapopulation lation evolution, evolution, as as shown shown by by other other chapters chapters of of this this volume volume (see (see also also Venable, Venable, 11993). 993). Migration Migration is is itself itself an an evolving evolving character, character, and and many many species species exhibit exhibit distinct distinct adaptations adaptations to to migration migration (e.g., (e.g., in in plant plant species, species, pappi pappi on on seeds, seeds, spines spines on on pods). pods). The The possible possible evolution evolution of of migration migration is is testified testified to to by by the the occurrence occurrence of of polymor polymorphisms for phisms for migratory migratory behavior behavior in in numerous numerous species species of of animals animals and and plants plants (see (see examples 1 979), Venable examples of of seed seed migration migration polymorphism polymorphism in in Venable Venable ((1979), Venable and and Law Lawlor 1 980), Olivieri Olivieri et 1 983), Olivieri Berger ((1985), 1 985), Schmitt 1 985), lor ((1980), et al. al. ((1983), Olivieri and and Berger Schmitt et et al. al. ((1985), and 1 982) and 1962), Dingle Dingle and Clay Clay ((1982) and of of wing wing dimorphism dimorphism in in insects insects in in Southwood Southwood ((1962), et al. al. ((1980), Kaitala ((1990), et al. al. ((1991), Karlson and and Taylor Taylor et 1 980), Kaitala 1 990), Ben-Shlomo Ben-Shlomo et 1 99 1 ), Karlson ((1992), 1 992), Hairston 1 993), and 994); see 1 980), Roff l 986), Hairston ((1993), and Tsuji Tsuji et et al. al. (( 11994); see Harrison Harrison ((1980), Roff ((1986), and 1 992) for insects and Pulido and Roderick Roderick and and Caldwell Caldwell ((1992) for reviews reviews of of insects and Berthold Berthold and and Pulido ((1994) 1 994) for migratory activity birds). We for aa study study of of heritability heritability of of migratory activity in in birds). We will will not not make make a undirected a distinction distinction between between migration migration and and dispersal dispersal and and will will consider consider both both as as undirected and and irreversible irreversible movement movement away away from from the the habitat habitat patch patch of of origin origin (e.g., (e.g., strict strict dispersal Boer ((1990)): 1 990) ) : the will dispersal as as defined defined by by den den Boer the migration migration rate rate of of aa genotype genotype will be the patch in be equal equal to to the the probability probability that that an an individual individual of of that that genotype genotype leaves leaves the patch in which which it it was was born. born. Why does does migration migration evolve? evolve? At At first first sight, sight, migration migration should should be be selected selected Why against. is a against. First, First, in in most most species, species, there there is a risk risk of of dying dying during during migration, migration, for for in instance stance because because of of predation. predation. There There may may thus thus be be an an intrinsic intrinsic cost cost to to migration. migration. Second, imagine patches, where where some Second, imagine aa landscape landscape covered covered with with aa suite suite of of habitat habitat patches, some patches for a species, while while others not. Suppose patches are are suitable suitable for a given given species, others are are not. Suppose also also that that migration is restricted to within this landscape and is uniform across the patches. migration is restricted to within this landscape and is uniform across the patches. lead individuals individuals to bad environments. It is is clear clear that that migration migration may may lead to bad environments. The The total total It number of i.e., those patches) should should be be the the same number of emigrants emigrants ((i.e., those who who leave leave the the patches) same as as or or larger if there i.e., larger ((if there is is mortality mortality during during migration) migration) than than the the number number of of immigrants immigrants ((i.e., those patches (those inhabited by the those who who arrive arrive in in patches). patches). Since Since only only suitable suitable patches (those inhabited by the
13 Evolution of Migration Migration Rate Rate and Other Traits 1 3 Evolution
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species in in question) question) produce produce emigrants, emigrants, while while all all patches, patches, including including unsuitable unsuitable species ones, receive receive immigrants, immigrants, it it is is clear clear that, that, in in aa suitable suitable patch, patch, the the number number of of emem ones, igrants is is on on average average larger larger (often (often much much larger) larger) than than the the number number of of immigrants. immigrants. igrants A gene gene enhancing enhancing migration migration should should thus thus leave leave patches patches more more often often than than it it is is reinrein A troduced by by immigration immigration and and should, should, on on average, average, decrease decrease in in frequency frequency within within troduced patches. Overall Overall there there exists exists therefore therefore aa selection selection for for residency residency within within each each local local patches. population. population. There are, however, however, also also factors factors that that select select for migration among among populations; populations; There are, for migration these are factors acting acting at at the the among-population among-population level: level: avoidance avoidance of of sib sib compecompe these are factors tition, improved improved conditions conditions elsewhere. elsewhere. Migration Migration is is especially especially favored favored when when the the tition, spatio-temporal variability variability in in population population sizes sizes is is large large (Levin ( Levin et et al., al., 1984; 1 984; SouthSouth spatio-temporal wood, 1987; 1 987; Venable Venable and and Brown, Brown, 1988). 1 988). An An extreme extreme case case of of temporal temporal variability variability wood, in population population sizes sizes occurs occurs if if local local populations populations go go extinct. extinct. Johnson Johnson and and Gaines Gaines in ( 1 990) recently recently reviewed reviewed the the main main models models on on the the evolution evolution of of migration. migration. Few Few of of (1990) these models models explicitly explicitly consider consider local local extinctions, extinctions, though though such such extreme extreme variability variability these is realistic realistic in in many many cases, cases, as shown in in this this volume. volume. In In many many species, species, in in fact, fact, is as shown popUlation extinction is unavoidable, unavoidable, either of stochastic disturbances, population extinction is either because because of stochastic disturbances, or process of succession, or ultimately because because of of demographic demographic or the the ecological ecological process of succession, or ultimately stochasticity. Global persistence persistence of of any any genotype genotype in in such such species species necessitates necessitates stochasticity. Global colonization after after local local extinctions. extinctions. Each particular population population will will eventually eventually go go colonization Each particular extinct therefore only only offspring offspring which have emigrated emigrated will will be be able to reprorepro extinct and and therefore which have able to duce. Contrary to the the short-term short-term cost cost of of migration, migration, there there may may thus thus be be aa long-term long-term duce. Contrary to benefit to to it. it. benefit Thus two two opposing opposing selection pressures, selection selection for migration during during recolrecol Thus selection pressures, for migration selection against population has has been been established, established, onization and and selection against migration migration once once aa population onization act on on the the migration migration rate rate when local extinctions extinctions are are the the sole sole source source of of environ environact when local mental mental variation. variation. These These two two antagonistic antagonistic selective selective forces forces create create a metapopulation metapopulation the effect. effect. This This metapopulation metapopulation effect effect was was observed observed in in the the very very first first studies studies on on the evolution 962, 11987; 987; Van Van Valen, Valen, 1971; 1 97 1 ; see also den evolution of of migration migration (Southwood, (Southwood, 11962, see also den Boer, 11990). Other traits traits apart apart from from migration migration rate rate may may also also experience experience variable variable 990). Other Boer, 1 982) has that, in in a selection Caswell ((1982) selection due due to to the the metapopulation metapopulation effect. effect. Caswell has shown shown that, a given variability in population size size would would create given population, population, the the variability in population create aa succession succession of of episodes of of r selection selection and and K selection. selection. In In this will consider consider in in more more this paper, paper, we we will episodes detail the in a Moreover, we detail the evolution evolution of of life-history life-history traits traits in a metapopulation. metapopulation. Moreover, we will will show that that selection selection on on migration migration may may interact interact with with selection selection on on other other life-history life-history show traits, so traits, so that that coevolution coevolution among among characters characters may may occur occur as as aa result result of of the the two-level two-level selective process. selective process.
II. HOW HOWDOES DOESAA METAPOPULATION METAPOPULATIONPERSIST PERSISTIN IN AA LANDSCAPE? LANDSCAPE?AN AN EXAMPLE EXAMPLEOF OFAA MODEL MODELWITH WITHLOCAL LOCALDISTURBANCES DISTURBANCESAND AND SUCCESSIONAL SUCCESSIONALPROCESSES PROCESSES In - IV.A) or In this this chapter, chapter, we we present present either either published published (Sections (Sections 11 II-IV.A) or new new (the (the rest rest of of the the chapter) chapter) results results based based on on aa deterministic deterministic model model which which has has been been de de1 985) and 1995). This scribed in in Olivieri Olivieri and and Gouyon Gouyon ((1985) and in in Olivieri Olivieri et al. ((1995). This model model scribed
Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-Henri Gouyon Gouyon Isabelle
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is based based on on aa Leslie Leslie matrix matrix approach, approach, equivalent equivalent to to the the continous continous time time model model is in in Levin Levin and and Paine Paine (1974), ( 1 974), and and the the discrete discrete version version used used by by Paine Paine and and Levin Levin ( 1 98 1 ). We We describe describe here here the the main main assumptions assumptions of of this this model, model, extended extended to to aa (1981). species with with overlapping overlapping generations. generations. This This model model will will be be used used in in the the following following species sections. sections. The metapopulation metapopulation is is assumed assumed to to evolve evolve in in aa landscape landscape made of an an infinite infinite The made of number number of of patches, patches, each each containing containing at at most most one one population population (Fig. (Fig. 1). 1 ). A A given given patch z, in 0, 1, I , 22,..... ... ,Z, in which which state state 0° represents represents an an patch may may exist exist in in any any of of states states 0, unoccupied patch patch as as aa result result of of extinction extinction through through disturbance, disturbance, state state i(iE{ i (i E { 1,z l ,z -unoccupied f ) aa patch patch that that has has persisted persisted through through i consecutive consecutive seasons seasons without without an an extinction extinction 1I }) late successional successional patch patch in in which which recruitment recruitment by by the the organism organism event, and and state state zZ aa late event, under study study is is prevented by later later successional successional species. species. We We assume assume that that extinction extinction under prevented by through disturbance occurs occurs at at aa rate rate which which may may depend depend on on the the number number of of seasons seasons through disturbance since the the most most recent recent extinction. extinction. Ai Ai (iE{0,z}) (iE { O,z f ) is is the the probability probability that that aa patch patch in in since state is not not disturbed disturbed (and (and proceeds proceeds to to state + 11 in in the the next next season season if if i << zz or or state i is state i + if ii = z). This holds for empty patches as well, well, so so that the remains in in state remains state zz if -- z). This holds for empty patches as that the probability that given disturbed is recolonized propagules probability that a a given disturbed patch patch is recolonized and and produces produces propagules is equal its probability probability of persistence Ao. Ao. is equal to to its of persistence A shown in in Olivieri Olivieri and ( 1 985), Horvitz Horvitz and and Schemske Schemske (1986), ( 1 986), Ass shown and Gouyon Gouyon (1985), Hastings and and Wolin Wolin ((1989), 1 989), and Olivieri et et al. ( 1 995), these these assumptions assumptions describe describe Hastings and Olivieri al. (1995), aa Markovian stationary age Markovian process, process, such such that, that, at at equilibrium, equilibrium, one one can can express express aa stationary age distribution Vi as as aa function of A; A, and and z. When When zz is is infinite infinite and distribution of of patches patches V;, function of and ' A for then V V == A A = = 1 1 -- e, e, where is the the local local extinction extinction rate and V, V, Ai = A; = A for all all i, then where e is rate and the proportion proportion of occupied patches, patches, is is the the sum of V; V; over over all all ii except except ii == 0. 0. In In the of occupied sum of all examples examples given given in this this paper, paper, we we assume assume that that disturbance disturbance in in different different patches patches occurs independently. Moreover, Ai = occurs independently. Moreover, we we assumed assumed that that A; = A A 1I for for all all ° 0 < < i< < z, but but assume A o , and prob assume that that the the probability probability of of recolonization recolonization of of empty empty patches, patches, A0, and the the probability of : , have distinct values. values. ability of persistence persistence of of late late successional successional patches, patches, A A_, have distinct At At the the beginning beginning of of the the season, season, residents residents of of aa given given patch patch reproduce. reproduce. Adults Adults
TYPES OF SITES TYPES SITES
and transition probabilities
~
~
~ A i l1-- A A ii+1 +l , ,-,,\ x,~ \
1 - Ai
11-A A z-l z-1
~ \
Az 11-Az
\ ~ ~.._......
\ \� (E)-9 -(8)·-6-( 8 ) -· · (S)-(S)� � �
1-Ao
~x...~/. '~'
Vo Vo
~
Ao Ao
1 -A 22 ~~1_A11_A l- Al ' Al A
~
Vl
\
AH ' ~ ' ' ' ~ " A A2z Ai_l
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Vi
AI ~ " A i .Ai+1 Ai l
Vi+1
~
Az
"~"Az-1 AZ-l '~".......~ Vz_1 VZ-1
VZ
Frequency per type of site
FIGURI: 1| FIGURE
Metapopulation dynamics dynamics in a fragmented fragmented landscape. landscape. Each Each patch of age i can can be be either Metapopulation disturbed disturbed (with (with probability probability I1 - A,) A;) or or left left undisturbed undisturbed (with (with probability probability A,) A;) and and then then proceed proceed to to the the I. (Reprinted following successional successional stage stage i + + 1. (Reprinted with permission permission of University of Chicago Chicago Press Press from from 995.) Olivieri Olivieri er et al al.,.. 11995.)
1133
Evolution Evolutionof of Migrotion Migration Rote Rate and and Other Other Traits Troits
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either persist. Juveniles remain in in the patch either die die after after reproduction reproduction or or persist. Juveniles either either remain the natal natal patch to 11 in or disperse at some fixed, or disperse at some fixed, genetically genetically based based rate. rate. A A proportion proportion q q (equal (equal to in our the migrant migrant pool. our simulations) simulations) of of dispersing dispersing juveniles juveniles survive survive to to form form the pool. Comins Comins ((1982) 1 982) has has shown shown that that the the ESS ESS of of migration migration rate rate in in aa two-dimensional two-dimensional stepping steppingstone model. In In contrast, stone model model was was very very similar similar to to that that in in an an island island model. contrast, Lavorel Lavorel et et 1 995) and 1996) have some condi al. a l . ((1995) and Lavorel Lavorel and and Chesson Chesson ((1996) have shown shown that that under under some conditions, to be tions, the the detailed detailed spatial spatial pattern pattern of of environmental environmental variation variation has has to be taken taken into into account. In some species at least, however, an island model of migration account. In some species at least, however, an island model of migration appears appears to actual functioning In to be be aa good good approximation approximation of of the the actual functioning of of the the metapopulation. metapopulation. In our assume such migration. Each Each patch patch receives receives our model, model, we we assume such an an island island model model of of migration. an compete with nondispersing an equal equal fraction fraction of of the the migrants, migrants, which which then then compete with local local nondispersing juveniles. assume local viability and juveniles. We We assume local density-dependent density-dependent regulation regulation of of juvenile juvenile viability and a which is is reached reached at rate a patch patch age-independent age-independent carrying carrying capacity capacity equal equal to to K, which at aa rate patch in in dependent dependent on on the the parameter parameter values. values. For For aa given given number number N N of of adults adults in in aa patch first let let adult adult survival place in season t, we season we first survival to to take take place in aa density-independent density-independent manner. manner. If s is is the the constant constant adult adult survival survival rate rate from from one one season season to to the the next, juveniles next, juveniles If compete places left If the compete for for the the K K - Ns N s places left empty. empty. If the number number of of competing competing juveniles juveniles is less than is less than K K - Ns, N s , they they all all establish establish as as adults adults in in the the following following season. season. Otherwise, Otherwise, there juveniles that 1 , and patch there are are exactly exactly K K - Ns N s juveniles that become become adults adults at at time time t + + 1, and the the patch has All patches, has reached reached the the carrying carrying capacity. capacity. All patches, including including empty empty ones, ones, receive receive migrants. migrants. Extinction Extinction or or persistence persistence of of each each population population then then occurs occurs according according to to no disturbance, migrants and the local probabilities the local probabilities A A;.i ' If If there there is is no disturbance, migrants and resident resident pro propagules (if (if any) equal competitive is disturbance, pagules any) establish establish with with equal competitive abilities. abilities. If If there there is disturbance, no no individual individual establishes establishes in in the the site. site. The local) level The metapopulation metapopulation must must be be regulated regulated either either at at the the patch patch ((local) level or or at at the metapopulation the metapopulation (global (global)) level. level. Without Without density-dependent density-dependent regulation regulation (see (see Hastings and and Wolin Wolin ((1989) for aa model model with with global global regulation and local expoHastings 1 989) for regulation and local expo nential nential growth), growth), either either the the metapopulation metapopulation asymptotic asymptotic growth-rate growth-rate is is greater greater than than 11,, in case there is metapopulation is less less than in which which in which which case there is metapopulation explosion, explosion, or or it it is than 11,, in case case, it nevertheless pospos case the the metapopulation metapopulation goes goes extinct. extinct. In In the the latter latter case, it is is nevertheless sible sible that that each each extant extant population population is is in in fact fact growing. growing. The The metapopulation metapopulation struc structure itself does number of individuals ture itself does not not create create any any overall overall regulation regulation of of the the number of individuals 973), but neces (contrary Wilson, 11973), (contrary to to Wilson, but some some kind kind of of density density dependence dependence is is necessary Hanski, sary to to prevent prevent the the metapopulation metapopulation from from demographic demographic explosion explosion ((Hanski, 11990; 990; Taylor, 990). In model, a Taylor, 11990). In our our deterministic deterministic simulation simulation model, a viable viable metapopu metapopulation lation reaches reaches aa demographic demographic equilibrium equilibrium through through within-patch within-patch density-dependent density-dependent regulation. regulation. In the one In aa density-independent density-independent context, context, such such as as the one modeled modeled by by Horvitz Horvitz and and Schemske 1 986) and 1 994), the the response metapopulation Schemske ((1986) and Cipollini Cipollini eett aal. l . ((1994), response of of metapopulation growth the necessary infor growth rate rate to to changes changes in in life-history life-history parameters parameters provides provides the necessary information de mation concerning concerning selective selective pressures pressures on on those those characters. characters. However, However, density density dependence is likely first, because it pendence is likely to to affect affect the the evolution evolution of of life-history life-history traits, traits, first, because it creates in the and, second, second, because creates frequency frequency dependency dependency in the evolution evolution of of migration migration and, because it unlikely that classes would it is is unlikely that in in species species with with overlapping overlapping generations generations all all age age classes would -
-
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298 298
Isabelle and Pierre-Henri Gouyon IsobelleOlivieri Olivieriand Pierre-HenriGouyon
be equally sensitive sensitive to model thus be equally to density. density. The The Cipollini Cipollini et et al. al. m o d e l of of patch patch ddynamics y n a m i c s thus cannot adaptive evolution. cannot be be really really used used to to study study adaptive evolution.
III. WITHIN-POPULATION SELEGION VERSUS WITHIN-POPULATIONSELECTION VERSUSCOLONIZATION COLONIZATIONSELEGlON: SELECTION:AN AN INSIGHT INSIGHT INTO INTO THE THE METAPOPULATION METAPOPULATIONEFFEG EFFECT A. A. Polymorphism Polymorphism for for Migration Migration Van Valen ((1971) 1 97 1 ) suggested that that migration polymorphisms could be main maintained populations). tained by opposing selection pressures pressures within within and and between between groups ((populations). Using an analytical model for limiting cases ((patch patch saturation at recolonization), recolonization), and the more more general general numerical numerical model described described in the previous section, we have shown that that two genotypes straddling straddling the evolutionarily stable stable migration migration rate rate can indeed be maintained together in a stable pattern, as suggested by Van Valen ((1971) 1 97 1 ) and Roff ((1975) 1 975) and shown recently by Roff ((1994b); 1 994b); (Olivieri et al. , et al., 11995; 995; see an example poly example in Fig. 2). W Wee will see iinn the next section that this polymorphism is in fact not evolutionarily stable. At equilibrium, equilibrium, and in the cases in which a stable stable polymorphism could be
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113 3
Evolution Evolutionof of Migration MigrationRate Rateand and Other OtherTraits Traits
299 299
maintained, had the same maintained, all patches patches of of a given age had same genotypic frequencies. frequencies. These These frequencies shows that frequencies differed differed among among patches patches of of different different ages. Figure Figure 2 shows that the the frequency increases with the age frequency of of the the genotype genotype with the lowest lowest migration migration rate rate increases of of the of the population. population. Therefore, Therefore, each population population is evolving evolving toward toward fixation of nonmigrating genotype. demonstrates that selection nonmigrating genotype. This pattern pattern demonstrates selection is indeed indeed oper operating at opposite directions colonization (when favored) directions at the time of of colonization (when migration migration is favored) and thereafter migration is selected against thereafter (when migration against in each each population). population). We We are are aware aware of of only three three empirical empirical studies of of the relationship relationship between between successional 1 985; Olivieri successional stage and and migration migration ability (Brown, (Brown, 1985; Olivieri and and Gouyon, Gouyon, 11985; 985; Peroni, 994). Only two them (Olivieri and 985; Peroni, 994) Peroni, 11994). two of of them and Gouyon, Gouyon, 11985; Peroni, 11994) consider ( 1 985) found found that consider within-species within-species variability. variability. Brown Brown (1985) that along along plant succes succession, the proportion proportion of Heteroptera species winged decreased of Heteroptera species that were fully winged decreased from from 95 to 65% (see her Fig. 2). The observed observed decrease decrease was, however, however, not statistically pro signifi cant. The thistles Carduus pycnocephalus significant. pycnocephalus and C. tenuiflorus show show a pronounced have a pappus and nounced seed heteromorphism: heteromorphism: in each capitulum, capitulum, inner inner seeds have may be dispersed dispersed by wind, wind, whereas outer seeds remain in the capitulum capitulum and and fall on the ground 983). By studying four four populations populations of ground with it (Olivieri et al., 11983). of C. pycnocephalus found that, as pre pycnocephalus and three populations populations of of C. tenuiflorus, we found prepappus (used for migration) dicted, the average proportion proportion of of seeds with a pappus for wind wind migration) decreased decreased in both both species species along a successional successional gradient. gradient. The differences differences were slight but signifi cant. More 1994) tested the theoretical significant. More recently, Peroni ((1994) theoretical predic prediction in the red maple, Acer rubrum, rubrurn, by collecting collecting samaras samaras from from five populations populations located five populations located in early successional successional environments environments and and five populations located located in late successional wing loading loading ratio successional environments. environments. She measured measured the wing ratio (samara mass/ mass/ samara area), assumed proportional to migration assumed to be inversely proportional migration ability. She found found that that samaras samaras from from the early successional successional red maples maples showed showed slightly but signifi signifi- cantly lower lower wing wing loading loading ratios ratios than than those those from from late successional successional environments, environments, thus confi rming the theoretical interpretation as confirming theoretical prediction. prediction. The evolutionary evolutionary interpretation aspart of observed in natural populations populations is herher sumes that at least a part of the variation variation observed itable. There There are good good reasons to believe believe that that this is the the case in the studies of of Olivieri and 1 985) and Peroni ( 1994); see also Olivieri and Gouyon Gouyon ((1985) Peroni (1994); Olivieri and and Berger, Berger, ((1985). 1 985). One might wonder wonder whether One might whether all migration migration is the result of of these two opposing opposing selection May ((1977) 1 977) have could selection pressures. Hamilton Hamilton and May have shown shown that that migration migration could uniform population, population, because random choice evolve in a uniform because of of sib competition competition and and random choice of individuals forming of individuals forming the next generation. generation. We We have suggested suggested (Olivieri and and Gou Gouyon, 11985) 985) that by-product that at least some some of of long-distance long-distance migration migration is in fact fact a by-product of within-population migration. of selection for for short-distance, short-distance, within-population migration. In several plant spe speincluding those with migration migration apparatus) cies, seeds ((including apparatus) are more more likely to remain remain long-distance migration migration [e.g. in thistles thistles close to the parent parent plant than to experience long-distance 985) and 1 993)]. Peroni ( 1 994) (Oliveri and and Gouyon, 11985) and in grey mangrove mangrove (Clarke, (Clarke, 1993)]. Peroni (1994) proposed that because rodent predation of maple seeds proposed that because rodent predation of red red maple seeds is intense intense and and seems seems to be density-dependent, migration within each density-dependent, moderate moderate levels of of migration each popUlation population may be advantageous. migration ability advantageous. She suggested suggested that that this might explain explain why migration ability in red maple from the sole consideration maple seems seems greater greater than than what what would would be expected expected from consideration
300 300
Isabelle Gouyon Isabelle Olivieri Olivieri and Pierre-Henri Pierre-Henri Gouyon
of of local extinction extinction probabilities. probabilities. It could even be that the evolution evolution of of migration migration structures 1 994) structures has has little to do with migration migration itself. For For instance, instance, Tsuji Tsuji et al. ((1994) have have proposed proposed that wing wing dimorphism dimorphism in Cardiocondyla Cardiocondyla ants was was maintained maintained through 1 99 1 ) through correlations correlations between between winglessness winglessness and and emergence emergence time. Motro Motro ((1991) and others 986) have migration and and sexual di others (e.g., Holsinger, Holsinger, 11986) have suggested suggested that migration dimorphism morphism for for migration may evolve evolve as a way of of avoiding avoiding inbreeding. inbreeding. Other Other reasons 1 980) and reasons why migration may evolve evolve can be found found in Harrison Harrison ((1980) and Johnson Johnson and 1 990). We and Gaines Gaines ((1990). We will discuss genetic genetic correlations correlations between between migration migration and and other other traits traits in Section Section V.
B. B. Polymorphism Polymorphism vs vs ESS ESS Although .e., Although polymorphisms polymorphisms such as the one one shown in Fig. 2 are stable (i (i.e., gene frequencies equilibrium values values following frequencies return to some equilibrium following perturbations perturbations away from values), they are May from these these values), are generally generally not evolutionarily evolutionarily stable states (sensu (sensu May1 995), there nard 1 982). In all the cases we have nard Smith, Smith, 1982). have studied (Olivieri (Olivieri et al., 1995), there exists ESS), and the introduction exists one one evolutionarily evolutionarily stable migration migration rate rate ((ESS), introduction of of a genotype with a migration migration rate rate closer to the ESS than than is any of of the previously previously existing ones previous types. If If the newly ones leads leads to the the loss of of at least one one of of the the previous newly introduced possibly coexisting introduced genotype genotype is the the ESS ESS itself, itself, then then all other, other, possibly coexisting types, types, are are lost. In other other studies, studies, in which which particular particular hypotheses hypotheses were were made made about about spatial heterogeneity McPeek and heterogeneity of of local carrying capacities, capacities, there there was was no ESS ((McPeek and Holt, Holt, 11992; 992; Cohen 99 1 ; Ludwig 99 1 ). For Cohen and and Levin, 11991; Ludwig and Levin, Levin, 11991). For instance, instance, Cohen Cohen and and Levin found found that that in some situations, situations, there there may exist a strategy which may other single type (evolutionarily (evolutionarily compatible strategy), but which which is invade any other also open polymor open to invasion by any type. The The population population is then then likely to be polymorphic. It may may also be that ESS are unreachable unreachable because because of of the genetic determinism. determinism. We We have have modeled modeled a situation in which which the the migration migration rate among the the offspring offspring ((proportion proportion of of those those offspring offspring which which do migrate) is determined determined by the genotype genotype of of the mother. mother. If If a given mother mother is able to produce produce both both offspring offspring which migrate migrate and and offspring offspring which which do not, an intermediate intermediate migration migration rate rate can then then be genetically determined determined through through the mother mother genotype, genotype, so that that mixed mixed strategies strategies can can evolve. In some mi some species, however, however, a given organism organism can can produce produce only offspring offspring which which migrate grate or or only offspring offspring which which do not. In other other species, the migration migration behavior behavior is determined determined by the the genotype genotype of of the the offspring; offspring; i.e., the the character character under under selection is not the individual probability of the proportion proportion of of migrating migrating offspring offspring but but the individual of migra migration. In both both last cases, even though though the genetic determinism determinism of of migration behav behavior might might be polygenic, polygenic, there there may be no available available intermediate intermediate strategy, and and a polymorphism polymorphism would would be maintained maintained between between individuals individuals who who disperse disperse and in individuals Roff, 11994b). 994b). For dividuals who who do not disperse disperse ((Roff, For instance, instance, according according to Roff Roff ((1994a), 1 994a), wing wing histolysis in insects, which which leads to winglessness, winglessness, is a threshold threshold character character determined determined by the individual individual genotype, genotype, so that that morphological morphological variation variation is discontinuous, discontinuous, with no intermediates: intermediates: individual probability probability of of migration migration is either either 0 0 or or 11.. The The heritability ooff migration tendency tendency iinn insects iiss usually quite quite high (see
1133
Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits
301 301
Roff, 11994a, 994a, for a review), suggesting that polymorph isms between winged and polymorphisms wingless genotypes result from balancing selection such as the one suggested in this chapter 990, for chapter (though see Roff, 11990, for many other other factors). Quite unexpectedly, however, Roff 1 994a) has shown that Roff ((1994a) that when winglessness is a polygenic but threshold threshold character, character, much of the variability variability in the trait trait is retained retained even in the presence of directional selection for one morpho morph. The number of loci involved had very little influence on this result. Therefore, the high polymorphism level cannot be readily interpreted as a consequence migra consequence of a balancing selection acting on migration behavior.
IV. DOES DOES SElEGION THE METAPOPULATION SELECTIONADJUST ADJUSTMIGRATION MIGRATIONRATE RATEAT AT THE METAPOPULATIONLEVEL? LEVEL? A. Influence and Species Influence of Landscape Landscapeand SpeciesCharacteristics Characteristicson on ESS ESS Various factors influence the evolution of migration. Many authors have shown that that local extinctions, extinctions, by enhancing enhancing selection among populations, populations, favor migration. 1 992; see also migration. For instance, instance, according according to Roderick and Caldwell Caldwell ((1992; et ai. al.,, 11991; frequent migration migration is common in insects in Denno et 99 1 ; Denno, 11994), 994), frequent temporary habitats, habitats, where variation in local carrying capacity is large; whereas less migration and more winglessness occur in stable or isolated habitats. In an extensive review on the evolution of 1 990) showed of flightlessness in insects, Roff ((1990) that ightlessness is associated with decreased that fl flightlessness decreased environmental environmental heterogeneity. His data ightlessness is also associated data broadly supported supported the prediction prediction that that fl flightlessness associated with habitat persistence, though he suggested that more quantitative data would be needed needed to fully support this prediction. In the case when local extinctions extinctions are due to deterministic causes related to succession, for instance when a given population has a maximal lifespan due to the invasion of ai., 1995) 1 995) that of its patch by other species, we found (Olivieri et et al., the ESS varies in a nonmonotonic manner, fi rst decreasing but then increasing increasing first with increasing popUlation population lifespan lifespan (Fig. 3). An increase in popUlation population lifespan increases the expected expected future lifetime of the home deme, and hence local selection against migration ((Fig. Fig. 2) lasts longer. However, if population lifespan lifespan is very large, then the proportion of patches that have reached the end of of succession is decreased, and and hence hence the cost cost of of migration migration due due to to falling in such such patches is is de decreased. Using numerical simulations, we found a quite strong effect of fecundity on 995). Other fe the ES migration rate rate (Olivieri et et al. al.,, 11995). Other things being equal, high feUnder cundity species should disperse more offspring than low fecundity species. Under the assumption that in saturated patches juveniles juveniles may become established only when some adults die, the ES migration rate also increases increases with adult survival rate. This prediction is explained by the negative influence of of adult survival on the establishment of offspring in the home deme. The main possibility for juve of juveniles to establish establish is then then to reach an empty site.
302 302
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When When the the population population lifespan lifespan is is very very long, long, and and patches patches are are saturated saturated immeimme diately diately at at recolonization, recolonization, making making it it more more important important to to reach reach an an unsaturated unsaturated patch, patch, the the ES ES migration migration rate rate is is given given by by d* d* == lifA0-->A~ I if Ao 2: A l and and d* = (1 - A 1 + A0)/(1 - s A1) ifA0 < At, where ss is is the the adult adult survival survival rate, rate, A0 Ao is is the the recolonization recolonization rate, rate, and and A~ A 1 is is the the where persistence persistence probability probability of of colonized colonized patches. patches. This This result result can can be be obtained obtained from from Eq. Eq. et al. al. (1995) ( 1 995) by by assuming assuming that that zz is is infinite. infinite. The The result result shows shows that that ( 14) in in Olivieri Olivieri et (14) in the the absence absence of of succession succession (z in (z large) large) and and of of mortality mortality during during migration migration (q (q == I ), maximal maximal migration migration rate rate is is selected selected for for as as soon soon as as the the probability probability A0 A o of of recolrecol 1), onization of of empty empty patches patches isis larger larger than than the the probability probability of of persistence persistence of of estabestab onization lished populations, populations, A~. A I ' The The result result also also shows shows that that the the ES ES d* d* increases increases with with the the lished
1133
Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits
303 303
adult survival probability A mi survival rate rate s, the increase increase depending depending on on the probability A~. The ES ES miI ' The gration gration rate rate is equal equal to 11 as soon soon as the survival survival rate rate is larger larger than than (A ( A1~ Ao)/A I ' i.e., rate is is below below Ao/A1. AoiA I ' Ao)/A~, i.e., as as soon soon as as adult adult mortality mortality rate It is often concluded that high fecundities are associated with high migration migration often concluded rates 965). For For inin rates in nature nature (the "colonizer" "colonizer" syndrome; syndrome; Baker Baker and Stebbins, 11965). stance, Peat and stance, in a survey of of ecological ecological characteristics characteristics of of British British angiosperms, angiosperms, Peat and Fitter 1 994) found with wind-dispersed Fitter ((1994) found that that species species with wind-dispersed seeds seeds produced produced more more seeds seeds than species with no specialized migration mechanism. Few studies have consid than species no specialized migration mechanism. Few studies have considered migration. Some ered the the association association of of high high adult adult survival survival rates rates with migration. Some data data appear appear to agree agree with our nonintuitive nonintuitive prediction prediction that that perennial perennial species species should should be selected selected to have than annuals. Venable and have higher higher rate rate of of long-distance long-distance migration migration than annuals. Venable and Levin Levin ((1983), 1 983), in an extensive Asteraceae, extensive review review of of some some 6000 6000 species species in the the family Asteraceae, found cantly smaller percentage found that a signifi significantly percentage of of annual annual than than perennial perennial species had had migration that could long-distance migration. They migration structures structures that could affect affect medium medium and and long-distance They concluded that their the idea idea that migration may be more concluded that their data data supported supported the migration may more important important for perennial than Moreover, as we next section for perennial than for for annual annual plants. Moreover, we will show show in the next (V.B), between life-history of mimi (V.B), there there are interactions interactions between life-history evolution evolution and and evolution evolution of gration: we that when when migration rate is increased, there there may may be be we will suggest that migration rate selection selection for for increased increased adult adult survival survival rate rate (detrimental (detrimental to fecundity), fecundity), thus thus rein reinforcing the effect forcing effect just described. described. -
Conditional Migration B. Conditional The would seem The result shown shown in Fig. 2 raises raises a paradox. paradox. Intuitively, it would seem optimal optimal to have have low migration migration rate in the colonizing phase phase and and to leave the patch patch when when successional successional events events are about about to drive drive the population population to extinction. extinction. What What happens happens in our our model model is exactly the reverse. reverse. Newly formed formed populations populations send send their their progeny progeny old ones ones keep keep their their offspring offspring in the the patch patch where where they have have no no future. future. out while old This result makes makes sense, given that the colonizers colonizers must must by definition definition be composed composed of migrators. migrators. This This is because because until now, we have have considered considered that that a given given mainly of genotype genotype is characterized characterized by a single single migration migration rate (an unconditional unconditional migration migration strategy, McPeek 992). In reality, it might migration behavior behavior McPeek and and Holt, 11992). might be be that migration varies well-known that in varies with some some environmental environmental factors. factors. For For instance, instance, it is well-known some some species species of of mammals, mammals, emigration emigration rate increases increases with density density (Johnson (Johnson and and 1 993) suggested Gaines, 990). In fi shes, Okland Gaines, 11990). fishes, Okland et al. ((1993) suggested that the migration migration deci decisions of of brown brown trout trout and and Atlantic Atlantic salmon salmon depend depend on individual individual growth growth rates. rates. In birds, 1 990) studied between habitat lim birds, Pruett-Jones Pruett-Jones and and Lewis ((1990) studied the interaction interaction between habitat limitation, influencing migration migration decisions itation, habitat habitat quality, and and sex ratio ratio in influencing decisions in a pop population ulation of of fairy-wrens. fairy-wrens. They They suggested suggested that that young young males males delayed delayed their their migration migration in response mates and response to a limited number number of of mates and secondarily secondarily to habitat habitat limitation. In insects, many factors as well as wing wing or many studies studies have have shown shown that that environmental environmental factors fl ight polymorphism uence behaviors flight polymorphism infl influence behaviors related related to migration migration (see Roderick Roderick and Caldwell, 992, for several references). Hastings Caldwell, 11992, references). Turning Turning to theoretical theoretical studies, Hastings ((1992) 1 992) considered under some considered age-dependent age-dependent migration. migration. He He found found that that under some conditions conditions
304 304
IsabelleOlivieri Olivieri and and Pierre-Henri Pierre-HenriGouyon Gouyon Isabelle
(stronger (stronger density density dependence dependence in in age age classes classes which which migrate migrate less), less), chaotic chaotic popu population lation dynamics dynamics might might appear. appear. Such Such patterns patterns are are likely likely to to influence influence the the evolution evolution of 1 989) showed of migration. migration. Kaitala Kaitala et et al. al. ((1989) showed that that winglessness winglessness conditional conditional on on pop population ulation density density is is an an ESS ESS when when compared compared to to an an unconditional unconditional strategy strategy in in aa model model applied to the 1 992), the evolution of of migration in aa waterstrider. McPeek and and Holt ((1992), using using a two-patch model, studied the the influence of spatio-temporal variability in in local carrying carrying capacities on on migration (thus (thus phenotypically plastic migration, which they called a conditional migration strategy). strategy). They They found found that that "local pop pop-
ulation sizes and the proportions of of local populations that disperse should be negatively correlated correlated if if populations populations are both at their evolutionary and demo demographic equilibria and iffitnesses if fitnesses are density-dependent. density-dependent."" Their model thus pre predicts that emigration from from low density patches should be larger than emigration rates from from high density patches, contrary to what is observed in mammals, for example. In our model, carrying capacity is the same for all patches. A strategy in which migration rate increases with population age or local density, should be advantageous: advantageous: low-migration low-migration rate rate following following arrival arrival in in an an empty empty patch patch would fa facilitate colonization, whereas migration would be a better strategy when the patch is crowded. We may define a plastic strategy as one in which migration rate increases with population age. By running pairwise contests between nonplastic and plastic genotypes, we found found that such plastic strategies would usually invade, provided provided that their average average migration rate rate was not too different from the ES migration 1 984) reached similar con migration rate rate for for a nonplastic nonplastic strategy. Levin et al. ((1984) conclusions.
C. Metapopulation Viability as a Function Function of Migration Migration One may wonder wonder whether whether all migration migration rates per One rates allow the metapopulation metapopulation persistence, and if not, how how far for the metapopulation metapopulation can selecselec and if far from from the optimum for push the Figure 4 shows shows the equilibrium metapopulation metapopulation sizes tion push the migration rate. rate. Figure the equilibrium for a single for single annual annual genotype genotype as as aa function function of of the the migration migration rate rate in the the two two landland L 1I and and L2 L2 (see Table Table I). I). For For aa given given landscape, landscape, there there exists aa limited range range scapes L of migration rates rates that that allow metapopulation metapopulation persistence. persistence. Below Below aa minimum minimum of the number number of of migrants establishing new new populations populations is not not sufficient to value, the migrants establishing sufficient to compensate for extinctions. Beyond Beyond aa maximum maximum value, the the number number of of progeny progeny compensate for extinctions. remaining in in aa patch patch may may not not be be sufficient sufficient to to allow allow population population establishment. establishment. For For remaining some some parameter parameter values values corresponding corresponding to to particularly particularly favorable favorable habitats habitats (few (few end end of succession succession sites) sites) the the maximal maximal value value may may be be equal equal to to 1. I . The The minimal minimal value value is of et al. al. (1994) ( 1 994) observed observed the the same same kind kind of of pattern pattern in in aa always positive. positive. Hanski Hanski et always parameterized butterfly butterfly metapopulation metapopulation model. model. Their Their model model suggested suggested that that the the parameterized migration rate rate of of the the butterfly butterfly Melitaea Melitaea cinxia cinxia should should lie lie between between 10 I O and and 40%, 40%, migration and indeed indeed the the observed observed migration migration rate, rate, as as measured measured from from mark-recapture mark -recapture studstud and ies, ies, was was about about 30%. 30%.
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Metapopulation size, as migration rate, in L L2 described Metapopulation equilibrium equilibrium size, as aa function function of of migration rate, in L I1 and and L2 described in Table Table I, with fecundity of of 5 ((L1) The ES migration rate indicated by an arrow. arrow. L l ) and and 7.5 (L2). The ES migration rate is indicated OOO)/Iog( I 0). Size patch X Size is is expressed expressed in in Log(average Log(average number number of of individuals individuals per per patch • I1000)/log(10).
D. ESS ESS versus versus Optimal Optimal Migration Migration Rate Rate Migration Migration is important for for evolution evolution (it ensures the genetic cohesion of of a species and allows its demographic demographic survival), and migration is an evolving char character itself. However, However, what are the effects of of a given migration rate on the survival of of the metapopulation? metapopulation? Structured Structured population population genetics models models (Wade (Wade and Mc McCauley, 11988; 988; Whitlock and McCauley, 11990) 990) show that genetic differentiation among populations may be maximized by some intermediate migration rates.
landscape LandscapeParameters ParametersUsed Used inin the the Numerical Numerical Examples Examplesof of the the Model Model Described Described in in This ThisChapter" Chapter"
TABLE TABLE I
zz
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aLandscape Landscape LI L1 is is characterized characterized by by aa longer longer population population lifespan, lifespan, but but smaller smaller recolonization recolonization rate rate and and smaller smaller disturbance disturbance rate rate of of late late successional successional (uncolonizable) (uncolonizable) sites sites (5, (S= sites). sites). Carrying Carrying capacity capacity K K is is the the same same in in the the two two landscapes. landscapes.
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Isabelle Isabelle Olivieri Olivieri and Pierre-Henri Pierre-HenriGouyon Gouyon
From From a purely demographic perspective, one may define the metapopulation den density as the average average number number of of individuals per per site. This is equivalent to the "site 1 977) and Comins et 1 980). One occupancy" used by Hamilton and May ((1977) et al. al. ((1980). may then wonder wonder how the metapopulation metapopulation carrying capacity, that is, its density at demographic uenced by the migration ne an demographic equilibrium, is infl influenced migration rate rate and defi define optimal optimal strategy (OS) as the migration migration rate rate which maximizes the metapopulation carrying capacity ((Roff, Roff, 11975; 975; Hanski et 1 994). This OS does not give any et al., al., 1994). indication about the direction of of evolution. However, knowledge of of the optimum value of of the migration rate rate could be of of interest in management programs, if one could manipulate the level of of exchanges exchanges between populations populations or the number number of of immigrants in newly founded al., 11980; 980; founded populations. Several authors authors (Comins et et al., Motro, 982) have shown that the Motro, 11982) the OS is indeed different different from the the ESS. In the present model, we found found that the OS was always larger than the ESS. Figure 4 shows two examples of of the relation between migration rate and metapopulation equilibrium density; the ESS are indicated by the arrows. The ES migration rate rate always lies within the range that permits persistence: a strategy leading to meta metapopulation extinction cannot be evolutionarily stable. Declines in the frequencies frequencies of of migration enhancers in older populations may reduce the migration migration rate below the level considered considered optimal at the among-pop among-population level. Such within-population mi within-population effect does not, however, depress the migration gration rate rate to to the the point point leading leading to to metapopulation metapopulation extinction, because because the the "non "nonrst, at least in our deterministic model. In a very viable" genotypes disappear disappear fi first, different 1 986) were able to produce different theoretical setting, Iwasa and Roughgarden Roughgarden ((1986) produce a model in which a species which could not invade an empty landscape could nonetheless nonetheless invade invade if another species was present, and and in some cases both both species went subsequently extinct. 1 993) and extinct. Hanski Hanski and Gyllenberg ((1993) and Hanski Hanski and and Zhang ((1993) 1 993) have discussed the processes processes by which species can bifurcate into either a core or satellite distribution, occupying a large or small fraction of of the suitable habitat, respectively. An OS would would be to retain just enough enough offspring offspring in the the local local population population to ensure maximum local population population size, and to export export all remaining propagules propagules in search of of other patches. Such a strategy is never evolutionarily stable, however, because "cheater" "cheater" genotypes genotypes that that would keep their offspring offspring at home would would in increase in frequency within populations. The situation might be even worse if migration migration behavior behavior is determined by the genotype genotype of of the offspring: as suggested by den Boer 1 990), local selection will often favor Boer ((1990), favor the individuals which stay in the natal population, population, irrespective of of whether whether this increases the chances of of survival of 983; Roitberg 993, for of the species (see also Motro, 11983; Roitberg and Mangel, 11993, for parent parentoffspring offspring conflict in the evolution of of migration). At the other other extreme, genotypes that would never export export offspring offspring could never become established in other patches patches and would disappear following following the extinction of of their local population. population. Our Our results that the OS is above the ESS is in agreement agreement with the numerical results of of Roff Roff ((1975) 1 975) and the analytical results 1 980). In contrast, the ES rate results of of Comins et et al. al. ((1980).
1133
Evolution and Other Evolutionof of Migration Migration Rate Rate and Other Traits Traits
307 307
exceeds 1 977) sib competition model. exceeds the optimal rate in the Hamilton and May ((1977) In their model, each patch patch can support a single adult, adult, with competition competition for for the patch mi patch occurring among among the nonmigrating offspring of of the resident resident and the miother patches. patches. Migrating offspring have a constant constant probgrating offspring from other prob p) during migration. In their fi rst model, Hamilton and May ability of of survival ((p) first assumed that all patches were occupied. The ES migration rate was then equal to l /(2-p). In their second model, they further 1/(2-p). further assumed that that some patches might remain vacant, because because of demographic demographic stochasticity. They then found found that that the rst model, and they described the ES migration rate was higher than in the fi first "optimum" "optimum" migration rate as the one which maximized the proportion of patches patches occupied. They found that the optimal migration rate was lower than the ES migration rate. Hamilton and May ((1977) 1 977) suggested that Roff's 1975) assump Roff's ((1975) assumption of several adults per per patch increased increased selection favoring those remaining in the natal patch, patch, causing the ES migration rate to fall below the optimal optimal rate. Additionally, the possibility of environmentally caused local extinctions in the al. but not in the Hamilton and May sib competition models may Comins et et al. promote migration as a mechanism for for increasing total population size, causing 1 980, p. 2 14). the optimal migration migration rate rate to rise above the ES rate rate (Comins et et at. al.,, 1980, 214). In fact, if if there are only a few individuals per per patch on average and local extinc extinctions are due to demographic stochasticity only, the optimal migration migration rate rate is always lower than than the ES migration migration rate. In contrast, contrast, if local extinctions are mainly due to environmental stochasticity, the optimum migration rate is higher higher than migration rate, whatever the number per patch. than the ES migration number of of individuals per patch. There There is not always an ES migration rate rate and an optimal rate rate to compare. compare. First, an optimal migration rate exists only if there are local extinctions, extinctions, otherwise the number of of individuals in the the metapopulation is constant. Second, as shown by Frank ((1986), 1 986), selection depends depends on the among-population among-population variance variance in gene frequency. An ES migration rate thus exists only if there is some gene frequency variance among populations. If there is no mortality during migration, many inin dividuals per per patch (no demographic stochasticity), and no environmentally caused local extinctions, gene frequencies frequencies quickly homogenize across local pop populations in a constant environment, and hence hence neutral neutral migration polymorphism is always maintained and there there is no ES migration rate. The analytical expression for for the ES migration rate rate under under such assumptions is indeterminate indeterminate (see, for for in instance, Comins et 980, p. 2 1 3 , or Eqs. (B6)-(B 8) with q et et at., al., 11980, 213, (B6)-(B8) q = = 11 in Olivieri et 995). If, however, there is some mortality during migration but no extinc at., al., 11995). extinctions, migration is always selected against, and the ES migration rate rate is zero (Comins et 980, p. 2 1 3). With local extinctions et at., al., 11980, 213). extinctions due to environmental sto stochasticity, there is always some variance of gene frequencies frequencies among populations, and hence there migra there is evolution of of gene frequencies. frequencies. In summary, an optimal migration rate exists only if there are some kind of of local extinction. extinction. An ESS exists if there is some cost to migration, or if there are are local extinctions, extinctions, or both. both. In the former case, the ES migration rate is zero.
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Isabelle Olivieri Gouyon Isabelle Olivieri and Pierre-Henri Pierre-Henri Gouyon
The fact that the metapopulation may not be viable for for too low or too high consequences for for population genetic genetic models of of migration rates has interesting consequences subdivided subdivided populations populations with local extinctions extinctions (with rate rate e). These These models usually assume McCauley, 1990), 1 990), that assume recolonization rates rates of of I1 or I1 - e (Whitlock (Whitlock and and McCauley, local carrying capacities capacities are reached reached in a single generation, generation, and very low effective effective migration rates rates between between extant populations. In other other words, words, these these models models assume assume that that a very very small number number of of emigrants emigrants is able able to recolonize any number number of of patches. patches. This long as cient This is is clearly clearly impossible impossible as as long as one one does does not not assume assume either either aa very very effi efficient habitat choice choice by migrants, so that that most most of of them them arrive in unoccupied unoccupied patches, patches, or a very high fecundity together together with a very low survival during migration, so that themselves in occupied patches, but saturate migrants have no chance chance to establish themselves occupied patches, presented in this newly colonized colonized patches patches in in aa single generation. generation. In In the the model presented chapter, migrants which, by chance, chapter, we have have assumed assumed that that colonizers colonizers are are simply migrants chance, arrived in an empty empty patch. patch. The The number number of of recolonizers recolonizers is then then a function of of the migration rate, whereas the rate of of recolonisation is equal to Ao A0 as soon as the number of of recolonizers is positive (viable metapopulation). The The result result that that the the optimum optimum migration migration rate rate is is not not equal equal to to the the ES ES migration migration rate, rate, could could possibly be be used used in in the the management management of of those those species species whose migration migration rate rate can be manipulated: since the optimum is usually higher higher than the ESS, we might enhance number of enhance the conservation of of a species species by increasing the number of colo colonizers nizers and and migrants. migrants. Actually, Actually, an artificially increased increased migration rate (as long as it beneficial for species in it remains remains reasonnable) reasonnable) will will probably probably be be beneficial for the the species in most most cases, cases, not not only only for for the the reason reason just just given, given, but but also also because because enhanced enhanced migration decreases decreases inbreeding, inbreeding, a fact fact which might be desirable desirable from a genetic point of of view.
E. Two landscapes, a Viable E. A Mixture Mixture of of.Two Landscapes,Each Each Sustaining Sustaininc/a Viable Metapopulation, Metapopulation, May Cause Metapopulation Extinction Cause Metapopulation ExtinctionWhen When Connected Connectedto Each Each Other Other Until patches sharing common Until now, now, we we have have defined defined aa landscape landscape as as aa set set of of patches common transition probabilities and demographic landscapes, however, demographic properties. properties. Many landscapes, include include different different kinds kinds of of components, for instance instance rocky rocky versus nonrocky patches. heterogeneity has patches. Landscape Landscape heterogeneity has been been considered considered by by several several authors. authors. For For instance, 1 99 1 ) have shown that spatial heterogeneity have instance, Cohen and Levin ((1991) have some features features in common common with positive temporal correlations; they found found that that the optimal dispersal rate rate was in general decreased decreased as compared compared to a homogeneous homogeneous environement. We nd that We consider consider here here a quite different different model, in which we fi find the increased landscape landscape heterohetero the ES ES dispersal dispersal rate rate may may increase increase or or decrease decrease with increased geneity. made of de geneity. Assume for for instance instance aa landscape landscape made of the the two two sublandscapes deof landscape landscape 2 scribed in in Table Table I, with with aa proportion a o~ of 2 and and aa proportion proportion ((11 - a) o~) of of landscape landscape I1.. Figure 5 describes describes the metapopulation equilibrium size as a func function I . It tion of of the the migration migration rate rate within within landscapes landscapes with a c~ varying varying from from 0 to to 1. It can can be be seen whatever the seen that that for for some some values values of of a, a, the the metapopulation metapopulation is not not viable viable whatever the migration rate and tremendous and that a low proportion of of landscape landscape 11 can can have have tremendous consequences consequences for for metapopulation viability. The sudden decline of of the metapopmetapop-
1133
FIGURE S5 FIGURE
Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits
309 309
Metapopulation Metapopulationequilibrium equilibrium size, size, in a landscape landscape made made of a sublandscape sublandscape L2 (fraction (fraction
a) LII (fraction a) (see a) and of sublandscape sublandscape L (fraction I1 � - c0 (see Table Table I), as a function function of migration migration rate rate and a. c~. Fecundity LI and 7.5 in L2. (size measured as in Fig. Fecundity was 5 in L1 (size is measured Fig. 4).
ulation ulation size size at at the the limits limits of of the the interval interval is is a a striking striking result result which which does does not not seem seem easy easy to to explain explain intuitively. intuitively. Figure Figure 6 6 describes describes the the same same pattern, pattern, for for a a different different species species or or a a different different landscape, in which fecundity is is slightly Note that that the landscape, in which fecundity slightly larger. larger. Note the sublandscape sublandscape 2 2 (L2) (L2) is L I ), since is is more more favorable favorable than than the the sublandscape sublandscape I1 ((L1), since metapopulation metapopulation density density is greater than in l . However, the two two landscapes mixed, L Ll1 seems greater in in L2 L2 than in L L 1. However, when when the landscapes are are mixed, seems to influence more strongly strongly than than L2 L2 the the range of adapted adapted migration migration rates rates that that allow allow to influence more range of the this result is that the persistence persistence of of the the metapopulation. metapopulation. Implication Implication of of this result is that one one should should be cautious interpreting source when the the evolution evolution be cautious in in interpreting s o u r c e-sink - s i n k systems. systems. In In particular, particular, when of to a of the the migration migration rate rate of of the the entire entire metapopulation metapopulation leads leads to a rate rate which which is is out out of of the range range necessary in one one of the necessary for for the the existence existence in of the the sublandscapes, sublandscapes, the the set set of of populations in sublandscape behaves like a populations in this this sublandscape behaves like a sink. sink. This This does does not not mean mean that, that, were this this sublandscape sublandscape alone, alone, it the species, provided were it could could not not be be favorable favorable for for the species, provided another was allowed to evolve. Evolution may may not always be be driven another migration migration rate rate was allowed to evolve. Evolution not always driven by adaptation the potentially most productive productive environment. by adaptation to to the potentially most environment. The in Figs. Figs. 55 and and 6 have important important consequences. The results results shown shown in 6 have consequences. Although Although aa species species may to adapt landscapes, it it may may not be able to may be be able able to adapt to to two two different different landscapes, not be able to persist in in a these two two landscapes. This occurs occurs when when the the ranges ranges of of persist a mixture mixture of of these landscapes. This migration in each each landscape landscape do not migration rates rates allowing allowing metapopulation metapopulation persistence persistence in do not overlap. in part part of overlap. In In terms terms of of land land management, management, it it follows follows that that a a change change in of a a land landscape example if if aa given proportion of landscape 1I is is transformed transformed into into land scape (for (for example given proportion of landscape landscape 2) may metapopulation extinction. extinction does does not scape 2) may lead lead to to metapopulation extinction. This This extinction not result result
Isabelle IsobelleOlivieri OlivierJand and Pierre-Henri Pierre-HenriGouyon Gouyon
3310 10
.� 'D en g
01 a -a :s Q. · c 0 ,&1
0. ... CIS :";:
Ii
5
4
3
2
FIGURE Sameas Fig. Fig. 5, but with with fecundity fecundity of 88 in L2 L2 instead instead of 7.5. 7.5. The evolutionarily evolutionarilystable stable FIGURE 66 Same migration indicated as well. migration rate rate is indicated well.
from either patches or from an increase either a decrease decrease in the availability of of suitable suitable patches increase in the fragmentation fragmentation of of the landscape. landscape. The The extinction is caused caused by there there being no longer cient migration of the metapopulation. None longer suffi sufficient migration allowing the persistence persistence of None of of the two two landscapes landscapes is unsuitable, each each of of them them being being able to sustain a viable viable metapopulation; but the the mixture is unsuitable. unsuitable. To To clarify such situations situations further, = 0 in Fig. 5 let us imagine imagine that a landscape starts with a a = 5,, and imagine that the proportion of fertilizers are of L2 is increased. For instance, assume that fertilizers are applied to increased, but but also some patches. patches. As a consequence, consequence, fecundity fecundity in these patches is increased, the successional life successional processes processes becomes becomes faster, thereby decreasing decreasing the population lifespan (z). We may because of increased may then observe that the the species species disappears disappears because of increased interspecific interspecific competition (lower z). In In contrast, contrast, consider another another landscape landscape in in ). Assume Assume that which all patches patches receive receive fertilizers fertilizers (a (a = = I1). that some farmers farmers decide decide to stop stop applying fertilizers fertilizers to their their soil. The The species species may go extinct, extinct, and and we will think think that that this is because because the environment has become become poorer poorer (fecundity has decreased). decreased). The The explanation explanation is is proximally proximally correct, correct, but but it it accounts accounts aa part part of of the the truth truth only. Increased Increased competition in a rich environment may not be lethal for for a species species which can can invade invade numerous numerous new new patches patches while being being eliminated eliminated from from old old ones. ones. Decreased Decreased fecundity fecundity may may not not be lethal if the the species species can can stay stay longer longer in each each patch. However, for for all these these conditions, there may or may not exist a migration When such migration rates exist, the rate allowing the metapopulation to persist. When metapopulation may or may not persist, persist, depending depending on its ability to adjust adjust the migration rate rate fast fast enough. Figure 6 also gives gives the ES migration rate rate as a function function of of a, a, the proportion of of landscape landscape L2 in the mixture. It is striking striking that, for for certain critical values very small small changes in a provoke large large changes in the the ES ES critical values of of a, a, very changes in a provoke changes in
13 1 3 Evolution Evolution of of Migration Migration Rate Rate and and Other Traits Traits
31 1 311
migration rate. rate. There There are are thus thus certain certain mixtures mixtures of of different different kinds kinds of ofhabitats habitats which which migration are critical critical for for the the ability ability of of the the metapopulation metapopulation to to adapt adapt its its migration migration rate, rate, given given are the available available additive additive variance variance that that might might occur occur for for this this character. character. the The evolution evolution of of migration migration rate rate in in particular particular landscapes landscapes is is thus thus aa critical critical The characteristic of of aa metapopulation metapopulation determining determining persistence persistence or or extinction. extinction. ExtincExtinc characteristic tion may may result result from from landscape landscape changes changes which which lead lead to to situations situations in in which which no no tion migration rate rate allows allows persistence, persistence, or or when when changes changes in in landscape landscape structure structure are are so so migration fast that that the the available available genetic genetic variability variability does does not not allow allow adaptation. adaptation. Moreover, Moreover, as as fast pointed out out by by Gomulkiewicz Gomulkiewicz and Holt (1995), ( 1995), when when aa metapopulation metapopulation adapts adapts to to pointed and Holt novel environment, environment, its density density may may fall fall below below a critically critically low low level level for for a period period aa novel of time time during during which which the the metapopulation metapopulation is highly highly vulnerable vulnerable to to extinction. extinction. of Metapopulation adaptation adaptation can can result result in constant evolution toward toward new new equiequi Metapopulation constant evolution librium values, which which are adapted to to (although (although not not optimal optimal in) novel novel landscapes. landscapes. librium values, are adapted When aa landscape landscape is changing, changing, conservation conservation decisions decisions on on species species management management When depend on on whether whether those those new new values are attainable or or not. If If they are are not, should depend are attainable management decisions decisions should be be mainly concerned concerned with with the the landscape landscape itself. If If management they are, it will not be sufficient, or even desirable, to reinforce or reintroduce they are, it will not be sufficient, or even desirable, to reinforce or reintroduce popUlations without without taking into consideration of such such actions actions populations taking into consideration the the consequences consequences of on the the evolution evolution of of the the species. species. For For instance, instance, let let us us imagine imagine a plant species species which which on a plant because its migration respect to increased increased is going going extinct extinct because migration rate rate is too low with respect habitat fragmentation. fragmentation. Conservation biologists biologists could try to prevent extinction of of such a species by by reintroduction. For this purpose, it will be necessary necessary to collect seeds and reproduce them. Care Care should be given not to favor the less dispersed dispersed seeds, which are are possibly easier easier to collect: this would create create a selection pressure pressure decrease metapopulation adaptation. If this point was against migration, and thus decrease not taken into account, the species could gain new individuals at the expense of its ability to maintain itself without human help. Den Boer ((1990) 1 990) suggested that the Murphy ((1985) 1 985) on the SLOSS SLOSS discussion discussion in in Wilcox Wilcox and and Murphy on the the design design of of nature nature reserves passed passed over the essential point: the differences differences in migration rate between species. We may go further and suggest that differences differences in migration rate rate between genotypes within a species may be a critical consideration in some conservation tasks. tasks.
V. V. THE THE METAPOPULATION METAPOPULATIONEFFECT: EFFECT:OTHER OTHERTRAITS TRAITS We have called the antagonistic two-level selection observed on migration migration behavior behavior as as the the metapopulation effect. effect. We We will will now now show show that that this this effect effect can can be be observed observed for for other other traits as well, for instance instance life-history traits and and properties of the the genetic genetic system. system.
A. A. Evolution Evolutionof of Dormancy Dormancyand and Diapause Diapause Ellner 1 98 1 ) suggested Ellner and and Shmida Shmida ((1981) suggested that that the the desert desert plants plants of of Israel Israel have have evolved aa variety variety of of migration-restricting migration-restricting seed-containers seed-containers that that protect protect the the seed seed
3312 12
Isabelle Isabelle Olivieri Olivieri and and Pierre·Henri Pierre-HenriGouyon Gouyon
from predation and flooding, regulate the within-season within-season timing of of germination, and spread 1 99 1 ) suggested spread germination over several years. Yeaton Yeaton and Bond ((1991) that differences differences in both migration in space and dormancy in seed bank bank could could promote promote coexistence coexistence of of two shrub shrub species in a habitat disturbed by fire (see also Ellner, 11987; 987; Shmida and 994). Recently, Tsuji 1 994) have have sugsug and Ellner, Ellner, 11994). Tsuji et et al. al. ((1994) gested, using an ESS model, that a trade-off trade-off between dormancy and migration may evolve even in stable environments. Migration in space is one way of of es escaping local adversity. Dormancy (in plants) and diapause (in animals), by al local and allowing escape in time, represent other ways of achieving a comparable result represent of comparable (Hairston, 993; Ellner 994). Models ((Venable Venable and Brown, 11988; 988; (Hairston, 11993; Ellner and Hairston, 11994). 984; Cohen and Levin, 1991; 1 99 1 ; L Levin et et al. al.,, 11984; I. Olivieri, unpublished) unpublished) as well as et al. 983) demonstrate the Venable and Lawlor, 11980; 980; Olivieri et observations ((Venable al.,, 11983) antagonism between the two strategies, which are often often two kinds of of responses responses to the same sort of uncon of environmental variation. All models assuming only unconditional strategies predict Nondis predict that dispersed dispersed seeds should not be dormant. Nondispersed persed seeds might be dormant dormant or nondormant. nondormant. The The metapopulation effect described described in the evolution of of migration rate is also operating in the evolution of dormancy. Consider Consider two competing strategies, with different different dormancy rates rates of of nondispersed nondispersed seeds. Within a patch, those seeds which germinate germinate will obviously be overrepresented overrepresented in the adult population. population. Thus Thus the gene frequency of of an enhancer enhancer of of dormancy decreases decreases through through time in the adult population. population. However, when conditions are such that a stable polymorphism is maintained maintained between two germination strategies, we observe observe (unpublished (unpublished simu simuof the more more dormant genotype increases through through lation results) that the frequency of time in the seed bank. In the absence absence of of disturbance, disturbance, this type will have little success since nondormant nondormant genotypes constantly produce produce seeds which occupy the available space. Following a complete disturbance disturbance in a given patch, no offspring offspring are produced produced in the patch, and recolonization occurs either through migration or through germination from from the seed bank. The dormant dormant strategy is then overrep overrepresented at recolonization. recolonization. Just like with migration, dormancy is favored at the population level. Although Although this metapopulation level, while selected against at the population process process clearly occurs occurs at the patch level, so that one actually need not consider consider several patches to study the evolution of of dormancy, it would be difficult, from a genetic point of of view, to consider the new population population as the same as the one previously occupying the patch. We thus still consider consider that the process occurs at the metapopulation metapopulation level, even though though the metapopulation metapopulation may occur in a single patch, patch, successively successively occupied occupied by by different different populations. populations.
B. Evolution of life Life Histories Classical models of life-history evolution usually consider consider the asymptotic population population growth growth rate rate as a measure of of fitness in species with age-structure age-structure (Char (Charlesworth, 1994b). 1 994b). This is reasonable reasonable only in the absence of of frequency dependency dependency
Evolution Evolutionof of Migration MigrationRate Rateand and Other OtherTraits Traits
113 3
3313 13
(Kawecki, (Kawecki, 11993) 993) and provided provided that populations populations ever ever reach reach this growth growth rate. IInn a populations are ones go metapopulation, new new populations are regularly regularly established and and old old ones go ex extinct. that the tinct. It It is is unlikely, unlikely, in in such such aa system, system, that the asymptotic asymptotic growth growth rate rate calculated calculated for a population population could could be of of any relevance. relevance. Kawecki Kawecki ((1993) shown that that in a for 1 993) has shown patchy environment, environment, where individuals individuals compete compete for resources within within patches, patches, increasing increasing mortality mortality always favors favors decreased decreased allocation allocation to late survival, if mor mortality tality is is caused caused primarily primarily by by destruction destruction of of entire entire patches. patches. This This means means that that an an increase in local extinction extinction rate rate decreases decreases evolutionarily evolutionarily stable stable adult survival. At equilibrium, the realized realized asymptotic asymptotic growth growth rate rate of of the metapopulation metapopulation is one, and equilibrium, in and the the metapopulation metapopulation reaches reaches aa dynamic dynamic equilibrium, in which which populations populations that that have capacity increase increase in our particular particular model have not not yet yet reached reached the the carrying carrying capacity in size. size. In In our model of local density dependence, dependence, while all populations populations undergo undergo similar phases phases (den (denof sity-independent unless extinc sity-independent growth growth followed followed by by density-dependent density-dependent regulation regulation unless extinction occurs), these equilibrium, aa stable tion occurs), these phases phases are are not not synchronized. synchronized. At At equilibrium, stable age age distribution metapopulation level, not necessarily necessarily at distribution is is attained attained only only at at the the metapopulation level, not at the the population populations of population level. An example is given in Fig. 7. All populations of a given age i have the individuals, but the age-class age-class distribution distribution have the same same age-class age-class distribution distribution of of individuals, but the of population may of aa given given population may vary vary between between years. Let us the question question of polymorphisms of genotypes Let us then then ask ask the of whether whether stable stable polymorphisms of genotypes with different histories may be maintained, maintained, and influence of pop with different life life histories may be and what what is is the the influence of population lifespan, fecundity, and extinction rates the evolution of life life ulation lifespan, fecundity, and local local extinction rates on on the evolution of histories. For instance, compare three genotypes with with similar migration rates rates of of histories. For instance, compare three genotypes similar migration 0.5. Assume no senescence. Let Let genotype P be a typical perennial no senescence. genotype P perennial (iteroparous) genotype, with viable seeds seeds per adult) and high genotype, with low low annual annual net net fecundity fecundity (four (four viable per adult) and high adult adult survival rate rate (0.9). (0.9). Let Let genotype II be be intermediate, with with net net fecundity fecundity of of 8, 8,
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FIGURE population age FIGURE 77 Age structure structure of individuals individuals in each each population age class class at metapopulation metapopulationequilibrium. equilibrium. The model model was was iterated iterated with with disturbance disturbance rates rates of landscape landscape L2 (Table (Table I) and population population lifespan lifespan of :z = 0, migration 0.5, constant rate of 0.9, fecundity of 2. = 110, migration rate of 0.5, constant adult adult survival survival rate 0.9, and annual annual fecundity
3314 14
Isabelle Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-Henri Gouyon Gouyon
and and adult adult survival rate rate of of 0.45. 0.45. Finally, let genotype genotype A be annual annual (semelparous), (semelparous), with net 2. The net fecundity fecundity of of 112. The results results are given Table Table II. The lifespan, the more peren The results results show show that the longer longer the population population lifespan, more the perennial genotypes genotypes are selected for. /, 4; A, 6), interesting By dividing dividing all fecundities fecundities by half half (P, 2; I, interesting differences differences emerge perennials emerge (Table (Table II). Notice Notice that that a polymorphism polymorphism between between annuals annuals and and perennials is possible 1 ), indicating indicating that possible (here (here for for zz = = 111), that disruptive disruptive selection selection on life histories histories high, the may occur. When When zz is small, the annual annual genotype genotype is favored. favored. When When zz is high, the may perennial perennial type is favored. favored. When When zz is intermediate, intermediate, the intermediate intermediate I/ may be be able to replace pairwise contests intermediate replace one or the other other genotype genotype in pairwise contests (here (here the intermediate 1 ) and natural selection wins against the annual annual for for zz = = 111) and still be be eliminated eliminated by natural selection when when both both other other types are are present. present. This This example example illustrates illustrates that that dynamics dynamics of of game game theory models with more Maynard-Smith, more than than two two strategies strategies may may be complex complex ((Maynard-Smith, 11982). 982). Such Such disruptive disruptive selection occurs occurs here here because because the empty empty patches patches are faster faster recolonized by the perennial type for recolonized the annual annual type, whereas whereas the the more more perennial type is selected for niches, the intermediate intermediate type does than the in old populations; populations; in both both niches, does less less welI well than best adapted adapted genotype. In this particular particular case, the dynamics dynamics are highly dependent dependent conditions, even though though the the only evolutionarily evolutionarily stable state is the oc ocon initial conditions, currence perennial types in frequencies frequencies of 2 : 3. This currence of of both both annual annual and and perennial of ca 2:3. This is illustrated 1 00 illustrated in Fig. 8, which which shows shows that when when the simulation simulation is initiated with with 100 seeds of of each each genotype genotype in each each empty patch, patch, the annual annual genotype genotype almost goes extinct because of of competition with the intermediate intermediate genotype genotype /, I, before before /I starts starts to decline because of alIowing the annual of competition with the perennial, perennial, allowing annual type to increase increase again. again. The The metapopulation metapopulation processes processes may thus thus lead lead to particular particular and and
Invading P, or or Pairs Invading Genolype Genotype in in Contests Contests Involving Involving Either Either the the Three Three Types, Types, A, A, I,I, P, Pairs of of Them, Them, as as aa Function Function of of Population Population lifespan, Lifespan, zZo~
TABLE TABLE IIII
Genotypes Genotypes intially intiaily present
A, I, I, P P A,
z
77 8 9 9 1100 I11I 1122
A, A, II
A, A, P P
I, I, P P
P 1"
1/2 c 1I2c
11
1/2 112
11
1/2 112
A A
A A A A II
I I I P P PP P P
I I I II P P P P
A A A A A A P P P P P P
A A A A A A A A A A ++PP P P
I I P P P P P P
I
A+P A+P P P
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1133
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Generations Generations FIGURE F[GURE 8 8 Evolution of the number of of individuals of of three genotypes with variable life-histories. The y-axis gives the numbers per patch of age class I1 (newly founded populations). Genotype A is of 6; genotype P P is perennial with fecundity of of 2 and adult survival rate of of 0.9; annual with fecundity of genotype I is intennediate intermediate with fecundity of 4 and adult adult survival rate of of 0.45. All three genotypes have the same migration rate of I I was assumed. In this of 0.5. Landscape L2 (Table I) with zz = = 11 example, the simulation was started with 1100 O0 seeds of of each genotype in each patch. patch. The evolutionarily stable state (A (A + + P) was insensitive to initial conditions.
unexpected metapopulation, it is likely u n e x p e c t e d ddynamics y n a m i c s of of gene g e n e frequencies. f r e q u e n c i e s . In In a a finite finite m e t a p o p u l a t i o n , it is likely that increase. that the the annual a n n u a l genotype g e n o t y p e would w o u l d go go extinct extinct bbefore e f o r e its its frequency f r e q u e n c y starts starts to to increase. In selects for In this this eexample, x a m p l e , decreasing d e c r e a s i n g fecundity f e c u n d i t y selects for increasing i n c r e a s i n g reproductive r e p r o d u c t i v e effort effort (annual is apparently the colonization of nnew (annual life life cycles). cycles). This This is a p p a r e n t l y bbecause e c a u s e the c o l o n i z a t i o n of e w patches patches limiting factor. bbecomes e c o m e s the the limiting factor. When When a a stable stable ppolymorphism o l y m o r p h i s m bbetween e t w e e n annual a n n u a l and and pperennial e r e n n i a l types types can can be be maintained, in yyoung m a i n t a i n e d , the the frequency f r e q u e n c y of of the the annual a n n u a l type type is is hhigh i g h in o u n g ppopulations, opulations, whereas Fig. 9). w h e r e a s the the frequency f r e q u e n c y of of the the perennial p e r e n n i a l type type increases i n c r e a s e s after after colonization c o l o n i z a t i o n ((Fig. 9). Within W i t h i n populations, p o p u l a t i o n s , selection s e l e c t i o n favors favors perennials, p e r e n n i a l s , while w h i l e between b e t w e e n ppopUlations o p u l a t i o n s an annuals are is exactly in nuals are selected s e l e c t e d for. for. The T h e two-level t w o - l e v e l selection s e l e c t i o n pprocess r o c e s s is e x a c t l y the the ssame a m e as as in
316 31 6
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metapopulation as a function of population age. Equilibrium gene frequencies in the metapopulation Landscape L2 (Table I) was assumed with zz = - I11. genotype has a fecundity of 2 and Landscape ! . The perennial genotype a survival rate rate of of 0.9. (a) Net fecundity of of the annual genotype genotype is 6, as in Fig. 8. (b) Net fecundity fecundity of the annual genotype is 6.3. of
the migration. This prediction prediction is consistent consistent with observations the case case of of migration. with abundant abundant observations about plant along ecological successions, successions, both among about plant life histories histories along both within within or or among species. species. In our could not single isolated our model, model, we we could not maintain maintain a polymorphism polymorphism in aa single isolated population. population. Crawley Crawley and and May May (1987) ( 1 987) showed showed that that within-population within-population polymorpolymor phisms phisms of of annuals annuals reproducing reproducing by by seeds seeds and and perennials perennials spreading spreading by vegetative vegetative reproduction reproduction could could be be maintained maintained if if ramets ramets had had aa competitive competitive advantage advantage over over seedlings. seedlings. In In our our model, model, perennials perennials have have aa competitive competitive advantage advantage over over annuals annuals (once (once aa patch patch has has reached reached its its carrying carrying capacity, capacity, seeds seeds may may germinate germinate only only after after death of of adult adult plants), plants), but but both both annuals annuals and and perennials perennials may may reproduce reproduce through through death seeds seeds only. only. Our Our results results are are consistent consistent with with interspecific interspecific studies studies of of life-history life-history changes changes with plant plant successional successional stage. stage. The The work work by by Huston Huston and and Smith Smith (1987), ( 1 987), Tilman Tilman with ( 1 990), McCook McCook (1994), ( 1 994), and and Huston Huston (1994), ( 1 994), suggest suggest that that "facilitory" "facilitory" and and "inhi"inhi (1990), bition" effects effects of of species species on on each each other, other, leading leading to to sequential sequential replacement replacement of of bition" species, are are best best explained explained in in terms terms of of correlations correlations of of life-history life-history traits. traits. In In insects, insects, species, Brown (1985) ( 1 985) found found that that along along aa plant plant succession, succession, in in which which annuals annuals were were propro Brown gressively replaced replaced by by perennials, perennials, aa decreasing decreasing fraction fraction of of species species of of insects insects had had gressively more than than one one generation generation per per year. year. Bengtsson Bengtsson and and Baur Baur (1993), ( 1 993), in in contrast, contrast, found found more that that pioneer pioneer species species of of terrestrial terrestrial gastropods gastropods had had the the same same average average longevity, longevity, age age at at first first reproduction, reproduction, and and clutch clutch size size as as nonpioneer nonpioneer species. species.
1133
Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits
31 3177
Another result of of our model (not shown) is that perennials are selected against when disturbance disturbance rate increased. increased. Although this result is intuitive, it is not always verified experimentally. 1 990), in an experiment de experimentally. McLendon and Redente Redente ((1990), experiment designed to study the influence of of disturbance disturbance on plant communities, found that, as expected, whereas expected, perennial perennial species dominated under low disturbance disturbance regimes, whereas annuals dominated under high disturbance experimental study of disturbance rates. The experimental of Fahrig Fahrig et al. ((1994) 1 994) showed, however, that at very high disturbance rates, perennials disturbance perennials with large migration rates were favored over annuals. We We will discuss these results in Section Section VI.
C. Coevolution 1. Interacting Interacting Species Other situations where the metapopulation effect effect may be expected include those in which two replicating entities (a host and its pathogen, pathogen, or two symbionts) -pathogen interaction will be highly interact. A A priori, priori, one could expect that that a host host-pathogen influenced by the processes of Frank, this volume). In of extinction-recolonization extinction-recolonization ((Frank, particular, particular, if the host possesses both general general resistance (polygenic, "horizontal" "horizontal" resistance) resistance) and specific resistance (oligogenic, "vertical"), "vertical"), the former will be very advantageous advantageous at the time of of colonization, when the host colonizes a patch patch with uncorrelated pathogens. pathogens. The latter latter resistance resistance type will become become progressively fa favored as a given patch grows older. Within each patch, the local pathogens create create for the specific resistance genes, while colonization selects a selection pressure for specific resistance for general general resistance. resistance. The same kinds of predictions predictions may be made about about predators predators and parasites, namely that generalist predators and parasites should be found in parasites, predators young host populations, whereas old host populations should harbor more spe populations, specialized 1 985) showed that cialized natural natural enemies. In support of this prediction, prediction, Brown ((1985) that phytophagous Heteroptera of of the young stages stages of of the succession were more more gen generalists than than those of of late successional stages. Metapopulation coexistence of Metapopulation structure appears to promote promote the coexistence of hosts and and pathogens, as well as genetic genetic variability in host-pathogen host-pathogen systems with gene genefor-gene interactions, 1 992), Frank ((1991 1 99 1 b, interactions, as suggested by Burdon and Jarosz Jarosz ((1992), this volume), Antonovics ((1994), 1 994), Antonovics and Thrall ((1994), 1 994), and Antonovics et al. ((1994). 1 994). 2. Interacting Interacting Genomes Genomes and Reproductive Reproductive Systems Systems
"Interactive "Interactive genomes" can be different compartments compartments of of the same genome. We do not know any demonstration demonstration of the dynamics of of transposons transposons or other repeated repeated sequences in a metapopulation, metapopulation, but one case of of intragenomic conflict highly influenced by extinction-recolonization nucleo extinction-recolonization processes is the case of nucleocytoplasmic confl ict in the determination of conflict of male sterility in plants. The most extensive study of this process process has been made in thyme Thymus Thymus vulgaris vulgaris and
3 18 318
Isabelle Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-HenriGouyon Gouyon
summarized in Gouyon and Couvet ((1985, 1 985, 1987) 1 987) and Atlan et ( 1 990). In this et al. al. (1990). perennial species, female female (male-sterile) (male-sterile) individuals coexist with hermaphroditic (male-fertile) (male-fertile) ones. The females produce produce more seeds than than hermaphrodites, as demonstrated by various authors. Darwin ((1877) 1 877) found a 70% increase, Assouad et 1 978) found a 200% increase. tness of et al. al. ((1978) increase. Consequently, as the fi fitness of cytoplasmic genes (maternally inherited) causing male sterility is directly related related to seed pro production, the frequency of these genes increases. When When there are are many females, females, nuclear nuclear genes of of hermaphrodites reproduce much much better better than those those of of females females (the former reproduce through their their own ovules, plus through pollen in seeds produced by hermaphrodites and by females). In thyme, the proportion of of females can reach reach very high values locally, more than 90%. In such situations, nuclear nuclear genes able to produce hermaphrodites are highly selected for. In thyme, there there exists a number number of of mitochondrial strains which produce male-sterility, and a number of of nuclear nuclear 1 99 1 , 11993). 993). genes which specifi cally restore specifically restore male fertility (Belhassen et et al. al.,, 1991, Considered at this level, the interaction is formally equivalent host-pathogen equivalent to a host-pathogen interaction (see also Frank, this volume). However, in the case of of male sterility, there there is probably no generalist restorer: each each cytoplasmic type can be restored by specific restorers only. A metapopulation of thyme undergoes a series series of of events events that can can be described described as follows. At the establishment of a new population, only few few individuals are present in a patch. They mate mainly with the nearest et aI. nearest neighbor (Couvet et al.,, 11986). 986). Consequently two situations may arise. ((i) i ) If If one of the individuals is a female and its nearest hermaphrodite hermaphrodite does not possess the specifi specificc nuclear restorer restorer genes, this female female progeny. This will create female will produce purely female create a "colony" (spatial unit) composed composed of of females exclusively. Such "colonies" "colonies" were were observed in natural Dommee and Jacquard, 1 985). Mated natural populations ((Domm6e Jacquard, 1985). Mated with the same her hermaphrodite, maphrodite, all these females will keep producing pure female female progenies. Due to the high seed production cient production of of such females, females, these "colonies" will be very effi efficient colonizers. (ii) If, in contrast, the few nd their few cytoplasmic types locally fi find their specific restorer genes, hermaphroditic progeny will be produced. produced. Since hermaphrodites hermaphrodites produce fewer fewer seeds than than females, "colonies" of this kind colonize colonize less efficiently the initially empty patch. Molecular Molecular markers have allowed a demonstration of of this process. After After disturbance disturbance in a patch, recolonization thus thus occurs from large colonies of of fefe of females and and small colonies of hermaphrodites. Very high proportions of patches ((Belhassen al.,, et al. males are indeed found in most recently colonized patches Belhassen et 11987). 987). Subsequent evolution of the population is much slower, because once the patch has been invaded, young individuals may establish only after the old ones have died. As females become more more numerous numerous in a female female "colony," the proba probability that some of them are fertilized by pollen grains bearing the restorer gene corresponding corresponding to their cytoplasmic type increases. As the popUlation population grows older, these nuclear in nuclear restorer genes invade it, and the proportion of of hermaphrodites hermaphrodites increases ((Belhassen Belhassen et al., 11990). 990). This metapopulation effect in this case results et al.,
Evolution and Other Evolutionof of Migration Migration Rate Rate and Other Traits Traits
1133
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FIGURE population age in the gynodioecious FIGURE 1| 0 0 Proportion Proportion of male-sterile plants as a function of population gynodioecious al., 11983). 983). species Thymus vulgaris vu/garis L. (modified from Dommee Domm6e el et a/.,
in the 1 0. This the evolution of female proportion as shown in Fig. 10. This process explains explains why, in disturbed habitats, habitats, the proportion of of females is higher higher than than in stable 983) and why the average proportion habitats (Gouyon et et al. al.,, 11983) proportion of of females can be as high as 60% in the region of Montpellier, southern France ((Belhassen Belhassen et of Montpellier, et al. al.,, 11991). 99 1 ). This example provides us with a relatively complete picture of of the complexity of the metapopulation effect. There There are are actually processes at three three levels (Couvet et al., 11985): 985): selection et al., selection within within a colony (favoring nuclear nuclear restorer restorer genes), among colonies within populations (favoring cytoplasmic genes determining male ste sterility as well as nuclear communica nuclear nonrestorer nonrestorer genes; Denis Couvet, personal communication), and among populations, as extinction and recolonization recolonization processes generate local founder effects. In this sense, this process is related to the shifting balance 1 993). process (Barton and Whitlock, this volume; Michalakis and Olivieri, Olivieri, 1993).
320 320
Isabelle Olivieri Olivieriand and Pierre-Henri Pierre-HenriGouyon Gouyon Isabelle
VI. INFLUENCE INFLUENCEOF OF MIGRATION MIGRATIONON ON THE THE EVOLUTION EVOLUTIONOF OF LIFE-HISTORY LIFE-HISTORYTRAITS TRAITS VI. In Section IV.A, we showed that in some cases, increasing the fecundity of sureach genotype by a constant factor had consequences for selection on adult sur vival. We also found found that the ES migration rate increased increased with fecundity and adult survival. This agrees with classical predictions and observations. One could ask further whether migration might influence selection on life histories. To answer perennials this question, we ran pairwise simulation contests between annuals and perennials for various values of the migration rate. Results Results are as described in Section V.B, for shown Table III. It is clear that the larger the migration rate, the more strongly the perennials (Baker are selected for. This is contrary to the classical "colonizer syndrome" (Baker 1 994) also showed, using fi eld ob and Stebbins, 11965). 965). Recently, Fahrig Fahrig et al. ((1994) field observations and simulations, that in highly disturbed habitats, perennials tended to dominate dominate over annuals. Using Using an ecological (optimization, not ESS) model, in
Winners of Pairwise Pairwise (ontests Contestsbetween between an an Annual Annual and Perennial, Winners of and aa Perennial, as aa Function Function of of Population Population Lifespan Lifespan and and Migration Migration Ratea Rate~ as
TABLE III
Population lifespan lifespan PopUlation z
< 99 <
9
10 10
11 II
12-13 1 2- 1 3
14-22 1422
>> 22 22
Migration rate Migration d d 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.50 0.50
Winners Winners of of pairwise pairwise contests between between annuals annuals and perennials perennials
Annual Annual Annual Annual Annual Annual Annual Annual Annual Annual Polymorphism Polymorphism Annual Annual Annual Annual Perennial Perennial Annual Annual Polymorphism Polymorphism Perennial Perennial Annual Annual Perennial Perennial Perennial Perennial Polymorphism Polymorphism Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial
"An annual with and aa perennial annual fe" An annual with fecundity fecundity of of 66 and perennial with with constant constant annual fe cundity and survival rate rate of of 0.9 0.9 were were assumed assumed in landscape landscape L2 L2 cundity of of 2 and (A" = = 0.4; 0.4; A A II == 0.9, 0.9, A_ k= = 0.95). 0.95). (Ao
1133
Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits
321 321
which they compared populations of compared the equilibrium density of of monomorphic monomorphic populations of either annuals annuals or or perennials, they suggested that "long-distance clonal spreading" spreading" of of herbaceous herbaceous perennials perennials might explain their finding, especially if seeds were were more sensitive to disturbances disturbances than stems. Our Our own own simulation results point to the same direction, though we provide a different different explanation. In highly disturbed disturbed habitats, high migration rates are expected to evolve. In such habitats, only species with high fecundity can sustain sustain a viable metapopulation. We showed showed in Section Section III that perennial life histories histories are favored when fecundity and and migration migration rates rates are high. For certain landscapes, landscapes, migration rates and fecundities may be such that perennials perennials are favored over annuals when disturbance disturbance rates increase, as found found by 1 994). Fahrig et al. ((1994). 1 99 1 ) conducted Ben-Shlomo Ben-Shlomo et al. ((1991) conducted a selection experiment on the the migration ability of the flour beetle Tribolium found that correlated Tribolium castaneum, castaneum, and they found correlated responses to selection included shorter generation times. Roff ( 1 986) has shown responses generation Roff (1986) that there are fitness costs associated with the ability to disperse. These correla These correlations are likely to affect the evolution evolution of of migration behavior, as formally shown shown by Cohen and Motro 1 989). Den 1 990), in contrast, found Motro ((1989). Den Boer Boer ((1990), found that dispersing production than nondis individuals of of some arthropod arthropod species had a higher higher egg production than nondispersing persing individuals. A review of studies of of genetic correlations correlations among migration characters tness components characters and other fi fitness components in insects may be found found in Roderick Roderick and Caldwell ((1992; 1 992; see also Roff, 990). The idea that life-history characters Roff, 11990). characters do not of course not new. evolve independently is of
VII. CONCLUSION CONCLUSION We have shown in this this chapter chapter that the processes processes determining the migration rate in a metapopulation specific to the very functioning metapopulation are specific functioning of of the metapopulation metapopulation (the metapopulation effect) and interacting with the processes determining the evolution cant life-history traits. These evolution of of most signifi significant These processes result in a partial adaptation of of the metapopulation to its landscape. landscape. This adaptation is incomplete because the processes processes involved act between between genes at both both the population population and and the metapopulation metapopulation levels. They thus necessarily involve frequency dependence dependence and ' s fundamental theorem of Fisher's of natural selection does not apply at the therefore Fisher metapopulation metapopulation level. One One of of the difficulties involved in the study study of of metapopulation metapopulation evolutionary of processes is the general confusion that characterizes the questions questions of of levels of selection. The controversy between group group selectionists (sensu (sensu Wynne-Edwards, Wynne-Edwards, 11971) 97 1 ) and individual selectionists (e.g., Williams, 11971) 97 1 ) has led to radical radical posi positions and cult to sort of view and made made the whole whole point diffi difficult sort out. Roughly, one point point of is the one developed 970s, which states developed during during the 11970s, states that group group selection acts necessarily against individual selection. A more recent point of of view defends defends the idea that all levels can be taken into account, whether they act in different or similar directions. This latter point of of view seems much more promising and
322
Isabelle Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-Henri Gouyon Gouyon
tractable than Lloyd, 11994). 994). It can be formalized in diverse ways, the than any other other ((Lloyd, best known 1 970), extended by Wade ((1985), 1 985), known being the one developed by Price ((1970), 1 992) to, and applied by Frank ((1986, 1 986, 11987, 987, 11994b) 994b) and Goodnight Goodnight et et at. al. ((1992) to, for for instance, migration, sex-ratio, and altruism [see also Heisler and Damuth 1 987) Damuth ((1987) and Damuth 1 988), for Damuth and Heisler Heisler ((1988), for extensions extensions to multiple characters characters evolution evolution and discussion). It consists of and further further discussion]. of a nested nested decomposition decomposition of of forces acting acting on allelic frequencies, assigned to each frequencies, with one component component being assigned each level at which which supposed to act. The levels may be within individual individual (e.g., repeated selection is supposed repeated sequences), populations, etc. sequences), within colony (or deme), within population, population, among populations, If If aa DNA DNA sequence sequence that that can can be be repeated repeated in in the the genome genome of of an an individual individual acts acts positively or negatively on its fitness, nobody nobody will deny that that the study of of the evolution evolution of of this sequence sequence has to take take into account account both both the the within- and and the among-individuals component of understand its fate. Oddly of selection to understand Oddly enough, enough, it seems much cult for evolutionary biologists to jump much more more diffi difficult jump one more more level. If a gene can be selected selected within a population population and a n d influence influence the probability that that a new patch is colonized, then, its fate se fate can be analyzed in terms of of two-level selection. Whether Whether this is done explicitly or not in the models is just a question of of words. Unfortunately, Unfortunately, the confusion confusion between this hierarchical approach approach to levels of of selection selection and and the the naive naive Panglossian Panglossian group group selection postulates postulates still survives. In their very clear review about about the evolution of of migration, Johnson Johnson and Gaines Gaines ((1990) 1 990) still mix in their "group "group selection selection models" section section a paper paper by Wynne WynneEdwards 1 962) with a model 1 97 1 ). This last model explicitly Edwards ((1962) model by Van Valen ((1971). distinguished distinguished the individual and and demic levels of of selection, while others others (e.g., Comins et 980; Levin et 984), who et al., al., 11980; et at. al.,, 11984), who have treated these levels implicitly, are classified in the "individual "individual selection" selection" section. The The fact that these these models models do do not not differ differ in their assumptions but in their wording is made clear by the the fact that they produce the same results! (Assuming survival rate during migration is very low, all these models find find that the ES migration rate is equal to the local extinction rate.) The same confusion confusion can be found found in a recent recent review about about metapopulation metapopulation ( Hastings and Harrison, Harrison, 1994), 1 994), where consequences of dynamics and genetics (Hastings of 1 985), Olivieri et 1 990), the metapopulation metapopulation effect described described by Rice and and Jain Jain ((1985), et al. al. ((1990), 1 992) are treated as interdemic selection, leading the and and Manicacci et et al. al. ((1992) the authors authors to conclude that "however, this is open open to explanation explanation in terms of of individual selection." The confusion, confusion, once again, comes from the lack of of distinction between between the entity that 976; or the information, that is selected selected (the replicator, Dawkins, 11976; Gouyon 988; Gliddon 989) and the level of Gouyon and Gliddon, Gliddon, 11988; Gliddon and Gouyon, 11989) of inte integration 976; gration at which differences differences in reproduction reproduction exist (the interactor, Dawkins, 11976; or 988; Gliddon 1 989). All formal or avatar, Gouyon and Gliddon, Gliddon, 11988; Gliddon and Gouyon, 1989). All formalized models of of migration assume that the selected entity is the genetic information. ex information. Sometimes the two levels (within and among populations) populations) are are explicitly treated treated and sometimes they are not. Both levels are nonetheless nonetheless always included in the analyses, leading leading to convergent convergent analytical and simulation results.
1133
EvoluTIon Evolutionof of MigraTIon Migration Rote Rate ond and Other Other Traits Traits
323 323
ACKNOWLEDGMENTS ACKNOWLEDGMENTS Sandrine Bernard Godelle, Sandrine Maurice, Yvain Dubois, Dubois, Stuart Baird, Baird, Bob Bob Holt, Bernard Godelle, and Stephanie St6phanie Brachet made of this chapter. made useful comments comments on a final version of chapter. Susan Mazer, Mazer, Yannis Yannis Michalakis, Michalakis, Ilkka Hanski, Simon Levin, and an anonymous anonymous reviewer raised raised very interesting interesting points points on the sub submitted version. Anne-Marie Duffour helped with the documentation, documentation, and and Stephanie St6phanie Brachet Brachet was very helpful in drawing 3-D figures and Fig. I1 was drawn by Jean-Yves Pontallies. The present present version benefited from the considerable considerable help of of Ilkka Hanski, who made extensive (useful! (useful!)) changes changes on the paper. This is publication publication ISEM95-097 ISEM95-097 of the Institut des Sciences Sciences de l'Evolution, Montpellier.
sdfsdf
14
Spatial Processes Processes in Host - Parasite Genetics Host-Parasite Genetics Steven Steven A. Frank
I. INTRODUCTION INTRODUCTION Host - parasite diversity can be described first is Host-parasite described in two different different ways. The The first simply the observed parasites in a particular particular observed variability among among the hosts hosts and and parasites population. For 1 99 1 ) classifi ed 67 wild flax plants plants population. For example, example, Burdon Burdon and Jarosz Jarosz ((1991) classified into 110 0 distinct distinct resistance genotypes when when tested against six races races of of flax rust. One One host genotype genotype was was completely resistant resistant to all six pathogen pathogen races, races, whereas whereas another ve of another genotype was susceptible susceptible to fi five of six races. races. The The second second type type of of variability is the range range of of potential potential genotypes genotypes that can can occur over 1 985) used field transplant ex occur over space space and time. For For example, example, Parker Parker ((1985) transplant experiments fungal pathogen periments to study the legume Amphicarpaea bracteata and its fungal pathogen Synchytrium decipiens. heavy in each of three locations. decipiens. Fungal Fungal infection infection was heavy each of three locations. However, a plant However, plant moved moved to a new new location location developed developed little or or no no infection, infection, sug suggesting among sites. In a second gesting that that the pathogen pathogen populations populations differ differ among second experiment, experiment, host their ability to resist host lines derived derived from from different different locations locations varied varied in their resist a single single pathogen host populations. populations. pathogen isolate, indicating indicating spatial spatial differentiation differentiation among among the the host Parker's Parker's study suggests suggests that that the potential potential range range of of diversity over over space space and and time is often observed in a single location. The The poten often greater greater than than the variability observed single location. potential diversity is limited by the biochemistry morphology of parasite biochemistry and and morphology of hosthost-parasite traits, whereas controlled by the local dynamics of whereas the observed observed diversity is controlled of disMl'tap(} Metapopulation l'u/alioll Biology
Copyright «) 1997 by Academic Press, Press. Inc. All rights of reproduction in any form reserved. reserved. 9 1997
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Steven A. A. Frank Frank Steven
ease and and the the global global processes processes of of extinction extinction and and colonization colonization in in the the metapopulametapopula ease tion. tion. The first first goal goal of of this this paper paper is is to to suggest suggest that that increasing increasing potential potential diversity diversity The causes aa qualitative qualitative shift shift in in metapopulation metapopulation dynamics. dynamics. Local Local processes processes dominate dominate causes when potential potential diversity diversity is is low. low. Colonization-extinction Colonization-extinction dynamics dynamics of of alleles alleles in in when the metapopulation metapopulation become become more more important important with with an an increase increase in in the the potential potential numnum the ber of of distinct distinct genotypes. genotypes. In In the the next next section section I present present aa simple simple model model to to illustrate illustrate ber the importance importance of of potential potential diversity. diversity. the After briefly briefly discussing discussing the the model, model, I review review evidence evidence that that many many host-parasite host- parasite After systems do do in fact fact have have high potential potential diversity. diversity. Examples Examples include include plant-pathplant- path systems ogen genetics genetics and and bacterial bacterial defense defense systems against viral parasites parasites and and conspecific conspecific ogen competitors. I also discuss the antagonistic interaction between cytoplasmic and competitors. also discuss the antagonistic interaction between cytoplasmic and nuclear genes in cytoplasmic male sterility. nuclear genes in cytoplasmic male sterility. Data from from these these studies studies suggest suggest that that spatial spatial variation and colonization-excolonization - ex Data variation and tinction important in the the observed patterns of of diversity. However, tinction dynamics dynamics are important observed patterns diversity. However, data are interpret because because of of limited limited sampling sampling over space the data are difficult difficult to interpret space and and time. This difficulty leads leads to my second goal: the emphasis of space-time space-time scaling scaling emphasis of interpreting host-parasite host-parasite diversity. Spatial when interpreting Spatial scales that that are are small relative distance have well-mixed populations populations dominated inter to migration distance dominated by local interactions. Local processes also dominate on temporal scales that are short relative to the expected times to extinction and recolonization of of genotypes. By contrast, observations aggregated over long spatial and temporal scales may ob obscure colonizations, extinctions, extinctions, and rapid rapid changes in genetic composition that ner scales. Thus the patterns of observed variability are strongly infl u occur on fi finer influenced by the spacetime scaling of colonizations and extinctions in the meta space-time metapopulation. population.
II. DIMENSIONALITY AND COLONIZATION - EXTINGION DYNAMICS DIMENSIONALITYAND COLONIZATION-EXTINCTION DYNAMICS In this section I describe more precisely the relationship between between potential variation and observed diversity. I defi n e the potential number of genotypes as define the dimensionality of the system. I begin with a verbal illustration of the link dimensionality verbal between dimensionality and colonization -extinction dynamics. I then tum and colonization-extinction turn to a simple simple model. model.
A. A. Verbal Verbal Description Description The The observed diversity of host-parasite host-parasite genetics depends on the range of of possible variants and and the the processes that that govern local extinction extinction or success success of each each genotype. genotype. For For example, example, suppose suppose that that the the host host has has just just two two alternative alternative geno genotypes, and the parasite has two genotypes, PI and P ' The host h I can types, hh Il and and hh2, and the parasite has two genotypes, pl and P2. The host h~ can 2' 2 recognize I but ' Likerecognize and and resist resist the the matching matching parasite, parasite, P p~, but hh~I is is susceptible susceptible to to P P2. Like2 '
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to P wise, h h 22 can c a n resist resist P P22 but but is is susceptible susceptible to p~. In this this case case strong strong frequency frequency de deI ' In pendence pendence will will favor favor rare rare genotypes, genotypes, and and genotype genotype frequencies frequencies will will fluctuate fluctuate around around 0.5. 0.5. Thus Thus diversity diversity is is controlled controlled by by the the local local dynamics dynamics of of frequency frequency de dependence. pendence. Now host-parasite interaction interaction but with more Now consider consider the same same pattern pattern of of host-parasite but with more genotypes. host genotypes genotypes. In In particular, particular, each each of of the the n n host genotypes h il . . . h h,n matches matches the the single single corresponding corresponding parasite parasite genotype genotype from from the the set set of of PI pl '9 9. 9. P P,. Thus Thus h Il resists resists h 2 resists pPIl but but is is susceptible susceptible to to all all other other parasite parasite genotypes, genotypes, h2 resists P2, so on. on. The The same same P2 , so frequency frequency dependence dependence occurs, occurs, favoring favoring equal equal abundance abundance of of all all genotypes. genotypes. How However, in. As ever, the the frequencies frequencies now now fluctuate fluctuate about about l1In. As the dimensionality dimensionality n n increases, increases, the declines, and to cause the average average frequency frequency declines, and small small fluctuations fluctuations are are more more likely likely to cause local extinction of a genotype. local extinction of a genotype. An An extinction extinction leads leads to to aa sequence sequence of of events events that that changes changes the the local local dynamics. dynamics. hi is locally extinct. Then matching For example, suppose that host genotype For example, suppose that host genotype is locally extinct. Then the the matching because it it can attack all parasite parasite P Pii has has an an advantage advantage over over other other parasite parasite types types because can attack all hosts by their matching hosts in in the the local local popUlation. population. The The other other parasites parasites are are resisted resisted by their matching host decline toward toward local host genotypes. genotypes. Thus Thus Pi Pi increases increases and and the the other other parasites parasites decline local extinction. recolonization and by hi, hi ' extinction. The The patch patch is is now now ripe ripe for for recolonization and rapid rapid increase increase by which point which would drive drive Pi pi and the other other host host types toward toward local extinction. extinction. The The point is that dynamics are now now controlled by the times to extinction extinction and and recolonization. recolonization. The population at will be The observed observed variation variation in in aa particular particular population at aa particular particular time time will be much much lower lower than than the the potential potential diversity. diversity. n '
B. B. The Model Model II now the same in the now tum turn to to the the formal formal model. model. The The ideas ideas are are the same as as in the verbal verbal model model Some readers readers may may prefer just given, but but the points points are are made made more more precisely. Some prefer to skip later to the details skip ahead ahead to to the the sections sections on on natural natural history history and and return return later to the details of of the the model. model. I focus focus on a single-patch single-patch model model with with extrinsic extrinsic colonizations colonizations rather rather than an explicit, explicit, multipatch multipatch metapopulation metapopulation analysis. analysis. In In the the next next section section II discuss discuss single singlepatch models. patch and and multipatch multipatch models. The locus. Each The model model has has aa single single haploid haploid locus. Each of of the the n n host host alleles alleles causes causes recognition recognition and and resistance resistance to only one one of of the n parasite parasite alleles. Thus Thus each host host is )In resistant and each resistant to to lin 1/n of of the the parasite parasite genotypes, genotypes, and each parasite parasite can can attack attack (n (n -- 11)/n of the host genotypes (Frank, 1 99 1 a, 1 993a). I call this the "matching-allele" of the host genotypes (Frank, 1991a, 1993a). I call this the "matching-allele" model. model. In In aa popUlation-genetic population-genetic context context the the different different alleles constitute constitute aa polymor polymorphic phic locus locus of of a single single species. In an ecological ecological context context each allele is associated associated with species. II will will use the population-genetics popUlation-genetics language language of with aa different different species. use the of allelic allelic polymorphism, ecological interpretation polymorphism, but but an an ecological interpretation of of species species diversity diversity is is equivalent equivalent for for these these assumptions. assumptions. II use -Volterra equations to describe system. These use Lotka Lotka-Volterra equations to describe the the system. These equations equations show show the rather than just the relative genotype the dynamics dynamics of of genotype genotype abundances abundances rather than just the relative genotype frefre-
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population sizes and quencies. Thus the model tracks epidemic fluctuations in population disease intensity in addition to changes in genotype frequency. The model is Ah; = = hh;[r(1 - H/K) H/K) - m(P m(P - p ;pJ] ) ] At Ahi ; lr( l Apj = p j [ - s
+ b(H-
((1) 1)
hj)] At.
The values of hhii and P pjj are the abundances abundances of hosts of of genotype i and parasites parasites of genotype j. The total abundance abundance of hosts is H H = = sLk� 1 h kk,> and the total abun abunof of parasites is P P = = Lk� s 1P Pkk' dance of host'' s intrinsic rate ooff increase, H/K H / K iiss the strength ooff density The term r iiss the host dependent competition competition among among hosts with carrying capacity capacity of K, m is the mor mordependent and mortality per parasite attack, attack, s is the parasite death death rate, and b is the bidity and parasite'ss intrinsic intrinsic birth birth rate rate per hosthost-parasite contact. The The At term term is the size of of parasite' parasite contact. which the interactions occur. occur. For For example, example, At may be the the time step over which of one host generation or one season season in a discrete-time discrete-time model. model. When When birth, birth, length of death and and disease cause continuous continuous change change of of the the abundances abundances of of hosts hosts and and par pardeath asites, At � ~ 0. O. The system in Eq. ((1) easier to analyze when when rewritten in nondimensional The 1 ) is easier nondimensional form (Segel, (Segel, 11972; Murray, 11989). Nondimensional analysis focuses focuses attention attention on form 972; Murray, 989). Nondimensional of parameters parameters and and highlights highlights relative magnitudes magnitudes (scaling relations) relations) a minimal set of among the processes processes that that drive the dynamics. This This is accomplished accomplished without without al alamong interpretation because because one can translate translate freely between between the tering the dynamics or interpretation biologically motivated motivated formulation formulation and the nondimensional nondimensional quantities. quantities. biologically The system can be rewritten rewritten with the following following substitutions substitutions The
= hi~K, h;!K,
hhii =
s/r, s = = sir,
mp/r, fPjj= = mpi/r,
7~'== rr At At
=
(2) (2)
l) Kb/r. b = Kb/r.
Dropping Dropping the hats yields the the nondimensional nondimensional system system
Ahi Ahi Apj Apj
p;)]7 = h i [ 1 -- HH -- (P (P -- Pi)]7 = hi[1
= pj - ss + + bb(H ( H -- hj hj )]'1". )]7. Pj [[-
=
(3)
The dynamics the system are are controlled by the the equilibrium all hosts The dynamics of of the controlled by equilibrium with with all hosts and parasites and p* / and parasites present, present, which which occurs occurs at h* = = s/[b(n s/[b(n -- 1)] 1 )] and = (1 ( 1 -- H H**))/ P* = * * 01 nh and, and, by by the the symmetry symmetry of of the the system, system, h* ht = = h* and and 1 ), where where H H* -= nh* (n -- 1), p* for all i and pj == p* and j. j. This This equilibrium equilibrium point point is unstable unstable when when there there are are discrete discrete P * for time lags lags in in the the competitive effects effects among among hosts hosts and and in in the the interactions interactions between between time host host and and parasite. parasite. This This equilibrium equilibrium is neutrally neutrally stable stable when when interactions interactions occur occur in in continuous + 0). A A detailed detailed analysis is given given in in the the Appendix Appendix of of Frank Frank continuous time time ((7r -� (1993a). ( l 993a). Figure shows the the dynamics Figure 1I shows dynamics of of this this system system with with two two hosts hosts and and two two parasites parasites (n = = 2). Each Each panel panel shows shows how how one one of of the the two two host-parasite host- parasite pairs pairs changes changes from from an an initial initial condition. condition. In In each each case the the abundances abundances follow follow aa stable stable limit limit cycle cycle that that
1144 II) u c: cu "0 c:
a
Spatial Processes Parasite Genetics Spatial Processesinin HostHost-Parasite Genetics
329
c
.E <
0.
h I Abundance Dynamics for the matching-allele parasites. (a,b) Limit Limit Dynamics matching-allelemodel model with with two hosts hosts and two parasites. cycles cycles in which which abundances abundances fluctuate fluctuate in a periodic periodic and stable stable way. way. (c) (c) Spiral Spiral from from an initial initial condition condition out to a limit limit cycle, cycle, where this case case the where parasite parasite abundances abundances repeatedly repeatedly drop drop very very close close to zero. zero. In this parasite become locally locally extinct, leading to colonization-extinction parasite is likely likely to become extinct, leading colonization-extinction dynamics. dynamics. The panels panels show Eg. (3), (3), with = 11.2, .2, show the changes changes in abundance abundance for one of the two host-parasite pairs pairs in Eq. with b = s = 0.4, and T . 1 25, 0.375, 0.625 for the three panels, with with increasing increasing rT moving moving from from left ~"= 00.125, 0.375, 0.625 three panels, left to right. right. FIGURE FIGURE 1I
=
repeats regular intervals. intervals. These These cycles cycles are because trajectories repeats at at regular are stable stable because trajectories away away from from the cycle then remain remain on the cycle. cycle. the cycle spiral spiral toward toward and and then on the All point, and All three three panels panels of of Fig. Fig. 11 share share the the same same parameters, parameters, equilibrium equilibrium point, and initial conditions conditions except size of time step, step, T. time steps steps desta initial except for for the the size of the the time ~-. Larger Larger time destabilize the system. As As "i" T increases from the the right right panel, the oscillations bilize the system. increases from the left left to to the panel, the oscillations increase in magnitude. magnitude. The very low low parasite that occur increase in The very parasite abundances abundances that occur in in the the right right panel suggest that in that system would would be be prone prone to which panel suggest that the the parasites parasites in that system to extinction, extinction, which would change subsequent course the dynamics. would change the the subsequent course of of the dynamics. The prone to ex The difference difference between between a a repeating repeating cycle cycle and and cyclic cyclic dynamics dynamics prone to extinctions two figures. figures. Figure for tinctions can can be be seen seen in in the the next next two Figure 2 2 shows shows time-series time-series plots plots for aa model two hosts hosts and and two parasites (n (n = is simulated simulated by by model with with two two parasites = 2). 2). Extinction Extinction is setting to zero any abundance than 0.0 1 . In In this gure abundances setting to zero any abundance less less than 0.01. this fi figure abundances never never drop drop that low low and is simulated adding 0.0 that and extinction extinction never never occurs. occurs. Colonization Colonization is simulated by by adding 0.011 to parasite if if a random number number is less than than the the to the the abundance abundance of of each each host host and and parasite a random is less colonization (see figure legend). These colonizations have have little little effect colonization rate rate (see figure legend). These colonizations effect on on the the dynamics because because the the system system follows follows aa stable limit cycle. dynamics stable limit cycle. An increase in the Figure 33 shows with nn = Figure shows the the same same system system with = 4. 4. An increase in the number number of of hosts parasites has the hosts and and parasites has two two effects effects on on the the dynamics. dynamics. First, First, larger larger n lowers lowers the equilibrium abundance of each host host and parasite type. shifts equilibrium abundance of each and parasite type. A A lower lower equilibrium equilibrium shifts the entire cycle Fig. 11). ). Thus Thus an increase in the entire cycle down down and and to to the the left left (see (see Fig. an increase in n shifts shifts the the cycle closer closer to cycle to the the p p = = 0 0 and and h h = = 0 0 boundaries. boundaries. The location of the cycle leads to second effect, The shift shift in in the the location of the cycle leads to the the second effect, a a tendency tendency for host genotype from the the for genotypes genotypes to to become become locally locally extinct. extinct. When When a a host genotype is is lost lost from local population, popUlation, the matching parasite parasite genotype fitness advantage because local the matching genotype has has aa fitness advantage because it can can attack Eventually the locally extinct host is is rein it attack all all local local host host genotypes. genotypes. Eventually the locally extinct host reintroduced and because it it can resist attack the locally dominant troduced and spreads spreads rapidly rapidly because can resist attack by by the locally dominant parasite. The causes a decline among the host's parasite. The spread spread of of the the resistant resistant host host causes a decline among the host's com competitors parasite genotypes. petitors and and an an increase increase among among all all nonmatching nonmatching parasite genotypes. These These ex extinctions by random immigration into the system system cause unpredictable tinctions followed followed by random immigration into the cause unpredictable fluctuations in host and and parasite parasite genotypes genotypes (Fig. (Fig. 3). 3). fluctuations in the the composition composition of of the the four four host
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These These theoretical theoretical examples examples show show the the qualitative qualitative shift shift in in dynamics dynamics caused caused by by colonization -extinction processes. colonization-extinction processes. Systems Systems are are more more prone prone to to extinctions of of genotypes genotypes when when local local population population sizes are are small, the the number number of of genotypes genotypes (di (dimensionality) uctua mensionality) is is high, high, or or nonlinear nonlinear dynamics dynamics cause cause large, large, deterministic deterministic flfluctuations. Colonization Colonization by by locally locally novel alleles depends on the frequency of immi immigration and and on the the spatial variation in genotypes genotypes among populations. gration Scale Scale is clearly important. Frequent migration migration on a particular particular distance scale leads to high immigration but little differentiation among among popUlations. populations. Very rare migration enhances enhances differentiation but increases the waiting time before locally extinct alleles are are reintroduced by immigration. To To complete the picture these spatial spatial scalings must be tied to the temporal scales of local dynamics and and extinc extinc(Frank, 11991 tions (Frank, 99 1 b).
C. Summary Summary C. It may seem rather disappointing to have only the simplest, single-patch Lotka-Volterra Lotka-Volterra model for a metapopulation theory of host-parasite host-parasite genetics. However, I believe this is the right way to seek theories that apply broadly. A brief justifi cation may be useful before turning to the observations from natural justification systems. What What would it take to produce a full full model model of of host-parasite host-parasite genetics genetics within the context of metapopulation dynamics? Since genetics is the question, we need context several loci, and several alleles per per locus. Natural Natural systems often have this genetic complexity, which may play an important important role in determining determining spatial and temporal dynamics. We must also consider consider sex and recombination and the interaction interaction be-
r
d~
<
I
T ime Time
FIGURE 2 FIGURE
Time series series for for the the matching-allele matching-allele model model with with two two hosts hosts and and two two parasites, parasites, from from Eq. Eq. Time (3) with with nn == 2, 2, bb == 2.4, 2.4, sS == 0.4, 0.4, and and rT -= 0.25. 0.25. The The dynamics dynamics are are shown shown over over aa time time period period of of 500 500 (3) T. Extinction Extinction is is simulated simulated by by setting setting to to zero zero any any abundance abundance less less than than 0.01. 0.0 I . Colonization Colonization of length length 7. steps steps of is simulated simulated by by adding adding 0.01 0.01 to to the the abundance abundance of of each each host host and and parasite parasite in in each each time time step step ~" T if if aa random random is number between between zero zero and and one one is is less less than than 0.01. 0.0 1 . Thus Thus the the average average time time between between colonization colonization events events number for for each each type type is is 100"i. l OOT.
114 4 Spatial Parasite Genetics SpatialProcesses Processesinin HostHost-Porasite Genetics
h~
331 331
Pl
h2
h3 <
P3
ha
Time Time FIGURI: 3 3 FIGURE
Time for the Time series for the matching-allele matching-allele model model with with four four hosts hosts and and four four parasites. parasites. The The papa rameters rameters and and methods methods are are the the same same as as in in Fig. Fig. 2 except except that that nn == 4. 4.
tween tween host host and and parasite. parasite. Mutation Mutation is is important important because because rare rare events events can can have~a have' a large large impact impact on on diversity. diversity. We We now now have have many many parameters, parameters, but but have have not not yet yet specspec ified ified ecological ecological processes. processes. So So we we must must add add in in birth birth and and death death rates, rates, and and explicit explicit descriptions descriptions of of spatial spatial movement movement in in the the metapopulation. metapopulation. We are are ready ready to to see see that that host-parasite host-parasite genetics genetics is is like like the the weather. weather. A A epiepi We demic demic arises arises seemingly seemingly without without warning warning in in the the northwest, northwest, caused caused by by aa rare rare mimi grant parasite parasite genotype genotype that that sweeps sweeps through through the the local local host host population. population. The The patch patch grant is is ravaged, ravaged, perhaps perhaps extinct extinct or or left left with with only only aa depauperate depauperate set set of of genotypes genotypes and and aa few few individuals. individuals. Colonizations Colonizations occur occur over over time. time. The The new new composition composition is is very very different from from the the original original composition. composition. And And so so on on over over space space and and time. time. Dial Dial the the different migration migration parameter, parameter, and and aa different different but but equally equally beautiful beautiful map map appears appears on on the the computer screen. screen. We We have have many many parameters, parameters, each each with with an an effect effect over over some somerange range computer of the parameter parameter space. space. of the Of course, course, what what we we would would really really like like to to know know about about isis invariance, invariance, regions regions Of where changes changes in in aa parameter parameter do do not not matter, matter, and and "bifurcation" "bifurcation" in in the the generic generic where sense, parameter parameterchanges changes that that cause cause aa qualitative qualitative shift shift in in the the dynamics. dynamics. We We want want sense,
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Steven Steven A. A. Frank Frank
to know about about qualitative properties of of invariance and and change change over this vast and immensely complex parameter space. Here is my conjecture. Previous Previous population population genetic models missed the most interesting point because because they always studied studied one or two loci with two alternative alleles per locus. At that dimensionality, one finds the usual nonlinear nonlinear dynamics of cycles and chaos. Local dynamics dominate because, in each patch, all possible possible genotypes genotypes are usually present. However, However, if one increases the number number of of loci and alleles, the system bifurcates, changes generically. Colonization -extinction dy Colonization-extinction dynamics namics matter, times to extinction and and recolonization recolonization dominate. Local Local dynamics are much less important. Having con Having discovered discovered one major axis along which qualitative aspects are controlled, one can now pursue other other interesting questions. questions. Scale always comes up, but one has to put the problem in the context of of the the biology and and the the first first major axis. In summary:
1. and bi bi1 . The goal is to search for invariance over an interesting domain and furcation between domains because because that is only way to learn learn something general about a complex problem. 2. Host-parasite dimen Host-parasite systems bifurcate bifurcate as they move from low to high dimension. At low dimension, dimension, local dynamics are probably more important important for for under understanding genetic diversity. At high dimension, spatial processes dominate. This shift appears inevitable. The purpose purpose of of the simple one-patch model is to illustrate this point. 3. What What about about real systems? There There is good good evidence that that many systems have have surprisingly presented in the following sections. surprisingly high dimension. The The evidence is presented Data about impor about spatial spatial dynamics dynamics is sketchy, but where where available, suggest the importance of extinction dynamics at the level of of colonizationcolonization-extinction of genotypes. 4. When When analyzing these systems one is inevitahly inevitably measuring diversity. One of scale. Observed Observed diversity can be understood understood only within the has to be aware of context of of potential diversity and the spatial and temporal temporal dynamics. In the tum to data and the following following sections I turn and theory for for natural systems. As expected, dimension extinc dimension and and scale scale are important. In addition, the details of of extinctions, migration, and the genetic system determine the particular particular attributes of of each case. When When one can measure these details, it may pay to consider consider a complex metapopulation metapopulation model tuned to that system, although although the size of of the parameter space will make the analysis difficult. I summarize the general conclusions conclusions that can be drawn drawn from from current empirical and theoretical studies, which are still in an early early stage stage of of development. development.
III. IINTRODUCTION NTRODUmON TO THE EXAMPLES TO THE EXAMPLES Researchers host- parasite systems have recently Researchers working on two different host-parasite turned plant- pathogen turned their attention to spatial variation in allele frequencies. frequencies. In plant-pathogen
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interactions interactions the hosts often have numerous numerous resistance resistance genotypes genotypes and the pathogens pathogens have have correspondingly correspondingly diverse host-range host-range genotypes. The limited data from from natural natural populations populations suggest spatial variation variation both in the frequency frequency of of successful successful infections infections and in allele frequencies. frequencies. Several authors authors propose propose metapopulation metapopulation dynamics dynamics as and 989; Thompson Thompson and Burdon, Burdon, the cause of of spatial variation variation (e.g., Burdon Burdon et et al., al., 11989; 11992; 992; Frank, 11992, 992, 11993b; 993b; Antonovics 994). Antonovics et et al. al.,, 11994). The hermaphroditic plants. I The second second system system is cytoplasmic cytoplasmic male male sterility in hermaphroditic ict will describe the details of of this system later. The important important feature feature is confl conflict between genes over between cytoplasmic cytoplasmic genes genes and nuclear nuclear genes over the production production of of pollen. pollen. There There are are different different cytoplasmic cytoplasmic genotypes, genotypes, each of of which which is "resisted" "resisted" by specific, specific, matching matching nuclear nuclear genes. genes. The interaction interaction is similar to a system with several match matching host host (nuclear) and parasite (cytoplasmic) (cytoplasmic) genotypes. Preliminary studies show spatial variation variation in the frequencies frequencies of of nuclear nuclear and cytoplasmic genotypes. Several authors have analyzed this variation in terms of of metapopulation metapopulation dynamics (e.g., 990). Gouyon 1 985; Van 986; Frank, 11989; 989; Olivieri et Gouyon and Couvet, Couvet, 1985; Van Damme, Damme, 11986; et al. al.,, 11990). In the the following following sections sections I summarize summarize the the natural history and observations observations for other hosthost for plant-pathogen plant-pathogen genetics and cytoplasmic cytoplasmic male male sterility. I then list other parasite for metapopulation parasite interactions interactions of of high high dimension dimension that are are candidates candidates for metapopulation dynamics. These These later examples include include bacterial defense defense against against viral pathogens pathogens and herbivore systems. Finally, I consider how to test and polymorphism of of plantplant-herbivore different variation. different explanations explanations for for the observed observed patterns patterns of of variation.
IV. PLANT - PATHOGEN IINTERACTIONS NTERAGIONS PLANT-PATHOGEN Genetic city is common geno Genetic specifi specificity common in plant-pathogen plant-pathogen systems. Each host genotype resists only specific pathogen pathogen genotypes; each each pathogen pathogen genotype attacks only specifi specificc host genotypes. In this section I describe describe the details details of of genetic spec specifi city, the dimensionality ificity, dimensionality of of the interaction, interaction, and and spatial spatial variation variation in natural pop populations. ulations. conducted the fi first detailed study of of genetic genetic polymor polymorFlor ((1956, 1 956, 11971) 97 1 ) conducted rst detailed polymorphisms for phisms for for resistance in plants plants and the complementary complementary polymorphisms for host range in pathogens. The interaction interaction between plant plant and pathogen pathogen genotypes genotypes turned turned out to have simple properties properties that that Flor referred referred to as a "gene-for-gene" "gene-for-gene" system. In an idealized resistance and susceptibility idealized gene-for-gene gene-for-gene system, each pair pair of of resistance alleles in the hos hosfthas a matching matching pair pair of of host-range host-range alleles in the pathogen. pathogen. Recent Recent biochemical biochemical models models suggest that resistance resistance occurs occurs only when a path pathogen allele produces produces a particular particular gene product (elicitor) (elicitor) that can be recognized recognized by a matching 990). If matching host host receptor receptor (Gabriel (Gabriel and Rolfe, Rolfe, 11990). If an elicitor-receptor elicitor-receptor match occurs, then the host host induces a defensive defensive response response and resists attack. If If the same pathogen host produces produces a nonmatching receptor, pathogen elicitor elicitor is present, present, but the host nonmatching receptor, then disease develops. develops. Infection Infection also occurs when when a pathogen pathogen lacks an elicitor elicitor that that matches matches the specifi specificc host host receptor. receptor. In multilocus multilocus interactions interactions each each host polymorphism is matched matched to a unique, unique, complementary locus in the pathogen. The host resists attack when at least one complementary pathogen. The attack when one
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Steven StevenA.A. Fronk Frank
of the matching matching pairs pairs of of host-pathogen host-pathogen loci leads to recognition recognition and and resistance. of pathogen succeeds succeeds only when when it escapes escapes recognition recognition at all the complementary The pathogen loci. The The relation between between plant and and pathogen pathogen factors is simple simple in a gene-for-gene gene-for-gene system, but the total interaction is complex because because many loci are involved. involved. Flor identified factors in fl flax, and others have identifi ed 29 separate host resistance factors ax, each with a 98 1 ). complementary host-range ax rust (Flor, 11971; 97 1 ; Lawrence et ai., al., 11981). host-range factor factor in fl flax Lawrence et Similar gene-for-gene gene-for-gene interactions interactions are now now known known or suspected for for over 25 dif different host-pathogen 1 987). These systems do not conform host-pathogen pairs (Burdon, (Burdon, 1987). conform ex exactly to the idealized idealized gene-for-gene 1 987), but et ai., al., 1987), but these these gene-for-gene assumptions assumptions (Christ et systems do do have complementary complementary major-gene major-gene interactions interactions between between hosts hosts and and path pathogens. ogens. These These genetic genetic analyses analyses have have been been conducted conducted in agricultural agricultural systems. They They establish establish the possibility that plant-pathogen plant-pathogen interactions interactions in natural natural populations populations have have genetic specificities of of very high high dimension. According According to the theory theory de described scribed earlier, earlier, high high dimensionality suggests suggests that that observed observed polymorphisms polymorphisms and and the influenced by colonizationextinction dydy the dynamics of of disease are strongly influenced colonization-extinction namics in a metapopulation. metapopulation. That That story story of of dimensionality and and metapopulation metapopulation dynamics is intriguing, intriguing, but but Data from populations are metapopulation dydy is it true? true? Data from natural natural populations are suggestive suggestive of of metapopulation but there not enough enough information information to to draw firm conclusions. conclusions. I briefly namics, but there is not draw firm summarize available data summarize the the available data in the the remainder remainder of of this this section.
A. Dimensionality Dimensionality The on wild wild populations populations suggest suggest widespread widespread genetic polymor The few few studies studies on genetic polymorphisms for host phisms for host resistance resistance (Burdon, (Burdon, 1987; 1 987; Alexander, Alexander, 1992; 1 992; Parker, Parker, 1992). 1 992). For For example, the the matrix matrix in Fig. 44 shows shows the the frequencies frequencies of of different host phenotypes phenotypes example, different host of wild flax when of when tested tested against against seven races races of of flax flax rust. This This matrix matrix implies implies
• susceptible 4) 0 co
c::
I:: 4)
01) 0 -5
�
AIH
N
AF
AG
l ..
�: ,
K
E
�
'
I I
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4) u
�
,' . ." , .. . 0.27 .�4) - 0.28 c:: .. -+--1 0.37 '0 •• I:: 0.73 ()' . • ... 4) --4--+--+--+� 97 O. ... .. .. ..,.., ::-:-:-, ".,.-:-:-: .. ..,..,. I " ,-I-,I I. I I ' '" � Frequency of Ho t Phenotypes
I
;
I
I
I I
I I
,
[
L1.
FIGURE 4 Qualitative Qualitative resistance resistance in a wild wild population population of flax. flax. The matrix matrix shows shows the frequency FIGURE distribution of resistant patterns from 67 different host plants collected from a single population when flax rust (races A and H are grouped together). Redrawn from tested against seven pathogen races of flax Burdon and Jarosz (1991 ( 199 1 ).
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Spatial - Parasite Genetics Spatial Processes Processesinin Host Host-Parasite Genetics
335 335
complementary major-gene effects at multiple loci with extensive extensive polymorphism in the host. Similar studies of pathogen isolates in both natural and agricultural systems show that pathogen populations are often highly polymorphic ((Wolfe Wolfe and Caten, 11987; 987; Burdon and Leather, 990). Leather, 11990). The pathogen system in natural pop The most detailed detailed study of a natural plantplant-pathogen popvulgaris (groundsel), and its fungal ulations has been on an annual weed, Senecio Senecio vulgaris 990). In a recent recent study the authors pathogen, Erysiphe Erysiphe fischeri fischeri (Clarke et aI., al., 11990). authors obtained 5 pathogen isolates from 0 isolates were from each of two locations. These These 110 known to have different genotypes for specific host genotypes. for the ability to attack specific The same two locations were used to obtain 1 993b). obtain 360 host plants (Bevan et al al., 1993b). Progeny from 2 1 5 plants were tested against 5 of the 215 the pathogen isolates, and 45 plants were tested against all 110 0 isolates. These two progeny from the other 1145 tests yielded large large matrices of of susceptible or resistant resistant interactions. interactions. In both cases 70% of of the hosts were susceptible to all pathogens tested. The case with five test races 2 different resistance phenotypes races of of pathogen yielded 112 phenotypes among the hosts. Each cation against Each phenotype has a unique resistance/susceptibility resistance/susceptibility classifi classification against the pathogen test races. The case with 110 0 test races races yielded 14 different host phe phenotypes. Variation Variation in natural natural isolates of of the pathogen pathogen was measured measured in a second study (Bevan et al 993a). Twelve isolates were obtained from each of lo al., 11993a). of the two locations used for for the host study described described above. These 24 pathogen pathogen isolates were were tested against 50 inbred inbred lines of of host plants. Pathogen Pathogen growth on each host was scored on a scale ranging ranging from 0 (complete resistance) resistance) to 4 (vigorous fungal growth and sporulation). For the purposes of classifying genotype, each host of hostpathogen pair pair was labeled labeled as either "resistant" "resistant" or "susceptible," "susceptible," by splitting the of fungal growth. continuous scale of Table I shows that the majority of of host and pathogen isolates have unique genotypes. The extensive variability in a limited sample suggests that natural populations are tremendously diverse for this particular plant-pathogen plant-pathogen system. Put another way, the community matrix matrix that describes describes the interactions interactions between plant and pathogen genotypes has very high dimension. Do other -pathogen systems have other plant plant-pathogen have high dimensionality, or is the ground groundsel system unusual? unusual? The data are too limited to draw firm conclusions. There are several hints that diversity is high, but also some apparent exceptions. path Multilocus genetic genetic diversity for for resistance to fungal, viral, and and bacterial pathogens is typical in agricultural varieties relatives of crops (Burdon, varieties and wild relatives 11987). 987). The The pathogens of of cultivated plants evolve quickly in response to changing host genotypes, suggesting complementary genetic complexity ((Vanderplank, Vanderplank, 11984). 984). Studies of natural plant-pathogen revealed high plant-pathogen populations have have often revealed high di diversity. Examples include the groundsel study summarized here and Burdon and Examples the study and Jarosz's 1 99 1 ) study of flax and flax rust ((Fig. Fig. 4). All analyses do not find find Jarosz's ((1991) of wild flax variability of both host and pathogen in every sample. A study of of a perennial herb, Silene alba and anther-smut fungus, Ustilago Ustilago violacea. violacea, found genetic vari..
..
Steven Steven A. A. Fronk Frank
336 336
Numbers and Host-Range 25 Numbers of of Resistance Resistance and Host-Range Phenotypes Phenotypes Inferred Inferred from from aa Test Test Matrix Matrix of of 25 Pathogen Isolates Isolates by by 50 50 Inbred Inbred Host Host Lines Lines~ Pathogen
TABLE I
a
Infection Infection type type category category used to define define resistance/host-range ance/host-range
No. of of differdifferent groundsel groundsel resistance resistance phenotypes phenotypes discriminated discriminated
No. of of different different fungus fungus hosthostrange range phenotypes discrimidiscriminated
Minimum Minimum number of of hypothetical resistance/host-range resistance/host-range gene pairs pairs required required to explain explain the observed variation· variation b
0 I1 2 3
112 2 2211 34 43
112 2 118 8 23 24
tl0o 114 4 23 24
U
shows results infection intensity is used define the binomial binomial split uEach Each row shows results when a particular infection used to define between resistance and susceptibility. susceptibility. For example. example, if category 2 is used used for the split. split, then categories 993a). 0-2 Bevan et 0-2 are defined as resistant, and categories 3-4 3-4 are defined as susceptible (from (from Bevan et al., 11993a). h hThe calculation assumes assumes that resistance occurs occurs when the host has a resistance allele allele that matches a particular host-range allele pathogen. One match allele at a complementary locus locus in the pathogen. match at any of the I 993a). complementary host-pathogen loci loci is sufficient sufficient to cause cause resistance. See Bevan Bevan et et al. ((1993a).
ability 989). However, ability among among hosts hosts for for resistance resistance to to the the pathogen pathogen (Alexander, (Alexander, 11989). However, no no variability variability of of the the pathogen pathogen was was detected detected when when six six isolates isolates from from a a single single lo loal., 11993). 993). cation cation were were tested tested against against 1155 host host lines lines (Alexander (Alexander et et al.,
Colonization-Extinction Dynamics of Alleles B. ColonizationExtinction Dynamics Given high will be be occasional Given high dimensionality dimensionality it it seems seems inevitable inevitable that that there there will occasional extinctions of extinctions of alleles alleles from from aa local local population population and and subsequent subsequent recolonizations recolonizations by by immigration. immigration. The The problems problems now now concern concern pattern, pattern, process, process, and and inference. inference. What What are allele frequencies, disease intensity, intensity, and are the the temporal temporal and and spatial spatial patterns patterns of of allele frequencies, disease and population uence (scaling) population sizes? sizes? What What is is the the relative relative infl influence (scaling) of of colonization c o l o n i z a t i o n -extinc -extinction tion processes processes compared compared with with other other ecological ecological and and genetic genetic processes? processes? What What mea measurable surable properties properties can can be be used used to to infer infer process? process? Only Only a a few few studies studies of of natural natural systems systems have have measured measured spatial spatial variation. variation. II briefl genetic variation. variation. brieflyy summarize summarize two two projects projects that that have have focused focused on on genetic Parker 1 985) used eld transplant experiments to variability in Parker ((1985) used fi field transplant experiments to study study variability in the the legume Amphicarpaea Synchytrium decipiens. legume Amphicarpaea hracteata bracteata and and its its fungal fungal pathogen pathogen Synchytrium decipiens. II describe describe the the details details of of his his work work because because transplant transplant experiments experiments are are a a relatively relatively simple method method of of measuring measuring the the scale scale of of spatial spatial variation variation in o s t - p a r a s i t e intersimple in hhost-parasite inter actions. rst experiment experiment analyzed the focal focal population, km away actions. The The fi first analyzed three three sites: sites: the population, 1I km away from population, and 1 00 km km away the focal population. Seeds from the the focal focal population, and 100 away from from the focal population. Seeds were in each three populations. populations. were collected collected from from two two self-fertilized self-fertilized plants plants in each of of the the three For For each each of of the the six six groups groups of of selfed selfed progeny, progeny, 1155 - 220 0 seedlings seedlings were were transplanted transplanted into into the the focal focal population. population. All inAll of of the the seedlings seedlings derived derived from from the the focal focal population population developed developed severe severe in-
1 4 Spatial Spatial Processes Processes inin Host-Parasite Host- Parasite Genetics Genetics 14
337
fection when when transplanted transplanted back back into into their their natal natal location. location. Progeny Progeny from from one one of of the the fection km away away was was free free of of disease disease when when transplanted transplanted and and plant lines lines derived derived from from 1I km plant grown in in the the focal focal population. population. The The other other line line from from 11 km km away away had had 88% 88% of of the the grown progeny infected, infected, but but the the average average intensity intensity of of infection infection was was about about one-fifth one-fifth that that progeny of the the native plants. plants. Infection Infection intensity intensity was was measured measured as as number number of of sori sori per per plant plant of sorus is is the the initial initial fungal fungal lesion). lesion). All All of of the the progeny progeny derived derived from from 100 1 00 km km away away (a sorus were were completely completely free free of of infection infection when when transplanted transplanted into into the the focal focal population. popUlation. This transplant experiment suggests suggests spatial spatial variation variation in in the the genotypes genotypes of of hosts hosts This transplant experiment and pathogens over over distances distances of of 11 km km or or greater. Fungal infection infection was was heavy heavy in in and pathogens greater. Fungal each of the three locations. When a plant was moved to a new location, it devel each of the three locations. When a plant was moved to a new location, developed little little or or no no infection, infection, suggesting suggesting that that the the pathogen populations differ differ between between oped pathogen populations the focal focal site site and and the the other other two two sites. sites. The The variation variation in in infection infection among among the the host host the lines derived derived from different locations locations and and transplanted transplanted into into the the focal focal site site suggests suggests lines from different spatial differentiation differentiation among among the the host host populations. populations. spatial In a second second experiment experiment Parker Parker (1985) ( 1 985) obtained obtained stronger stronger evidence evidence for spatial In for spatial variation over 11 km. km. He He tested tested one one pathogen isolate from from the the focal focal population population variation over pathogen isolate against 13 1 3 plant plant families families from from the the focal focal population popUlation and and 11 1 1 families families from from 11 km km against away. All 13 1 3 local local families families developed developed infection, infection, but but 10 1 0 of of the the 11 1 1 families families from from km away away were were completely completely resistant resistant to to this this pathogen pathogen isolate. isolate. 11 km The final experiment analyzed variation variation on on aa smaller smaller spatial the The final experiment analyzed spatial scale scale within within the 1 00-km population. lines were collecting along along a linear linear trantran 100-km population. Plant Plant lines were established established by collecting sect from from six six sites sites separated separated by by 30 30 m. m. The The sites sites were were labeled labeled in in order order from one sect from one end of of the the transect transect to to the the other. A A pathogen isolate isolate from from site site 5 was tested against against end each plant plant line. line. II describe describe the the details details to to show show the the difficulties difficulties that that often arise when when each often arise measuring variability in the the interactions interactions between between host host and and parasite. parasite. Three different measuring Three different measures of resistance provide provide information about genetic variation. measures of resistance information about genetic variation. First, when when resistance or susceptibility susceptibility was was measured measured as the presence presence or or ab absence infection, there sence of of initial infection, there was was no no significant significant variation variation among among sites, with aa mean frequency of of 74%. 74%. Second, Second, if resistance resistance was measured mean infection frequency measured by perper that abort reproduction, then plants from centage of of sori that abort before before fungal reproduction, then all plants from site 6 00% resistant. The ve sites aborted were 1100% The other other fi five aborted 0 0 --220% 0 % of of sori. Third, Third, the sites sites varied significantly when of sori per per plant plant was was used used to measure measure re revaried when the number number of sponse. example, site 11 was the sponse. For For example, the least resistant, resistant, with a mean mean _±_+ SE SE of of the most with 2.3 2.3 ± 0.4, but 111.8 1 .8 ± _ 3.2. 3.2. Site Site 6 6 was was the most resistant, resistant, with ___ 0.4, but neighboring neighboring site site 5, 5, where ± 2.5. where the the pathogen pathogen was was derived, derived, was was the the second second highest, highest, with with 8.0 8.0 -+2.5. These These results results suggest suggest that that quantitative quantitative components components of of resistance resistance may may be be race race specific, specific, the groundsel groundsel study discussed above. In Parker's Parker's study, details about race raceas in the specifi specificc quantitative variation would require tests of the plant plant lines with different pathogen pathogen isolates. isolates. ' s work Parker Parker's work shows shows that that genetic genetic variation variation can can occur occur over over short short distances. distances. pathogens are highly successful on plants near near the location location at which In this case, pathogens they were found, but had poor poor success on plants plants from other other locations. locations. It appears that immigrant host genotypes, with resistance to local pathogens, pathogens, could increase in frequency and change the spatial patterns of of differentiation. Experimental
338 338
Steven A. A. Frank Frank Steven 1 .0
a
0.5
! JI
• •
1 .0
1-
•
rb:-------,
0.5
1 .0 r-------, c
0.5
A
B
E
K
N
U AF AG
Pathogen Race
FIGURE variation in pathogen pathogen genotypes genotypes and host resistance among wild populations populations of FIGURE S5 Spatial variation (Linum marginate) marginale) and flax rust (Melampsora lini). lini). Both host and pathogen isolates were obtained flax (Linum pathogen population and from several different sites. Each panel shows the racial composition of the pathogen the frequency of host resistance to each pathogen race when summarized over a different geographic scale. (a) (a) Data Data from from aa I1-ha plot for for 67 67 host host lines and and 94 94 pathogen isolates. (b) Combined Combined data data for for 40 40 scale. -ha plot host lines lines and and 37 37 pathogen isolates isolates from two populations populations 300 300m and 2.7 2.7 km away away from the the plot host m and summarized in in the first first panel. (c) Combined Combined data data for for 1108 and 80 pathogen pathogen isolates from from summarized 08 host lines and six populations populations 113.8-75 km away away from from the the plot plot summarized summarized in in the first first panel. panel. Redrawn Redrawn from from Jarosz Jarosz six 3 .8-75 km and 1 99 1 ). and Burdon Burdon ((1991).
of genotypes genotypes followed followed by by time-series time-series monitoring monitoring of of consequences consequences may may movement of provide aa method method for for inferring inferring the the joint joint roles roles of of selection selection and and colonization colonizationprovide extinction extinction dynamics. dynamics. The second second major major study study of of spatial spatial variation variation in in natural natural populations populations was was con conThe ducted on on flax flax (Linum (Linummarginate) marginale) and and its its pathogen, pathogen, flax flax rust rust (Melampsora (Melampsoralini) lini) ducted (Jarosz and and Burdon, Burdon, 11991; Burdon and and Jarosz, Jarosz, 11992). A summary summary of of spatial spatial vari vari(Jarosz 99 1 ; Burdon 992). A ation in in genotype genotype isis shown shown in in Fig. Fig. 5.5. To To study study the the role role of of metapopulation metapopulation dynam dynamation ics, the the authors authors measured measured the the composition composition of of nine nine pathogen pathogen popUlations populations over over 22 ics, to to 44 consecutive consecutive years. years. This This isis the the most most extensive extensive study study of of temporal temporal and and spatial spatial variation in in natural natural populations, populations, but but limitations limitations of of the the data data must must be be considered considered variation
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Spatial - Parasite Genetics SpatialProcesses Processesinin Host Host-Parasite Genetics
339 339
before before drawing drawing any conclusions. conclusions. First, the host host plant plant is perennial, so the time span of cover genetic of the study does does not cover genetic changes changes in the host host populations. populations. Second, Second, it is not move each not clear how how far far the wind-borne wind-borne spores spores can move each year, in other other words, words, the known. In In spite the scaling scaling of of spatial spatial distance distance relative relative to to migration migration distance distance is is not not known. spite of of these constraints, constraints, a few few tentative tentative conclusions conclusions are are interesting. interesting. Four pathogen pathogen races races dominated dominated the metapopulation metapopulation over over all 4 years years of of the ((i) i ) Four populations contained study. (ii) (ii) The The majority of of host host populations contained little or or no no resistance to any of of the four four dominant dominant pathogen pathogen races. Thus Thus host host resistance resistance alone alone cannot cannot ex explain iii) Pathogen plain temporal temporal and and spatial variation variation in the the pathogen. pathogen. ((iii) Pathogen races races occa occasionally became became locally extinct in a particular particular population rein population but were were often often reintroduced iv ) Fluctuations troduced within within a year year or or two. ((iv) Fluctuations in the the genetic composition composition of of local pathogen uenced by the dynamics of pathogen populations populations may be strongly infl influenced of pop population size. Twenty-two host populations were host populations were sampled sampled for for the the presence presence or or ab absence of of infection infection in 2 consecutive consecutive years. One One population population had had no no pathogen pathogen infec infections tions in the first year year but but was infected infected in the next next year. Another Another population population had had infections in the first year but was was free of of disease in the next next year. Finally, two popUlations were populations were free of of infection infection in both both years. Burdon 1 992) suggest that, over Burdon and and Jarosz Jarosz ((1992) over the temporal temporal and and spatial scale ' s genetic structure of of their study, the observed observed fluctuations fluctuations in the pathogen pathogen's structure were were driven driven by colonization-extinction colonization-extinction events and drift. Thus Thus the populations populations in that that region region may act as a cohesive unit linked linked by frequent frequent migration, with selection dynamics of of allele frequency. frequency. Put Put another another way, the playing a limited role in the dynamics time scale of of pathogen pathogen movement among among these these populations populations may may be on the order order of extinction dynamics of host host generation generation time and thus thus too too short for for colonizationcolonization-extinction dynamics of of alleles among among these these populations populations to exert strong strong coevolutionary coevolutionary pressures. pressures. Per Perhaps of haps at a larger larger spatial scale the the migration rate is small relative relative to the length of generations -the time scale over which selection At that host generations-the selection is effective. effective. At that scaling scaling between - extinction dynamics between migration migration and and selection, the colonization colonization-extinction dynamics of of alleles may cause composition. cause occasional occasional major major shifts shifts in genotypic composition. To To summarize summarize these two two studies studies on plant-pathogen plant-pathogen systems, it is easy to imagine imagine how metapopulation metapopulation dynamics dynamics can can influence influence genetics, but but very difficult difficult to measure measure space-time space-time variation variation over over the proper proper scales. How How can can convincing convincing data data be be obtained? obtained? One One way is to observe observe colonizations colonizations of of locally absent absent alleles and and the subsequent subsequent local local dynamics. It may be be difficult difficult to observe observe such such rare rare events, but but there there is one one suggestive suggestive study study of of cytoplasmic male male sterility that that I describe describe in the next next section. section.
V. V. CYTOPLASMIC CYTOPLASMICMALE MALESTERILITY STERILITY Most DNA from their mother, with no no input Most organisms organisms inherit mitochondrial mitochondrial DNA from their input from from their their father. father. By contrast, contrast, most most other other genetic material is obtained obtained equally from modes of transmission, from the mother mother and and father. Typically these these different different modes of transmission, matrilineal versus versus biparental, biparental, have have no no consequences consequences for for the direction direction of of evoluevolu-
340 340
Steven A. A. Frank Fronk Steven
tionary tionary change change favored favored by by selection. selection. For For example, example, efficient efficient respiration respiration increases increases both matrilineal matrilineal and and biparental biparental transmission. transmission. both The allocation allocation of of resources resources to to sons sons and and daughters daughters affects affects matrilineal matrilineal and and The biparental transmission transmission differently. differently. Traits Traits that that enhance enhance the the production production of of daughdaugh biparental ters at the expense expense of of sons always always increase increase the the transmission transmission of of matrilineally matrilineally inin ters herited genes. For For example, example, in some some hermaphroditic hermaphroditic plants the mitochondrial mitochondrial herited genes. plants the genes may inhibit inhibit pollen pollen development and and simultaneously enhance enhance the production production genes of seeds seeds (Edwardson, (Edwardson, 1970; 1 970; Hanson, 1991). 1 99 1 ). Selection of of genetic genetic variants variants in the the of mitochondria would would favor favor complete complete loss of of pollen pollen production production in exchange exchange for for a mitochondria increase in seed seed production production because because the the mitochondrial mitochondrial genes genes are transmitted transmitted small increase through seeds seeds (Lewis, 1941). 1 94 1 ). only through of resources resources from from pollen pollen to seeds seeds can can greatly reduce reduce the the Reallocation of transmission of of nuclear nuclear genes genes because because biparental biparental transmission transmission depends depends on the the transmission of the the success success through seeds and pollen. pollen. Thus Thus there conflict of of inin sum of through seeds there is a conflict terest between between the mitochondrial (cytoplasmic) and nuclear nuclear genes genes over over the the terest mitochondrial (cytoplasmic) of resources to male male (pollen) reproduction (Gouyon allocation of (pollen) and female (ovule) reproduction and 1 985; Frank, 1989). 1 989). Consistent with this idea of conflict, nuclear nuclear and Couvet, 1985; of conflict, genes often restore restore male fertility by overcoming the male-sterility effects genes often effects of of the the cytoplasm. cytoplasm. The nuclear- cytoplasmic conflict is very similar host-parasite system: The nuclear-cytoplasmic similar to a host-parasite reproduction, cytoplasmic ((parasite) parasite) there is antagonism over resources for for reproduction, genes determine determine the host-range for in for exploitation, and cytoplasmic genes interact with nuclear specificity teract nuclear ((host) host) resistance resistance genes to determine determine the specifi city of of the interinter inheritance influences the patterns patterns of action. Cytoplasmic inheritance of "parasite" "parasite" transmission population dynamics of hosthost but, on the whole, the the genetics genetics and and population dynamics are are typical typical of parasite interactions (Gouyon and Couvet, 11985; 985; Frank, 11989; 989; Gouyon et et al. al.,, 11991). 99 1 ). The reduction of pollen caused by cytoplasmic genes is called cytoplasmic reports of of CMS in 140 male sterility (CMS). Laser and Lersten ((1972) 1 972) list reports 1 40 species from from 47 genera across 20 families. More than one-half one-half of of these cases occurred occurred intraspecificc crosses, and the rest were naturally, about 20% were uncovered by intraspecifi observed in interspecific crosses. Moreover, this listing is an underestimate of the true extent of of CMS because detecting a cytoplasmic component to a male sterile phenotype requires genetic analysis of polymorphism (Frank, 11994a). 994a). Wild populations of of CMS maintain several several distinct distinct cytoplasmic genotypes (cytotypes). Each cytotype is capable of causing male sterility by an apparently different mechanism because each is susceptible to a particular particular subset of nuclear restorer alleles. Nuclear restorer alleles are typically polymorphic at several loci, with each allele specialized for for restoring pollen fertility when associated with particular cytotypes. The observations are summarized in Frank ((1989), 1 989), Couvet et 1 990), and Koelewijn l 995a,b). et al. al. ((1990), Koelewijn and Van Damme Damme ((1995a,b). CMS has reciprocal genetic specifi city of nucleus and specificity and cytoplasm and wide widespread polymorphism. The basic questions of dimensionality and colonization colonizationextinction dynamics are similar - parasite systems. What similar to those of other host host-parasite are the temporal and spatial patterns of female and hermaphrodite ((phenotype) phenotype)
1144
Spatial Processes - Parasite GeneTIcs Spatial Processesinin Host Host-Parasite Genetics
341 341
Summary Summary of of Available Available Evidence Evidence on on Number Number of of Cytoplasmic Cytoplasmic Genotypes Genotypes and and Nuclear Nuclear loci Loci That Have Been Been Detected Detected in Various Agricultural and Wild Wild Species That Have in Various Agricultural and Species~
TABLE II
a
Species Species
Agricultural speciesh species h Agricultural Beta Beta vulgaris vulgaris Daucus Daucus carota carota Helianthus Helianthus spp. spp. Nicotiana Nicotiana spp. spp. Oryza Oryza spp. spp. Solanum Solanum spp. spp. Triticum Triticum spp. spp. Zea Zea mays mays Wild Wild species species Beta Beta maritima' maritima' Origanum Origanum vulgared vulgare J Nemophila Nemophila menziesi( menziesii' Thymus vulgaris! vulgarisf Thymus Plantago Plantago lanceolala' lanceolata,~ Plantago Plantago coronopush coronopus h
Cross Cross type
Within Within Within Within Between Between Between Between Between Between Both Both Between Between Both B oth ~ Within Within Within Within Within Within Within Within Within Within
Molecular Molecular evidence evidence
Cytoplasmic Cytoplasmic genotypes genotypes
Nuclear genes
+ + + +
22 2 2 Many Many 88 22 4 4 22 3-4 3- 4
2-7 2-7 33 ?9 ?9 33 Many Many 2 2 5 5
2 2 2 2 2 2-Many 2-Many 2 2 22
?9 2- 7 2-7 2 2 ?9 3-5 3- 5 3- 5 3-5
+ + + + + + + + + + + +
+ + + + -
. . . . . . . . . .
""Copied Copied from l995a). from Koelewijn Koelewijn and and Van Van Damme D a m m e ((1995a). hh Compiled 1 985), Kaul 1 988). Compiled from from Hanson Hanson and and Conde Conde ((1985), Kaul ((1988). 1 987). ' Boutin Boutin et et al. ((1987). d 98 1 ). J Kheyr-Pour Kheyr-Pour (( 1980, 1980, 11981). Ganders ((1978). 'e Ganders 1 978). f Belhassen et 1 99 1 ). rBelhassen et al. al. ((1991). 'g Van Van Damme 1982), Van 1 983). D a m m e and and Van Van Delden Delden ((1982), Van Damme D a m m e ((1983). hh Koelewijn Koelewijn and 1995a,b). and van van Damme D a m m e ((1995a,b). <
frequencies, influ frequencies, allele frequencies, frequencies, and population sizes? What What is the relative influence of -extinction processes compared of colonization colonization-extinction compared with other ecological and and genetic processes? processes? What What measurable measurable properties can be used to infer infer process? As before, the data are not suffi cient to answer sufficient answer all these these questions, questions, but the literature literature provides intriguing -extinction dy intriguing hints about dimensionality and colonization colonization-extinction dynamics.
A. Dimensionality Dimensionality and Spatial Spatial Variation Variation Two par Two or more more different different cytoplasmic cytoplasmic genotypes genotypes may cause cause eMS CMS within a particular ticular species. species. The The cytoplasms cytoplasms are are recognized recognized as as distinct distinct because because they react react dif differently to particular nuclear nuclear restorer genotypes. The The dimensionality of of the system increases with the number number of of different different cytoplasmic types that cause cause male male sterility, increases each with its own associated set of of specifi specificc nuclear nuclear restorer restorer loci. Table Table II II summarizes summarizes data data on on the the dimensionality dimensionality of of agricultural agricultural and and wild wild spe species. The "cross type" type" describes describes whether whether variability was was discovered with intra intraspecifi specificc crosses crosses or with hybridizations hybridizations between between species. species. Molecular Molecular evidence evidence
342 342
Steven A. A. Frank Fronk Steven
matches different different mitochondrial mitochondrial markers markers to to genetic genetic and and phenotypic phenotypic properties properties matches observed observed in in crosses. crosses. The The "nuclear "nuclear genes" genes" column column lists lists the the total total number number of of loci loci involved in in male male sterility. sterility. Although Although the the existence existence of of nuclear-cytoplasmic nuclear-cytoplasmic specspec involved ificity is is clear, clear, the the details details are are very very difficult difficult to to work work out. out. The The numbers numbers must must be be ificity considered minimum estimates because because a cytoplasmic polymorphism can can be be dede considered minimum estimates cytoplasmic polymorphism tected only when when present present in a study that also also has has matching matching nuclear nuclear polymorphism polymorphism tected only study that for restoration. restoration. Similarly, nuclear nuclear polymorphism polymorphism requires requires matching matching cytoplasmic cytoplasmic for polymorphism. Each Each study study requires requires tedious tedious crosses crosses and nurturing of of many many progprog polymorphism. and nurturing eny to to draw draw unambiguous unambiguous conclusions. conclusions. As As mentioned mentioned above, above, CMS eMS is widespread. widespread. eny The table table shows shows only only those those studies studies in which which attempts attempts have have been been made made to to analyze analyze The the number number of of genotypes. genotypes. the The data data in Table Table III show show that that the frequency of of females females varies varies widely widely among among The the frequency popUlations of of the the same species. The The column column for for "genetics" describes describes how how inforinfor populations was obtained obtained on on the the spatial variation of of cytoplasmic and and nuclear nuclear genes. genes. mation was Plantago lanceolata lanceolata because because the the two two cytoplasmic cytoplasmic genogeno Evidence "direct" for for Plantago Evidence is "direct" associated with different morphological abnormalities of failed pollen types are are associated different morphological abnormalities of failed pollen production and and anther crosses were were performed meaproduction anther development. In addition, addition, crosses performed to to mea frequency of associated restorer restorer alleles. Spatial Spatial varivari sure the frequency of the cytoplasms and and associated lanceolata will be be discussed below. below. For For Beta maritima, crosses crosses were were ation in P. lanceolata performed to infer infer the frequency of of cytoplasmic types and restorer restorer alleles for each performed to for each appeared that cytoplasmic frequencies frequencies did not not vary between between the the population. It appeared two populations. popUlations. The frequency was was the result result of of varivari two The large large difference difference in female frequency ation frequency of of restorers restorers between between the two locations. ation in the frequency Spatial variation variation was inferred inferred from crosses between between different different populations populations in from crosses Thymus vulgaris and P. coronopus. coronopus. These These long-distance crosses yielded vulgaris and long-distance crosses yielded higher higher frequencies fre frequencies of of females than than were observed observed within each population. High frequencies population, restorers restorers quencies of females in the crosses crosses imply that, within each population, are common for for the locally common cytoplasm but relatively rare for for other other cy cytoplasms. different cytoplasms, then toplasms. If different different populations populations are dominated dominated by different then the
TABLE III III
Spatial Spatial Variation Variation in in Wild Wild Populations Populations with with Cytoplasmic Cytoplasmic Male Male Sterility Sterility~ a
N N
Study
Genetics Genetics
?? > > 50 50 8
1100 00 1110 10 27
Kheyr-Pour ((1980) 1980) Gouyon and Couvet ((1985) 1 985) Van Damme Damme and Van Delden ((1982) 1982)
Inferred Inferred Direct Direct
115 5 113 3 31 31
27 27 88 88 22
Koelewijn ((1993) 1 993)
Inferred
Boutin-Stadler et al. ((1989) 1 989)
Direct
Range
Median
Origanum vulgare Thymus vulgaris vulgaris Plantago lanceolata lanceolata Plantago
11-62 - 62 5-95 5-95 11-23 -23
With With IN IN Plantago coronopus coronopus With IN Beta maritima
11-34 - 34 0-35 0-35 113-61 3-61 119-62 9-62
Species
a
The second and third columns "The columns show the range and median median in percentage of females per per population population for for samples samples from from N popUlations. populations.
14 1 4 Spatial Spatial Processes Processes in Host-Parasite Host - Parasite Genetics
343 343
crosses will will expose expose cytoplasmic cytoplasmic genotypes genotypes from from the the female female parent parent to to nuclear nuclear crosses backgrounds of of the the male male parent parent that that have have aa low low frequency frequency of of matching matching restorers. restorers. backgrounds species, the the "IN" "IN" rows rows show show the the frequency frequency of ofpartially partially malemale For the the Plantago species, For sterile (IN) (IN) plants. plants. Partial Partial male male sterility sterility also also depends depends on on an an interaction interaction of of cytocyto sterile plasmic and and nuclear nuclear genes. As As noted noted by by Koelewijn Koelewijn (1993), ( 1 993), partial partial male male sterility sterility plasmic common phenomenon phenomenon in in CMS, CMS, but but phenotypes phenotypes are are often often reported reported with with didi is aa common chotomous classification. classification. This This is is similar similar to to the the partial partial resistance resistance that that is common common chotomous in plant-pathogen plant-pathogen interactions, interactions, as as noted noted in in the the previous previous section. section. In In both both CMS CMS in and plant-pathogen plant -pathogen interactions, interactions, the the intermediate intermediate phenotypes phenotypes often often depend depend on on and specific interactions interactions between between genetic genetic polymorphisms polymorphisms of of the the host host and and parasite. parasite. specific
Colonization - Extinction Dynamics Dynamics B. Colonization-Extinction The frequency frequency of of females females in in a population population is the the frequency frequency of of unrestored unrestored malemale The sterile cytoplasms. The data suggest that the frequency of females varies among sterile The data that the of females varies among variation appears be associated associated with widespread widespread genogeno populations. Phenotypic variation appears to be restorer frequencies. frequencies. typic variation in cytoplasmic types and restorer Two related related metapopulation metapopulation scenarios scenarios have have been been proposed to explain phe Two explain phevariation. The The first theory concerns concerns the the colonization-extinccolonization -extinc notypic and and genetic variation. tion dynamics of 985; of alleles among among existing popUlations populations (Gouyon and Couvet, 11985; Frank, 11989). 989). The second theory focuses on the colonizationextinction colonization-extinction dynamics of populations 985). I will briefl populations (Gouyon and Couvet, 11985). brieflyy outline the eld study that hints at how natural populations may allelic theory, along with a fi field be influenced by these processes. At the end of this section I mention the popupopu lation -extinction theory. lation colonization colonization-extinction theory. To understand the colonization - extinction dynamics of alleles one must colonization-extinction imagine imagine aa sequence sequence of of events. events. ((i) i ) Initially, one of the cytoplasmic type is lost from a local population. Loss may occur by drift or because the alternative types have higher fi tness. fitness. Increasing dimensionality (more types) raises the probability that one or more cytoplasms will be absent locally. al((ii) ii ) When a cytoplasmic type is absent, the associated nuclear restorer al leles do not have any benefi cial effects. These specifi beneficial specificc restorer alleles may be lost from the the local population by a variety variety of processes. If there are are no fitness differ differences between restorer and alternative nonrestorer alleles, then the restorers may ences restorer and alternative nonrestorer alleles, be lost by drift. If the restorers, which must in some way infl u ence pollen devel be lost drift. restorers, influence development, ciency when opment, reduce reduce effi efficiency when their their matching matching cytoplasm cytoplasm is absent, absent, then then the specifi c restorers will specific will be lost lost by selection. selection. ((iii) iii) After steps i ) and steps ((i) and (ii), (ii), aa cytoplasmic cytoplasmic genotype genotype and and its specific specific re restorers storers are are absent locally. locally. If an an unrestored unrestored cytoplasm cytoplasm arrives arrives by by immigration, immigration, it tness it will have have aa fitness fitness advantage advantage and and spread spread quickly quickly in in the the population. population. The The fifitness advantage advantage occurs occurs because because an an unrestored unrestored cytoplasm cytoplasm causes causes aa male-sterile male-sterile pheno phenotype. type. Male-sterile Male-sterile plants plants typically typically produce produce more more seeds seeds than than hermaphrodites hermaphrodites ((Lloyd, Lloyd, 11976; 976; Van 984; Van 984). Because Van Damme, Damme, 11984; Van Damme Damme and and Van Van Delden, Delden, 11984). Because
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Steven Fronk Steven A. A. Frank
cytoplasmic fitness depends depends only on success through through the maternal line (seeds) and not on pollen pollen success, the male-sterile male-sterile plants plants have greater cytoplasmic fitness than than hermaphrodite hermaphrodite plants. Thus Thus the cytoplasms cytoplasms that that cause male sterility spread in the local population, population, causing an increase in the frequency of of females. ((iv) iv) Cytoplasmic genotypes are essentially alternative alleles at a hap hapgeno loid locus. When When one genotype increases in frequency, then the other genotypes necessarily decline in frequency. In the case of of mitochondria, an increase in the frequency of mito of one mitochondrial type will cause a decline in other other mitochondrial chondrial types. Thus the selective spread of of an unrestored unrestored cytoplasm may cause the local extinction of of alternative cytoplasmic genotypes. Loss of of cyto cytoplasmic genotypes may be associated with loss of of matching restorers, restorers, as in step step (ii). (ii). (v) The The population now has a high frequency of of females and a dominant cytoplasmic genotype. The restorers restorers matching the dominant cytoplasm are locally extinct. If a matching matching restorer restorer arrives by immigration, it will combine with the dominant dominant cytoplasmic type to produce produce hermaphrodites. hermaphrodites. The restorer allele spreads spreads rapidly because pollen is rare rare locally, thus the few few hermaphrodites hermaphrodites are the source of paternal alleles for for all members of of the population. population. As the restorer restorer spreads, spreads, the frequency of of females declines. The frequency of of cytoplasmic genotypes may be unaffected by the initial spread of restorers. unaffected (vi (vi)) A locally absent cytoplasmic genotype can invade and spread if its spe specific restorers are absent. The cycle then repeats, with a genotypic turnover in the local population. -the number population. The greater the dimensionality dimensionality-the number of of cytoplasmic genotypes and the more and matching matching specific restorersrestorers-the more likely an immigrant cyto cytoplasmic type will be locally absent absent and can start a new round round of genotypic turn turnover. Van Damme 's study of P. lanceolala Damme's of P. lanceolata provides just enough enough detail to show show how parts of population. Van Van Damme of the above scenario may work in a natural population. Damme and Van 1 982) distinguished two cytoplasmic genotypes in P. lanceo/ala Van Delden Delden ((1982) P. lanceolata each with its own set of of nuclear restorers. restorers. Table IV shows phenotypic frequencies frequencies in 12 populations pop populations in two habitat groups; the original paper paper lists data data for for 27 poppopUlation are abbreviations ulations in five categories. The labels for for each population abbreviations for for locations. locations. The cytoplasmic genotype MS I when when genotype R causes the male-sterile phenotype MS1 unrestored unrestored and IN IN1I when partially restored. The cytoplasm P P causes MS2 when unrestored four types are unrestored and and IN2 when partially restored. All four are morphologically distinct and can be scored scored by direct direct examination. examination. Restored Restored cytoplasms of of either either type are hermaphroditic, H. The cytoplasmic type of of a hermaphrodite hermaphrodite can be determined only by crossing crossing until the cytoplasm is exposed exposed in an unrestored unrestored nuclear background. background. The two population population groups shown in Table IV are the most differentiated of of the five groups populations either groups listed in the original paper. Five of of the hayfield populations lacked the R cytoplasm or were fixed for for the R restorers. In the pasture pasture popula-
1144 TABLE TABLE IV IV
Population Population Hayfield Hayfield Dr Ze An Re Me Me l1 Ve Ve Br Br Pasture Pasture Wd Wd Bm2 Bm2 Pa Pa Ac2 Ac2 Ju
Spatial - Parasite Genetics Spatial Processes Processesinin Host Host-Parasite Genetics
345 345
Phenotype PhenotypePercentages Percentagesinin Natural Natural Populations Populationsof of Plantago Plantago lanceolataa lanceolata ~ MSI MS 1
IN! IN 1
MS2 MS2
IN2 IN2
H H
Sample size
0 0 0 0 0 0 0 0 0 112.2 2.2 23.0 23.0
0 0 0 0 0 0 0 0 2.2 2.2 7.0 7.0
0.2 0.2 5.0 5.0 8.2 8.2 5.0 5.0 3.9 3.9 0 0 0.3 0.3
0.2 0.2 0.8 0.8 11.3 .3 3.6 3.6 5.5 5.5 0.6 0.6 0.2 0.2
99.6 99.6 94. 94.11 90.5 90.5 9 1 .4 91.4 90.6 90.6 85.0 85.0 69.5 69.5
8811 11 742 742 754 695 695 688 688 623 623 601 601
4.6 4.6 7.3 7.6 7.6 111.8 1 .8 2 1 .5 21.5
0.6 0.6 3.9 3.9 7.8 7.8 110.8 0.8 7.0
0.5 0.5 0 0 0.5 0.5 0 0 0
0.9 0.9 11.3 .3 0.9 0.9 11.0 .0 0.5 0.5
93.4 87.5 87.5 83.2 83.2 76.4 76.4 7 1 .0 71.0
6902 6902 386 386 437 437 305 305 4 14 414
""From From Van 1982). Van Oamme Damme and and Van Van Oelden Delden ((1982).
tions, rare or -specific restorers tions, either either the the P P cytoplasm cytoplasm was was very very rare or the the P P-specific restorers were were com common. population groups groups were for MS mon. The The other other three three population were relatively relatively more more mixed mixed for MS 11 and and MS2 phenotypes. phenotypes. Van Damme 1 986) made variation within within the Van Damme ((1986) made an intensive study of of spatial variation the Westduinen (Wd) population listed listed in picture of the fi eld at West Westduinen (Wd) population in Table Table IV. IV. A A picture of the field at Westduinen Fig. 6, 6, with listed in V. Females were duinen is is shown shown in in Fig. with some some of of the the data data listed in Table Table V. Females were rare whole population, MS 11 more common than than MS2. rare over over the the whole population, with with MS more common MS2. However, However, larger in a few few locations locations the the frequency frequency of of MS MS1I was high high (Fig. 6). Within Within the the larger clusters -p4, the zero at clusters of of MS MS I1,, pJ pl-p4, the frequency frequency of of MS MS I1 phenotypes was was close close to zero at the borders borders and and rose to 60% 60% near the center. The eld as a whole MS2) cytoplasm, with The fi field whole was dominated dominated by the P P ((MS2) with an overall frequency frequency of of 0.94. The The frequencies frequencies of of the P-specific P-specific restorer restorer alleles alleles were also high. Thus also high. Thus most most plants plants were were hermaphrodites hermaphrodites with with aa P P cytoplasm cytoplasm and and P P restorers. restorers. The overall frequency frequency of of the R cytoplasm was 0.06, 0.06, and the R-specific restorers 0.02 and restorers at at the the two two restorer restorer loci loci had had frequencies frequencies of of 0.02 and 0.08. 0.08. Genotypic Genotypic composition composition was was very very different different in in those those few few areas areas that that had had high high frequencies V). The MS 1) I ) cytoplasm, frequencies of of the the MS MS 1I phenotype phenotype (Fig. (Fig. 6 6 and and Table Table V). The R R ((MS cytoplasm, rare in the population population as a whole, had had frequencies frequencies ranging ranging between between 26 26 and and 39% 39% in -p4. The restorers, also in populations populations pJ pl-p4. The R-specifi R-specificc restorers, also rare rare in in the the whole whole field, field, were were more cult more frequent frequent in the MS I1 clusters, clusters, although although the exact frequencies frequencies were diffi difficult to estimate. ' s interpretation Van Van Damme Damme's interpretation agrees agrees with with the the scenario scenario outlined outlined above. above. Initially Initially most eld was most of of the the fi field was dominated dominated by by P P cytoplasms cytoplasms and and P-specific P-specific restorers. restorers. R Rbearing MS I1 spots -specific restorers bearing colonists colonists founded founded the the MS spots and, and, since since the the R R-specific restorers were were initially rare, the MS I1 females spread from a central focus. MS plants produce produce MS 11 plants
346 346
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FIGURE 6 Distribution pasture of of areas areas with with MS1 IN1! plants plants of of P. lanceolala. lanceolata. Single FIGURE Distribution in a pasture MS! and and IN Single plants circles. The is the the plants are are represented represented by by dots dots and and groups groups of of plants plants by by circles. The number number within within each each circle circle is area covered by by the the local plants. The four largest are labeled labeled pl plarea in in square square meters meters covered local group group of of plants. The four largest groups groups are p4. The Redrawn from from Van 1 986). The shaded shaded areas areas are are pools pools that that cattle cattle use use for for water. water. Redrawn Van Damme Damme ((1986).
more hennaphrodites more seeds that that are are larger and and survive better than seeds from hermaphrodites ((Van Van Damme and Van Delden, 11984), 984), so the females have a competitive advan advanet al., al., 11986), welltage locally. Seeds disperse at a slow rate (8 cm/year; Bos et 986), thus well patches can fonn. form. As the frequency of of unrestored unrestored R cytoplasms cytoplasms rises in defined patches area, selection favors an increase increase in R-specific restorers. In an area area with a high an area, concentration concentration of of R cytoplasms, the main pollen donors donors will be R-restored R-restored her hermaphrodites. maphrodites. The low restorers in the overall population popUlation sug low frequency frequency of of the R-specific restorers suggests that that these these alleles are are at a selective disadvantage when when the the R cytoplasm is P cytoplasm is likely to lack absent. If so, then a population population dominated by the the P the R restorers, as at Westduinen. That genotypic composition is susceptible to invasion invasion by R cytoplasms, followed followed by a subsequent subsequent change change in genotypic com composition. position. rst lanceolata require require further further study. However, this fi first Many details details about about P. lanceolata glimpse does suggest that colonization -extinction dynamics and the strong secolonization-extinction
1144 TABLE V V
Spatial - Parasite Genetics Spatial Processes Processesinin Host Host-Parasite Genetics
347 347
Percentage Percentageof of Phenotypes Phenotypesof of Plantago Plantago lanceolata lanceolota at at aa Westduinen Westduinen Field Fielda a
Description
MSI MS1
INI IN1
MS2 MS2
IN2 IN2
MS3 MS3
H H
Sample Sample size
Total field pI pl p2 p3 p4 p4 Remainder Remainder
4.6 4.6 28.6 28.6 25.6 2 1 .3 21.3 22.7 11.9 .9
0.6 5.4 5.4 11.6 .6 3.0 11.6 .6 0.4
0.5 0 0 0 0.3 0.3 0 0 0.6 0.6
0.9 0 0 11.2 .2 4.7 4.7 0.9
0.5 0 0 0 0 0 0.5 0.5
92.9 66. 66.11 72.8 74.3 771.1 1.1 95.7
6902 1112 12 188 695 688 6 140 6140
a
MS3 is a rare rare phenotype phenotype controlled controlled by variation variation at autosomal autosomal loci. The locations locations of of populations populations p l1 -p4 are shown in Fig. 6. Data from Van 1 986). Van Damme Damme ((1986).
lective pressures pressures on cytoplasm cytoplasm and and nucleus may be responsible responsible for the observed spatial variability. The P. lanceolata lanceolata example example emphasizes emphasizes the the colonization and and spread of of a lo locally novel genotype into into an existing existing population. Cytoplasms may also "escape" "escape" their restorers when a empty patch patch is founded by one or a few few colonists (Gouyon and Couvet, 11985). 985). Species that are are subject subject to local population extinctions extinctions and colonizations of of empty habitat patches may be particularly variable variable in the fre frequency of of females females and the spatial variation in genotypes. Studies of of Thymus Thymus vulgaris vulgaris in southern France suggest that disturbed disturbed populations and recently col colonized patches undisturbed, patches are are likely to have higher higher frequencies frequencies of of females than than undisturbed, 1 989; Olivieri older populations (Gouyon and Couvet, 11985; 985; Belhassen et al al.,. 1989; and Gouyon, this volume). However, it is difficult difficult to obtain obtain convincing data data on processes that cover large temporal temporal and geographic scales. All studies do not find evidence evidence of a dynamic ( 1 993) dynamic process. Koelewijn Koelewijn (1993) summarized summarized his work on Plantago Plantago coronopus coronopus by noting that the frequency of of phenotypes in each of 0 years. of four locations was "remarkably constant" constant" over 110 The evidence evidence also suggested that both cytoplasmic male-sterile male-sterile genotypes oc occurred at intermediate 1 993) com intermediate frequency in all four locations. Koelewijn's Koelewijn's ((1993) comments serve as a reminder reminder that we have only the haziest picture picture of a few few cases, empirical guidelines about about appropriate appropriate temporal and spatial scales at with no empirical which nonequilibrium fluctuations may be important. .
VI. OTHER OF HIGH VI. OTHERSYSTEMS SYSTEMSOF HIGH DIMENSION DIMENSION II have discussed dimensionality and spatial variation for plant diseases and cytoplasmic male male sterility. Are these systems unusual, unusual, or are other host-parasite host-parasite interactions similarly diverse? diverse? There There are are very very few few systems systems with with good good data data available available on on both both the the host host and parasite. parasite. II briefly describe a few cases to illustrate the kind of of information that has been collected. collected. My interpretation interpretation is that high dimensionality occurs often, there will certainly be many exceptions. although there
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Steven Steven A. A. Frank Frank
Bacteria have a simple recognition-based recognition-based immunity system that that protects them from invasion by foreign 1 99 1 ). There are two com foreign DNA (Wilson and Murray, 1991). components ponents to the system. Restriction enzymes cut DNA DNA molecules that that carry a par particular sequence of of nucleotides. Modification enzymes recognize the same nucle nucleotide sequence sequence but, instead instead of of cutting the DNA, DNA, these enzymes modify the recognition site in a way that protects that molecule from from restriction. A bacterial ' cell s own DNA is modifi e d, otherwise the restriction enzymes would cut the cell's DNA modified, DNA and kill the cell. Restriction-modifi cation ((RM) RM ) enzymes are known Restriction-modification known for over 200 different different recognition 990; Roberts, 990). Circumstantial evi recognition sites (Kessler (Kessler and Manta, 11990; Roberts, 11990). evidence dence suggests that defense against bacteriophage bacteriophage viruses has been been a power powerful force promoting 1 ) RM promoting diversity. ((1) RM can protect protect host cells from invading phage 952; Arber, 965). (2) Phage phage (Luria and Human, 11952; Arber, 11965). Phage that that develop in bacteria bacteria with a particular ed for particular RM type are modifi modified for the associated recognition recognition sequence. These modifi ed phage modified phage can attack attack other bacteria bacteria of of the same RM type, but are sensitive to restriction by different RM RM systems. Rare RM types are favored be because few phage ed for phage will be modifi modified for their recognition recognition sequence. This frequency frequencydependent dependent selection promotes promotes diversity of of RM RM as a defense against against phage phage (Levin, 11986, 986, 11988). 988). (3) Phage Phage carry a variety of antirestriction mechanisms (Kruger (Kruger and Bickle, 1983; Sharp, 11986; 986; Korona 993). For example, many phage lack Korona et et al., al., 11993). particular RM RM recognition sequences. sequences. The probability of of having these recognition particular sequences is very high if no selective pressure were acting on sequence compo composition. These These details suggest that the interaction interaction between RM and phage phage is of of high dimension. Not enough enough sampling has been done to draw draw any conclusions conclusions about spatial variation, but it seems likely that the genotypic composition of of commu communities varies widely among among different different locations. locations. A second second bacterial system acts in a very different way from RM but is also highly diverse. This allelopathic system affects competition between between bacterial strains rather How rather than what is usually thought thought of of as a host-parasite host-parasite interaction. Howcity of attack and defense promotes ever, the genetic specifi specificity promotes widespread widespread poly polymorphism -parasite dynamics. morphism in much the same way as in host host-parasite In this system of of bacterial allelopathy, cells often carry plasmids that that encode encode a bacterial toxin ((bacteriocin) bacteriocin) and immunity to that toxin ((Reeves, Reeves, 1972; 1 972; Lewin, 11977; 977; Hardy, 11975). 975). Immunity works works by neutralizing the toxin after it has entered the cell. Bacteria may also be resistant to bacteriocins bacteriocins because they lack a com compatible patible receptor receptor through which which the toxin can enter enter the cell. Many distinct distinct bacteriocin types are are found found within a population. A type is defi ned by its susceptibility to a set of defined of toxin-producing toxin-producing test test strains. With With n n test strains, there bac there are 2" possible types. Epidemiological studies frequently use bacteriocin typing to identify and follow pathogenic strains of of bacteria. These These studies provide information about suscepti about the diversity of of bacteriocin bacteriocin production production and and susceptibility in populations. For example, Chhibber 1 988) summarize data Chhibber et et al. al. ((1988) data on the number 0 studies of of number of of isolates, test strains, and bacteriocin susceptibilities for for 110
1144
Spatial Processes Parasite Genetics Spatial Processesinin HostHost-Parasite Genetics
349
Klebsiella Klebsiella pneumoniae. pneumoniae. The fewest number number of of observed types occurred occurred in a study 200 isolates, four four test strains, and 1111 types of of a possible 24 2 4 = - - 16; with 200 1 6; the most occurred isolates, seven of a possible occurred in in aa study study with with 553 553 isolates, seven test test strains, strains, and and 64 types types of a possible 27 1128. 28. Similar levels of 2 7 = of diversity have been reported reported for for a variety of of species (Gaston et al. al.,, 11989; Senior and V6r6s, Rocha and and de Uzeda, 11990; Traub, (Gaston 989; Senior Voros, 11989; 989; Rocha 990; Traub, 11991; 99 1 ; Riley and Gordon, 992). Gordon, 11992). The The next example example is disease resistance in vertebrates. There is great diversity at the loci that encode specific recognition that encode recognition of of parasites, the major major histocompati histocompatiMHC) genes. However, bility complex ((MHC) However, this host diversity has rarely been been matched to specifi c polymorphisms of parasites. specific of The molecules that bind intracellular protein fragments and bring them to protein fragments the surface are coded by genes that reside within the MHC region. Each coded reside Each antigen antigenpresenting molecule from the MHC has a groove that accommodates nine amino acids. Each present on the cell sur Each particular particular MHC molecule can recognize recognize and present surface only a subset of of protein fragments. An individual has several different MHC types that, taken together, determine the set of of protein fragments that can be recognized recognized and carried to the cell surface surface for for presentation. The 0 and 80 different The MHC MHC loci are highly polymorphic, with between 110 alleles known for for each locus. Two Two lines of of evidence evidence suggest that resistance to particular particular diseases can strongly affect the frequency of of MHC MHC alleles. First, most of the variation variation among alleles occurs occurs in the groove that that binds protein protein fragments fragmentsof the specific recognition recognition area. Second, Second, a few cases are known known in which there is a strong spatial correlation between endemic diseases and MHC alleles that are associated with resistance to those diseases. For For example, the allele HLA-B53 HLA-B53 is associated with resistance to a severe strain strain of of malaria malaria that that oc occurs in children children in The Gambia. HLA-B53 HLA-B53 occurs occurs at a frequency of of 25% in this west African nation; by contrast, 1% contrast, the frequency of of this allele in Europe Europe is 1% ((McMichael, McMichael, 1993). 1 993). Other Other MHC MHC alleles are implicated in resistance to HIV, the -Barr virus, the cause of cause of of AIDS, AIDS, and to Epstein Epstein-Barr of various cancers. Disease correlations uence the ev correlations with MHC MHC alleles suggest that that selective pressures pressures infl influence evolution isms (Thomson, 99 1 ; Mitchison, olution of of the immune immune system polymorph polymorphisms (Thomson, 11991; 11993). 993). concerns genetic variation in plant resistance to herbivores. herbivores. The final example concerns The resistance 1 992) lists 37 studies resistance may be biochemical or or structural. Karban ((1992) that that show show evidence evidence of of genetic genetic variability in resistance resistance to herbivore herbivore attack. attack. These These studies usually demonstrate genetic variability by growing different plant geno genotypes in a common environment and measuring variation in herbivore herbivore damage. The number of The details of of variable resistance and the number of independent independent traits involved involved (dimensionality) are typically unknown. unknown. Insect herbivores herbivores are are often genetically variable in their ability to attack dif different plant varieties (Gould, 983). Edmunds Edmunds and Alstad ((1978) 1 978) suggested that (Gould, 11983). insect species often differentiate into populations populations that are locally adapted for for the Karban ((1992) 1 992) summarizes studies that examine host genotypes in their area. Karban geographic herbivores. He concludes that the data geographic specialization of of insect herbivores. data are are not
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Steven Steven A. A. Fronk Frank
convincing because of limited sampling, but the hypothesis that herbivores herbivores are geographically specialized specialized remains an important idea that deserves further study.
VII. VII. THEORIES THEORIESAND AND TESTS TESTS host-para The evidence summarized in the previous sections suggests that host-parapopula site interactions interactions can be very diverse. The few careful studies of of natural populations indicate indicate a spatial component of diversity when the system is viewed on the appropriate spatial scale. In this final section I review the processes that that can explain spatial variation in host-parasite host-parasite allele frequencies. I then summarize the plantpathogen and cytoplasmic male sterility studies in light of the alternative plant-pathogen alternative explanations explanations for for spatial variation. Five factors may influence spatial variation variation in host -parasite genetics. host-parasite ((i) i ) Migration-drift Migration-drift dynamics occur when selection is a relatively weak force and allele frequencies frequencies fluctuate fluctuate stochastically. stochastically. Locally extinct alleles can return return to a population population by immigration if populations populations are are connected connected in a metapop metapopulation. Drift Drift is relatively more important than selection in causing fluctuations when local populations are are small or have frequent bottlenecks. Migration Migration can se overcome selection when the movement of alleles occurs more quickly than selection can change local allele frequencies. ((ii) ii ) Local, nonlinear dynamics cause spatial spatial variation when populations fl uctuate in an uncoupled manner. fluctuate manner. Selection causes changes in allele frequencies, frequencies, of local dynamics. Migration and migration does not cause major perturbations of ciently rare to prevent synchronization of uctuations. must be suffi sufficiently of population fl fluctuations. ((iii) iii) Environmental Environmental heterogeneity can favor different allelic combinations in particular particular locations. This will be particularly important in inbreeding or asexual host-para species, where where chance chance linkage will occur between alleles involved in host-parasite interactions and alleles affecting success in different habitats. determine the role of of linkage in ((iv) iv) The sexual system will, in general, determine changing allele frequencies. frequencies. With With low recombination, selection at one host-para host-parasite locus can change allele frequencies frequencies at many other loci. caused by selection coupled coupled with with global migration lead (v) Local extinctions caused to colonization - selection -extinction dynamics. These are the processes that I colonization-selection-extinction emphasized in my descriptions of of plant plant-pathogen CMS systems. In this case emphasized -pathogen and eMS immigration of of locally extinct aIleles alleles will sometimes cause a major perturbation of of local dynamics. Spatial variation may be dominated by the timing of of local aIleles and the waiting time until those alleles are reintroduced by extinctions of of alleles immigration. The The complicated details details of nonlinear dynamics dynamics (limit cycles, chaos, etc.) may be relatively unimportant in systems of of high dimension because the timing of of extinctions and colonizations determines local and regional variation. These five processes can all occur in a single system when measured measured over
1144
Spatial Spatial Processes Processesinin Host Host-- Parasite Parasite Genetics Genetics
3S 1 351
different spatial and temporal scales. For example, example, in the Burdon Burdon and Jarosz Jarosz ((1992) 1 992) study of of a plant-pathogen plant-pathogen system, the evidence suggested that migration migrationof pathogen genotypes over drift dynamics dominated the spatial distribution of experience frequent approximately 75 km. Local pathogen populations apparently experience immigration or recolonization from neighboring neighboring populations. bottlenecks, with immigration The movement of of pathogen alleles occurs on a time scale that that is shorter than than host generation time, suggesting that that migration is more more powerful than than coevolutionary selection pressures in determining the spatial dynamics of of pathogen pathogen allele fre frequencies. quencies. The rate rate of pathogen migration will be low at some sufficiently long spatial scale. The colonization of of a region by a long-distance migrant allele could cause a major local perturbation. For example, 00 years example, once in 1100 years a locally novel re resistance allele may land land in a region, changing changing the selective pressures pressures on the path pathogens and favoring the immigration of new host-range alleles. These perturbations may be rare compared to the usual scale of study, but could be a major cause of of regional variation. variation. Other Other systems, such as eMS CMS in P. lanceolala, lanceolata, may have rel relatively low rates selection rates of of migration migration over short short distances. distances. Thus colonizationcolonization-selectionextinction dynamics dynamics may occur over smaller smaller scales that are easier easier to study (Van Damme, 11986). 986). Problems of of inference inference can can be severe. On the the measurement measurement side, side, polymor polymorphism in coevolutionary systems can be difficult difficult to detect Frank, 1994a). 1 994a). For detect ((Frank, For example, two different male sterile cytoplasms both yield the same hermaphro hermaphroditic phenotype when the study sample contains matching nuclear nuclear loci that are fixed for fixed for restorer alleles. Thus the potential diversity (dimensionality) of of systems is diffi c ult to measure. On the statistical side, very different processes may yield difficult statistical different the same patterns of hostparasite polymorphism when the observer uses a parpar same of host-parasite when observer ticular sampling scheme. For example, drift models and strong selective, selective, coevo coevolutionary models often often have similar patterns patterns when when sampled sampled without long time timeseries data ((Frank, Frank, 11996). 996). The The only remedy remedy is thorough understanding understanding of of both the patterns expected under alternative alternative processes and and the consequences consequences of of differ different sampling schemes and methods of of data analysis. Two standards standards of of empirical empirical progress will help. help. First, manipulation experi experiments in the field are an easy way to discover 1 985) discover spatial variation. Parker's Parker's ((1985) study of a plant -pathogen system is an excellent example of plant-pathogen excellent example of how transplant transplant experiments can be used to document document the extent and scale of of genetic variability. Van 1 986) did not Van Damme Damme ((1986) not manipulate his populations of of cytoplasmically male sterile plants. However, it is easy to imagine an experiment experiment in which locally absent absent cytoplasms or restorers restorers are introduced into fields in which those alleles are are extinct. Then, over several years, the natural spread of of the alleles could be monitored. The second avenue of of progress will come come from molecular molecular methods of of sam samed by laborious methods pling. At present, host and parasite genotypes are identifi identified of of genetic crossing experiments and phenotypic testing. The time required se severely restricts Molecular probes will eventually restricts the scope of of data data collection. collection. Molecular
352 352
StevenA. A. Fronk Fr0nk Steven
widespread sampling sampling of of host and and parasite parasite genotypes over different spatial spatial allow widespread temporal scales. The preliminary preliminary work work on plant plant diseases diseases and and male male sterility sterility and temporal host-parasite systems will be highly highly variable variable and strongly strongly influ influsuggests that host-parasite enced by metapopulation metapopulation dynamics. dynamics. enced
ACKNOWLEDGMENTS ACKNOWLEDGMENTS thank P. Amarasekare, Amarasekare, R. M. Bush, Bush, and D. D. R. Campbell Campbell for helpful helpful comments. comments. My research research I thank supported by NSF NSF Grant Grant DEB-905733 DEB-9057331I and NIH Grants GM42403 GM42403 and GRSG-S07-RR0700S. GRSG-S07-RR07008. is supported NIH Grants
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IV CASE CASESTUDIES STUDIES
Empirical metapopulation 99 1 , at metapopulation studies studies were were scarce in 11991, of publication publication of of our our previous volume on metapopu metapoputhe time of lation dynamics. dynamics. Today, Today, the situation is changing changing rapidly, with valuable fi field studies appearing appearing monthly monthly if not We diseld studies not weekly. We dis though there tinguish between between two two kinds kinds of of empirical studies, though is also also substantial substantial overlap: studies studies driven driven by a desire desire to test theories and and concepts concepts and and studies studies motivated motivated by conservation conservation theories management management questions. questions. The The former former metapopulations metapopulations are are more more likely to be at equilibrium equilibrium and to fi fitt the classical metapopulation classical metapopulation ideas. The The latter latter metapopulations, metapopulations, most most often created created by human human fragmentation fragmentation of of the landscape, landscape, may may not not be at equilibrium equilibrium and often have have other features features that make them problematic problematic from from the perspective perspective of of testing theory. The The studies studies we we have chosen chosen to feature feature in this section section are are entirely of of the the former former type, that that is, directly oriented oriented toward toward testing theory theory and and elucidating elucidating con concepts. This This is not not to say that metapopulation consermetapopulation studies in conser vation vation biology would not ultimately prove prove equally rewarding rewarding scientifically, if for for no no other other reason reason than than that they are often backed backed by significantly better better funding funding than purely academic academic research. research. One One obstacle at present present is that much much of of the conserconser-
vation-oriented research on metapopulations metapopulations is reported reported in the gray literature. The four case studies in this section represent represent four taxa: butterflies (Thomas (Thomas and Hanski), mammals (Smith and Gilpin), plants (Giles and -parasitoid and Goudet), Goudet), and and plant-herbivore plant-herbivore-parasitoid metapopulations (van der Meijden and van der Veen-van Wijk). The in The focus varies from single-species studies to interspecific interspecific interactions genet teractions and from population dynamics to population genetics. The mechanisms of extinction and colonization in these metapopulations metapopulations are diverse. Especially in Europe, the taxon which is presently receiving the greatest attention is butterflies, which of metapopulation which now now play a role role in the the development of theory similar to that played by birds in the development of of the dynamic theory of of island biogeography. Butterflies Butterflies are attrac attractive study animals not only because they fly in nice places on sunny days, but also because many species in many landscapes landscapes are structured into proper metapopulations of of many but small local populations and and because because butterflies present many many advan advantages for field studies. Furthermore, many butterflies especially for field in northern northern Europe Europe have greatly declined declined over the past decades, apparently because the amount amount of of suitable suitable habitat habitat has dimin diminished, and there there is therefore much conservation interest interest in but butterflies. Other Other taxa taxa that that have been been prominent in metapopulation studies include birds and small mammals on true and habitat habitat islands, frogs in their naturally patchy environments, and many other insect species apart patch-oriented apart from from butterflies. The patch-oriented metapopulation approach approach is apparently apparently suitable for for only a small number number of plant plant species, mostly short-lived species in sparse vegetation, as exemplified exemplified by the chapter chapter of of van der Meijden Meijden and van der Veen-van Wijk. Returning to butterflies, they have produced much empir empirical data data because it is relatively easy to define, for many though not all species, what is suitable habitat, often based on the prespres ence of one or two relatively scarce and patchily distributed host plants. It is easy to delineate habitat patches patches for for many butterflies, it is possible to map large areas areas for for suitable habitat and the presence of of local populations, and it is convenient to study many aspects of of population biology of butterflies, butterflies, includ including individual movement behavior. So far, far, the work on butter butterrestricted to ecology and metapopulation flies has been largely restricted dynamics (Thomas and Hanski), but very soon we should have related related studies on metapopulation genetics and life-history life-history ev evolution. The quality of of the empirical data data has been sufficient to
parameterize parameterize metapopulation models, and model predictions have been successfully tested in the field. Nonetheless, Nonetheless, even with all these virtues of butterflies, not all is as clear as might first appear. appear. The causes and mechanisms mechanisms of population Extinctions are population extinction is a case in point. Extinctions are often often simply attributed to environmental stochasticity, but exactly what is "environmental "environmental stochasticity"? stochasticity"? The standard answer answer is: Changing environmental conditions affecting many individuals population in similar manner. manner. A drought drought causing mass mor morin a population tality of of caterpillars is environmental stochasticity, but if all plants happen to die, then we have a "deterministic" "deterministic" extinction. Yet, if conditions improve by next year, was it not, after all, a minor catastrophe, extreme environmental stochasticity? Or take small populations, which go extinct because small popu populations in small habitat patches have a high risk of of stochastic extinction. y away extinction. Butterflies may, however, be very likely to fl fly from very small patches, with large perimeter to surface surface ratio, and hence popUlations in small patches may go deterministi hence populations deterministically extinct due to emigration losses. Thomas and Hanski discuss such ecological issues with the large data base that butterflies now provide. As with spatial popUlation population structures, the processes of population extinction and establishment establishment are more diverse than the models might imply. Though at some level the details may not matter matter very much, the reward reward to an ecologist is a mechanistic understanding of what is actually happening. The mammalian case study by Smith and Gilpin is on the American pika, a metapopulation that Smith has followed for more than 20 years. Habitat patches consist of rock tailings from previous mining operations. operations. These patches are of the same quality and they are located located in the midst of of a practically uniform sagebrush vegetation, which is entirely unsuitable for pikas. Local populations are small, mostly less than 2 animals, with than 112 extinction largely attributable attributable to demographic stochasticity. Movements among the patches patches are infrequent, as shown by both behavioral and genetic studies. This fragmented landscape and the pika metapopulation living in it satisfy closely the assump assumptions of the classical metapopulation metapopulation concept, and the pika metapopulation does indeed exhibit classical metapopulation dynamics with population turnover, effect of patch size on ex extinctions and effect of isolation on colonizations. A spatially correlated correlated pattern of patch occupancies has evolved over the past 20 years, possibly driven by regional stochasticity, in other
words by by year-to-year year-to-year variation variation in in the the extinction extinction and and colonicoloni words zation rates. zation rates. In In the the pika pika metaapopulation, metaapopulation, mustelid mustelid predators predators may may sigsig nificantly increase increase the the risk risk of of local local extinction. extinction. Van Van der der Meijden Meijden nificantly and van van der der Veen-van Veen-van Wijk Wijk describe describe another another 20-year 20-year field field and study, study, in in which which interspecific interspecific interactions interactions are are evidently evidently critically critically important: important: the the ragwort ragwort plant plant on on aa Dune Dune area area in in the the Netherlands, Netherlands, its specialist specialist herbivore herbivore the the cinnabar cinnabar moth, moth, and and aa braconid braconid parpar its asitoid attacking attacking the the latter. latter. In In contrast contrast to to the the pika pika metapopulametapopula asitoid tion, among-patch among-patch movements movements are are common common in the the ragwortragwort tion, cinnabar moth-parasitoid moth -parasitoid system, with with the the moth moth apparently be cinnabar apparently being ing the most most dispersive dispersive species species of of all. all. Movement Movement patterns patterns go a long way in explaining the the dramatic dramatic spatial spatial and and temporal temporal patpat terns in plant plant biomass and herbivore herbivore numbers numbers that that van van der der terns Meijden Meijden and and van van der der Veen-van Veen-van Wijk Wijk have have observed. observed. What What they they describe is not a classical classical metapopulation metapopulation scenario, scenario, but but spatial spatial describe structuring of populations, which which nonetheless nonetheless has critical concon of populations, The tempting tempting (but possibly difficult) difficult) sequences for for dynamics. The experiment of ragwort experiment would be to create create one very large patch of in this system, and observe the dynamics of of the three three species species in the absence absence of of habitat Meijden and habitat subdivision. As van der der Meijden van der Veen-van Wijk point out, the outcome is difficult difficult to predict, partly because, in the absence of habitat fragmentation, the poorly dispersing parasitoid might interact interact much more strongly with the moth and fundamentally change the now so dramatic -herbivore oscillations. dramatic plant plant-herbivore geThe fourth case study by Giles and Goudet is entirely ge netical. Their study is focused on a structured metapopulation in which patch age is known and in which the history of local population size can be estimated. They study a plant species on a system of approximately 50 islands off off the coast of Sweden. population undergoes colonization and growth, fol folThe plant popUlation lowed by a slow decline due to successional factors; otherwise there there is no significant turnover in this system. The key question extinction-colonization which they ask is whether extinctioncolonization dynamics in a metapopulation tend to increase increase genetic differentiation among populations relative to the situation where there there is no population turnover. Using electrophoretic electrophoretic data, Giles and and Goudet estimate F ST for FST for various age and size classes of islands and and popUlations, populations, and and they find significant significant genetic structuring structuring both both within and and between islands. They also nd that ST structure depends also fifind that the F FST depends patch size size and and patch age in ways that accord accord with theoretical theoretical on patch predictions. predictions. Their Their results results underscore underscore the the conclusion conclusion of of Hedrick Hedrick
and Gilpin that a pattern of F ST variation can be produced by Fsv different combinations of metapopulation parameters, in this this and ef efcase propagule size, colonization frequency, gene flow and fective population sizes. This chapter points to the next gen generation of metapopulation fi eld studies, where the ecology, ge field geare investigated in a netics, and ultimately evolution of species are natural landscape.
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Butterfly Metapopulations Butterfly Metapopulations Chris Chris D. Thomas Thomas
IIkka Ilkka Hanski
I. INTRODUmON INTRODUCTION Some taxa have played a disproportionate disproportionate role in the development of of eco ecological concepts and theories. The dynamic theory Mac theory of of island biogeography biogeography ((MacArthur and Wilson, 11967) Arthur 967) was illuminated by numerous examples on birds, which are sufficiently well known even on remote oceanic islands. At present, we feel that butterflies have gained a somewhat similar status in the study of of metapopu metapopulation dynamics, especially in Europe Hanski, 11996b). 996b). There Europe ((Hanski, There are several reasons reasons for manner that for this. First, butterfly populations are often often structured in space in a manner is broadly consistent consistent with the metapopulation metapopulation concept. Second, Second, the ecology of of butterflies is well known known in most countries in Europe Europe and elsewhere. Third, Third, an exceptionally large large fractions of of butterflies, in northern northern Europe Europe in particular, particular, have declined, become endangered, endangered, or gone gone extinct already. Athough Athough the examples used in this chapter chapter exclusively concern butterfly metapopulations, metapopulations, we expect many of wider of the patterns patterns and processes processes described described to be applicable to a much much wider range range of of organisms. Despite a great deal notable conservation deal of of effort and and some some notable conservation successes, at attempts to conserve rare butterflies at the scale of of entire countries have largely failed ((New, New, 1991; 1 99 1 ; New et 995; Pullin, 1995). 1 995). In Britain, despite extensive et al. al.,, 11995; knowledge of of the ecological requirements requirements of of individual species, rates rates of of popupopuMelapO[Julatioll Metapoptdation Biology Biology Copyright 997 by Academic reproduction in any form fonn reserved. Copyright © 9 11997 Academic Press. Press, Inc. All rights of of reproduction reserved.
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Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski
lation extinction have been nearly as high on nature reserves as in the rest of of the 984; J. A. Thomas, Thomas, 1984, 1 984, 1991; 1 99 1 ; Emmet and Heath, et al. al.,, 11984; landscape (Heath, et 11990; 990; New, 11991; 99 1 ; New, et 995; Warren, 1992, 1 992, 1993). 1 993). The et at. al.,, 11995; The traditional con conservation approach approach has been to protect protect and and manage manage habitat habitat in isolated reserves as of the landscape effectively as possible, but it now now appears appears that the fate fate of of the rest of landscape may play an important populations important role in the long-term maintenance maintenance of of butterfly populations within protected areas. As a result, the metapopulation approach approach has been widely adopted by butterfl y ecologists and conservationists, and adopted butterfly and conservationists, and has spawned spawned studies on metapopulations butterfly species. In this chapter, we review the metapopulations of of a number number of of butterfly ecological insights that have been gained from from these studies. The The conservation conservation implications have been reviewed elsewhere (C. (c. D. Thomas, 1994a,b,c; 1 994a,b,c; New, New, et 1 995). et al. al.,, 1995). We commence by describing describing the general features of of spatial population population struc strucWe ture in butterflies, and the effects of landscape structure on butterfly populations. of landscape populations. The two critical elements in the metapopulation metapopulation framework are the effects of of The patch area area and and isolation on the distribution distribution of of species. We then then tum turn to the habitat patch two key metapopulation processes, processes, extinction and colonization, colonization, which are better better studied for for butterflies than than perhaps perhaps for for any other other comparable taxon. The next section discusses discusses empirical tests of of theoretical predictions, another another area where butterflyy studies have played a critical role in recent years. With rapidly accubutterfl accu mulating knowledge, knowledge, certain model assumptions appear appear increasingly suspicious, suspicious, and particulars that are now missing from and we have have a long long section on various various particulars metapopulation models but should probably be included in the next generation generation of models. This leads to the final section of of conclusions conclusions and what the future may of look like.
II. CONSEQUENCES CONSEQUENCESOF OF LANDSCAPE LANDSCAPESTRUaURE STRUCTUREFOR FOR POPULATION POPULATIONSTRUaURE STRUCTURE A. A. Population PopulationStructure Structure More than (Ford, 1945; Ford than 50 years ago, E. B. Ford Ford and colleagues (Ford, Ford and Ford, 11930; 930; Dowdeswell et 1 940) recognized et at. al.,, 1940) recognized that most individuals of of many patch), but that a few inin species of of butterfly remain within their natal habitat ((patch), dividuals may stray some kilometers from the breeding breeding popUlations, populations, and could of new new local populations. Ehrlich and col colbe influential influential in the establishment of populations. Ehrlich leagues developed this understanding understanding in a long-term study of of three three local popu populations of of the checkerspot checkerspot Euphydryas Euphydryas editha editha bayensis bayensis on a serpentine outcrop outcrop at Jasper 1 984; Ehrlich 975; Ehrlich Jasper Ridge, in California (Ehrlich, (Ehrlich, 1984; Ehrlich et et at. al.,, 11975; Ehrlich and and Mur Murphy, 11987; 987; Singer and Ehrlich, 1979). 1 979). Mark - release - recapture experiments Mark-release-recapture showed that that 97% of of individuals remained within the local population population in which first marked and and where most of of them them must have emerged. The The smallest they were fi rst marked local popUlation population became extinct, but the level of of exchange exchange among among populations populations was adequate adequate for for this site to be recolonized recolonized rapidly from the surviving local
15 1 5 Butterfly Butterfly Metapopulations Metapopulations
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populations. Subsequent Subsequent work work on on E. E. editha editha bayensis bayensis on on other other serpentine serpentine outcrops outcrops populations. confirmed that that small small local local populations populations are are most most susceptible susceptible to to extinction extinction and and confirmed revealed that that potential potential breeding breeding areas areas of of serpentine serpentine grassland grassland have have aa good good chance chance revealed of becoming becoming recolonized recolonized if if they they are are within within 5 kkm of an an existing existing large large local local poppop of m of et al., at., 1988; 1 988; Harrison, Harrison, 1989; 1 989; Weiss Weiss et et al., at. , 1988). 1 988). ulation (Harrison ( Harrison et ulation The foregoing foregoing research research programs programs were were conducted conducted or or initiated initiated before before Levins Levins The ( 1 970) coined coined the the term term "metapopulation," "metapopulation," and and before before metapopulation metapopulation ideas ideas bebe (1970) came popular popular in in the the late late 1980s 1 980s (Hanski ( Hanski and and Gilpin, Gilpin, 1991). 1 99 1 ). Nonetheless, Nonetheless, the the auau came thors of of these these early early studies studies recognized recognized the the same same components components of of population population strucstruc thors ture which are are now now part part of of the the metapopulation meta population concept: concept: distinct distinct breeding breeding areas areas ture ( habitat patches) patches) within within which which large large numbers numbers of of adult butterflies butterflies spend spend their entire entire (habitat life, with with limited limited migration between between local local populations. populations. About About three-quarters three-quarters of of butterflies and and 60% of of Finnish Finnish butterflies butterflies appear appear to to have have aa population population British butterflies structure of this this general general type, and and most most endangered that have have been been studied studied structure of endangered species that detail conform pattern (Arnold, (Arnold, 1983; 1 983; New New et et al., at., 1995; 1 995; Hanski Hanski and and in detail conform to this pattern 1 995; C. D. Thomas, 1 996). Local Local extinctions and and colonizations colonizations have have Kuussaari, Thomas, 1996). Kuussaari, 1995; been observed observed in most most studies of metapopulations, but but we do do not not regard regard been of butterfl butterflyy metapopulations, prerequisite before before using using the term metapopulation metapopulation to describe describe a particular particular this as a prerequisite ( Hanski and and Kuussaari, Kuussaari, 1995); 1 995); nor nor do do we wish to imply that local extincextinc system (Hanski tions and and colonizations rare in many tions colonizations are are necessarily in balance, as this is probably probably rare Harrison, 11991, 99 1 , 1994b; 1 994b; Hanski, 1994b, 1 994b, 1996b, 1 996b, this volume; modem landscapes landscapes ((Harrison, at., 11996b; 996b; C. D. Thomas, 994a,b,c). Butterfly metapopulations metapopulations come Hanski et et al., Thomas, 11994a,b,c). in all sorts of of shapes and sizes, but the term term metapopulation metapopulation is nonetheless nonetheless helpful in that it focuses focuses the attention of of researchers researchers and conservation conservation managers managers on pro processes at regional scales. Despite a wide range of of specifi specificc differences differences among metapopulations, we are encouraged that a rather small number of patterns and processes appear to be important in many cases that have been intensively studied. This is illustrated by the empirical results reviewed in the following following sections.
B. B. Effects Effects of Habitat Patch Patch Area and Isolation Isolation on Distribution Distribution A serious concern in any empirical metapopulation study is to recognize suitable habitat independently of the presence of the focal focal species. If this cannot be done, the appropriate habitat network cannot be identifi ed. The skipper but identified. butterfly Hesperia Hesperia comma comma is mostly restricted to southerly facing dry calcarous grass grasslands in in southern southern England, England, lays lays its eggs on on one species of grass (Festuca (Festuca ovina), ovina), lands and and only only on on plants plants of of certain certain size size growing growing in in aa particular particular microhabitat microhabitat (J. (J. A. A. 986). Depending on Thomas Thomas et al., 11986). on their their aspect, aspect, slope, slope, local local grazing, grazing, and and dis disturbance, turbance, only a subset subset of calcareous grassland grassland fragments represent suitable suitable hab habitat patches for this butterfly. It is the the distribution of these special habitat patches, not not the the overall overall distribution distribution of of calcarous calcarous grassland, grassland, which which forms forms the the habitat habitat patch patch network 1 988) were network for for this species. species. Using Using aa similar similar approach, approach, Harrison et et al. al. ((1988) were able able to to define define which which patches of of serpentine serpentine grassland could could be be regarded regarded as as potential potential habitat habitat for for E. E. editha editha bayensis. bayensis.
362 362
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potential habitat habitat has has been been identifi identified, possible to map map the the distri distriOnce potential ed, it is possible bution bution of of occupied occupied and and empty habitat. habitat. Studies of of many butterfly metapopulations metapopulations reveal that large habitat habitat patches patches are usually occupied, especially if they are close to other patches are other patches, patches, whereas whereas relatively small and/or and/or isolated patches are the ones ones Harrison et et al., and Harrison, Harrison, al., 11988; 988; C. D. Thomas Thomas and most likely to be vacant (Fig. 11;; Harrison 992; C. D. Thomas 1 993; Hanski, Hanski, 1994a,b, 1 994a,b, 11992; 992; C. D. Thomas Thomas et et al. al.,, 11992; Thomas and and Jones, Jones, 1993; Arnold, 11983). These patterns patterns are are consistent consistent with with the idea idea that this volume; cf. Arnold, 983). These small local populations prone to extinction patches populations are prone extinction and that isolated isolated habitat patches are least likely to be (re)colonized. The same pattern extends networks of extends to networks of habitat habitat patches; patches; the fraction fraction of of suitable suitable habitat habitat occupied occupied by Glanville Glanville fritillaries fritillaries (checkerspots) patches are (checkerspots) Melitaea Melitaea cinxia cinxia is greatest greatest in patch patch networks networks in which patches 995a, this volume). Hanski et large and close together together (Table (Table I) ((Hanski et al., al., 11995a, volume). Both pat patterns are consistent Hanski, 11991, 99 1 , 11994a,b, 994a,b, this volume; volume; consistent with model model predictions predictions ((Hanski, Hanski 994). Hanski and and Thomas, Thomas, 11994). The distribution distribution of of patch sizes in a fragmented fragmented landscape landscape is thus of of great importance Harrison and Taylor, this volume). volume). Metapopulations placed importance ((Harrison Metapopulations can be placed on a continuum - island" structure structure in which one habitat continuum from a "mainland "mainland-island" habitat patch patch is much much larger larger than than all the others, others, and long-term long-term persistence persistence is dominated dominated by the Boorman and Levitt, 1973), 1 973), persistence of of the largest (mainland) population population ((Boorman through to systems in which all local populations are equally important ( Levins through which all populations important (Levinstype metapopulations). metapopulations). At one extreme, the Morgan Morgan Hill metapopulation metapopulation of of E.
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and Regional Number of TABLE I Effects Effectsof of Average Average Patch Patch Area Area and Regional Density Density ((Number of Patches in in Squares Squares of of 44 km2) km2) on on the the Fraction Fraction of of Patches PatchesOccupied Occupied (PJ (P) by by the the Patches Glanville Glanville Fritillary Fritillary Melifaea Melitaea cinxia cinxia on on the the Aland Aland Islands Islands in in SW SW Finlanda Finlanda Average Average patch area (ha) (ha)
< 0.01 < 0.01 0.0 1 -0. 1 0.01-0.1 0. 1 0- 1 .0 0.10-1.0 > .0 > 11.0
Occupancy
n n
P P
No. of patches per 4 4 km2 km 2 per
23 23 1 38 138 88 88 6 6
0.24 0.24 0.24 0.24 0.40 0.40 0.56 0.56
I1 2-3 2-3 4-7 4-7 > >7 7
Occupancy n n
P P
6 611 70 70 58 58 66 66
0.21 0.21 0.32 0.32 0.25 0.25 0.41 0.41
a
a Effects Effects of of both both average average patch patch area area and and regional regional density density on on occupancy occupancy are are highly (from Hanski highly significant significant (from Hanski et et al., al., 1995). 1995).
editha editha bayensis bayensis is dominated dominated by one very large and and apparently persistent "main "mainland" population with transient "island" "island" populations found nearby in smaller smaller hab hab1 988). At the other extreme, the metapopu itat patches patches (Fig. 2; Harrison et et at., al., 1988). metapopuA land islands in the Baltic is an extensive lation of of M. M. cinxia cinxia on the the/~land extensive system of of hundreds of of small local populations, each of which is potentially susceptible susceptible to at., 11994, 994, 11995a,b). 995a,b). Important extinction (Fig. 11 in Hanski, this volume; Hanski et et al., differences differences between these two metapopulations metapopulations appear appear to be due to differences differences of habitat patch areas in the landscape, landscape, rather rather than to a fun funin the distribution of damental difference Hanski et difference in the biology of these closely related species ((Hanski et at., al., 11994). 994). Indeed, different Ptebejus argus different metapopulations of of the the blue butterfl butterflyy Plebejus argus show nearly as much variation in their spatial structure as do all comparisons of metapopulation structure across all species studied to date. Differences Differences in the rates and patterns patterns of of local extinction and colonization in different metapopulations of P. P. argus argus are apparently largely due to differences in the distributions of of patch sizes and vegetation dynamics, not to differences in the butterfly (c. D. Thomas differences (C. and Harrison, 1992). 1 992). Most metapopulations occupy an intermediate intermediate position along this continuum, with some relatively large, but not necessarily permanently pop populated patches, and other small and/or higher turnover and and/or isolated patches with higher probabilities of being occupied. lower probabilities One diffi culty in making difficulty making deductions deductions about about the persistence persistence and and dynamics of of metapopulations hab metapopulations from a "snapshot" "snapshot" distribution of of "occupied" "occupied" and "empty" "empty" habitat is that empty habitat might not be suitable of some subtle suitable after all, because because of unrecognised attribute attribute of particular particular habitat patches. patches. Although this potential prob problem should be considered carefully in every empirical study, we believe that it is empirical issue in most most butterfly studies. A few misclassified patches are a relatively minor issue unlikely to change the overall conclusions in any study of which we are aware. Introductions of of butterflies to empty habitats have succeeded succeeded in establishing new local populations on numerous occasions (Oates and Warren, 990; C. D. Warren, 11990; Thomas, 11992; 992; C. D. Thomas and Harrison, 11992; 992; Neve 1 995), and other N~ve et et at., al., 1995), studies have reported natural colonization of patches which had previously been
Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski
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FIGURE FIGURE 2 The Morgan Morgan Hill metapopulation metapopulation of Euphydryas Euphydryas editha editha bayensis. bayensis. Black areas areas are are ser serpentine outcrops. "Mainland" Morgan Hill supported ~ 11 0066 butterflies, with "island" outcrops sup supporting 0 1 - 1 0' butterflies each (arrowed). porting 1101-102 (arrowed). Eighteen of of the more isolated outcrops outcrops were suitable but unoccupied. Reprinted with permission ai., 1988, 1 988, University of permission from Harrison Harrison et al., of Chicago Press. =
identified unoccupied but suitable habitat 3; C. D. Thomas Jones, identified as as unoccupied but suitable habitat (Fig. (Fig. 3; C. D. Thomas and and Jones, 11993). 993). Introductions Introductions and and natural natural colonizations colonizations both both demonstrate demonstrate that that at least least some empty empty habitat habitat was was indeed indeed suitable. suitable.
III. POPULATION BUTTERFLIES POPULATIONTURNOVER TURNOVERIN BUTTERFLIES AA.. Local Local Extinction Extinction Empirical incomplete, Empirical evidence evidence on local extinctions extinctions and colonizations colonizations is still incomplete,
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given the Excluding extinctions extinctions caused given the need need for for long-term long-term study. study. Excluding caused by by complete complete destruction cultivation, and urbanization, rela destruction of of habitat, habitat, such such as as deforestation, deforestation, cultivation, and urbanization, relatively high in small populations tively high rates rates of of butterfly butterfly extinction extinction are are generally generally observed observed in small populations Ehrlich in in E. in small small habitat habitat patches; patches; this this pattern pattern has has been been found found in E. editha editha bayensis bayensis ((Ehrlich et 975; Ehrlich 987; Harrison 1 988), M. Fig. et al. al.,, 11975; Ehrlich and and Murphy, Murphy, 11987; Harrison et et al. al.,, 1988), M. cinxia cinxia ((Fig. 994, 11995b), 995b), P. 992; C. 3, et al. al.,, 11994, P. argus argus (c. (C. D. D. Thomas Thomas and and Harrison, Harrison, 11992; C. 3, Hanski Hanski et D. 994b, 11996), 996), and 993). Local Local D. Thomas, Thomas, 11994b, and H. H. comma c o m m a (c. (C. D. D. Thomas Thomas and and Jones, Jones, 11993). populations which which contain butterflies are populations contain hundreds hundreds of of adult adult butterflies are frequently frequently seen seen to to become become extinct extinct during during studies studies lasting lasting 33 to to 30 30 years years (though (though some some such such populations populations survive tens of survive for for decades), decades), whereas whereas populations populations of of thousands thousands or or tens of thousands thousands rarely rarely become extinct during studies of provided the become extinct during studies of the the same same duration, duration, provided the environment environment remains relatively stable. stable. Historical popu remains relatively Historical records records indicate indicate that that extremely extremely large large populations become extinct extinct over over longer periods, usually usually associated lations can can become longer periods, associated with with declines declines in 984, 11991; 99 1 ; J. Morris, 1994; 1 994; in habitat habitat quality quality (1. (J. A. A. Thomas, Thomas, 11984, J. A. A. Thomas Thomas and and Morris, see below). E. editha editha below). see also also E. Population i ) Demo Population extinction extinction appears appears to to be be affected affected by by many many factors. factors. ((i) Demographic graphic stochasticity, stochasticity, the the chance chance variation variation inevitably inevitably associated associated with with death death and and birth, may even in constant environ birth, may cause cause the the extinction extinction of of aa small small population population even in aa constant environment. most likely likely to cant where ment. This This form form of of stochasticity stochasticity is is most to be be signifi significant where habitat habitat patches species with patches are are very very small, small, and and especially especially in in species with gregarious gregarious caterpillars, caterpillars, because large extent because the the fates fates of of caterpillars caterpillars in in one one group group are are to to aa large extent correlated. correlated. Small just one Small local local populations populations of of the the Glanville Glanville fritillary fritillary M. M. cinxia cinxia often often have have just one or or aa few few groups groups of of caterpillars caterpillars (50 (50- 11000 0 individuals individuals per per group group in in late late summer, summer, but aI., 11995a; 995a; Fig. Fig. 3). must be but fewer fewer in in spring; spring; Hanski Hanski et et al., 3). Pure Pure chance chance must be a a major major
(b)
;--
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;--
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tv1esSlI"e of butterfly density In the SlI"fOUldlngs of a patch
FIGURE Rates of popUlation empty meadows meadows in the large meta FIGURE33 Rates population extinction extinction and colonization colonization of empty large metapopulation of Melitaea (Fig. I1 in Hanski, Hanski, this this volume). volume). (a) Fraction of populations population Melitaea cinxia (Fig. populations that went extinct from 993 until until 11994 994 as a function extinct from 11993 function of the size size of the population, population, measured measured by the number of larval 993. (b) Rate isolation. Isolation Isolation was larval groups groups detected detected in 11993. Rate of recolonization recolonization as a function function of isolation. measured by the sum of the sizes sizes of the surrounding surrounding populations, weighted by their their distance distance to the measured populations, weighted focal meadow (see Eq. (4) in Hanski, Hanski, 11994a). 994a). meadow (see
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Chris Thomas and and IIkka Chris D. D. Thomas Ilkka Hanski Hanski
factor in the dynamics of of such small populations, which have a high probability of extinction even if there is nothing wrong with the habitat patch. patch. (ii) (ii) Environ Environof mental stochasticity; drought, aseasonal cold, and other extreme weather weather events, is known to have generated extinctions extinctions in populations populations of of the blue Glaucopsyche Glaucopsyche iygdamus, Aphantopus hyperantus, frit lygdamus, E. E. editha, editha, M. M. cinxia, cinxia, the ringlet Aphantopus hyperantus, and the fritillary Boloria Ehrlich et 972, 1980; 1 980; Harrison et al., ai., 1988; 1 988; Han Boloria aquilonaris aquilonaris ((Ehrlich et ai., al., 11972, Hanski et al. 995a, unpublished; Pollard and Yates, 11993; 993; T. Ebenhard, al.,, 11995a, Ebenhard, personal personal communication). iii ) The interaction communication). ((iii) interaction between environmental stochasticity and habitat heterogeneity can cause extinctions. extinctions. Large Large patches usually contain several microhabitats, microhabitats, not all of which will become equally inhospitable inhospitable as a result of of a single environmental event. Variation in local topography and turf height appear appear H. comma, comma, respectively, from to buffer populations of E. editha editha bayensis bayensis and H. ai., 1988; 1 988; Murphy local extinction in large habitat patches (Singer, 11972; 972; Weiss et al., et ai., 990; C. D. Thomas, I1994a; 994a; see Hanski, this volume, and below). ((iv) iv) al., 11990; When When a metapopulation metapopulation inhabits more than one type of of habitat, rare environmental events may cause widespread extinctions in some habitat types but not in others. Ehrlich et al., ai., 11980; 980; C. D. Thomas et Again, E. editha editha provides an example ((Ehrlich et ai., al., 11996). 996). Rare events may be particularly likely to cause extinction from suboptimal habitat. One local population of the blue butterfl y, Lysandra butterfly, Lysandra bellargus, bellargus, became 976, while two extinct from relatively poor habitat patch, patch, during a drought in 11976, nearby populations survived in better quality habitat (J. A. Thomas, 11983a). 983a). Meanwhile, Meanwhile, A. hyperantus hyperantus populations appeared to survive the the same drought bet better in areas of of intermediate intermediate shade than in the open or in denser shade (Sutcliffe (Sutcliffe 996c). (v) Vegetation dynamics /grazing patterns et al. al.,, 11996c). dynamics/grazing patterns and other habitat changes changes render render particular particular patches patches unsuitable, and the respective populations decline decline de deterministically to local extinction. This appears appears to be the most common cause of of local extinction for for medium-sized and large local populations and may therefore be of particular particular importance to the persistence persistence of of metapopulations. There are ex examples amples of of this mechanism for all species covered covered in this chapter chapter (e.g., Harrison, 11991, 99 1 , 11994b; 994b; J. A. Thomas, 1 99 1 ; Warren, 992, 1993; 1 993; C. D. Thomas, 1 994a,b,c, Thomas, 1991; Warren, 11992, Thomas, 1994a,b,c, 11996; 996; see Section V.B below). Because Because vegetation dynamics dynamics can occur occur at a variety of population) is more likely of scales, loss of all suitable habitat from a small patch ((population) than than complete complete loss of of suitable suitable habitat from an initially large patch. (vi) Isolated (c. D. Thomas and local populations may suffer increased rates of extinction (C. Jones, 11993; 993; Hanski et ai. 995b) as a result of al.,, 11995b) of a reduction in the number of of immigrants. When 977) become When such rescue effects (Brown and Kodric-Brown, Kodric-Brown, 11977) weak, populations in poor (sink) habitat pop habitat may become isolated from source pop1 994), and poppop ulations and become extinct (cf. Warren, 994; Rodriguez Warren, 11994; Rodriguez et ai., al., 1994), ulations in small habitat patches may disappear disappear because immigration no longer replaces individuals lost to emigration emigration (we discuss this possibility in detail below). As more evidence evidence becomes available, it is becoming apparent that environ environmental stochasticity, isolation, vegetation dynamics, and both subtle and extreme extreme human activities interact interact to generate generate patterns patterns of of extinction and that we cannot
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expect aa single single factor factor to to explain explain all all extinctions extinctions even even within within aa single single metapopumetapopu expect lation. lation.
Colonization B. Colonization Empirical of colonization colonization present present compelling compelling evidence evidence that that mimi Empirical observations observations of gration occasionally takes takes place place over over distances distances considerably considerably greater greater than than those those gration occasionally normally normally revealed revealed in mark-release-recapture mark-release -recapture studies studies conducted conducted within within existing existing metapopulations (C. (c. D. Thomas Thomas et et al., al., 1992; 1 992; Hanski Hanski and and Kuussaari, Kuussaari, 1995). 1 995). PatPat metapopulations terns of of colonization colonization have have been been particularly particularly clear clear when when entire entire networks networks of of empty empty terns habitat patches patches have have been been available for for colonization colonization (C. (c. D. Thomas Thomas et et al., al. , 1992; 1 992; habitat C. D. D. Thomas, Thomas, 1994a,b,c). 1 994a,b,c). Hesperia Hesperia comma comma invaded invaded habitat habitat patches patches which which had had recently become become suitable suitable for for this this species in in part part of of southern southern England. England. Over Over aa 9recently year period, period, population population establishment establishment occurred occurred with with aa high high probability probability in large large year were close close to existing existing populations; populations; beyond beyond 11 km, km, the the probability probability patches which were of colonization declined, and the maximum maximum observed observed colonization colonization distance distance was of colonization declined, and the Srymonidia pruni, pruni, km (C. (c. D. Thomas Thomas and and Jones, 1993). 1 993). The black hairstreak hairstreak Srymonidia 8.6 km The black the (checkerspot) Mellicta Mellicta athalia, athalia, the skipper skipper Thymelicus Thymelicus acteon, acteon, the heath heath fritillary (checkerspot) and P. P. argus argus have have similarly showed showed decreased decreased probabilities probabilities of colo M. cinxia, M. cinxia, and of colonizing relatively isolated habitat Thomas, 1983b; 1 983b; C. D. nizing relatively isolated habitat patches patches (Fig. 3; J. A. Thomas, 1 992; C. D. Thomas 1 992). These These results indicate Thomas Thomas et et al. al.,, 1992; Thomas and and Harrison, Harrison, 1992). results indicate rates of of migration migration with with increasing increasing distance. Over Over short short distances, distances, imim declining rates migration ciently high high that successful colonization colonization is practically practically migration rates rates are are suffi sufficiently that successful assured H. comma, this was was attained within 11 km, km, after after 9 years). At At somewhat assured (for (for H. comma, this attained within somewhat greater distances, colonization observed in empirical studies but but with decreasing greater distances, colonization is observed empirical studies decreasing probability. probability. At longer longer distances distances still, colonization colonization is not observed observed on the time timescale scale of of empirical empirical studies. been deliberately deliberately released released and have have become become established established Several species have been in a new 992; C. D. Thomas new region region (see C. D. Thomas Thomas and and Harrison, Harrison, 11992; Thomas et et al. al.,, 11992; 992; C. D. Thomas, 11992, 992, 11994a, 994a, 1996, 1 996, for Neve et al. ((1995) 1 995) report for examples). examples). Nbve the sequential Polygonum bistorta host plant) meadows sequential colonization colonization of of 24 Polygonum bistorta ((host meadows by bog fritillary Proclossiana Proclossiana eunomia, following the butterfly's introduction to the bog eunomia, following butterfly ' S introduction the Morvan region in France. Within Within a network of of habitat patches, patches, there was a noisy, but approximately approximately linear, relationship between between distance distance achieved achieved and time since introduction. introduction. Diffusion Diffusion models of of colonization colonization predict a linear linear relationship relationship between distance and time if there are no major major barriers barriers to colonization colonization (Andow et al. al.,, 11990; 990; Nash et al. al.,, 11995), 995), suggesting that the patch structure may not have been a major constraint metapopulation, though constraint on colonization in this metapopulation, though the scatter scatter of of data points points leaves the question open. Migration distances recorded recorded in mark markrelease - recapture studies on P. eunomia release-recapture eunomia revealed substantial movements movements of of in individuals among patches (Baguette 994; Neve 995), again (Baguette and Neve, N~ve, 11994; N~ve et al. al.,, 11995), metapopulations. In other indicating that patch isolation may not be great within metapopulations. species, migration rates between local populations .4% in a populations vary between about 11.4%
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Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski
metapopulation Wales (O. (0. T. Lewis Lewis metapopulation of of the the particularly sedentary sedentary P. P. argus argus in North Wales et al., unpublished) et al., unpublished) and and 3% in Jasper Jasper Ridge Ridge E. E. editha editha to > > 20% in several 996c; H. H. comma, comma, Hill et 1 996; species species (e.g., A A.. hyperantus, hyperantus, Sutcliffe Sutcliffe et et al., al., 11996c; et al., al., 1996; see Warren, 987a; C. D. Thomas, 994a; Hanski and Kuussaari, 1 995, for Warren, 11987a; Thomas, 11994a; Kuussaari, 1995, for other exchange exchange rates). Overall, high exchange exchange rates among patches patches and and rapid colonization butterfly meta colonization suggest that the rescue effect is common in butterfly metapopulations populations and and that that isolation isolation may not be a major major constraint constraint on colonization colonization of of empty habitat within empty habitat within many many existing existing metapopulations. metapopulations. Often Often only a small small fraction fraction of unless patch of habitat habitat patches patches is empty within within a connected connected patch patch network, network, unless patch sizes are (c. D. Thomas 992). High migration are extremely extremely small small (C. Thomas and and Harrison, Harrison, 11992). migration rates and consequent consequent strong rescue effects effects may generate generate alternative stable equilibria equilibria in metapopulation metapopulation metapopulation dynamics; i.e., almost all patches patches occupied occupied or metapopulation extinction. 1 995b; extinction. A putative example is described for for M. M. cinxia cinxia by Hanski et et al. al. ((1995b; see Fig. 4 in Gyllenberg Gyllenberg et et al., al., this this volume). In contrast contrast to the the situation situation within within patch patch networks, networks, isolation isolation is an extremely important important reason why empty habitat habitat does does not not become become colonized colonized beyond beyond the rec rec(c. D. Thomas Harrison, ognized boundaries boundaries of of existing metapopulations metapopulations (C. Thomas and Harrison, 1992). Groups of of habitat patches patches are are often separated separated by distances distances which 1 992). Groups which will prevent colonization colonization from another another patch patch network network ((1I to > > 20 km separation separation may may provide provide an effective effective barrier, depending depending on the the species). species). Once Once such metapopula metapopulations become become extinct, reestablishment reestablishment can be very slow, which highlights the importance importance of of obtaining obtaining an empirical and theoretical theoretical understanding understanding of of the factors factors that contribute to the persistence of populations. of entire meta metapopulations.
IV. IV. THEORETICAL THEORETICALPREDIGIONS PREDICTIONSTESTED TESTED Despite metapopulations, empirical Despite many complications complications in specifi specificc butterfly metapopulations, empirical data lend considerable considerable support support to the central central tenets of of metapopulation metapopulation theory, namely that extinction /population size and extinction is related related to patch patch area area/population and colonization colonization is distance-dependent distance-dependent and that these these relationships contribute contribute to the observation that populations populations are most most likely to be present present in habitat habitat patches patches which are large 994b ). These general confidence that and close Hanski, I1994b). close together ((Hanski, general results results give us confidence the approach approach is valuable, valuable, even if if more more specifi specificc and complex models may be required required to predict predict the dynamics of of a particular particular system. The potential potential ability to test model assumptions assumptions and and predictions predictions is one of of the attractions c, spatially attractions of of using using specifi specific, spatially explicit explicit models, models, of of the the type outlined outlined in Hanski l 994a, this volume; 994; Sjogren Hanski ((1994a, volume; Hanski Hanski and and Thomas, Thomas, 11994; Sj6gren Gulve and and Ray, 11996). 996). In these models, habitat patch is specifi ed, models, the location location and and area area of of each habitat specified, with rules governing governing the probability of of local extinction extinction (mainly dependent dependent on population isola population size) and the probability of of colonization colonization (mainly dependent dependent on isolation and predictions of and source source popUlation population sizes). In this section, we deal deal with the the predictions of existing following existing models models and and consider consider some some of of the model model assumptions assumptions in the following section. section.
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able to test the predictions of a spatially explicit metapopu metapopuWe have been able H. comma c o m m a ((Hanski lation model with data for H. Hanski and Thomas, 11994). 994). This model explicitly iterates local dynamics in each occupied patch, which are connected to each other via distance-dependent migration (modeled at the population rather than individual level). Colonization is a mechanistic consequence of immigration to an empty patch. Altogether, the model has nine parameters. Where possible, parameter values should be estimated independently, but some parameters are pavery difficult or extremely time-consuming to estimate reliably. In our case, pa rameter values which could not be measured independently were estimated by simulation from a metapopulation which was assumed to be at equilibrium; this was done by choosing parameter parameter values which generated model predictions that matched empirical patterns of patch occupancy. The model was then applied, using these parameter parameter values, to a different different metapopulation, which was clearly not at equilibrium, to test whether the model could successfully predict dynamics metapopulation. in a nonequilibrium metapopulation. H. comma c o m m a occupies short, sparse, dry grasslands. Myxomatosis In England, H. rabbit grazing grazing in the midmid-1950s habitat became became over overremoved rabbit 1 950s and the grassland habitat butterflyy became very localized between grown, with the result that the skipper butterfl localized between (J. A. Thomas et et al. al.,, 11986). c o m m a became 986). In East Sussex, H. H. comma became 11960 960 and 11975 975 (1. restricted to one large. large population. In 1982, restricted 1 982, the skipper was still thriving in this refuge population and had colonized two small nearby habitat patches. patches. By large refuge was clear clear that substantial areas of of habitat were again suitable for breeding 11982, 982, it was by H. H. comma, comma, as rabbits had partially recovered from from myxomatosis and conser consermanagement on some vation organizations had begun to undertake active grazing management of the previously overgrown grasslands. This represented represented an ideal situation in of which to test the predictions predictions of of the model; model; a network of of empty patches which which could be mapped could mapped and a known known distribution distribution of of the skipper skipper in 1982. 1 982. The The predictions predictions of the models of models could then be compared compared with the observed observed skipper skipper distribution after after of colonization colonization (C. (c. D. D. Thomas Thomas and and Jones, Jones, 1993). 1 993). The The results were a 9 years of qualified qualified success. The model model predicted predicted that the skipper skipper would spread spread in this region and that it would not spread spread in other regions regions where where it failed, failed, in reality, to expand expand its distribution, distribution, but but the model underestimated underestimated the real rate of of expansion (Hanski ( Hanski and Thomas, 1994). H e s p e r i a ccomma o m m a actually occupied occupied 21 after 9 years 2 1 patches patches after 1 994). Hesperia ((Fig. Fig. 7 in Hanski and Thomas, 1994), 1 994), which which was more than than the model model predicted predicted patches occupied 1 1 , in 100 1 00 replicate replicate simusimu (mean 8.6 patches occupied after after 9 years, maximum 11, lations). quantitative mismatch prediction and observation prompts lations). The The quantitative mismatch between between prediction and observation prompts a series of of new new questions. questions. Was Was the the region region used to parameterize parameterize the the model model really equilibrium (not quite; quite; C. D. D. Thomas Thomas and and Jones, Jones, 1993)? 1 993)? Was Was the the negative at equilibrium exponential exponential distribution distribution used for for migration/colonization migration/colonization appropriate appropriate (new markmark release-recapture data were migration disrelease-recapture data were collected collected to resolve resolve this issue, and and migration dis tances found to tances were were found to fit a negative negative power power function function better, better, implying more more longlong distance distance migrants migrants than than in the the original simulations; simulations; Hill et et al., al., 1996)? 1 996)? et al. al. (1996c) (1 996c) have have attempted attempted to to predict predict the the distribution distribution of of M. M. cinxia cinxia Hanski et Hanski Aland in the over the Baltic, Baltic, using using an an incidence incidence function function model model over some 1000 1 000 km km22 on ,~land
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Chris and IIkko Chris D. D. Thomas Thomas and Ilkka Hanski Hanski
((Hanski, Hanski, 11994a). 994a). Again Again the the results results are are encouraging encouraging but but mixed. mixed. The The model model was was parameterized 1 99 1 (Hanski parameterized using using data data collected collected from from aa small small part part of of Aland Aland in in 1991 (Hanski et aI., 11994) 994) and et al., and model model predictions predictions were were tested tested with with independent independent data data collected collected in 993. Over fractions in 11993. Over large large parts parts of of Aland, Aland, the the model-predicted model-predicted and and observed observed fractions of Fig. 8 in in Hanski, Hanski, this volume), but of occupied occupied habitat habitat were were in in good good agreement agreement ((Fig. this volume), but this level this began began to to break break down down in in drier drier areas areas of of the the island island and and in in areas areas where where the the level of y ' s distribution. distribution. As As with with of grazing grazing was was an an important important determinant determinant of of the the butterfl butterfly's H. comma, testing the model revealed that there were further aspects of the bi H. c o m m a , testing the model revealed that there were further aspects of the biology of species that ology of the the species that needed needed to to be be understood. understood. Another fragmented Another kind kind of of problem problem for for predicting predicting species' species' distribution distribution in in fragmented landscapes is which there is landscapes is posed posed by by the the possibility possibility of of mUltiple multiple equilibria, equilibria, for for which there is M. cinxia Fig. 4 this empirical case of empirical evidence evidence in in the the case of M. cinxia ((Fig. 4 in in Gyllenberg Gyllenberg et et al., al., this 995b). Multiple impose inherent inherent uncertainties volume; volume; Hanski Hanski et et al., al., 11995b). Multiple equilibria equilibria impose uncertainties on particular netnet on our our ability ability to to predict, predict, accurately, accurately, the the distribution distribution of of species species in in particular works. role when when habitat is itself itself dynamic. dynamic. Al works. History History can can also also play play aa crucial crucial role habitat is Although meta though aa habitat habitat network network may may be be extensive extensive enough enough to to support support aa persistent persistent metapopulation now, now, the be too been colonized. population the whole whole network network may may be too isolated isolated to to have have been colonized. In some some areas areas where where H. H. comma c o m m a used to occur, occur, and and where where it became extinct extinct when when used to it became In the the habitat habitat was was overgrown, overgrown, the the habitat habitat has has now now recovered. recovered. Simulations Simulations suggest suggest be reestablished least one that that aa substantial substantial H. H. comma c o m m a metapopulation metapopulation could could be reestablished in in at at least one such recolonize this this area such area, area, but but the the butterfl butterflyy has has failed failed to to recolonize area naturally naturally and and simu simu(c. D. D. Thomas lations 1 00 years lations indicate indicate that that it it is is likely likely to to take take more more than than 100 years to to do do so so (C. Thomas and Jones, 11993; 993; Hanski Thomas, 11994). 994). and Jones, Hanski and and Thomas,
Metapopulation Persistence Persistenceand Establishment Establishment A. Metapopulation One One of of the the major major potential potential uses uses of of models models is is to to predict predict whether whether metapopu metapopulations are in specific specific networks habitat patches. lations are likely likely to to persist persist in networks of of habitat patches. The The models models referred Hanski, 11994a; 994a; Hanski 1 994) can be used referred to to above above ((Hanski, Hanski and and Thomas, Thomas, 1994) can be used for for this that one one has has been been able them. this purpose, purpose, assuming assuming of of course course that able to to parameterize parameterize them. One models to size and persistence of One may may use use the the models to explore explore how how the the size and persistence of aa particular particular metapopulation loss of is likely likely to to metapopulation is is affected affected by by further further loss of suitable suitable habitat, habitat, which which is increase the risk of local extinction and to decrease the probability of recoloni increase the risk of local extinction and to decrease the probability of recolonization. 1 994a,b) and 1 994) give zation. Hanski Hanski ((1994a,b) and Hanski Hanski and and Thomas Thomas ((1994) give examples. examples. More More generally, generally, one one may may ask ask how how the the expected expected metapopulation metapopulation lifetime lifetime depends depends on on the the number local populations populations that connected to to each number of of extinction-prone extinction-prone local that are are connected each other. other. 1 996c) have explored this question with with the the incidence incidence function function Hanski Hanski et et al. al. ((1996c) have explored this question islands. The model using the model using the observed observed patch patch networks networks for for M. M. cinxia cinxia on on the the Aland ,~land islands. The expected to the expected time time to to metapopulation metapopulation extinction extinction is is closely closely related related to the product product px/~, where where p p is is the the fraction fraction of of occupied occupied habitat habitat patches patches at at stochastic stochastic steady steady state state pJH, patches ((Hanski, Hanski, this volume). A A rough of and and H H is is the the number number of of suitable suitable patches this volume). rough rule rule of thumb metapopulation is is likely likely to thumb is is that that if if this this product product exceeds exceeds 33 the the metapopulation to persist persist much of a the much longer longer than than the the expected expected lifetime lifetime of a local local population. population. Assuming Assuming that that the
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species patch network which it species occupies occupies most most patches patches in in the the patch network in in which it is is present, present, a a minimum of some 1 5 20 well-connected patches are required for long-term minimum of some 1 5 - 2 0 well-connected patches are required for long-term per persistence. in local local dynamics dynamics and sistence. With With spatial spatial correlation correlation in and much much environmental environmental sto stoal., 1996b). 1 996b). Hanski et chasticity, even this this would would not not be enough ((Hanski et al., Empirical tests tests of of the the persistence persistence of of entire entire metapopulations metapopulations are are very very limited Empirical limited and, and, understandably, understandably, difficult difficult to to accomplish. accomplish. The The extinction extinction of of an an entire entire meta metapopulation is is much scale than population much rarer rarer and and takes takes place place on on a a longer longer time time scale than does does the the extinction populations. As As Harrison Harrison ((1991, 1 99 1 , 11994b) 994b) and and C. C. D. D. extinction of of individual individual local local populations. Thomas 1 994a,b) have noted, metapopulation metapopulation extinction is generally generally related Thomas ((1994a,b) have noted, extinction is related to to an the amount to widespread an overall overall decline decline in in the amount of of suitable suitable habitat habitat due due to widespread human humancaused in the landscape. Here, Here, we caused changes changes in the landscape. we are are concerned concerned whether whether a a metapopu metapopulation is likely to persist in a network of habitat patches which is not becoming lation is likely to persist in a network of habitat patches which is not becoming further In studies studies of of P. D. Thomas 1 994b) further degraded. degraded. In P. argus argus and and H. H. comma, comma, C. C. D. Thomas ((1994b) found 5 to tended to populated, but but found that that regions regions with with over over 115 to 20 20 habitat habitat patches patches tended to be be populated, that 1 0 patches populated, suggesting the latter latter that regions regions with with < < 10 patches were were rarely rarely populated, suggesting that that the may shows just just may be be inadequate inadequate for for long-term long-term metapopulation metapopulation persistence. persistence. Figure Figure 4 4 shows the M. cinxia. cinxia. T. T. Ebenhard Ebenhard (personal (personal communication) communication) was was able able the same same pattern pattern for for M. to extinctions directly, the cranberry to examine examine metapopulation metapopulation extinctions directly, when when working working on on the cranberry fritillary B. aquilonaris wet year. year. Boloria Boloria aquilonaris aquilonaris in in an an extremely extremely cloudy cloudy and and wet aquilonaris fritillary B. occurs in within remnant with occurs in bogs bogs within remnant forest forest patches patches in in southern-central southern-central Sweden, Sweden, with of cranberry-containing cranberry-containing bogs per per forest fragment fragment representing representing the the number number of
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number meta number of of habitat habitat patches patches per per network. network. Ebenhard Ebenhard found found that that many many of of the the metapopulations fewer than than 20 patches patches became populations with fewer became extinct while those those with > > 20 patches population collapse. patches survived a massive population collapse. At accord. At this stage, theory and and empirical empirical data data appear appear to be in approximate approximate accord. A metapopulation metapopulation rarely persists for for very long at < < 1100 local populations populations and and usually does so for populations. There for extended extended periods periods at > > 20 local populations. There is obviously obviously a trade-off between the number patches and patch, with fewer trade-off between number of of patches and area area per per patch, fewer patches patches needed xed length length of needed for for some some fi fixed of metapopulation metapopulation survival when when each patch patch is large; one persistence if if it is vast and heterogeneous (the theory one patch patch alone alone may ensure ensure persistence and heterogeneous theory described above metapopulation lifetime in relation to the lifetime of described above predicts predicts metapopulation of local interpreted). When habitat local populations, populations, and and the the results results should should be thus thus interpreted). When the habitat consists consists of of transient transient or successional successional vegetation, much much larger areas areas may be required required ensure that appropriate appropriate habitat continuously available, available, the amount amount of of extra extra to ensure habitat is continuously vegetation required the natural vegetation and vegetation required depending depending on the natural dynamics of of the vegetation and on patterns of of human management. Finally, if there regional stochasticity stochasticity patterns human management. there is much regional metapopula (spatially correlated correlated environmental environmental stochasticity), stochasticity), even even very large large metapopulations may go extinct, extinct, the extreme case case being a catastrophe catastrophe of of some some kind kind sweeping sweeping away species from Given away the species from a large large area area (see below below and and Hanski, Hanski, this volume). Given these and other other uncertainties, uncertainties, we must that the patch numbers numbers mentioned mentioned these and must stress that the patch above rules or planning. Much above should should not not be taken taken as rules or guidelines in conservation conservation planning. Much larger required in some larger numbers numbers of of patches patches will be required some circumstances, circumstances, but but one one vast vast patch ce in some conservation, each must patch may suffi suffice some cases. In the context context of of conservation, each case case must be considered considered individually. One most valuable uses of One of of the most valuable uses of specific specific models models is in predicting predicting the con consequences management options. Predicting the actual proba sequences of of possible possible future future management options. Predicting probability of bound to be fraught uncertainties, but but prepre of long-term persistence persistence is bound fraught with uncertainties, dicting response to changes changes in the distribution dicting the likely direction of of response distribution of of habitat periods of few decades is potentially potentially of over periods of a few of immense immense value to managers, and and may be be possible explicit metapossible to achieve. We We have already seen that that a spatially explicit meta population model predicted population predicted correctly, in qualitative qualitative terms, where where H. H. comma comma would would expand its distribution distribution in response response to increased increased habitat quality and and where it would nements, such models would not. Even without without any further further refi refinements, models could be of of use to conservation been possible habitat conservation managers. managers. It has has been possible to draw draw maps maps of of existing existing habitat around around surviving surviving H. H. comma c o m m a populations populations and to assess assess the potential for for further further spread. present spread. In East Sussex, Sussex, managers managers can can relax in the sense that that continuing continuing the present management is likely to permit permit continued continued expansion of of H. H. comma. c o m m a . In three other other expansion is not not predicted (c. D. regions in southeast southeast England, England, further further expansion predicted (C. D. Thomas Thomas and 993; Hanski, 11994a; 994a; Hanski and 994). When and Jones, 11993; and Thomas, 11994). When the species species is unable unable to spread spread within within the existing patch patch network, network, the consequences consequences of of many many different different management management options options can be be explored explored by changing changing the distribution distribution and sizes of of specifi specificc habitat patches in the model. model. For For example, example, would enhanced enhanced management management to increase increase existing existing population population sizes facilitate further further spread, spread, and and would management would management of of the surrounding surrounding areas areas to increase increase target target patch patch areas or or to decrease spread? In a model, decrease distances distances between between patches facilitate spread? model, many different different
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options options may may be be compared compared quickly, quickly, before before embarking embarking on on time-consuming time-consuming and and expensive expensive management management work. work. Even Even if if the the predictions predictions are are not not quantitatively quite quite correct, correct, the the relative relative merits merits of of different different practical practical options options may may be be robustly robustly assessed. assessed. An An additional additional important important application application is is likely likely to to be be in in predicting predicting the the potential potential of species translocation translocation projects. Hanski and and Thomas ((1994) that success of 1 994) found that one one set of parameter values correctly correctly predicted the existing existing distribution of of P. argus argus on limestone grassland, and its invasion of two new networks of habitat habitat introduced in the the past. For H. comma, patches to which it had been successfully introduced the the model has been used to identify identify at least one network which which could potentially support support a viable metapopulation metapopulation of the skipper, but which is too isolated for colonization to occur naturally. This approach may help to eliminate some of the effort wasted on releasing rare butterflies in single or very small groups of of habitat patches where long-term persistence is improbable 990; improbable (Oates and Warren, 11990; explicit models, the the target species species can can be C. D. Thomas, 11992). 992). Using spatially explicit introduced to the the same same patch network repeatedly, and and the the value value of releasing a fixed number of individuals at one versus many sites can be explored. There are number of individuals several lessons to learn. For example, managers should not necessarily give up even if the fi rst attempt fails; the introduced first introduced population may have to exceed some threshold threshold before before establishment is likely. This is especially true if there are multiple equilibria ((Hanski Hanski et ai., 995b; Gyllenberg al., 11995b; Gyllenberg et ai., al., this volume). volume). In most cases, the largest, largest, high quality and and least isolated habitat patches patches in the network should should the isolated habitat the network be targeted Hanski, 11994b), 994b), even targeted for for releases releases ((Hanski, even if if these these patches patches fall outside outside existing existing nature reserves reserves in the network. network. nature
V. ADDING CONCEPT ADDING REALISM REALISM TO TO THE THE METAPOPULATION METAPOPUlATION CONCEPT At this relatively relatively early stage stage in the the development development of of specific specific and and predictive predictive At metapopulation models, field field studies studies are needed to test model model assumptions assumptions and metapopulation are needed and to identify additional additional behavioral behavioral and and ecological factors factors which which may may need need to to be be to incorporated into into the the next next generation generation of of models. models. Having Having identified identified existing existing shortshort incorporated comings, comings, it it is important important to to find find out out whether whether the the predictions predictions of of refined refined models models differ adding many differ substantially substantially from from the the predictions predictions of of simpler simpler models, models, because because adding many more parameters parameters to to models models creates creates new new problems. problems. In In this this section section we we address address more model assumptions assumptions and and complications complications that that we we believe believe will will have have important important imim model plications for for metapopulation metapopulation persistence, persistence, patterns patterns of of distribution, distribution, and and rates rates of of plications colonization. colonization.
A. Migration Migration In Hanski Hanski and and Thomas Thomas (1994, ( 1 994, p. p. 170), 1 70), we we highlighted highlighted "the "the need need for for good good In In the the model, model, empiricai data data on on emigration emigration and and immigration immigration rates rates in in butterflies." butterflies." In empirical we assumed assumed that that the the distribution distribution of of dispersing dispersing individuals individuals could could be be fitted fitted to to aa we negative exponential exponential function, function, which which is is typical typical of of metapopulation metapopulation models models (Har( Harnegative
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rison et et al., al., 1988; 1 988; Hanski, Hanski, 1994a; 1 994a; Hanski Hanski and and Thomas, Thomas, 1994; 1 994; Akqakaya, Ak�akaya, 1994; 1 994; rison Sjogren Gulve Gu1ve and and Ray, Ray, 1996). 1 996). This This distribution distribution clearly clearly gives gives the the right right general general Sjrgren pattern, of of many many short-distance short-distance movements movements and and aa few few long-distance long-distance movements, movements, pattern, but needs needs to to be be tested tested more more rigorously with with empirical empirical data. data. but When P. argus argus were were released released into an extensive extensive area area of of empty empty habitat, habitat, When adult adult P. it was found found that that the the distribution distribution of of migration distances distances was was a close close fit fit to to a exponential (O. (0. T. Lewis, Lewis, C. D. Thomas Thomas and and J. K. Hill, Hill, unpublished). unpublished). negative exponential However, individuals individuals that that moved moved the longest distances distances may have left left the whole However, the longest may have the whole area, and and hence hence the the tail tail of of the the distribution distribution is likely likely to to have have been been underesunderes study area, timated. Mark-release-recapture Mark- release-recapture work work has has shown shown that that the the between-patch between-patch distridistri timated. H. ccomma ( Hill et et al., al., 1996) 1 996) fits a negative negative power power bution of of distances distances moved moved by H. bution o m m a (Hill function well, function M = zD -~,
(1) (1)
M iiss the the fraction fraction of of individuals individuals reaching reaching distance distance D and zz and and k are are concon where M where D,, and For H. H. ccomma, negative exponential stants. For o m m a , the negative exponential distribution distribution gives a slightly poorer poorer fit to the the data ( 1 ); in particular, particular, the negative negative exponential exponential underesunderes fit data than does Eq. (1); timates the proportion of of butterflies butterflies that fly relatively long distances. We We surmise timates that this tail of of the distribution distribution is generated generated either either by individuals that change change habitat, behavior during migration, after after they have initially failed to locate locate new habitat, or by individuals that are are inherently dispersive. When appropriate, this alternative alternative individuals that When appropriate, assumption about about migration could easily be incorporated into spatially explicit explicit models, and would presumably generate generate more long-distance colonizations than and would negative exponential exponential distribution. the negative distribution. 1 994) assumed, with In the absence absence of of good field data, Hanski and and Thomas Thomas ((1994) proportion of of individuals emigrates from each local misgivings, that a constant proportion population. Empirical data now show that the fraction of individuals emigrating are relatively high when patches are small and have high perimeter-to-area perimeter-to-area ratios 996; Sutcliffe et 996c; Kuussaari et et al., al., 1996; 1 996; M. Baguette and ((Hill Hill et et al. al.,, 11996; et aI. al.,, 11996c; G. Neve, N~ve, personal personal communication; see Kareiva, 1985). 1 985). Therefore, it would be desirable to incorporate area-dependent area-dependent emigration rates in spatially explicit meta metacult to calculate calculate the actual population models. Unfortunately, it is very diffi difficult actual (as opposed to relative) rate of emigration in relation to patch area, because emigrants are rarely detected unless they immigrate into another habitat patch; substantial numbers of of emigrants may fail to arrive in any patch. Since the spread spread of of mi miindividuals is neither entirely random (migrants may be attracted to new grating individuals patches from some distance, and may then then stay stay there) nor nor entirely directed, it may not be feasible to estimate the fraction lost accurately. C. D. Thomas, O. T. Lewis, and and 1. J. K. Hill (unpublished) have used a sim simulation approach in order to estimate emigration rates (see also Buechner, 11987; 987; 987). We took imaginary habitat patches with a range of areas Stamps et et al. al.,, 11987). and placed butterflies in random locations within those patches. Butterflies were then allowed to migrate, by making each move away from its origin at a random = per generation) angle, and for a distance chosen at random from the empirical ((~ distribution of migration distances that we had already recorded recorded for P. P. argus a r g u s and
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H. H. comma. comma. The The fraction of individuals leaving the the patch was then then recorded. recorded. These estimates i ) In reality, individuals may estimates of emigration emigration rate rate have two main biases: ((i) perceive perceive patch patch boundaries boundaries and be reluctant to leave (we assumed that boundaries were fully permeable), ii ) permeable), leading to an overestimate of of the emigration rate, and ((ii) the proportion of of individuals moving long distances may bbee under-represented under-represented in the empirical empirical distributions of of migration distances, distances, leading to an underestimate underestimate of of the the emigration fraction. These two biases biases act act in opposing directions. In any case, case, this exercise produced a strong and and negative relationship between between patch area and the fraction of of individuals emigrating, with most individuals individuals emigrating from from small patches with high perimeter-to-area perimeter-to-area ratios. For a given area, the emigration rate differed considerably between ecting differences between the two species, refl reflecting differences in dis dispersiveness. persiveness. The implications -area relationships implications of of such emigration emigration-area relationships have not yet been explored explored in the context context of metapopulation models, but they are likely to be im important. Within Within central parts parts of metapopulations, high perimeter-to-area perimeter-to-area ratios ratios also result in high per-unit-area per-unit-area rates of of immigration into small patches and potentially 994; H. H. comma, 1 996; to high local densities ((Hanski Hanski and Thomas, 11994; comma, Hill et et at. al.,, 1996; 994). However, the consequences M. M. cinxia, cinxia, Hanski et et al. al.,, 11994). consequences are are much more more sig signifi cant when nificant when habitat patches are are relatively isolated. Changes in population size in isolated patches (c. D. Thomas, patches with no immigration can be described described by (C. O. T. Lewis, and J. K. Hill (unpublished) O. T. Lewis, and J. K. Hill (unpublished) N,+ 1 = e N t e'(1-RN'/K),
(2)
where where Nt N, is the number number of of individuals in generation t, rr is the intrinsic rate of of population population increase, R is the proportion of of individuals which are resident in the local population (proportion 11 - R emigrates), and K is the local carrying ca ca+ In R/r) pacity. The equilibrium population size in this model is (K/R)( (K/R)(11 + R/r) and extinction takes place reproduction fails place when In R/r R/r < < - 11,, that is, when local reproduction to replace replace losses due to emigration. Furthermore, Furthermore, even when isolated populations can sustain emigration losses, depression of of local population size by emigration would make the populations more susceptible to other causes of of extinction. Incorporating the emigration -area relationship emigration-area relationship described described above in the model, independent estimates of rr and K, allows us to predict the expected expected and using independent population population size of of H. H. comma comma in habitat habitat patches patches which are isolated enough for immigration to have little effect on local population size, but not so isolated that colonization is unlikely ((Fig. Fig. 5). In Fig. 5, we assume that emigrants for that generation have already left at the time of of census, but this assumption makes little difference difference to the overall pattern. pattern. The good match match between between the model predictions of apparently apparently suitable and empirical data suggests that many isolated patches of habitat are not populated because the losses due to emigration are too high for 985). The match is equally good for local reproduction to match (see Kareiva, 11985). P. argus, P. argus, though much smaller smaller patches can be populated populated in isolation by this less dispersive dispersive insect. These These results are are encouraging and and suggest that that the immigration immigration- emigration balance balance may determine patch occupancy patterns to a previously un unexpected extent. At the same time, as these patterns can also be explained explained by -
Chris and Ilkka IIkka Hanski Chris D. D. Thomas Thomas and Hanski
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Predicted logj() patch area (ha) Cha) for Hesperia Predicted and actual population population sizes in relation to log~o for Hesperia comma. Thin line, predicted predicted population population size in the absence absence of emigration emigration and stochastic stochastic extinction. extinction. Thick line, predicted predicted patch patch occupancy and population sizes, with carrying capacity reduced reduced by area areadependent emigration, emigration, according to Eg. Eq. (2) (adult butterflies were assumed to have emigrated at the popUlation sizes to allow zero values time of of census). census). One was added to all measured measured and predicted population values - ) and 982 (A). (A). Patches plotted to be plotted. Measured population sizes in 1991 plotted. Measured 1991 C(11) and 11982 plotted were were > > 0.6 to 5 km from the nearest Cother) (other) population, population, which are regarded regarded as sufficiently isolated that immigration is likely to have negligible effects on local density, but not so isolated isolated that sites could not be colonized migrants (from C. D. Thomas, O. T. Lewis, and J. K. Hill, unpublished). by occasional migrants unpublished).
alternative hypotheses (area-dependent extinction rate for reasons other other than em emigration), critical field experiments would be welcome. Area-dependent Area-dependent emigration rates have important dynamic implications. For example, successful invasion of empty patch patch networks is expected expected to proceed proceed by the colonization of large patches first first (as observed in H. H. comma; c o m m a ; C. D. Thomas and Jones, 11993), 993), or by stepwise colonization of small patches, gradually eroding isolation before before popUlation population establishment is possible. Loss of of a part of an isolated patch may result in rapid population extinction. Finally, the area of habitat needed to support a single isolated population is much larger than the minimum area of of habitat habitat that that can be populated within a metapopulation where immigration roughly equals emigration in each patch.
B. Deterministic Deterministic Population Population Responses Responsesto Habitat Change Change Perhaps the most serious shortcoming of metapopulation theory has been been the general assumption that the distribution of suitable habitat remains constant through time. Many butterfl butterflyy species occupy successional vegetation. Even when habitats are potentially permanent, landscape changes brought about by human activities may drive signifi cant changes in species distributions (Arnold, 11983; 983; significant J. A. Thomas, 11991; 99 1 ; C. D. Thomas, 11994a,b,c; 994a,b,c; New 1 995). For example, New et et al., al., 1995). many local populations of fritillary butterflies inhabiting woodland clearings clearings in
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the United United Kingdom Kingdom have have gone gone extinct extinct because because of of successional successional changes changes in in the the the vegetation, not not because because of of stochastic stochastic fluctuations fluctuations in in local local population population size size (Fig. ( Fig. 6; vegetation, Warren, 1991; 1 99 1 ; Warren Warren and and Thomas, Thomas, 1992). 1 992). Reviews Reviews of of local local and and regional regional exex Warren, tinctions have have argued argued that that successional successional changes changes in in vegetation, vegetation, changes changes in in human human tinctions management of of surviving surviving habitat habitat fragments, fragments, and and outright outright habitat habitat loss loss are are principrinci management pally responsible responsible for for most most extinctions extinctions of of substantial substantial butterfly butterfly populations populations in in modmod pally em landscapes; landscapes; extinctions extinctions are are frequently frequently aa deterministic deterministic consequence consequence of of the the dede em terioration of of local local breeding breeding conditions conditions (Harrison, ( Harrison, 1991, 1 99 1 , 1994b; 1 994b; Warren, Warren, 1993; 1 993; terioration 1. A. A. Thomas, Thomas, 1991; 1 99 1 ; J. A. Thomas Thomas and and Morris, Morris, 1994; 1 994; C. C. D. D. Thomas, Thomas, 1994a,b,c, 1 994a,b,c, J. 1 996). Similarly, most most colonizations colonizations appear appear to take take place place when when environmental environmental 1996). improve locally (C. (c. D. D. Thomas, Thomas, 1994a,b,c, 1 994a,b,c, 1996). 1 996). The The spatial spatial dynamdynam conditions improve of many species appear appear to be driven driven by the the changing changing distribution distribution of of their their ics of habitats. These These insects insects are habitat mosaic. mosaic. habitats. are tracking a shifting shifting habitat Examples described this chapter chapter clearly illustrate the the importance importance of of stosto Examples described in this clearly illustrate chastic extinctions "traditional" metapopulation dynamics are extinctions too, but these "traditional" are su superimposed on a dynamic habitat habitat mosaic. mosaic. When When the the dynamics of of the the butterfly are fast relative to vegetation dynamics, ignoring the latter may still leave a good fit between between observed and predicted predicted species distributions distributions in the short short term. This may insects, with their fast dynamics, have become popular popular subjects be one reason why insects, for for metapopulation studies. However, these models may not be particularly useful for predicting long-term trends or long-term probabilities of persistence if longlong term changes are determined by underlying vegetation dynamics. The task of of superimposing stochastic dynamics of a butterfly on top of vegetation dynamics has has hardly hardly begun. begun. We give two examples examples to show how butterfly metapopulation dynamics dynamics may lag behind behind changes changes in the spatial distribution of suitable suitable habitat/vegetation. After earlier declines, H. H. comma c o m m a has recently enjoyed an increase in the amount and extent of suitable habitat in southern England, particularly in the late 11970s 970s and early 11980s 980s (described above). After After 9 years of documented colonizations, this butterfl y had still not colonized all of the "new" habitat available to it, and it is butterfly likely that the distribution will take several more decades to reach an equilibrium (C. D. Thomas Thomas and and Jones, 11993; and Thomas, 11994). (c. 993; Hanski and 994). Over the past 1100+ 00 + years, there has been no period of 330 0 years or more when the distribution of H. comma c o m m a habitat has remained even approximately stable, stable, and and it is hard to imagine that it will remain stable over the next 30 years. At least in modem landscapes, specialized specialized species may be continuously continuously chasing after after their their habitats. habitats. Another Another common common scenario scenario may be continuing continuing loss of of habitat. In parts parts of of A land, M. Aland, M. cinxia cinxia habitat has been lost in recent decades, decades, and and predictions of spa spatially explicit explicit models suggest suggest that the dynamics may not be at equilibrium equilibrium in all areas. O in Hanski areas. Figure Figure l10 Hanski (this volume) illustrates such such nonequilibrium dynamics with land. Within with an an example example from from M. M. cinxia cinxia on on A Alan& Within an an area area of of ca ca 25 25 km km22,, the the total total area area of of suitable suitable habitat habitat has has declined declined to to one-third one-third and and the the number number of of habitat habitat patches patches has 5 - 20 years. has decreased decreased from from 55 55 to to 42 42 over over the the past past 115-20 years. The The metapopulation metapopulation of of M. M. cinxia cinxia is is predicted predicted to to have have followed this this decline rather rather rapidly, apparently apparently
Chris D. D. Th0m0s Thomas 0nd and Ilkk0 IIkko H0nski Honski Chris
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o ,
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FIGURE66 FIGURE
the distribution of Mellicta athalia in response to rotational cUlling cutting (cop (copChanges in the pieing) of deciduous woodland in Blean Woods, Kent, England. Numbers show year of cUlling; picing) cutting; shaded areas areas indicate indicate the distribution distribution of adult adult bUllerfties. butterflies. Glades (GL) and wide rides (WR) were sporadically used for for breeding. Reprinted from from Warren, 11991, from Elsevier only sporadically 99 1 , with kind permission from Kidlington OX5 11GB, Science Ltd, The Boulevard, Langford Lane, KidlinglOn GB, UK.
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because because it occupied most of of the habitat habitat before patch patch reduction, reduction, and most of of the dynamics has occurred in small patches patches with fast turnover. turnover. However, an entirely different picture is predicted predicted if if the area area of of each patch patch is halved again over the next 20 years. In this case, further loss of habitat is further of habitat predicted predicted to lead to a network Fig. 10 1 0 in Hanski, this vol smaller than that required required for for long-term persistence ((Fig. volume). However, However, the actual extinction of the metapopulation metapopulation is predicted predicted to take a long time, and for for tens of of years we would see a metapopulation metapopulation slowly but inevitably oscillating to extinction. The final decline to extinction extinction. extinction is slow because the last populations to go are typically the largest ones with the smallest risk of of local extinction. This latter latter finding is especially worrying, in that that the status of of many or even most species in landscapes which are are gradually being degraded may presently presently be "better" Hanski, 11996b; 996b; Hanski et "better" than than the expected expected status status at equilibrium equilibrium ((Hanski, et al., 11996b,c). 996b,c). Nonequilibrium systems of of this type may lead us to conclude conclude that that po potential tential colonization distances distances are are greater than than they really are are and to overestimate local population persistence persistence in small patches. These biases result in overestimates overestimates of of metapopulation metapopulation lifetimes. Some currently surviving metapopulations metapopulations may be doomed doomed even if all further habitat loss is prevented.
C. Habitat Heterogeneity Habitat heterogeneity is probably crucial to the persistence persistence of of many local butterfly populations populations and metapopulations. metapopulations. For convenience, convenience, there there is a strong tendency in metapopulation metapopulation ecology to define some parts of of the landscape as "habitat" "matrix" - and to ignore the latter. "habitat" and the remainder as "nonhabitat" "nonhabitat" or "matrix"--and There are several problems with this (see also Wiens, this volume). ((i) i ) The en environment may exist as a series of of habitats which vary in suitability, and it may be difficult to distinguish between Rodriguez et al., 11994). 994). between habitat habitat and nonhabitat nonhabitat ((Rodrfguez Typically, not Hochberg et al. not all suitable habitat will be of of equivalent quality ((Hochberg al.,, 11992, 992, 11994; 994; C. D. Thomas, 11996). 996). (ii) (ii) Vegetation dynamics and human human activities change the suitability of of a given habitat patch; some of of these changes changes will be can change predictable (succession in a woodland clearing clearing resulting in a gradual decline decline in patch quality for for clearing clearing butterflies), butterflies), but other other changes changes may be unpredictable unpredictable and reversible. iii) Temporal environmental variability may affect an entire en reversible. ((iii) environmental gradient. For example, example, the warmest and driest parts of of a habitat habitat patch may represent represent the environmental optimum in normal years, but be inhospitable in a drought year, requiring the drought-affected veg drought-affected species to move into taller vegetation (H. comma, 994a) or to more mesic slopes (E. editha comma, C. D. Thomas, 11994a) editha bayensis, 988; the large blue Maculinea Maculinea arion, bayensis, Weiss et aI. al.,, 11988; arion, J. A. Thomas, 1 988) found that personal communication). Singer ((1972) 1 972) and Weiss et et al. ((1988) that com complex spatial variation in the microdistribution of of disturbance disturbance and of two host plant species, and the aspect of of the slope, interacted interacted with climatic variation to affect population persistence editha bayensis pop persistence and changes in population size in E. editha bayensis populations on serpentine serpentine grassland. Since a greater range of of microhabitats microhabitats is more
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Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski
likely to be present present in large large than than in small areas, areas, this may be an extremely im important habitat fragfrag portant reason reason why why populations populations are are often often most most persistent persistent in in large large habitat ments. ((iv) iv) Some occupied habitat patches patches may not be suitable for for population because of persistence; populations populations may be present present in these habitats habitats (sinks) only because of 994; immigration from source populations (Pulliam, 988; Rodriguez (Pulliam, 11988; Rodrfguez et et al., 11994; Warren, 994). Warren, 11994). Adding Adding realistic variation in habitat quality is likely to be one one of of the the key key issues facing empirical and theoretical metapopulation biologists in coming facing coming years. The 996) is almost certainly unusual, unusual, The following example (C. D. Thomas Thomas et et al., 11996) but nonetheless nonetheless highlights the the potential potential complexity of of metapopulation dynamics when more than than one habitat habitat type is present. A metapopulation of of the checker checkerspot E. editha editha (an unnamed race which differs differs from bayensis) bayensis) occurs occurs at 2000 2000 to 3000 m elevation in openings in coniferous coniferous forest in Sequoia National Forest and Sequoia 967, the butterflies butterflies were restricted Sequoia National National Park Park in California. Before 11967, to natural of their eggs on Pedicularis Pedicularis semibarbata natural rocky outcrops and laid most of semibarbata (Scrophulariaceae). 967, clearings (Scrophulariaceae). Around Around 11967, clearings in the forest were made made by logging. Pedicularis in Pedicularis semibarbata semibarbata disappeared disappeared from clear-cut clear-cut areas, areas, but the the butterfly invaded this habitat and colonized Collinsia Collinsia torreyi, torreyi, which is also in the Scrophu Scrophulariaceae, 1 983; Singer 1 993, lariaceae, but which is not used on outcrops outcrops (Singer, 1983; Singer et et al. al.,, 1993, 11994). 994), By 11985, 985, a patchwork of 00 + km km22,, of host use use had been been established established over over 1100+ with P. semibarbata semibarbata as the principal host on unlogged outcrops and C. torreyi torreyi as the principal host in clear-cuts. Clear-cuts 980s. Although Clear-cuts acted acted as as population population sources sources during during the the 11980s. Although the the clear clearcut habitat received received fewer eggs, it generated generated more adults due to higher higher survival there 983; Moore, 11989). 989). The there than than on outcrops outcrops (Singer, 11983; The butterflies moved from clear-cut to outcrop about twice as frequently as they moved in the opposite direction. Biased Biased movement generated generated a gradient in insect density, such that em emigration from clear-cuts (c. D. Thomas clear-cuts raised insect densities on nearby nearby outcrops outcrops (C. Thomas et 996). et al. al.,, 11996). Then, Then, a severe summer summer frost killed virtually all of of the C. torreyi torreyi in the clear clear992 (Singer 994). Although E. editha cut habitat habitat in 11992 (Singer et al. al.,, 11994). Although E. editha eggs and larvae larvae were were not damaged damaged by the cold, the larvae starved. The populations populations in this habitat de de1 992 to two in 1993, 1 993, even though C. torreyi clined from - - - 11 04 0 4 egg batches batches in 1992 torreyi regenerated 1 993. Pedicularis Pedicularis semibarbata semibarbata was regenerated in abundance abundance in all clear-cuts clear-cuts in 1993. unaffected 992 frost, unaffected by by the the 11992 frost, E. editha editha survival survival was was apparently normal normal on on out outcrops, and there there was no mass mass extinction in this habitat. habitat. Extinction of of the source populations populations set up a fascinating fascinating natural experiment. In the sourcesink theory, sources are areas source-sink areas of of habitat which generate generate individuals and sinks consume consume them; sinks are are areas which are are populated populated because because there is a net influx of of migrants migrants into the habitat, and they are predicted to become extinct in the absence 988). "Pseudosinks" absence of of immigration (Pulliam, 11988). "Pseudosinks" are are areas areas which can support support a population population without immigration, but where where immigration immigration increases increases population population density density above above the the local local equilibrium; equilibrium; removal removal of of immigration immigration should should result in a decline in density to the local carrying capacity rather rather than than in extinction
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(Holt, 985; Watkinson 1 995). Following sudden extinction extinction (Holt, 11985; Watkinson and Sutherland, Sutherland, 1995). Following the sudden of overall egg egg densities of the population population sources, sources, overall densities fell on P. semibarbata, semibarbata, the decline decline was greatest 993 densities greatest close close to former former population population sources, sources, and and the 11993 densities on out outcrops crops were were no no longer longer correlated correlated with with isolation isolation from former former sources. sources. In this case, we know pseudosinks, and know that that natural natural outcrops outcrops were were pseudosinks, and not not true true sinks, because because E. editha editha populations populations occurred occurred on outcrops outcrops before before the the clear-cut clear-cut habitat habitat was was created, created, they survived on on outcrops outcrops after the frost, frost, and and they persisted persisted throughout throughout the study period period at moderate moderate abundance abundance on undisturbed undisturbed outcrops outcrops in Sequoia National National Park, Park, et al., 1 996). to the south of our disturbed study sites (c. D. Thomas south of our disturbed (C. Thomas 1996). Source Source populations populations are often considered considered especially important important for for metapop metapopulation ulation persistence persistence -they m they clearly are are when when long-term long-term survival is impossible impossible in sink habitats, as has has been been shown shown for for the blue blue butterfly Cyanaris Cyanaris semiargus semiargus (Rod (Rodriguez 994). However, matters to rfguez et al. al.,, 11994). However, it is resistance resistance to extinction extinction that that really really matters persistence, persistence, not not the balance balance of of birth birth and and death death in "normal" "normal" years. In this particular particular pseudo sinks were example, pseudosinks were more more resilient resilient to a particular particular type of of extreme extreme envi environmental ronmental event, although although it may well be that that in most most metapopulations metapopulations the the sources sources are are usually the more more resilient resilient to environmental environmental extremes. extremes. Empirical Empirical evidence is lacking. for lacking. Both Both source source and and pseudosink pseudosink populations populations may may be prone prone to extinction extinction for all of of the reasons reasons given in Section Section lILA. III.A. Unfortunately, Unfortunately, there is almost almost no no em empirical information between local population population information with which which to assess assess the relationship relationship between productivity in a "normal" "normal" year, and ability to survive environmental environmental extremes. extremes.
D. Spatial Spatial Synchrony Synchrony in Population Population Dynamics Dynamics When uctuate in synchrony, When populations populations fl fluctuate synchrony, and and particularly particularly when when they they become become extinct much extinct in synchrony, synchrony, the probability probability of of metapopulation metapopulation persistence persistence may be be much lower Hanski, 1991). 1 99 1 ). When When the chance lower than than predicted predicted by standard standard models models ((Hanski, chance of of extinction the probability extinction is completely independent independent in each each local population, population, the probability of of metapopulation metapopulation extinction extinction rapidly rapidly declines declines with increasing increasing number number of of local pop populations Fig. 4; Hanski, 11991, 99 1 , this volume), ulations ((Fig. volume), but but if if extinction probabilities probabilities are correlated, correlated, for for instance instance because because local popUlations populations are responding responding in a similar similar way to climatic variability, metapopulations popu metapopulations with even even large large numbers numbers of of local populations may Hanski, 11991, 99 1 , this may be susceptible susceptible to extinction extinction ((Hanski, this volume). volume). There There are examples examples of of synchronous synchronous butterfly extinctions extinctions in response response to single climatic Ehrlich et al., 11980; 980; C. D. Thomas 1 996; above) events events in E. editha editha ((Ehrlich Thomas et al. al.,, 1996; above) and and Aphantopus hyperantus Pollard and Yates, 993; Sutcliffe 996b), and and Aphantopus hyperantus ((Pollard Yates, 11993; Sutcliffe et al. al.,, 11996b), Hanski, evidence evidence that that extinction extinction probability probability varies varies between years in M. M. cinxia cinxia ((Hanski, this volume). As As yet, these these events events have have rarely been been shown shown to cause cause metapopula metapopulalarge metapopulations metapopulations or tionwide extinction, extinction, but but the above above examples concern concern large or ones habitat ones which contain contain either either some some very large patches patches or or some some relatively safe safe habitat number of type. The The example in which which B. aquilonaris aquilonaris became became extinct from from a number of forest patches, after forest fragments fragments that that contained contained fewer fewer than than 20 habitat habitat patches, after a wet and and cloudy summer summer (T. Ebenhard, Ebenhard, personal personal communication, communication, above), above), shows shows that that ex extreme environmental environmental events events can cause cause entire entire metapopulations metapopulations to become become extinct.
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Over Over longer longer time time periods, periods, landscape landscape and and habitat habitat changes changes that that cause cause deterministic deterministic Hesperia extinctions have have often often been been relatively relatively synchronized synchronized over over large large areas. areas. Hesperia extinctions comma comma and and Lysandra Lysandra bellargus bel/argus both both showed showed aa period period of of rapid rapid decline decline in in England England when when myxomatosis myxomatosis killed killed rabbits, rabbits, and and their their short-grass short-grass habitats habitats disappeared disappeared throughout the the UK UK (J. A. Thomas, 1983a; 1 983a; C. D. Thomas Thomas and and Jones, Jones, 1993). 1 993). SimSim throughout economic pressures pressures and and technological technological innovations innovations that that cause cause changes changes in ilarly, economic farming or or forestry forestry practices practices usually usually do do so so over over very large areas areas in a relatively farming very large short short space space of of time, causing causing widespread widespread changes changes in the the fortunes fortunes of of associated associated The information information that that exists at the the moment moment suggests suggests that that extinctions extinctions are are species. The least partially synchronized. synchronized. normally at least the absence absence of of better better data data on on the the spatial spatial synchrony of extinctions and and In the synchrony of colonizations (see also also Hanski, Hanski, this this volume), volume), we we must must rely on on analyses analyses of of extant extant colonizations populations, and and presume presume that that different different levels of of synchrony synchrony in in their their dynamics dynamics populations, provide some some insight insight into into the the extent extent to to which which local local extinctions extinctions might might be be synsyn provide chronized Analyses of of butterfly, butterfly, moth, and aphid population popUlation chronized over over wide areas. Analyses and aphid over wide wide areas (Britain) suggest popUlations fluctuate in synsyn dynamics over areas (Britain) suggest that populations chrony over areas of at least 105 1 05 km km22 ((Pollard Pollard and Yates, Yates, 1993; 1 993; Hanski Hanski and and chrony over areas of ' s metapopuWoiwod, 1993), 1 993), which which is orders orders of of magnitude magnitude greater than anyone metapopu Woiwod, greater than anyone's For many of of these species, correlated correlated population popUlation fluctuations fluctuations lation study areas. For occur over over areas much larger distances occur areas that that are are so much larger than than their potential migration distances of year-to-year that climate must be a major major determinant determinant of year-to-year population population variability ((Pollard Pollard and Yates, 1993, 1 993, and references therein). These These conclusions conclusions are based and references are based on counts counts of others) from from on of insects (transect (transect counts counts for for butterflies, traps traps for for the others) scattered locations locations across However, each point widely scattered across the landscape. However, each sampling point together insects from more population, may to some extent extent lump together more than than one one local population, and local popUlations populations may be fluctuating partly out of of synchrony. In the case of of the = ~ 11-- to 3-km butterfly transect transect walks, several habitats habitats are are sampled. For aphids aphids and moths, traps traps may attract insects from more more than than one habitat. If If each sampling location counts insects from more than than one habitat habitat patch, local population population vari variability may have been been averaged out, leaving only residual large-scale large-scale variability caused caused by by the climate climate to be be detected. detected. Studies of of population population fluctuations at a smaller scale provide provide a rather different different picture. Small-scale analyses are possible for but for butterflies because the British butterfly transects are usually divided into about 10 1 0 sections, and separate counts are made for each section. Populations Populations in individual sections often fluctuate in parallel with regional fluctuations, apparently because of of weather weather effects, but changes in local habitat management can cause deviations from regional trends (Pollard and Yates, 11993). 993). When local habitats improve in quality, the change in local popu population size is upward upward relative to the overall regional trend, and downward downward when local habitats deteriorate ((Pollard Pollard and Yates, 11993). 993). Such deviations are partic particularly clear in species which are associated with successional vegetation. Plebejus argus fluctuates out of synchrony in areas which are only 500 m apart on suc successional habitats 99 1 ), and M. athalia, which habitats in heathland (c. (C. D. Thomas, 11991), inhabits inhabits freshly made made clearings in a forest in southeast southeast England, increases in new
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others as they become overgrown ((Warclearings, but simultaneously declines in others War ren, 11987b, 987b, 11991). 99 1 ). There are good and bad years for these species over wide areas, associated associated with climatic fluctuations, fluctuations, but local populations behave idiosyn idiosynareas, response to local habitat conditions. cratically in response Even in nonsuccessional species, analyses of local population fluctuations show that there is a great deal of heterogeneity in local dynamics, often but not (mialways associated with different responses of local populations in different (mi cro)habitats to annual variation in the climate (Ehrlich et 975; Sutcliffe et et al., al., 11975; et al.,, 11996a,b; al. 996a,b; see Fig. 2 in Hanski, this volume). Such heterogeneity in local E. editha editha above, dynamics may be crucial to long-term persistence. As found with E. habitat in most years may be crucial to per perwhat appears to be relatively poor habitat sistence after some extreme environmental event The scale over which systemwide extinction is likely to take place as a result of extreme events is one of the most crucial, but least well understood, aspects of metapopulation biology and needs to be addressed by long-term and large largescale field studies.
Conclusions VI. Conclusions Many of the predicted predicted patterns patterns and processes are widely observed in studies of butterfly metapopulations. to realize that a key of metapopulations. However, we have also come cometo identify the critical habitat habitat requirements requirements of of different different empirical challenge is to identify species and the factors causing changes in the distribution of habitats. Species individualistic habitat habitat and host plant plant requirements, requirements, hence the habitat patch have individualistic networks available to each species are specific and not generally congruous with human of general for each human definitions of general vegetation type. The specific habitat habitat mosaic for modem, human-dominated human-dominated landland species is likely to be dynamic, especially in modern, scapes, distributions of scapes, and changes in the distributions of species species are often often driven by spatial changes in the distribution of suitable distribution of suitable habitats. habitats. Populations in large habitat habitat patches patches have have low rates of of extinction for Populations for several reasons, including size, high habitat heterogeneity, and including large initial population population size, low risk of extinction from habitat dynamics. There is also some evidence of extinction evidence to show that isolated isolated local populations populations are relatively prone to extinction. extinction. ColoniColoni zation probability probability is determined determined by isolation, zation isolation, by the the sizes of of source populations, populations, and of the the patch be colonized colonized (large patches more likely and also by the size of patch to be patches are are more become colonized). colonized). The The dynamic dynamic processes processes of of extinction and colonization colonization can can to become extinction and thus generate generate the the widely widely observed observed pattern pattern in which which large large patches patches that that are located located close to each each other other are are likely to to be populated populated but but in which which small small and and isolated isolated patches are are usually usually empty. It appears appears that that the the flow of of migrants in and and out out of of patches habitat habitat patches patches is also an important important determinant determinant of of patch patch occupancy occupancy and and local population sizes, sizes, and and this needs to to be be addressed addressed more more specifically specifically in the the next population generation of of models. models. Considerable spatially realistic Considerable progress progress has has been been made made with with spatially realistic simulation simulation modmod els, which which have have successfully successfully predicted predicted the the observed patterns patterns of of patch patch occupancy
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based on the dynamic processes of of extinction and colonization. colonization. These models models based have predicted persistence persistence where where metapopulations metapopulations do survive and extinction extinction where they do not, and qualitatively, the models have predicted which empty networks networks of habitat habitat patches patches butterflies will invade invade and which networks they will between model predictions and fi field not invade. Quantitative differences between eld data have been useful in revealing where further biological information is required, required, for for example example on migration and the effects of habitat habitat quality. Use of the models in specific systems has also helped us to identify some general problems, including the need for for butterfly population population dynamics to be superimposed superimposed on shifting habitat mosaics. We return now to two serious issues which which have not yet been settled and pop where much more information is required. required. The first relates to migration and population structure, and the second to the importance of of specifi specificc habitats to persist persistence. Metapopulation Metapopulation research research has stimulated a considerable considerable reappraisal of of the migration capacities of butterflies. Results that are presently available suggest that the notion of of a local population, which is at the core of metapopulation ideas, may be under under threat. Exchange rates of individuals among among adjacent but distinct habitat habitat patches may be so high (sometimes > > 20%) that that local populations have only limited demographic independence. independence. If the population population in an individual patch includes many immigrants, stochastic breeding failure will not result in extinction of i ) the habitat im of that "local" "local" population population unless ((i) habitat changes in such a way that immigrants no longer enter or remain in the patch or (ii) regional stochasticity produces patches, thus interrupting produces simultaneous breeding breeding failures in a group of of patches, immigration. Observed local extinction rates may therefore be much lower than 995b; Hanski, this volume). Yet, even when the underlying rate (Han ski et (Hanski et at., al., 11995b; dea metapopulation consists of an assemblage of such populations with a high de gree of popUlations at of connectance, connectance, direct interactions interactions (migration) between between local populations the opposite ends of the same network im network may never occur. In such systems, immigration and emigration are important important determinants determinants of of local dynamics, but the whole network network is certainly not one panmictic population. Migration is vital to local dynamics as well as to metapopulation-wide metapopulation-wide processes. Most current meta metapopUlation population thinking thinking (if not modeling) limits the role of migration to one of of seeding empty habitat patches (with little effect of of immigration on abundance abundance after col colonization), and to a lesser extent extent as propping up small local populations which are under threat threat from stochastic extinction (rescue effect). The role of migration in local dynamics needs to be explored more fully in structured structured metapopulation metapopulation 1 992; Hanski and Gyllenberg, 1993; 1 993; Gyllenberg models (Gyllenberg and Hanski, 1992; et popUlation variavaria et at. al.,, this volume) and through analyses of of spatial patterns in population bility, Hanski, this volume). bility, colonization, colonization, and and extinction extinction ((Hanski, volume). Because Because emigration and and immigration rates vary with patch size and and isolation, real metapopulations metapopu metapopulations do not fall easily into the various categories categories of of metapopulation types ((Harrison, Harrison, 11991; 99 1 ; Harrison and Taylor, this volume). In some parts of of a patch patch network, persistence may largely depend on the existence of one or a few large blocks of habitat habitat (mainlands), but other parts of the same system may persist
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because there there is a high density of small patches. patches. Parts Parts of of metapopulations with a high density of of small patches, each with a high emigration and immigration rate, resemble scaled-down versions of of the population structure of of highly mobile spe species ((patchy patchy populations, sensu 99 1 ). In mobile species, such as the sensu Harrison, 11991). nettle-feeding nymphalids in Europe, practically no single patch could support a nettle-feeding local population for for more than a few generations in isolation, and each individual will enter and leave several such patches. The distinction between between relatively sed sedentary species which are regarded as existing as metapopulations metapopulations and the more mobile species with "patchy populations" is becoming increasingly vague. In some cases, it is just a matter of of scaling. The relative contribution of of local versus "populations" within metapopuregional population processes in different local "populations" metapopu lations and in different metapopulations is a much more important issue than trying to pigeonhole each system and give it an approved name. General models and fi eld studies exploring these notions would be very useful. field The contribution im contribution of of specifi specificc habitats habitats to persistence persistence is also becoming an important question. Most metapopulation models and fi eld studies examine proba field probabilities of extinction in relation to patch area, local population popUlation size and isolation, but pay limited attention attention to variation variation in habitat habitat quality. Nonetheless, there there is already enough evidence to suggest that the impor the type of habitat can be just just as important. If populations respond differently to environmental stochasticity in different habitats or microhabitats, popUlations against microhabitats, habitat habitat heterogeneity can buffer buffer populations large fluctuations fluctuations and extinction. We should ask whether whether large populations in large habitat patches survive best because they are large, or because large patches patches usually contain several microhabitats; microhabitats; and whether large metapopulations persist because they have much habitat habitat or because they have more kinds of of habitats habitats and microhabitats than 0 patches, than small patch networks? networks? A small metapopulation metapopulation in 110 patches, each of of a slightly different different habitat type, might possibly persist for longer than a metapopulation patches. meta population in 50 identical patches. Allied to this is the question of of whether whether some habitats habitats always hold the key L. bel/argus to persistence. J. A. Thomas ((1983a) l 983a) suggested that L. bellargus may spread spread in good years, but is confi ned to population refuges in bad years. The same argument confined has been put forward for for several mobile species which may breed over large areas at favorable times of year, but retract to specifi specificc habitats at other times (Shapiro, 11979; 979; Jordano et al., 11991). 99 1 ). If this phenomenon et al., phenomenon is widespread, widespread, the existence existence of of specifi specificc habitats habitats wit:lin within patches patches or patch patch networks may be more more important to persistence persistence than patch size or number. An entire program of of empirical empirical research is required to evaluate to what extent populations in different different habitats vary in their responses to environmental stochasticity, whether whether habitat heterogeneity buf buffers populations populations against extinction, extinction, and whether specific habitats habitats hold the key to persistence. Finally, a metapopulation metapopulation approach approach is becoming important important in several other areas of of butterfly population biology. A metapopulation approach approach provides the potential to bridge the gap between between studies of of local population dynamics and species distributions. Densities, Densities, sizes, and average suitabilities suitabilities of of habitat habitat patches
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may vary geographically, but this aspect of metapopulation biology has barely been considered for butterflies. Comparing central and marginal parts of species ranges is almost bound to reveal interesting results (J. A. Thomas, 1993; 1 993; J. A. Thomas et 994). Another area of interest is the extent to which metapopu et ai., al., 11994). which metapopulation structure and migration affect local adaptations to different habitats (C. (c. D. Thomas and Singer, 11987; 987; Thompson, 11993, 993, 11994; 994; Singer and Thomas, 1996; 1 996; Barton and Whitlock, this volume) and levels of of genetic variation (Descimon and Napolitano, 11993a,b; 993a,b; Hedrick and Gilpin, this volume), and whether habitat gege 1 976; Dempster, Dempster et ometry itself affects the evolution of of migration ((Dempster et ai., al., 1976; 11991; 99 1 ; Olivieri and Gouyon, this volume). In an ever changing landscape, evo evolutionary changes may play an increasingly important role in popUlation persist population persistence and extinction.
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TTritrophic ritrophic Metapopulation Metapopulation Dynamics Dynamics A A Case CaseStudy Study of Ragwort, Ragwort, the Cinnabar Cinnabar Moth, Moth, and the Parasitoid ParasitoidCotesia Cotesiapopularis popularis Ed van der Meiiden Meijden
Catharina Catharina A. M. van der Veen Veenvan Wiik Wijk
I. INTRODUGION INTRODUCTION Many short-lived monocarpic plant species have a markedly patchy distri distribution. These species reproduce only once in their lifetime, they typically exploit disturbed habitats, and they often have a high rate of local extinction (Harper, 11977; 977; Gross and Werner, 11978; 978; van Baalen, 11982; 982; Reinartz, 1984; 1 984; Grubb, 11977; 977; de long 988; see van der Meijden et 1 992, for a review). Jong and Klinkhamer, 11988; et al. al.,, 1992, We II-known examples are biennial plant Well-known plant species colonizing colonizing windfalls in wood woodlocalIy grazed or otherwise disturbed vegetation on sand lands, species exploiting exploiting locally dunes and chalk grasslands, and species of "old fields." To survive over long periods of of time, such extinction-prone extinction-prone biennials depend on regional regeneration through interacting local populations populations (metapopulations). (metapopulations). Critical processes for long-term persistence persistence include seed dispersal and dormancy in variable and patch patchdistributed environments (Kuno, 11981; et al. al.,, 11987). of po98 1 ; Klinkhamer et 987). Also of po ily distributed tential importance are biotic interactions with species at higher trophic trophic levels. levels. Often the herbivores and their parasites are monophagous, monophagous, and their populations populations too may function as metapopulations. metapopulations. Dynamics of of species at the higher trophic levels are necessarily affected by dynamics of the host plant, but the herbivores and predators can also play a more active role by modifying the extinction prob probabilities abilities at other trophic levels (Nee et et al. al.,, this volume; Holt, this volume). Metapoplliariofl Metapopulation B/(J/ogy Biology 1997 by Academic Prc�s. Copyright � 9 1997 Press. Inc. All rights of of reproduction reproduction in any form reserved.
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In this chapter, we analyze whether whether and to what extent interactions interactions within a tritrophic system are affected by the spatial distribution of patches. In of habitat patches. doing so, we follow the suggestion of 1 995): "many of Harrison Harrison et al. ((1995): "many populations appear patchy to the human eye, but [that] critical examination examination is required to deduce the dynamic consequences of cally, we review of this patchiness." patchiness." Specifi Specifically, our long-term data data (two decades) on the relationships between between the plant plant ragwort (Senecio jacobaea), jacobaea), its most important herbivore, the monophagous cinnabar cinnabar moth (Tyria jacobaeae), jacobaeae), and the specialist parasitoid of of the herbivore, Cotesia Cotesia
popularis. popularis. Ragwort has been the subject of of intensive studies in several countries over a long time, starting in 11935 935 when Cameron summarized his early biocontrol proj proj"Natural Control of of Ragwort." Ragwort seeds had been ect in a study entitled "Natural accidentally introduced introduced into New Zealand, and the plant had grown into a major pest 874 and 1900. 1 900. pest by colonizing the entire country in a few decades, between between 11874 The wort is undoubtedly based The powerful colonizing and and weedy behavior of rag ragwort on its capacity to efficiently exploit scattered scattered disturbed habitat habitat patches. patches. Ragwort is a "pest" thanks to the alkaloids that it produces, which are toxic to cattle but not to the cinnabar 1 957), which was used as a bioconbiocon cinnabar moth (Harper and Wood, 1957), trol agent. Subsequently, two additional additional long-term population studies of this plant - moth system have been carried plant-moth carried out in the United United Kingdom Kingdom (Dempster, (Dempster, 11982; 982; Crawley 989). Dempster concluded moth's Crawley and and Gillman, 11989). concluded that that "the moth's population is buffered against extinction by the heterogeneity within the habitat," habitat," indicating that that some sort of of spatial effects are important. In the early 1970s, 1 970s, we commenced our studies of ragwort in The Netherlands on three small local dune populations. Within 3 years, two of the three populations had become extinct. As ragwort density in the dune area area as a whole did not continue to decrease, decrease, we became convinced that population dynamics of this species should be studied on a much broader broader scale and that that spatial aspects are crucial for understanding the mechanism of of persistence, persistence, which is the focus of the present chapter. chapter. Apart from the patchy distribution of the plant and, consequently, of the herbivore herbivore and its parasitoid, typical features of of this tritrophic tritrophic system on sand dunes include frequent complete defoliation of plants by the cinnabar cinnabar moth (not only leaves are consumed, but also buds and fl owers, thus reducing ragwort flowers, seed production to zero). The lifetimes of of local plant populations are restricted, restricted, with the cinnabar cinnabar moth and its parasitoid continuously tracking these ephemeral ephemeral populations (van der Meijden, 11979a). 979a). chapter, we will first give an outline outline of of the the population dynamics of of In this chapter, 1 99 1 , 11992) 992) and describe how they the three three organisms (van der Meijden Meijden et al. al.,, 1991, interact interact with each other. Next, we will calculate parameters describing the degree of of synchrony between between local populations and the metapopulation to reveal to what what extent the dynamics of local populations differ differ from each other. We also pay attention to spatial correlations within and between between the three species. Finally, we will discuss the mechanisms that appear of the appear to play a role role in the persistence of the
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plant, the herbivore, and the parasitoid in their patchy patchy environment. This infor infor(Han ski and Gilpin, 11991, 99 1 , mation will be used to infer the type of of metapopulation (Hanski this volume) that best describes the organisms of this tritrophic interaction. tritrophic interaction.
II. MATERIALS AND METHODS MATERIALSAND METHODS A. A. Study StudySystem System Ragwort is a facultative facultative biennial plant. It is native to Europe and has invaded overgrazed 957; Dempster, 11982; 982; overgrazed areas throughout the world (Harper (Harper and Wood, 11957; van 979b). Its weedy character van der der Meijden, 11979b). character is largely due due to its extremely powerful powerful reproductive reproductive potentials. Individual plants may produce up to 20,000 seeds with pappus 1 940) refer to pappus that that enable wind dispersal. dispersal. Poole and and Cairns ((1940) per plant plant ranging from 50,000 to 1150,000 seed numbers per 50,000 in New Zealand. Mowing, plowing, and other other such conditions conditions that that reduce reduce the opportunities opportunities of of generative reproduction stimulate vegetative reproduction (Poole and Cairns, 11940; 940; Harris et al. , 1 978). Even in vivo, small root fragments may develop into mature plants al., 1978). (van der Meijden, 11979b). 979b). Seeds may remain viable for more than 8 years (Poole and Cairns, 11940). 940). Ragwort is a common weed weed of sand sand dunes, dunes, roadsides, and and waste lands. Local disturbances disturbances create suitable circumstances for for establishment, establishment, whereas vegetation of suitable growing sites. Often, however, populations succession may lead to loss of disappear disappear without any changes in the vegetation. Such sites may become recol recolonized at a later later time. The cinnabar cinnabar moth is a univoltine insect and monophagous on ragwort (else (elsewhere vulgaris; where it has been reported reported to occasionally use the closely related related Senecio Senecio vulgaris; Aplin and Rothschild, 11972). 972). It lays its eggs in small batches of of ca. 30 eggs. (Dempster, 11982). First or second second instar Fecundity varies from l100 Oa to 400 eggs (Dempster, 982). First po larvae may become parasitized by the specialist braconid parasitoid Cotesia Cotesia popularis. S parasitoid larvae may develop per host larva. These larvae leave pularis. Up to I15 their host shortly before before it would otherwise have pupated, and the host then dies. From the the third instar onward larvae show a tendency to disperse. This This is especially and fifth instar instar when many larvae larvae leave their original original food plant so in the fourth and before it is fully defoliated. defoliated. This dispersal tendency tendency is related to the numbers of of larvae on the plant and the plant size (van der Meijden, 1 976; Sjerps and Haccou, larvae plant Meijden, 1976; Sjerps 11996). 996). Population data on the the three species were collected collected in a coastal sand dune area near The Hague in The Netherlands. Netherlands. The patchy distribution distribution pattern pattern of of local ragwort populations is brought about by the geomorphology of of the dunes in com combination bination with grazing activity of of rabbits. rabbits. The landscape is a mosaic of of north northof trees, shrubs, and grasses, poorly vegfacing slopes with a closed vegetation of veg etated etated south-facing slopes, and and valleys with a vegetation depending depending on the
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groundwater groundwater level. level. Ragwort Ragwort can can potentially grow grow anywhere anywhere in in this this landscape landscape pro provided that that neither the the grass grass layer layer nor nor the the shrub/tree shrub/tree canopy canopy is is closed. closed. vided An An area area of of about about 66 km2 km 2 of of aa much much larger larger system system was was searched searched for for local local ragwort 973. Plants ragwort populations populations in in 11973. Plants are are considered considered to to belong belong to to one one local local popu population lation if they they are are not spatially separated from each each other other by ragwort-free distances distances of 50 local populations, 1102 02 were selected, based on differ of more than than 5 m. Of 1150 differences ences in population density and habitat characteristics, such such as the amount amount of of Formica polyc shade by woody perennials and the presence of the predatory ant shade and the of the ant polyctena. Local populations covered areas 974 (mean areas ranging from from 8 to 3000 m2 m 2 in 11974 2 ). Plant numbers (from small vegetative rosettes to large fl owering plants) m2). flowering 900 m per population varied from 11.5 .5 to 62 per m2• m 2. The distance between populations ranged 0 to over 200 m, with at least 5 m without any ranged from 110 any ragworts. Local populations were usually separated from each other by barriers like scrubs, for forested areas, dune lakes, or blowouts.
B. Census CensusData B_ Census 974 to 11994. 994. Relevant Census data data were collected from 11974 Relevant data data for this paper are:
wort biomass per 11.. The amount of of rag ragwort per local population sample. During During the the per period of of cinnabar cinnabar moth egg laying, in May-June M a y - J u n e in each year, the same same permanent manent squares (4 m m 22)) in each local population were visited three to four four times, and ragwort cover cover in dm22 was estimated estimated (a measure measure of of biomass; biomass; van der der Meijden, Meijden, and prior to oviposition by the the cinnabar cinnabar moth. The The highest value per per local 11979a) 979a) just prior population in in each was used used as the fi nal estimate. Herbivory by by cinnabar cinnabar population each year year was as the final estimate. Herbivory moth larvae larvae may may reduce reduce biomass (and seed production) from from June June onward. DeDe moth foliated plants plants often produce regrowth foliage shortly after after herbivory. A A popupopu foliated often produce regrowth foliage lation of ragwort was supposed to have gone gone extinct if (either of ragwort if no no living plants plants (either plant, or or mature mature plant) plant) were seedling, rosette plant, were present present on any any census census dates dates 11 year year the cinnabar moth. Based Based on on observations observations made made after cinnabar moth. after the the last last herbivory herbivory episode episode by by the in the the vicinity vicinity of of the the sampling sampling squares, squares, we we concluded concluded that that disappearance disappearance from from aa in sample typically typically meant meant extinction of of that that particular particular local population. popUlation. sample number of of cinnabar moth egg egg batches per local population. Egg Egg 2. The The number cinnabar moth batches per local population. batches were were counted counted in in the the above-mentioned above-mentioned 44 m m22 squares squares at four four to to six visits visits batches with with weekly weekly intervals intervals to to each each local local population population in in May-July May-July each each year. year. A A popupopu lation of of the the cinnabar cinnabar moth moth was was supposed supposed to to have have gone gone extinct extinct if if no no eggs eggs were were lation found on on the the plants plants following following aa year year in in which which eggs eggs were were present present in in that that population. population. found 3 . Percentage Percentage parasitism parasitism by by Cotesia. Cotesia. Percentage Percentage parasitism parasitism by by Cotesia Cotesia was was 3. determined (from (from 1988 1 988 onward) onward) by by collecting collecting fourth fourth or or fifth fifth instar instar cinnabar cinnabar moth moth determined larvae in in five five census census populations popUlations at at three three moments moments during during the the larval larval season season (from (from larvae the end end of of May May until until the the beginning beginning of of August) August) because because of of aa seasonal seasonal trend trend in in the percentage parasitism parasitism (Soldaat, (Soldaat, 1991). 1 99 1 ). At At every every date, date, 50 50 larvae larvae were were collected collected percentage per site, site, yielding yielding 750 750 larvae larvae which which were were reared reared to to pupation pupation every every year. year. per
1166
Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics
391 391
III. III. POPULATION POPULATIONDYNAMICS DYNAMICSOF OF RAGWORT RAGWORT Local recolonization of Local extinction extinction of of ragwort ragwort and recolonization of empty patches patches are are frequent frequent populations over events events on sand sand dunes. dunes. Figure Figure 11 (A) shows shows the number number of of extant extant populations over time. In two 1 975 - 1 976 and 1 98 1 - 1 982), 40 two extreme extreme seasons seasons ((1975-1976 and 1981-1982), 40 populations populations dis disappeared appeared in a single single year year (Fig. 11,, middle). middle). The The cumulative cumulative extinction extinction curve curve over over time demonstrates demonstrates that that not all local populations populations are are equally vulnerable. vulnerable. Eighteen Eighteen 02 populations populations never never became of of 1102 became extinct during during the period period of of 20 20 years, years, whereas whereas 56 populations recolonized) once even populations disappeared disappeared (and were were recolonized) once or twice and 26 even three to five times. Apparently Apparently ragwort ragwort has refuge refuge populations populations with extincthree with a low extinc 992) revealed revealed that tion probability. probability. A A habitat habitat analysis analysis (van (van der der Meijden Meijden et et ai. al.,, 11992) this group of populations is located in areas with a mixed vegetation of populations located areas mixed vegetation of of trees and and shrubs and a not fully closed ground vegetation with grasses and/or mosses and and not closed ground vegetation grasses and/or mosses lichens. lichens. The The populations populations with the highest highest extinction extinction risk are situated situated in open sandy sandy areas. areas. To test whether whether the probability probability of of popUlation population extinction was was related related to their their To 974 until 1976 1 976 (Table size, we analyzed analyzed data data for for the period period from from 11974 (Table I). Contrary Contrary to the general expectation had a higher expectation (Hanski, (Hanski, this volume), volume), small popUlations populations had higher probability of of survival survival than than large large populations. populations. The The reason for this is that that popu popuprobability reason for lations in areas areas with trees and/or and/or shrubs, shrubs, with a low extinction extinction risk, tend tend to be small (Table I). Apparently Apparently vegetation structure structure detennines determines the size of of suitable suitable habitat habitat patches. patches. In the the open open dune dune areas areas suitable suitable habitat habitat patches patches are considerably considerably larger larger than in areas areas with trees and/or and/or shrubs. shrubs. Even shows huge uctuations Even on a regional regional (metapopulation) (metapopulation) scale, ragwort ragwort shows huge fl fluctuations ground cover The difference difference in ground ground cover cover between between 1980 1 980 in total ground cover (Fig. 11,, A). The
Extinction, and Habitat Extinction,Survival, Survival, and Habitat Type Type of of Ragwort Ragwort Populations Populations between between in Relation to Patch Patch Size Size in in 11974 11974 974 and 11976 976 in Relation to 974
TABLE I
No. No. of of populations populations Fate Fate
Patch Patch size size m') 2) ((m
< < 110 0 110-100 0- 1 00 1101-1000 0 1 - 1000 11001 00 1 -2000 - 2000 > > 2000 2000 X2 X'
Significance Significance level level
Habitat Habitat
Extant Extant
Extinct Extinct (( % % ))
Trees Trees or or shrubs shrubs
9 9 113 3 117 7 113 3 33
2( 18) 2(18) 4(24) 4(24) 117(50) 7(50) 1 0(43) 10(43) 112(80) 2(80)
10 110 0 23 9 88
111.03 l .03 0.026 0.026
Open Open sandy sandy habitat habitat
7 7 I11I 114 4 7 7
1
15.68 0.003 0.003
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(A) Time Time course course of of the the number number of of extant extant ragwort ragwort populations populations (black (black squares) squares) and and the the metapopulation metapopulation fluctuation fluctuation in ragwort ragwort ground ground cover cover (summed (summed over over all all local populations; populations; open open squares). course of course of squares). (B) (B) Time Time course of ragwort ragwort colonization colonization of of empty empty patches. patches. (C) (el Time Time course of extinct extinct ragwort ragwort current year) ragwort populations populations (= ( sites sites without without rag wort in the the current year) and and the the cumulative cumulative function function of of populations that disappeared least once. populations that disappeared at least once. =
16 16 Tritrophic TritrophicMetapopuiation MetapopulationDynamics Dynamics 1 00
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and 11987 987 was almost l100-fold. Oa-fold. There There is a fair fair correlation between total ground ground populations, which suggests cover of of the metapopulation and the number number of of extant populations, that it is unlikely that that there there is much asynchrony in fluctuations fluctuations among among local popthat pop ulations. As a measure of of synchrony synchrony we used the correlation coefficient coefficient calculated 2) of between the time series of of ragwort ground ccover o v e r ((dm d m Z2//m m 2) rag wort ground of every local popupopu No significant significant negative correlations metapopulation (Fig. 2, top). No lation and and the metapopulation correlations were found. significantly found. Fifty-eight local populations populations were were signifi cantly positively correlated correlated with the overall fluctuation, ground cover cover in 44 44 local populations populations was was not not fluctuation, but ground significantly correlated correlated with metapopulation metapopulation fluctuations. fluctuations. Sixteen populations of significantly populations of this are located this latter latter group group are located in sites sites that that were were completely overgrown overgrown by by dense dense vegetation (the (the percentage percentage cover cover of of trees, trees, shrubs, shrubs, and and grasses grasses increased increased from 1 973 vegetation from 1973 1 994 from from 71 to to 100%). 1 00%). Mechanical Mechanical reduction reduction of of immigrating immigrating seeds, seeds, reduction reduction to to 1994 of "safe "safe sites" for germination, and and reduction reduction of of penetrating penetrating light light by by the the closed closed of for germination, vegetation, may may have have rendered rendered these these sites unsuited unsuited for ragwort germination, germination, vegetation, for ragwort growth, growth, or or survival. survival. To test test whether whether sites sites where where ragwort ragwort had had disappeared disappeared had had indeed indeed become become To unsuitable for plant growth, growth, seeds seeds were were added added experimentally experimentally (Table (Table II). II). Seeds Seeds unsuitable for plant germinated successfully successfully in in all all sites sites and and aa number number of of the the rosettes rosettes survived survived in in most most germinated sites, indicating indicating that that ragwort ragwort is is seed seed limited. limited. However, However, twin twin plots plots that that were were cleared cleared sites, of of the the grass-herb grass-herb vegetation vegetation showed showed considerably considerably higher higher germination germination and and sursur vival, demonstrating demonstrating that that the the suitability suitability of of growing growing sites sites was was affected affected by by vegevege vival, tation tation development. development. Two main main factors factors are are responsible responsible for for the the fluctuations fluctuations in in biomass biomass and and the the Two shortage of of seeds seeds contributing contributing to to local local extinction: extinction: herbivory herbivory by by cinnabar cinnabar moth moth shortage
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FIGURE22 (Top) (Top) Pearson's Pearson's correlation correlation coefficient coefficient (r) (1")between between the the amount amount of of ground ground cover cover of of FIGURE ragwort in each each local local population population and and the the metapopulation metapopulation (the (the sum sum of of all all local local populations), populations), reflecting reflecting rag wort in the level level of of synchrony. synchrony. (Bottom) (Bottom) Pearson's Pearson's correlation correlation coefficient coefficient (r) (r) between between the the numbers numbers of ofcinnabar cinnabar the moth eggs eggs per per dm2 dm 2of of ragwort ragwort in in local local populations populations and and in in the the metapopulation metapopulation (the (the sum sum of of all all local local moth populations). populations).
16 1 6 Tritrophic Tritrophic Metapopulation Met(]popul(]tion Dynamics Dyn(]mics
395 395
Experimental Seed Seed Addition Addition to to 25 25 Sites Sites in in TABLE IIII Experimental TABLE Different Habitat Habitat Types Types without without Ragwort Ragwortaa Different Rosette Rosette plants plants
Seedlings Seedlings Habitat Habitat
Open woodland woodland Open Grass-herbs Grass-herbs Moss-herbs Moss-herbs
A A
B B
A A
B B
1 93 193 1 23 123 1 75 175
624 624 531 531 348 348
12 12 44 12 12
21 21 37 37 19 19
a
Three thousand thousand seeds seeds were were sown sown per per aa plot plot o 30 by by "Three off 30 30 cm. em. The The table table gives gives the the mean mean numbers numbers of of seedlings seedlings 30 and rosette rosette plants plants per per plot plot after after 11 year. year. Plots Plots were were either either and undisturbed (A) (A) or or cleared cleared of of the the grass-herb grass- herb vegetation vegetation undisturbed (B). (B).
larvae quantitative effect effect of of herbivory herbivory will be below) and drought larvae (the quantitative be described described below) and drought der Meijden Meijden et et al., ai. , 1985). 1 985). Defoliation Defoliation may may not not only mean mean complete complete loss of of (van der above-ground biomass, but but also also lack of of seed seed production over periods periods of of 2 or or even even above-ground biomass, production over more more years. The effects effects of of drought drought on plant performance performance were were also mentioned mentioned by 1 982) and 1 989). Dempster Dempster ((1982) and Crawley Crawley and and Gillman Gillman ((1989). In Figure 11 (C) colonization wort of patches is plotted plotted over colonization by rag ragwort of empty patches over time. Years Years in which which total total defoliation defoliation in the entire metapopulation metapopulation was observed observed are 11975 975 and 1976, 1 976, 11981 9 8 1 and 11982, 982 , and 986 and 1 987. During and 11986 and 1987. During such such periods periods seed production production is reduced reduced to zero. This figure immediately reveals that colonization colonization of empty patches may take place in years following following total defoliation defoliation and destruc destruction of of the seed crop. Consequently, Consequently, colonization colonization probably occurs through through ger germination mination from from the seed bank. However, relatively high rates of of colonization were found also in the second and third year following following defoliation, defoliation, when plants in the extant populations produced produced again seeds, suggesting that seed dispersal is also important important to to colonization. colonization.
IV. IV. MECHANISMS MECHANISMSOF OF PERSISTENCE PERSISTENCEIN IN RAGWORT RAGWORT Ragwort has three mechanisms that enhance the probability of of survival on local scale: local scale:
11.. Regrowth Regrowth capacity. capacity. Ragwort has has spectacular powers powers of of regeneration, defoliated plants may which make it such aa powerful weed. Even completely defoliated 988). Plants that regenerate regenerate (van der der Meijden et et ai. al.,, 11988). that were were artificially artificially damaged damaged by eld experiment by removing removing the the whole shoot shoot in in aa fifield experiment suffered suffered only 5% more more mor mortality tality than than control control plants. Biomass Biomass was was reduced reduced by by 35%. 35%. However, However, the the negative effect effect of of herbivory herbivory may may be be much much greater greater in in combination with with adverse adverse weather weather
396
Ed and Catharina Ed van van der der Meijden Meijden and Catharino A. A. M. M. van van der der Veen-van Veen-van Wijk Wijk
conditions (drought) during or after defoliation (van der 01., 1985; 1 985; der Meijden Meijden et et al., Prins 990). Repeated Repeated defoliation reduces regrowth as Prins and Nell, 11990). defoliation reduces as well (McEvoy, 11985). 985). Despite Despite the the fact that that ragwort is a biennial biennial plant, and and normally dies dies after after a Anal single production of of seeds, the life cycle may be prolonged by herbivory. Analogous to to aa seed seed bank bank we can can talk about about aa rosette bank bank in this this species. species. 2. Dormancy. Dormancy. Ragwort has only a small small seed bank, bank, but it seems seems to be effec effective, judging judging from germination in many many open open sites after at least least 2 years years in which which seed production was reduced to zero zero by cinnabar cinnabar moth moth herbivory (van der der Meijden Meijden et 985). Fifty 983, after et 01., al., 11985). Fifty populations were sampled in 11983, after two successive successive seasons seasons of total defoliation, by taking eight circular (diameter 1 0 cm) soil cores per pop of taking circular (diameter 10 cores per population. Samples of 3 1 populations did not contain any viable seeds. In 9 popu of 31 populations one seed was found, than found, in 4 two seeds, seeds, in I1 three three seeds, seeds, and and in 5 more than five seeds (which means . 1 4 m22 soil samples). These numbers five means 55 viable seeds in 33.14 numbers 20,000 seeds owering are small compared compared with a production of of 20,000 seeds per per individual fl flowering plant. Many Many local populations do not seem to have a seed bank bank at all, though one one cannot cannot be sure based based on small small samples samples of of soil. Experiments on the the longevity of of buried buried seeds showed that that viability was positively related related to the depth depth at which they were buried. Viability of 0 cm was hardly of seeds buried buried at a depth of of 110 hardly reduced reduced after 5 years (E. van der Meijden, Meijden, unpublished unpublished results). 1 940) showed 3. Seed Seed dispersal. dispersal. Experiments by Poole and and Cairns Cairns ((1940) showed that that the the majority of of ragwort seeds landed landed within a few meters meters from the the parent parent plant. plant. Yet aa considerable considerable number number of of seeds seeds was was found found to to have have been dispersed dispersed up up to 20 m, and especially in the direction direction of of prevailing winds the dispersal curve seemed seemed to der Meijden Meijden et et 01., al., 1985) demonstrated that that both have a long tail. We (van der 1 985) have demonstrated of dormant seeds and colonization through seed dispersal were imgermination of im portant mechanisms mechanisms in the reestablishment reestablishment of of local populations. populations. Because Because no seeds seeds were were formed in the 2 earlier earlier years, years, the occurrence occurrence of of seedlings seedlings in empty patches patches after a period of indicates germination from a seed bank. However, However, of defoliation indicates the level of of germination in empty patches patches in subsequent subsequent years years was much too high high explained by germination of of dormant seeds alone, and and much of of this gerto be explained ger mination must have been the result of of seed dispersal. dispersal. Unfortunately, our data data do not allow us to judge refuge populations played an judge whether whether the small number number of of refuge essential essential role role as as sources sources of of dispersing dispersing seeds_ seeds.
V. OF THE THE CINNABAR AND THE THE PARASITOID V. POPULATION POPULATIONDYNAMICS DYNAMICSOF CINNABARMOTH MOTHAND PARASITOID COTfSIA COTESIAPOPULARIS POPULARIS The The cinnabar cinnabar moth moth is conspicuous conspicuous not only only because because of its appearance appearance (a bright bright red and black colored moth and and even brighter yellow and black black colored larva), but also because of of its behavior. Periodically, larvae completely defoliate defoliate their food plants over large areas (in the Dutch coastal dunes dunes defoliation seems to be highly synchronized). During During such years, thousands thousands of of larvae can be observed to disperse fth instar disperse in in search search of of food. food. Fourth Fourth and and especially especially fi fifth instar larvae larvae can can cover cover
1]66
Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics
397 397
hundreds motor hundreds of of meters and and even leave leave their their habitat by crossing crossing beaches beaches and and motorways. Eventually they may defoliate all local populations populations of of ragwort. During such episodes Dempster ((1982) 1 982) and Crawley episodes thousands thousands of of larvae die from starvation. Dempster and Gillman ( 1 989) also found food shortage to be the key factor and (1989) found food shortage factor in population population dynamics in the United Kingdom, determining the magnitude of of population population fluc fluctuations first sight, the frequent tuations of of the cinnabar cinnabar moth. At first frequent total total defoliation defoliation and and the enormous uctuations in plant and insect populations populations might give the impression enormous fl fluctuations that this insect-plant insect-plant relationship relationship is one of of the best examples for for refuting coad coadaptation aptation and regulation. Nevertheless, both ragwort and the cinnabar cinnabar moth are still common species in sand dunes and some other other habitats of western Europe. The pest characteristics of wort, especially its regenerative powers, might well of rag ragwort, be the result result of of selection selection by the the insect. The The insect, on the other other hand, hand, sequesters the alkaloids of of its food food plant, probably as defense defense against potential natural natural en enlen and van der 99 1 ; Ehmke et 990). emies (van Zoe Zoelen der Meijden, 11991; et al. al.,, 11990). Figure 3 shows fl uctuations in the metapopulation size cinnabar moth fluctuations size of of the cinnabar expressed uctuated more expressed as numbers of of eggs per plant biomass. Egg numbers fl fluctuated than 200-fold. For comparison biomass fl uctuations of fluctuations of ragwort are plotted in the same graph. Above a level of 0 eggs per dm22 of of 110 of ragwort, total defoliation will take 99 1 ). The numbers of of take place (van der Meijden Meijden et et al. al.,, 11991). The moth moth lays the largest largest numbers eggs per per unit of of plant plant biomass biomass in open open areas, without without ants ants (F. polyctena; polyctena; Table III). These are also the areas that receive most eggs in absolute terms, because there there are many more plant populations in open open than fully shaded sites. Experi Experiments on oviposition oviposition demonstrated demonstrated an almost absolute preference preference for for open areas
30 30
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, 3 90 84 88 8~4 86 8'6 90 9'2 94 92 year year FIGURE number of [:[~gRl: 3 ~ Metapopulation dynamics of of the number of cinnabar moth eggs (mean number of eggs per population sample of of 4 m2; open squares) and ragwort ragwort ground cover (black squares). 76 7"6
78 7'8
80 80
82 8'2
398 398
Ed Meijden and and Cathmina Ed van van der der Meijden Catharina A. A. M. M. van van der der Veen-van Veen-van Wijk Wiik Mean dm22 Mean Yearly Yearly Egg Egg Load Load of of the the Cinnabar Cinnabar Moth Moth per per dm of of Ragwort Ragwort in in Local Local Populations Populationsof of Ragwort Ragwort in in Different Different Habitat Habitat Types 974 to 994aa Types from from 11974 to 11994 TABLE TABLE III
Habitat Habitat type
No. of of populations populations
Egg numbers/ numbers/ dm2 dm 2 ragwort ragwort
Standard Standard error error
- SS- F- F S S -- F F - S S F F S S F F
65 65 14 14 1166 7 7
9.08 9.08 3.88 3.88 4.79 4.79 0.99 0.99
0.83 0.83 11.03 .03 11.57 .57 0.41 0.41
a
shaded, Formica-free; a_ S S -- F, F, unshaded, unshaded, Formica-free; S S -- F, F, shaded, habitat. Effects Effects of - SF, SF, unshaded, unshaded, Formica; SF, SF, shaded, shaded, Formica Formica habitat. of shade shade and and Formica are are both both significant significant (P< ( P < 0O.OOI . 0 0 1 in in two-way two-way ANaYA); not sig ANOVA); the the interaction interaction between between shade shade and and Formica is is not significant. nificant.
over fully shaded areas al., 11991). 99 1 ). A similar result was found areas (van der Meijden et et al., with respect respect to daily light intensity: the moth does not oviposit in cloudy days. The amount amount of ragwort biomass in open open habitat is probably the the most impor important parameter parameter determining the population change of the herbivore. The correla correlation coefficient between the amount of of ragwort in the open populations and the number cant (Table IV). number of of cinnabar cinnabar moth eggs in the next year is highly signifi significant This correlation was not signifi cant in the shaded plant populations. significant Cinnabar Cinnabar moth population population growth is followed by an inevitable reduction of of plant plant biomass as explained explained above. The The correlation coefficient coefficient between the the number + 11 is negative and significant of eggs in year t and rag ragwort wort biomass in year t + of for the open populations (Table IV), indicating the effect of of herbivory on plant production. The correlation is negative but not signifi cant in the shaded plant significant populations. The interaction between between the plant plant and and the moth produces the striking cycles
Correlations Expressed as Correlationsbetween between the the Amount Amount of of Food Food ((Expressed as Ragwort Ragwort Ground Ground Cover! Cover)inin Year Year tt and and the the Number Number of of Cinnabar Cinnabar Moth Moth and between Amount of Herbivory Eggs Eggs in in Year Year tt + + I1 (A) (A) and between the the Amount of Herbivory ((Expressed Expressed as Eggs) in Year Year tt and as the the Number Number of of Cinnabar Cinnabar Moth Moth Eggs)in and Ragworl Ragwort Ground Ground Cover Cover in in Year Year tt + + I1
TABLE IV IV
B B
A A Ragwort Ragwort population population
Open Open populations populations Shaded Shaded populations populations
r r
p P
r r
p P
0.74 0.74 0.34 0.34
0.00 0.00 NS NS
- 0.74 0.74 - 0.46 0.46
0.02 0.02 NS NS
1166
Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics
399 399
in their their populations shown shown in Fig. 3, with plant plant biomass biomass followed by insect (egg) 2 ragwort in the numbers. 1 0 eggs/dm2 numbers. After overshooting the egg/biomass ratio ratio of of 10 the open populations, populations, a complete complete defoliation follows. Next larvae larvae disperse disperse to the still undefoliated undefoliated plants plants in other other populations. Finally this leads leads to regional defoliation and total loss of seed production. A second season of of total defoliation usually follows, resulting in a strong reduction reduction of of egg numbers in the following year. Reduced plant biomass is now driving insect numbers. The final increase increase in insect numbers in Fig. 3 was unexpectedly ended in 1991, 1 99 1 , without the moth overshoot overshooting the carrying capacity and without further reduction in plant biomass. It was only in 11995 995 that that the ragwort population became again completely defoliated. There is no effective regulatory mechanism for for the insect on the scale of of a local plant population. With the numerical advantage for an individual to lay for more eggs than its competitors, this makes food shortages among larvae inevi than larvae inevitable. Food shortages in tum reduc turn result in mass migration migration and starvation. The reduction in plant biomass leads to local disappearance of the herbivore. Local ragwort disappearance populations with relatively more food are selected by the adult moth for ovipo oviposition. Figure Figure 4 (top) illustrates the dynamics of the numbers numbers in local cinnabar cinnabar moth populations over time and (bottom) local moth extinctions. Within 6 years from 980, all local insect populations populations had become from the beginning of this study, in 11980, extinct at least once. Contrary to its food food plant, the cinnabar moth has no refuge populations that never became extinct. Figure 3 shows that that after the crash crash in herbivore herbivore (egg) numbers numbers due to food shortage and starvation it took another another year before egg numbers started to increase again ((1978, 1 978, 11984 984 and 11989). 989). This is unexpected because plant biomass did not appear 978 and 1989. 1 989. This lag appear to be limiting in these years, especially not so in 11978 period implies that during these periods insect numbers were not driven by avail available plant plant biomass. It also implies (Fig. 3) that that as herbivory is relaxed for 11 year after a period of heavy herbivory, local ragwort populations can recover recover and biomass can increase. The absence of of moth population growth in the year following the crash crash in egg numbers is probably due to two factors. In the fi rst place adult females are first much smaller than in a normal year (Dempster, 982), because (Dempster, 11982), because they suffered food shortage as a larva in the previous year. Mean pupal size is reduced reduced from 0.52 to 0.45 cm, which corresponds to a reduction in fecundity of of more than 60%. Sec Secpopularis is inversely related ond, the effect effect of of the parasitoid C. popularis related to the density of of the cinnabar 982). The highest percentage cinnabar moth (Dempster, 11982). percentage parasitism is found after 1 935) found after a population population crash of of the cinnabar cinnabar moth. Similarly Cameron Cameron ((1935) the highest percentage percentage parasitism after a population crash of of the cinnabar cinnabar moth in the United Kingdom in 11932. 932. We have hypothesized (van der Meijden et al., 11991) 99 1 ) that the the mechanism mechanism underlying this this inverse relationship is that Cotesia Cotesia has has a lower lower fecundity and and a lower lower dispersal capacity than its host. Especially in the years of crash in cinnabar 1 977, 1983, 1 983, and 11988), 988), few new patches cinnabar moth numbers ((1977, are colonized by the moth (Fig. 4, bottom). This allows a buildup of of parasite parasite numbers and consequently an increase in parasitism. parasitism. In the following years the host colonizes other local ragwort populations populations and temporally escapes from from its
Ed van van der der Meijden Meijden and end Cotharino CatharinaA.A. M. AA.von van der der Veen-van Veen-vanWijk Wijk Ed
400 400
parasitoid. If If the the rate rate of of increase increase of of the the host host is is higher higher than than that that of of the the parasitoid, parasitoid, parasitoid. percentage percentage parasitism parasitism should should decrease decrease with with the the rapidly rapidly increasing increasing host host numbers. numbers. Dempster's 1 982) data Dempster's ((1982) data suggest suggest such such aa lower lower rate rate of of increase increase in in the the parasitoid. parasitoid. Table Table V V gives further further information information on on parasitism, parasitism, colonization colonization of of new new sites sites by by the the cinnabar moth, and population change of the moth. The 1 988 crash resulted in aa cinnabar moth, and population change the 1988 crash
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FIGURE 4 (Top) (Top) Time Time course course of ofthe the number number of ofextant extant populations populations of ofthe the cinnabar cinnabar moth moth (= (= ragwortragwort FIGURE growing sites sites with with cinnabar cinnabar moth moth eggs eggs in in the the current current year). year). (Middle) (Middle) Time Time course course of of the the number number of of growing extinct extinct populations populations of of the the cinnabar cinnabar moth moth (populations (populations that that went went extinct extinct in in the the current current year; year; open open squares) and and the the number number of of ragwort-growing ragwort-growing sites sites colonized colonized by by the the cinnabar cinnabar moth moth in in the the current current year year squares) (black (black squares). squares).
116 6 Tritrophic Metapopulotion Dynamics TritrophicMetapopulation Dynamics TABLE TABLEV
Parasitism Cinnabar Moth Parasitismof of the the Cinnabar Mothby by Cotesia Cotesia popu/arisa popukil'Js a
Year
Percentage Percentage parasitism parasitism
Moth population population turnover rate
Change in moth population
11988 988 11989 989 11990 990 11991 99 1 11992 992 11993 993 1994
37.9 3.2 0.0 0.4 11.0 .0 .1 55.1 110.5 0.5
25 75 92 53 43 39 37 37
0.08 0.33 117.12 7.12 2.55 11.05 .05 0.52 11.10 . 10
a a
401
Population 1 00 X Population turnover turnover in the cinnabar cinnabar moth moth was calculated calculated as 100 • number number newly newly colonized colonized sites/(number sites/(number of persistent persistent popUlations populations + + newly newly colo colonized sites). Change size was calculated nized sites). Change in host host population population size calculated as the number number of cinnabar cinnabar moth moth eggs eggs in year year t - I/number 1/numberof eggs eggs in year year t. -
high percentage percentage of due to the small number of new sites high of parasitism, parasitism, apparently apparently due to the small number of new sites colonized 1 989 and 1 990 colonized by by the the moth moth and and the the reduction reduction in in its its population population size. size. 1989 and 1990 show increase in show a a fast fast decrease decrease in in percentage percentage of of parasitism parasitism together together with with an an increase in population population turnover turnover and and general general increase increase of of the the moth. moth. The The increase increase in in parasitism parasitism in 993 and 994 occurs host population population turnover in 11993 and 11994 occurs concurrently concurrently with with a a reduction reduction in in host turnover rate host. rate and and a a reduction reduction in in population population growth growth of of the the host. The level of The level of synchrony synchrony between between egg egg numbers numbers of of the the cinnabar cinnabar moth moth per per plant plant biomass biomass in in local local plant plant populations populations and and the the pooled pooled egg egg numbers numbers in in the the metapop metapopulation (Fig. ulation was was calculated calculated as as the the correlation correlation coefficient coefficient between between these these variables variables (Fig. 2, will only 2, bottom). bottom). As As eggs eggs will only be be found found on on ragwort ragwort plants plants we we skipped skipped those those sites sites that I %. Half Half of that had had no no ragwort ragwort or or had had aa ragwort ragwort cover cover of of < < 1%. of the the correlation correlation coefficients cant, indicating indicating considerable coefficients in in Fig. Fig. 2 2 are are positive positive and and signifi significant, considerable syn synchrony local chrony in in the the local local fluctuations fluctuations of of the the numbers numbers of of cinnabar cinnabar moth moth eggs. eggs. The The local populations invariably populations with with a a low low correlation correlation with with metapopulation metapopulation fluctuations fluctuations invariably had values due low ragwort less suitable had many many zero zero values due to to low ragwort density density or or less suitable habitat habitat (shaded (shaded ragwort populations or ragwort populations or populations populations with with the the predatory predatory ant ant F. polyctena). These These results is able to track results confirm confirm the the notion notion that that the the cinnabar cinnabar moth moth is able to track efficiently efficiently suit suitable plant. able local local populations populations of of its its host host plant.
VI. MECHANISMS OF PERSISTENCE PERSISTENCE IN IN THE THE CINNABAR VI. MECHANISMSOF CINNABARMOTH MOTH There five mechanisms There are are at at least least five mechanisms that that enhance enhance the the probability probability of of persistence persistence cinnabar moth: moth: in the cinnabar 11.. Dispersal of of adult adult moths and and the capacity to locate isolated isolated spots of of rag ragwort. The be strong yet the The female female moths moths do do not not seem seem to to be strong flyers, flyers, yet the distribution distribution of of eggs plant biomass across local ragwort shows good eggs in in relation relation to to plant biomass across local populations populations of of ragwort shows good synchrony, only be be explained casynchrony, which which can can only explained by by an an effective effective dispersal/searching dispersal/searching ca-
402 402
Ed Edvan van der der Meijden Meijdenand and Catharina CatharinaA.A. M. M. van van der der Veen-van Veen-vanWijk Wijk
pacity pacity of of the the female female moths. moths. The The infonnation information in in Fig. Fig. 44 on on newly newly colonized colonized local local sites is is aa clear clear illustration illustration of of the the dispersal dispersal capacity. capacity. Harrison Harrison et et al. al. ((1995) came sites 1 995) came to aa similar similar conclusion. conclusion. to 2. 2. Dispersal Dispersal of of larvae. larvae. During During periods periods of of food food shortage, shortage, larvae larvae are are observed observed to nd food. fth ins tar larvae to move move long long distances distances to to fifind food. Especially Especially fourth fourth and and fififth instar larvae can can cover hundreds hundreds of of meters meters and and reach reach yet yet undefoliated undefoliated sites. sites. A A remarkable remarkable feature feature of of larval behavior behavior is is that that some some of of them them leave their their food food plants before before defoliation defoliation has caused caused any any food food shortage. shortage. Using Using aa gametheoretical gametheoretical model model Sjerps Sjerps and and Haccou Haccou has ((1994) 1 994) demonstrated demonstrated that that such such behavior, behavior, especially when sibs sibs are are involved, may have an evolutionary advantage. advantage. 3. 3. The temporal temporal distribution of of oviposition. oviposition. Oviposition is extended from early May until early JUly. July. There are marked differences in the the beginning of the oviposition season between years which undoubtedly relate to effects of temper temperature on the development of pupae. The wide distribution of oviposition guar guarature antees that some egg batches have a considerable head start over others. During years of of food shortage shortage larvae from the first egg batches will have the highest of reaching the threshold weight for pupation. chance of 4. Flexible size for for pupation. During periods of of food food shortage pupal size can be considerably reduced 982). Successful pupation is possible only reduced (Dempster, 11982). Successful pupation reached a threshold threshold weight of of 140-150 Meijden, when the larva has reached 1 40- 1 50 mg (van der Meijden, reach in a year of of abundant abundant 11976), 976), which lies far below the weight that a larva may reach food supply (300-500 food (300-500 mg). Distribution of of eggs over expressed as 5. Distribution over different habitats. The The egg load expressed the number of of plant biomass varies varies significantly significantly and the number of eggs per per unit unit of plant biomass and consistently between different habitat types (Table (Table III). Open without the between different Open habitat without the predatory predatory ant is preferred, preferred, shaded shaded habitat habitat with with the the ant is avoided. avoided. This This suggest suggest that that the the less less ant favored habitats habitats may may act act as short-term short-tenn refuges refuges in periods periods of of food food shortage. shortage. As As favored mentioned moth has mentioned earlier, earlier, the the cinnabar cinnabar moth has no no refuge refuge populations populations over over long long periods periods of of time. time.
VII. DISCUSSION DISCUSSION A. Population Population Dynamics Dynamics and Persistence Persistence Populations of ofragwort ragwort show show tremendous tremendous fluctuations fluctuations in in biomass biomass at at both both local local Populations and regional regional scales, scales, with with local local fluctuations fluctuations frequently frequently resulting resulting in in extinction. extinction. HerHer and bivory by by the the cinnabar cinnabar moth, moth, especially especially in in open open habitats, habitats, contributes contributes to to the the local local bivory extinction extinction risk. risk. The The metapopulation metapopulation of of ragwort, ragwort, however, however, does does not not seem seem to to be be in in great danger danger of of extinction. extinction. Figure Figure 11 shows shows that that we we never never observed observed more more than than aa great half of of the the local local populations populations to to have have gone gone extinct extinct in in 11 year. year. The The lowest lowest percentage percentage half ground cover cover was was observed observed in in 1987, 1 987, when when the the plant plant was was still still found found in in 62 62 of of 102 1 02 ground patches. On On the the other other hand, hand, the the extinction extinction risk risk of of ragwort ragwort is is not not independent independent of of patches.
16 1 6 Tritrophic Tritrophic Metapopulation Metapopulation Dynamics Dynamics
403 403
location: the the plant plant has has refuge refuge populations populations that that never never went went extinct extinct during during 20 20 years, years, location: and itit has has local local populations populations with with aa very very high high risk risk of of extinction. extinction. and On aa local local scale scale the the cinnabar cinnabar moth moth has has an an ephemeral ephemeral existence existence of of one one or or On only aa few few years. years. Its Its survival survival is is closely closely linked linked with with the the presence presence of of ragwort. ragwort. By By only numerically overshooting overshooting the the local local carrying carrying capacity, capacity, larvae larvae are are subject subject to to numerically scramble competition competition for for food, food, which which frequently frequently leads leads to to mass mass starvation and scramble starvation and local extinction. extinction. Even Even on on the the regional regional scale, scale, the the cinnabar cinnabar metapopulation metapopulation seems seems local to be in great danger of extinction during such events. In 1 984 and 1 989, we to be in great danger of extinction during such events. In 1984 and 1989, we found only only six six and and five five egg egg batches, batches, respectively, respectively, which which could could have have been been propro found duced by a single female, female, in all samples (408 (408 m2). m2). Dempster Dempster (1982) ( 1 982) who who studied studied duced similar system in the United United Kingdom Kingdom found found only one one egg egg batch batch in 1969 1 969 in in a similar 1 50 m m22 samples. samples. He He observed observed immigration immigration of of adult adult moths moths from from outside his study 150 outside his more isolated isolated population, Dempster Dempster (1971) ( 1 97 1 ) observed observed an an system. In a smaller, more extinction of of the the moth moth in 1968, 1 968, and and it was was only 10 1 0 years years later later that that the site site was was extinction reoccupied. Apart of reoccupied. Apart from the risk of of extinction extinction one would expect extreme extreme loss of genetic population bottlenecks genetic diversity to result from such severe population bottlenecks (Hedrick (Hedrick and and Gilpin, this this volume). volume). Gilpin,
Extended Dynamics Probability of Survival? B. Do Spatially Extended Dynamics Affect Affect the Probability Survival? Do these these organisms survive because assemblages assemblages of of local populations populations persist in a balance balance between local extinctions and colonizations, as in the classical meta metapopulation scenario (Hanski, this volume)? In some biennial plants with a fairly constant appearance appearance and decay of of growing sites that might indeed be the case (van der Meijden ai., 11992). 992). In species like ragwort, with refuge populations Meijden et et al., and rather synchronous fl uctuations in biomass, the situation appears fluctuations appears to be dif different. Although some localities became unsuitable for ragwort, the majority re remained suitable and could have been recolonized from the seed bank or through seed dispersal from extant local populations. Survival over long periods of time is enhanced enhanced by these two buffer mechanisms, by the low vulnerability to extinc extinction in the refuges and by the capacity capacity of individual plants to recover from ex extreme treme damage. damage. The metapopulation type (Hanski and Gilpin, this volume) that fi ts ragwort best would be the sourcesink metapopulation consisting of patches fits source-sink with mostly negative population growth rate in the absence of seed dispersal and patches with a positive growth rate. Even so, seeds from open populations may occasionally disperse disperse to the refuge populations and and contribute contribute to their stability. stability. Regional survival rst of survival of of the cinnabar moth is brought brought about, about, fi first of all, by the the capacity capacity of of its its food plant to recover recover soon after complete complete defoliation. defoliation. Heterogeneity of of the ragwort ragwort habitat, habitat, which leads to to differences differences in egg load per unit of of plant biomass, biomass, and and the the temporally temporally extended extended oviposition oviposition period period also increase increase the the chance chance of of at at least least aa few few larvae larvae to to reach reach the the threshold threshold weight for for pupation pupation ev�n eve.n w!len when food food becomes 979a, becomes completely completely exhausted exhausted on on the the regional regional scale scale (van (van der der Meijden, Meijden, 11979a, 99 1 ; Dempster, 982). The van van der der Meijden, Meijden, et et ai., al., 11991; Dempster, 11982). The positive, positive, and and often often high, high,
404
Ed ond Cothorino Ed von van der der Meijden Meijden and Catharina A. A. M. M. von van der der Veen-von Veen-van Wijk Wijk
correlations between between temporal fluctuations in local populations populations and the metapop metapopulation indicates that the searching capacity of the female moth is high and ap apparently not much hampered by interpatch distances. In this respect, the cinnabar cinnabar moth may be considered to live in only one, but patchy, population and not in a classical metapopulation (Harrison, 11991; 99 1 ; Harrison and Taylor, this volume). An important question in metapopulation theory is whether an interacting persist system of of local populations is more stable or has a higher probability of of persistence than the separate 1 99 1 ). This case study has separate local populations (Hanski, 1991). demonstrated that many sites where ragwort went extinct are recolonized recolonized through seed dispersal from existing local populations. populations. Such dynamics must lead to higher regional densities densities and biomass production and, consequently, to a higher proba probability of persistence. persistence. This study has also demonstrated that habitat heterogeneity in local ragwort ragwort populations popUlations adds to the probability of of cinnabar cinnabar moth survival. Finally, it seems very plausible that that the parasitoid C. popuiaris, popularis, with its limited limited powers of of dispersal between between local popUlations, populations, causes a delay in the recovery of of the cinnabar cinnabar moth and consequently enables ragwort populations to grow for for one season without herbivory. In conclusion, spatial effects related related to habitat habitat hetero heterogeneity and parasitoid dispersal probably contribute contribute greatly to cinnabar moth survival. We would expect that that even if ragwort were were not patchily distributed, distributed, but all patches patches were were combined combined to one large population, population, it would still survive given its specific characteristics characteristics and the present level of spatial heterogeneity. Because dispersal of of cinnabar cinnabar moth moth larvae larvae would not be affected by distances in such a uniform environment, food would become exhausted sooner. This would lead to greater fluctuations in insect numbers and probably in plant plant biomass as well ai., 11991). 99 1 ). As its survival in a network (Sabelis et et al., network of connected connected patches patches seems to be risky, the survival of the cinnabar cinnabar moth would be even more more questionable. questionable. However, in such a system without any interpatch distances distances the parasitoid parasitoid C. popuiaris popularis might might become much much more more effective, effective, because dispersal, its weak point, would be less critical. We thus conclucde conclucde that that every possible outcome, from cinnabar cinnabar moth extinction to regulation by the parasitoid, remains a possibility in a large uniform habitat. Harrison et al. ((1995) 1 995) studied ragwort and a guild of of its herbivorous insects during 3 years in the United Kingdom to test whether whether coexistence coexistence of of these com competing species could be explained explained by any spatial effects. They experimentally demonstrated that insect dispersal across local patches was not seriously limited by interpatch interpatch distances. From From these experiments experiments they concluded that that it was un unlikely that patches patches were acting acting as separate dynamic entities entities with respect to com competition. petition. In other words, they refuted the idea that spatial effects were essential for for coexistence. Our study demonstrates demonstrates that that spatial effects are not limited to those caused by distances distances between local populations: differences differences in habitat quality among local patches may be critical. Although we prefer simple models to test ( 1 99 1 ) that "there ecological principles, we agree agree with Hanski (1991) "there is an urgent urgent need to develop metapopulation models that include variation in habitat quality." This
116 6
Tritrophic Metapopuiation Dynamics TritrophicMetapopulation Dynamics
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study metapopulation-level study has has also also demonstrated d e m o n s t r a t e d that that many many m e t a p o p u l a t i o n - l e v e l processes p r o c e s s e s cannot c a n n o t be be easily easily detected d e t e c t e d in in short-term s h o r t - t e r m studies. studies.
ACKNOWLEDGMENTS ACKNOWLEDGMENTS We enjoyed discussing discussing these ideas with Tom de Jong and Peter Klinkhamer, Rinny Kooi is acknowledged for technical technical assistance, assistance, and we especially thank Ilkka Hanski for his extensive com comments on the first version of this chapter.
sdfsdf
17
Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulaton Andrew T.T. Smith
Michael Gilpin
I. INTRODUGION INTRODUCTION Because interest in modeling fragmented landscapes landscapes and associated meta metapopulation dynamics is relatively recent ((Hanski Hanski and Simberloff, this volume; Wiens, this volume), few long-term empirical empirical studies studies of of metapopulations metapopulations are available to guide theoretical analysis and exploration exploration (but see Thomas and Han Hanski, this volume). In this chapter chapter we present the results of of a long-term study of of a metapopulation metapopulation that appears ideal with regard regard to the measurement of of parameters of of a metapopulation. In this study the patches are of roughly the same size, and interpatch spacing is fairly regular. The metapopulation is large relative to lifetime movements of the animals. Not all patches patches have been occupied occupied in any census period. Both numerous numerous extinctions extinctions and recolonizations have been recorded over the more than 20 years of observation of of the metapopulation. metapopulation. The study organism 1 32 g) alpine lagomorph; the is the American American pika (Ochotona (Ochotonaprinceps), princeps), a small ((132 study site is the abandoned gold-mining area of of Bodie, Mono Mono County, California. One of 1 969 (Smith, 11974a,b, 974a,b, of us (A.T.S.) has been studying pikas at Bodie since 1969 11978, 978, 11980), 980), and here here we present present data data from four complete complete population population censuses made at varying intervals intervals since that that time. Results from the first two censuses ((1972 1 972 and 11977) 977) were were interpreted in terms of of area-dependent extinction rates and distance-dependent distance-dependent colonization rates bor-
Metapopularion Metapopulation Biology Biology 1997 by Academic Press. reproduction in any form reserved. Copyright © 9 1997 Press, Inc. All rights of of reproduction
407
408
Andrew T. T. Smith Smith and and Michael Michael Gilpin Gilpin Andrew
rowed directly directly from from island island biogeographic biogeographic theory theory (MacArthur (MacArthur and and Wilson, Wilson, 1967; 1 967; rowed Smith, Smith, 1974a, 1 974a, 1980). 1 980). The The populations populations on on patches patches of of habitat habitat at at Bodie Bodie apparently apparently represented aa dynamic dynamic equilibrium equilibrium between between extinction extinction (which (which was was inversely inversely rere represented lated lated to to patch patch size) size) and and recolonization recolonization (which (which was was inversely inversely related related to to interpatch interpatch distance; Smith, Smith, 1974a, 1 974a, 1980). 1 980). These These results suggested suggested that that a two-dimensional two-dimensional distance; "stepping-stone" metapopulation model, i.e. one one for which interactions interactions occur occur bebe "stepping-stone" metapopulation model, for which tween neighboring patches, could be parameterized to explain these patterns. We tween neighboring patches, could be parameterized to explain these patterns. We introduce this chapter such a spatially spatially explicit explicit model model that that successfully successfully integrates integrates introduce in this chapter such patch-specific population population growth growth with with these these size and and distance distance effects effects for for the the Bodie Bodie patch-specific pika metapopulation. metapopulation. pika addition, qualitative qualitative inspection inspection of of the the map-based map-based pattern pattern of of patch extinc In addition, patch extinctions and and recolonizations that occurred between the the first first and and the the second second censuses censuses tions recolonizations that occurred between suggested these events events did did not not occur occur randomly randomly across across the the Bodie Bodie landscape. landscape. suggested that that these Instead, there there appeared appeared to to be be a clustering clustering of of extinction extinction and and recolonization recolonization events; events; Instead, neighborhoods patch patch occupancy markedly, while while in other in some some neighborhoods occupancy declined declined markedly, other neigh neighborhoods patch increased (Smith, 1 980). Our Our stepping-stone stepping-stone model model borhoods patch occupancy occupancy increased (Smith, 1980). inherently incorporates observations by accounting accounting for certain correlations correlations inherently incorporates such observations for certain population growth growth on patches patches within within neighborhoods, neighborhoods, particularly particularly as a result result of of in population recolonization from from neighboring neighboring patches. patches. Thus, Thus, the consideration recolonization the model model takes takes into into consideration that that immigration immigration onto onto vacant vacant patches patches and and recurrent recurrent colonization colonization of of occupied occupied patches effect;" Brown Brown and Kodric-Brown, 1977; 1 977; Smith, Smith, 1980) 1 980) in patches (the "rescue "rescue effect;" and Kodric-Brown, regions of high average average patch occupancy can can lower lower the the probability probability of of extinction extinction regions of high patch occupancy neighboring patches patches and result in the increased persistence of of clusters clusters of of on neighboring and result increased persistence patches. patches. We empirical investigation conducting a third third and and fourth We renewed renewed the empirical investigation by conducting fourth census 1 989 and 11991) 99 1 ) with the intention intention of census ((1989 of examining examining closer closer the spatial structure structure of of local population dynamics in the system. These These recent censuses censuses paint a picture of rather is different of the Bodie Bodie pika metapopulation metapopulation that rather different from from that presented presented earlier by Smith ((1974a, 1 974a, 11980), 980), with the scale of turnover events dramatically of correlated correlated turnover populations in one subregion half widened. Almost all populations subregion at Bodie (covering almost almost half of the area of of the metapopulation) metapopulation) had had gone extinct, while the remaining remaining subregion subregion of of poppop hardly suffered suffered any change change in distribution distribution or abundance. abundance. The "collapse" of ulations in the one subregion subregion represents represents a departure departure from from the supposed dynamic equilibrium and clustering effects effects seen in the earlier censuses. censuses. Thus, the full data data set (four censuses censuses and and three census census intervals) challenges challenges Thus, our ability to construct construct a metapopulation metapopulation model built exclusively around the ideas of island biogeography: area-dependent area-dependent extinctions and distance-dependent distance-dependent re recolonizations. The inability of the stepping-stone stepping-stone model to explain completely the regional effects observed in our map-based map-based observations observations of pika populations populations at Bodie led us to consider more fully elements of "regional stochasticity" of"regional stochasticity" (Hanski and Gilpin, 11991) 99 1 ) that have been explored theoretically by Gilpin ((1988, 1 988, 11990) 990) and Hanski 1 989). We explored these Hanski ((1989). these properties of the Bodie metapopulation metapopulation quantitatively with an examination examination of the autocorrelational autocorrelational properties of our spa-
117 7
Spatially SpatiallyCorrelated Correlated Dynamics Dynamicsinin aa Pika Pika Metapopulation Metapopulation
409
tially referenced data. The results demonstrate the need to understand the land landmetapopulations in nature to understand fully their behavior. scape properties of metapopulations
RELEVANTPIKA PIKA NATURAL NATURALHISTORY HISTORY II. RELEVANT Characteristics of American American pikas allow measurement measurement of the most important important Characteristics variables 974a,b, 11978, 978, 11980, 980, 11987; 987; variables needed needed for metapopulation metapopulation analysis (Smith, 11974a,b, and Ivins, Ivins, 11984; Smith and 984; summary in Smith and Weston, 11990). 990). Pikas are diurnally hibernate, are active all year, and spend active, thus easy to observe. Pikas do not hibernate, gathering vegetation that is stored in haypiles to serve as much of their summer gathering food overwinter. Haypiles form form the figurative center of activity for each individ individround feces that that can be distinguished distinguished ual. Pikas, being lagomorphs, have small round readily from those of of all other mammals. mammals. They also deposit deposit soft soft feces (black and readily animal.. Unlike Unlike most lagomorphs, lagomorphs, pikas are elongate) unlike those of any other animal and utter utter both short and long calls calls (males (males only) that can can be be heard highly vocal and distances. Vocalizations, haypiles, feces, and urine urine stains stains can can each each over long distances. Vocalizations, fresh haypiles, feces, and used to assess current current occupancy occupancy of of pika pika habitat. habitat. Because alpine alpine climates climates are be used haypiles and the round round fecal pellets pellets do not decompose decompose readily; at Bodie Bodie dry, both haypiles they feces may they may may persist persist for many many years. years. Old haypiles haypiles and and old feces may be used used to determine sites that have have been been occupied occupied previously by pikas, even if if they they no no determine pikas, even longer occur occur there. there. longer American pikas pikas are are habitat-specifi habitat-specificc to talus or piles of of broken rock adjoining adjoining American suitable vegetation vegetation for for grazing grazing and and gathering gathering forage forage for for their their haypiles. haypiles. As talus talus suitable characteristic habitat habitat type which which is easily distinguished distinguished from from surrounding surrounding is a characteristic vegetative define vegetative habitat, habitat, it is possible to defi ne precisely the the habitat habitat area area available available for for pika occupancy. occupancy. Additionally, distributed patchily, Additionally, talus talus is usually distributed patchily, and and at the Bodie Bodie site is highly highly fragmented fragmented (see (see below). below). American are individually individually territorial. and females sepAmerican pikas pikas are territorial. Males Males and females maintain maintain sep arate and there significant difference arate territories, territories, and there is no no significant difference in territory size by gender. gender. Population and averages averages about Population density is low and about four four to eight eight animals animals per per hectare hectare on suitable on suitable habitat habitat throughout the geographic range range of of the the American American pika. WithinWithin patch (Smith and and Weston, patch nearest-neighbor nearest-neighbor distances distances average average approximately approximately 20 20 m m (Smith Weston, 1990). 1 990). As As a result result of of these these factors factors it is possible to obtain obtain a good good estimate estimate of of carrying capacity, capacity, or of each each pika or percentage percentage saturation saturation by pikas, pikas, of pika habitat habitat patch. patch. It is likely likely that that carrying carrying capacity capacity can can be be determined determined more more accurately accurately for for pikas pikas than any than any other other animal. animal. Spacing Spacing among among territories territories does does not not vary vary among among years, years, and and most are traditional and built after year most haypile haypile localities localities are traditional and built on on the the same same site site year year after year (Smith (Smith and and Weston, Weston, 1990). 1 990). The sex-ratio pikas is near near unity, unity, and and animals The sex-ratio of of adult adult pikas animals tend tend to to reside reside on on territories an animal and Ivins, territories adjoining adjoining an animal of of the the opposite opposite gender gender (Smith (Smith and Ivins, 1984). 1 984). Male and body mass Male and female female pikas pikas are are remarkably remarkably similar; similar; body mass is not not significantly significantly different different and and even even their their external external reproductive reproductive morphology morphology is barely barely distinguishdistinguish-
410 410
Andrew T. Smith and Michael Michael Gilpin Andrew
able (Smith (Smith and and Weston, Weston, 1990). 1 990). American American pikas pikas are are relatively relatively long-lived long-lived for for small small able mammals. Survivorship Survivorship normally normally exceeds exceeds 50% 50% per per year, year, and and almost almost 10% 1 0% of of the the mammals. Bodie population population was was 5 or or 6 years years old old (Smith, (Smith, 1978). 1 978). Bodie Throughout the the range range of of the the American American pika, pika, all all females, females, including including all all yearyear Throughout lings, initiate initiate two two litters litters per per summer summer breeding breeding season and and successfully successfully wean wean only lings, one of of these these litters litters (Smith, (Smith, 1978; 1 978; Smith Smith and and Ivins, Ivins, 1983a). 1 983a). Litter Litter size size (determined (determined one from embryo embry'o counts counts of of pregnant pregnant females) females) is relatively small for lagomorph and and from for a lagomorph averages 3 throughout throughout the range range of of the the American American pika pika (Smith (Smith and and Weston, Weston, 1990). 1 990). averages The mean mean litter the highest yet studied The litter size size of of 3.7 3 .7 at at Bodie Bodie is the highest of of all populations populations yet studied (Smith, 1978). 1 978). Litter Litter size does does not not vary vary significantly with with age age of of mother, mother, habitat habitat (Smith, productivity, or or between between first first and and second second litters. Litter Litter size at weaning weaning is normally productivity, than litter litter size size determined determined by embryo embryo counts counts (Smith and and Weston, Weston, 1990). 1 990). less than Weaned young grow grow rapidly rapidly and and reach reach adult adult size their summer summer of of birth. birth. Weaned size in their Juveniles must must successfully colonize colonize a vacant vacant territory to survive to become become Juveniles adults. As adults adults tend tend to be be long-lived, long-lived, vacancies vacancies are are rare rare and and occur occur only sporadsporad ically, and and one one would expect pikas to exhibit high vagility in their search search for for available territories. However, there there are two severe constraints constraints on the movements movements available territories. of juvenile juvenile pikas. First, it is difficult for juveniles to move freely in saturated saturated pika pika of for juveniles habitat; apprehend and unfamiliar juveniles juveniles from their territories habitat; adults adults apprehend and chase chase unfamiliar their territories 1 983b, 1984). 1 984). Second, Second, pikas are cold-adapted and cannot cannot toltol (Smith and and Ivins, 1983b, are cold-adapted erate warm warm temperatures. temperatures. At At Bodie, which which is near near the lower distributional distributional boundbound erate ary of temperatures severely restrict restrict the of pikas in the Sierra Nevada, high daytime temperatures movements of of pikas (Smith, 1974b). 1 974b). movements Dispersal has been been observed in two investigaDispersal by marked animals animals at Bodie Bodie has investiga tions. In one study 58 animals were individually marked animals were marked (J. (1. D. Nagy, personal personal communication). Of Of 34 adults, 25 were resighted the following year. Only one 1 8 m to a neighboring neighboring patch. Of Of 24 adult dispersed, and this movement was 18 juveniles, juveniles, only 5 were were resighted the following year. Two Two of of these 5 juveniles juveniles dispersed dispersed from their their natal patch, patch, in each each case to the next closest patch (60 m and 150 of 105 1 50 m, respectively). In a second study, only one of 1 05 marked adults dispersed; it moved to a neighboring patch patch (Peacock, 1995). addition, Peacock Peacock ((1995) 1 995). In addition, 1 995) observed dispersal by 15 marked juveniles juveniles that occurred during the summer summer months. Three been Three juveniles dispersed dispersed from their their natal natal patches patches after they had been tagged; the other 112 2 were trapped postdispersal, and their natal patches were were identified by genetic identified genetic paternity exclusion analysis. Four of of these animals dispersed within a large patch (moving an average of 45 m from their natal home range), the other 1111 dispersed between patches. Nine of of the 1111 originated from saturated patches, the other 2 came from unsaturated patches. Nine colonized the the next closest unsaturated patch, while 2 "passed up" available neighboring patches to settle on patches farther farther away. The average distance moved by these 1111 juveniles that dispersed between patches was 1132.5 32.5 m (range 70396 m; Peacock, 11995). 995). 70-396 These results indicate that juvenile dispersal among patches occurs at Bodie, but that these movements are usually limited to nearby and/or neighboring neighboring patches.
1177
Spatially Dynamics in a Pika Metapopulotion SpatiallyCorrelated Correlated Dynamics Pika Metapopulation
4 11 411
This restricted restricted ability of of pikas to disperse between between patches has profound impli implications for the metapopulation dynamics at this site. the
III. III. THE THE BODIE BODIE SITE SITE American American pikas live in a patchy distribution throughout their their geographic range in the mountains of of western North America. America. Most expanses expanses of of talus are disjunct and of of of a size to harbor harbor pika populations in the neighborhood of of tens of 00 animals. It is rare rare for for continuous populations of of pikas to contain contain more than 1100 animals. animals. The The habitat habitat at Bodie presents presents an ideal situation situation for for assessing the metapopu metapopulation dynamics of ). Bodie is an old mining area, and most of of pikas (Fig. 11). of the habitat occupied occupied by pikas there are the tailings and scree left by prior mining mining activity. Average Average size of of rock rock in each tailing patch patch is similar, hence potential aspects of habitat habitat selection by substrate these tailing substrate are minimal. In general, these patches habitat -for patches are smaller and contain fewer fewer pikas than than natural natural habitat m for example example in the Sierra Nevada 1 859 and Nevada mountains 35 km away. Mining at Bodie began began in 1859 has continued, with bursts of of activity, until the present day. Roughly 1100 00 mine mine tailing tailing patches dot the landscape landscape at Bodie, and they vary in size and and distance distance from one another. another. These patches are separated separated by a sea of of sagebrush sagebrush and other other typical Great Great Basin sage community vegetation. Additionally, pikas have been found in several of of the higher natural natural bluffs surrounding surrounding Bodie, Bodie, habitat that was Pleistocene when it was cooler in the area most likely colonized during the Pleistocene area (Smith, 11974a). 974a). We have no indication of of any recent recent dispersal dispersal from these these natural habitats to the mine tailing patches at Bodie; in fact, the nearest nearest natural natural population on Sugarloaf 0) went extinct between Sugarloaf Hill (estimated (estimated K = 110) between the second and third censuses. In summary, the Bodie pika metapopulation system fits the patchpatch matrix paradigm paradigm discussed by Wiens (this volume). The intervening sagebrush habitat habitat is quite homogeneous, and animals dispersing through it would not face a series of of choices between between subhabitats. The pikas at Bodie were "discovered" "discovered" by the biologist Joye Harrold Severaid in the mid1 940s. We know from Severaid' 1 955) 4 years of mid-1940s. Severaid'ss ((1955) of intensive obser observations, historical accounts, and more recent 974a, 11980) 980) that recent censuses (Smith, 11974a, of available pikas at Bodie have at one time occupied occupied every patch of available habitat at Bodie. Severaid ((1955) 1 955) reported reported that that three long-time residents residents of of Bodie knew knew about the pikas, going back to their boyhood around the tum turn of of the century. We also know from inspection of of maps, photographs, and descriptions of of the mine tailing patches (Severaid, 11955; 955; Bodie California State Historical Park records), as well as direct observation over the past 25 years, that these patches are permanent permanent and have not changed quantitatively or qualitatively since the mid1 940s. mid-1940s. Severaid 1 955) was the fi rst to observe Severaid ((1955) first observe that that the pika population at Bodie "was never equal to the carrying capacity of of the habitat." habitat." He trapped out some =
412 412
Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin Andrew
Bodie Town. hip
•
. •
� Bodie B l u ff •
(::7, High • Peak
•
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•
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-
- . .
_ Red Cloud oonday •
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FIGURE FIGURE 1! The configuration of mine tailing patches occupied by pikas at Bodie, California. The filled rectangles rectangles represent represent habitable patches, with the size of the rectangle rectangle proportional to the carrying Major dirt roads roads are portrayed with the stippled lines. Principal natural topo topocapacity of the patch. Major graphic features and mine tailings are named.
of the tailing patches and observed that they were colonized slowly over the course of his studies. Thus, there were early indications that pika populations populations on patches patches at at Bodie Bodie were were linked linked by by occasional occasional dispersal. dispersal.
IV. IV. METHODS METHODS Censuses 972 (Smith, 974a), Censuses of the pika pika population at at Bodie were made in 11972 (Smith, 11974a), (Smith, 11980), and 11991. census was conducted conducted in in late late summer, summer, 11977 977 (Smith, 980), 11989 989 and 99 1 . Each census for for two two reasons. reasons. First, First, at at this time time those those juveniles with with aa chance chance of of surviving surviving to to
1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulation Metopopulation 17
413 41 3
become adults adults have have reached reached adult adult size size and and become become established established on on available available terter become ritories. ritories. Thus, Thus, at at this this time time the the resident resident population population is is at at its its highest highest level. level. Second, Second, in in late late summer summer there there is is more more available available sign sign (haypiles, (haypiles, feces, feces, etc.) etc.) to to facilitate facilitate an an accurate population census. accurate population census. The The four four censuses censuses included included almost almost all of of the the available available pika pika habitat habitat in the the Bodie region. region. More More than than 70 70 isolated isolated small small patches patches (effectively (effectively islands) islands) of of talus talus Bodie were included included in each each census. census. In In the the area of High High Peak Peak (Fig. 1), 1 ), where where mining mining were area of was extreme, extreme, we we sampled sampled three large expanses expanses of of tailings, tailings, but but did did not not activity was three large take take a complete complete census. census. We We estimate estimate that that this this large large area area harbors harbors aa population population of of approximately 50 pikas pikas (see (see also Peacock, Peacock, 1995). 1 995). Regrettably, Regrettably, one one area area at at Bodie Bodie approximately was not not censused: censused: the the expanse expanse near near the the top top of of Bodie Bodie Bluff Bluff (several (several small small patches) patches) was and the downslope downslope from from the the bluff in the the direction of the the Syndicate Syndicate Mine Mine (a (a small small and the bluff in direction of number of of medium-sized medium-sized patches; patches; Fig. Fig. 1). l ). Access Access was was denied denied to to this this area for the the number area for first two two censuses. censuses. first Patch size (measured by perimeter perimeter in meters), meters), degree degree of of isolation (measured (measured Patch size (measured by the nearest or more more pikas pikas in meters), meters), and and by distance distance to the nearest patch patch inhabited inhabited by three three or number of of resident resident pikas pikas were determined for each patch patch (following Smith, Smith, the number determined for 1 974a, 1980). 1 980). Patch Patch perimeter perimeter provided provided the the most most meaningful meaningful assessment assessment of of patch patch 1974a, because territories territories on the mine mine tailings are spaced linearly and and adjacent adjacent to size, because are spaced rock- vegetation interface. interface. None None of of the the patches are are so tall as to contain contain a the rock-vegetation of pikas. Interpatch Interpatch distances distances were were measured measured between between patches patches "second story" of inhabited by three three or or more more pikas pikas because those those with with fewer fewer animals were were unlikely inhabited of colonizing colonizing individuals individuals (Smith, 1974a, 1 974a, 1980). 1 980). to be sources sources of Percent saturation of pikas pikas on patches patches was was defined defined as Ni/Ki, NJK; , where where N; the Percent saturation of Ni is the number of of pikas pikas found found on the ith patch, patch, and and K; is the carrying carrying capacity capacity of of each each number based on patch size and the number number of of potential territories that would fi fitt patch, based into this this area 974a). into area (following (following Smith, Smith, 11974a). model that A spatially explicit structured metapopulation model that incorporates incorporates our data on patch patch locations, population population change change on patches patches between between census census intervals, data observations of of extinction of of populations and recolonization of of and observations populations on patches patches and vacant patches independent sto vacant patches was created. created. Conceptually the model assumes assumes independent stochastic growth, A, A, on each each of of the habitat habitat patches patches and the possibility of of rescue rescue chastic nearby patches. (sensu Brown and Kodric-Brown, 11977) 977) or recolonization from nearby patches. patch has has a population ceiling, K, based based on its size (perimeter in meters), Each patch patch populations populations that that rise above K in a single time step are are truncated truncated back back and patch to to K K at at the the end end of of the the time time step. step. For For the the one one area area of of High High Peak Peak that that was was sampled, sampled, rather than completely censused, we extrapolated our population samples to in incorporate corporate the the entire entire patch patch area area (new (new K K = = 50). 50). We We calibrated calibrated our our stochastic stochastic growth growth on the turnover seen between the 11972 972 and the 11977 977 censuses censuses (when the meta metapopulation was assumed to be in equilibrium, given that the occupancy remained near 60%), and accordingly our time step was set at 5 years. Immigration from nearby is based the number nearby patches patches is based on on the number of of individuals individuals living living within within an an "effects "effects of the target patch. We utilized a rectangular rectangular dispersal function such that radius" of dispersal dispersal probabilities probabilities drop drop to to zero zero beyond beyond the the maximal maximal dispersal dispersal (effects) (effects) radius. radius.
4 14 414
Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin
The steps ((100 l 00 years) this was was The model model was was run run for for 20 20 time time steps years) into into the the future, future, and and this radius. A A sensitivity investigation repeated repeated 20 20 times times for for each each A A and and effects effects radius. investigation was biologically perfonned performed by varying both A A and the effects effects radius radius over over a range range of of biologically plausible values, based pikas at Bodie. plausible based on the ecology ecology of of pikas Bodie. While While there are are a number number of directions directions that that such a model model may may take, we limited ourselves ourselves to an investigation investigation of of within 1100 00 years, pop of the the following following question: question: what what is the probability probability that, within years, all populations in a subregion will go extinct? In other words, our model is designed ulations subregion go extinct? other words, our model designed to allow us to detennine, for a variety of combinations of model the determine, for variety of combinations of model parameters, parameters, the extent extent to which the landscape landscape properties properties of of the patches patches at Bodie may may contribute contribute to regional regional persistence persistence or or collapse collapse of of pika pika populations. populations. Copies of of the compiled compiled model model are are available available by contacting contacting M.E.G. M.E.G. at [email protected]. [email protected]. We popu We have have used used two two approaches approaches to quantify quantify spatial spatial autocorrelation autocorrelation of of population-level lation-level events events on patches patches and, most most important, important, integrated integrated population population changes changes in neighborhoods neighborhoods or or subregions subregions centered centered on each each patch. First, First, we analyzed analyzed net population population growth growth or or decline in neighborhoods neighborhoods surrounding surrounding each each focal focal patch, excluding excluding the change change on the focal patch patch itself. Positive spatial autocorrelations autocorrelations were population size increased increased on both focal patch were said to have have occurred occurred when when population both the focal patch and neighborhood, or if both and in the surrounding surrounding neighborhood, or if both the focal patch patch and and surrounding surrounding patches patches decreased decreased in population population size. Negative Negative spatial autocorrelations autocorrelations occurred occurred when population growth when population growth or decline decline experienced experienced on a focal patch patch was was the the opposite opposite of We ran this analysis of the net population population trend in the surrounding surrounding neighborhood. neighborhood. We for .0, and for three three "effects "effects radii" radii" (0.5, 11.0, and 2.0 km) surrounding surrounding each focal patch. patch. Results Results of of these analyses analyses include include only those those focal focal patches patches that that exhibited exhibited a change change in population population size during during the respective interval. Second, we we quantified quantified spatial autocorrelation autocorrelation of of discrete discrete growth growth rates among among neighboring 990). neighboring patches patches more more precisely using Moran's Moran's I statistic (Haining, (Haining, 11990). autocorrelation occurs occurs when when occupancy occupancy patterns patterns of of pikas pikas on on patches within Spatial autocorrelation patches within specific distances distances are are signifi significantly associated. For we examined examined the specific cantly associated. For this analysis, we the spatial autocorrelation autocorrelation of of the discrete discrete growth growth rate rate per per census interval. This This growth growth A, is defined as (n/ patches. The rate, A, ( n , ++ I1 - n/)/n/ n , ) / n , for for each each of of the i patches. The general general approach approach is to look fonn look at a kind kind of of a spatially weighted weighted cross cross product product tenn term of of the form -
1I = 2: WijLl;j', = 2: EEw;/x, Ii
j J
where Llij is a measure where A~j measure of of the proximity of of the the variate, variate, the discrete discrete growth growth rate, between the ith and jth jth spatial positions, positions, and where where wij w;j is an arbitrary spatial weighting function. function. On On aa regular regular grid, grid, wij w 0 is is sometimes sometimes taken as as 1I for for nearest nearest neighbors and structure was irregular, irregular, we neighbors and 0 otherwise. otherwise. Because our our patch patch structure we ap approached the problem differently. differently. We We set wij equal to 1I if if the Euclidean Euclidean distance distance between points i and and jj was less than d, an "effects "effects radius," radius," and 0 otherwise. otherwise. For For the Moran (A) is the mean mean value value Moran statistic, Llij Aij is taken taken as (A;-(A») (A;-(A)) (\-(A»), (Aj-(A)), where (A) of Moran of the discrete discrete growth growth rate for for all patches patches during during the the time period. period. The The Moran coefficient tenn divided divided by the coefficient is then calculated calculated as this weighted weighted covariance covariance term variance of of the growth growth rates.
17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics inin a Pika Metapopulation Metapopulation
415 41 5
We computed computed the the Moran Moran statistic statistic for for each each of of the the census census period period intervals intervals and and We for aa range range of of different different effects effects radii, radii, from from 0.1 0. 1 to to 1.5 1 .5 km, km, in in 0.1-km O. l -km increments. increments. for The 95% 95% confidence confidence limit limit against against which which our our data data were were contrasted contrasted was was produced produced The by creating creating 200 200 independent independent realizations realizations of of uncorrelated uncorrelated discrete discrete growth growth over over our our by grid of of habitat habitat patches. patches. grid
RESULTS V. RESULTS Patch Occupancy Occupancy A. Patch As was was initially initially observed observed by by Severaid Severaid (1955), ( 1 955), not not all all habitable habitable mine mine tailing tailing As patches at at Bodie Bodie contained contained pikas pikas during during any any of of our our four four censuses. censuses. Approximately Approximately patches 60% of of the the patches patches were were occupied occupied during during the the first first two two censuses censuses (Smith, (Smith, 1974a, 1 974a, 60% 1 980), but but this this level level of of occupancy fell to to about 45 % for for the the latter latter two two censuses censuses 1980), occupancy fell about 45% (Table Correspondingly, more pikas were located on on the study area (Table I). Correspondingly, more pikas were located the study area during during the the two earlier censuses than latter two two (Table I). two earlier censuses than the the latter (Table I). Typical of a metapopulation system, there was frequent Typical of metapopulation system, there was frequent turnover turnover (extinctions (extinctions of populations on patches and of vacant vacant patches patches from from of populations on patches and subsequent subsequent recolonization recolonization of existing local populations) populations) between between censuses (Table II). In In addition, addition, roughly roughly 50% 50% existing local censuses (Table of of patches patches varied varied in in popUlation population size size during during each each of of the the three three census census intervals intervals (Table (Table II). II). The patches at Bodie The most most striking difference difference in the the pattern pattern of of occupancy occupancy of of patches Bodie among among the the censuses censuses was was the decline decline and and near near collapse collapse of of pika pika populations populations on patches Bodie patches in the southern southern half half of of the study area. area. The The distribution distribution of of patches patches at Bodie is is roughly roughly in in the the shape shape of of an an hourglass; hourglass; aa saddle saddle in in the the area area of of aa dirt dirt road road extending extending up from the Bodie township guratively divides the northern township fi figuratively northern and southern southern half half of the study area (Fig. 11). there was a mixture of of occupied occupied of ). In the first two censuses, there and and unoccupied unoccupied patches patches in in both both the the northern northern and and the the southern southern parts parts (Fig. (Fig. 2). 2). By By 11989 989 we noted a significant drop drop in the percentage percentage of of occupied occupied patches patches in the southern half (Fig. 2). This decline included the extinction of the pika population population on the Red Cloud tailing, a site that harbored 972 and 11977. 977. Only harbored nine pikas in 11972 22 years years later later pikas pikas were were absent absent from from nearly nearly all all patches patches in in the the southern southern half. half. In In the the extreme southern southern part of the study study area, only only one animal was found (on the relatively relatively large large Noonday Noonday tailing; tailing; Fig. Fig. 2). 2).
B. B. Area Area Effects Effects Size Size of of habitat habitat patch patch appeared appeared to to be be the the most most important important factor factor governing governing the the occurrence occurrence of of pikas on the habitat patches at at Bodie (Table I), an effect effect apparently due due to to the the relatively relatively low low probability probability of of extinction extinction of of popUlations populations on on large large patches patches (Smith, 980). In (Smith, 11980). In all all censuses, censuses, average average size size of of occupied occupied patches was greater greater than than the cant differences the average average size of of vacant patches. patches. There There were, however, however, signifi significant differences in rst two in the the apparent apparent effect effect of of patch patch size size among among the the four four censuses. censuses. In In the the fifirst two
4 16 416
Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin
TABLE I Descriptive DescriptiveMeasurements Measurements and and Correlational Correlational Statistics Statistics of of Pikas Pikas on on Mine Tailing Tailing Patches Patches at at Bodie Bodie Mine 1972" 1972 a
A verage size Average size of of patches' patches" (perimeter m) (perimeter in in m) Occupied patches Occupied patches Vacant Vacant patches patches A verage inter-patch Average inter-patch distances' (m) distances" (m) Occupied Occupied patches patches Vacant Vacant patches patches Correlation Correlation of of percentage percentage saturation with patch saturation with patch size size All All patches patches Occupied Occupied patches patches Correlation Correlation of of percentage percentage saturation saturation with with interpatch interpatch distance distance All All patches patches Occupied Occupied patches patches Number N u m b e r of of patches patches censused censused Percentage Percentage occupied occupied of of mine tailing tailing patches mine patches Total number of Total number of pikas pikas censused censused
1977b 1977 ~
1989 1989
1991 1991
96.0 96.0 29. 29.11
90.5 90.5 4 1 .3 41.3
96.9 96.9 48.2 48.2
85.9 85.9 55.2 55.2
1101.5 O l .5 1184.8 84.8
1119.1 19. 1 238.8 238.8
1102.6 02.6 267.7 267.7
193.4 193.4 11065.2 065.2
r, r~"a = =
r, rs = =
0.53*** 0.53*** 0.43** 0.43**
r~ = = r,
0.37*** 0.37*** - 00.07 .07
r, = = r,
r, rs = =
- 0.47*** 0.47*** 0.03 0.03
r, r~ = =
- 0.57*** 0.57*** - 00.48** .48**
r, r, = =
r, rs = =
r, rs = =
0.65*** 0.65*** 0.47** 0.47**
- 0.30** 0.30** 0.02 0.02
r, rs = =
r, rs = =
r, r~ = =
r, r~ = =
r, rs
0.18 0. 18 = - 00. . 1144
=
r, r~ = =
r, rs = =
78 78
78 78
77 77
78 78
60.3 60.3
57.7 57.7
44.2 44.2
43.6 43.6
1164 64
140 140
1118 18
1129 29
- 00.69*** .69*** - 00.01 .01
a
Data Smith ((1974a). l 974a). "D a t a from from Smith from Smith from Smith ((1980). 1 980). '' Excluding Excluding three three High High Peak Peak samples. samples. d d Spearman Spearman rank rank correlation correlation coefficient coefficient corrected corrected for for tied tied observations. observations. **p < 1. **P < 0.0 0.01. ***P ***P < < 0.001 0.001.. h Data Data h
censuses 1 972 and 977) there cant correlations censuses ((1972 and 11977) there were were signifi significant correlations between between patch patch size and percentage percentage saturation saturation for for both both all patches patches and just those those that that were were occupied. occupied. This cant) for This correlation correlation was weaker weaker (although (although still highly highly signifi significant) for all patches patches in 11989, 989, but there was no 1 989. In 1991, 1 99 1 , no correlation correlation for for occupied occut, ied patches patches only in 1989. neither 1 989, neither all patches patches nor nor occupied occupied patches patches silOwed si~owed a correlation correlation to area. In 1989, when metapopulation was was when the "collapse" "collapse" of of the southern southern half half of of the Bodie Bodie pika pika metapopulation underway, underway, a number number of of large large patches patches harbored harbored very few few pikas. The The resulting resulting low percentage resulted in the lack of percentage saturation saturation values on large large patches patches apparently apparently resulted of a correlation 99 1 , there correlation between between these these variables variables in that year. In 11991, there was no no correla correlation between between size and and percentage percentage saturation saturation for for all patches patches because because nearly all of southern half unoc of the patches patches in the southern half of of the Bodie metapopulation were were unoccupied, independent of remained no correlation cupied, independent of their their size. Interestingly, Interestingly, there remained correlation
17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Metapopulotion Metapopulation
417 417
Between-Census Extinctions Extinctions and ond Recolonizations Recolonizations and and Changes Changes in in Percentage Percentage Saturation Saturation on on TABLE III I Between-Census Mine Tailing Patches at Bodie Mine Tailing Patches at Bodie 1972- 1977 1972-1977
Patch extinctions extinctions Patch Patch recolonizations recolonizations Patch Percentage saturation saturation Percentage patches increased increased No. patches patches decreased decreased No. patches N o change change (including ( including unoccupied unoccupied No patches) patches) Percentage of patches that that experienced experienced a Percentage of patches change change
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between percentage saturation saturation for the occupied occupied patches the northern northern between size and and percentage for the patches in the half of of the the study area. While While the the southern southern half half was was nearly nearly void of of pikas, pikas, the the half study area. absolute number number of of pikas pikas increased increased in 1991 1 99 1 over over 1989 1 989 (Table (Table I). Thus, Thus, while while there there absolute were fewer fewer occupied 1 99 1 (Table (Table I), those those that that were were occupied occupied were occupied sites at Bodie in 1991 had relatively high percentage percentage saturation saturation values values independent of size of patch. patch. all had relatively high independent of size of
C Isolation Isolation Effects Effects C. There was correlation for There was a strong strong negative negative correlation for all patches patches of of percentage percentage satusatu ration ration with interpatch interpatch distance distance across across all years (Table (Table I), apparently apparently because the percentage percentage of of patches patches that that were were unoccupied unoccupied increased increased with with interpatch interpatch distance distance (see Smith, 11980). This correlation each subsequent subsequent 9 80). This correlation was more more pronounced pronounced in each census year year (Table (Table I). The The strongest strongest correlation, correlation, in 11991, resulted from from the long census 99 1 , resulted long inter-patch unoccupied patch inter-patch distances distances from from each each unoccupied patch in the southern southern half half of of the study area to the closest patch patch with three three pikas pikas in the northern northern half half (Table (Table I). area At the same time, in 3 of of 4 years there was was no no relationship relationship between between interpatch interpatch At distance distance and percentage percentage saturation saturation for for occupied occupied patches patches (Table I). Apparently, Apparently, once occupied, occupied, factors factors other other than than isolation play a major major role in the determination determination once of 989 when of percentage percentage saturation saturation on tailing patches. patches. The The one exception was was in 11989 when there was a signifi cantly negative correlation between these two significantly two variables (Table I). At this time the collapse of of the southern southern half of of the Bodie pika metapopulation metapopulation percentage saturation among among occupied patches patches apparently was had started, and percentage not being determined determined by patch size as in other other years (see above). Instead, the effect effect of of isolation isolation was was beginning beginning to to show show among among occupied occupied patches; patches; those those that that occupied yet declining in percentage percentage saturation were not receiving new were occupied propagules propagules (i.e. no "rescue effect"). Thus Thus the the data data on correlation of of percentage percentage saturation with patch size and interpatch distance portrayed in Table I show a "wave" effect from 972 and from 11972
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The Spatially Spatially Explicit ExplicitStructured StructuredMetapopulation Metapopulation Model Model DD.. The There are are many ways of constructing and parameterizing a metapopulation metapopulation There model (Hanski and Gilpin, 11991). 99 1 ). We developed a spatially explicit structured metapopulation model because we were interested in exploring a specific ques quessouthern region region of the Bodie metapopulation metapopulation more more extinction extinction prone prone tion: is the southern northern region? To approach approach this we have examined A, A, which governs than the northern "rescue" and recolonization, recolonization, rate of extinction, and the effects radii, which govern "rescue" reasonable, ranges of of values (Fig. 3). Our response response over wide, but biologically reasonable, of pikas in the northern region to the number number of pikas in the variable is the ratio of southern region at the end of 20 time steps, or 1100 00 years (Fig. 3). The outcome
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FIGURE 3 The The difference FIGURE difference between between projected projected population population behavior behavior of of pikas pikas in the north north and and south south subregions two parameters parameters of subregions at Bodie versus versus the two of the the stepping-stone stepping-stone model. One One independent independent axis axis shows shows aa set set of of values values for for the the "effects "effects radius" radius" (see (see text text for for explanation), explanation). while while the the second second independent independent axis from nearby axis shows shows values values for for the the expected expected discrete discrete growth growth rate rate (which (which is absent absent any any immigration immigration from nearby patches). Over Over the the full set set of of parameter parameter values, values. the the full full metapopulation metapopulation behavior behavior goes goes from full patches). from full occupancy occupancy and and full full saturation saturation to to complete complete regional regional extirpation. extirpation. The The dependent dependent axis, axis. plotted plotted vertically vertically as as histogram histogram cells, cells. shows shows the the ratio ratio of of the the number number of of animals animals in the the northmost north most half half of of the the metapopumetapopu lation to to the the number number of of animals animals in the the southmost southmost half half of of the the metapopulation metapopulation after 1 00 years years (20 (20 time time lation after 100 steps) based based on on 50 50 replications. replications. At At full full saturation (high effects effects radius radius and and high high expected expected growth growth rate), rate). steps) saturation (high there there are are about about 20% 20% more more animals animals in the the northern northern subregion. subregion. As As the the parameters parameters are are each each reduced reduced in in value, so that the value. so does does this this north-to-south north-to-south ratio, ratio, which which indicates indicates that the northern northern half half of of the the metapopulation metapopulation is predicted predicted to to be be the the more more vulnerable vulnerable subregion subregion to to subregional subregional extirpation. extirpation.
420
Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin
shows shows that the the north-to-south north-to-south ratio declines declines with declining declining parameter parameter values. The model appears appears most sensitive to variation in the effects effects radii. The north and south subregions paramet.er values values are subregions at Bodie Bodie appear appear to behave similarly only when when parameter are jointly high (leading (leading to (leading to full occupancy occupancy of of patches) or jointly low (leading regional regional extinction). extinction). At intermediate intermediate values values the north-to-south north-to-south ratio ratio falls below unity, indicating indicating that that the northern northern half half of of the metapopulation metapopulation is the more vulner vulnerable subregion subregion to subregional subregional extirpation extirpation (Fig. 3). This finding is the exact exact op op1 99 1 . posite of of our our empirical empirical result in 1991. One One potential potential reason reason for the discrepancy discrepancy between between the model and empirical empirical results is that average nearest-neighbor distance between each pair of patches that average nearest-neighbor distance between pair of patches is shorter 1 69 km) than Perhaps shorter in the southern southern region (0. (0.169 than in the north north (0.204 km). Perhaps more more important important is Fig. 4, a demonstration of of the degree degree of of connectivity connectivity (number (number of of potential potential patches patches within within sweeps sweeps of of each each of of four four effects effects radii) for for each each patch. patch. Clearly, au Clearly, there there are are significantly fewer fewer patches patches available available to influence influence spatially autocorrelated longer tocorrelated changes changes at the shortest shortest distance distance (0.25 km) than than at each each of of the longer .0, and 2.0 km, respectively (Fig. 4). At distances, 0.5, 11.0, At an effects effects radius of of 0.25 0.25 km, kin, very few of of the the patches patches show connectivity, connectivity, and extinction extinction of of popUlations populations on patches regionally or throughout the Bodie Bodie metapopulation metapopulation is not a surprising surprising result. At 0.5 km km many of of the patches patches in the southern region region are are connected, connected, and and there there are two distinct distinct clusters clusters with high connectivity. At this distance distance the the north is still very loosely connected. connected. The 2.0-km 2.0-km effects radius figuratively "joins" "joins" each each patch patch with a very large large subset of of patches patches within the Bodie metapopulation metapopulation system (Fig. 4).
E. Regional Effects Effects (Spatially Autocorrelated Patterns) The The spatial pattern pattern of of patch patch occupancy, occupancy, and and area area and isolation effects effects (Fig. 2; Tables 980) indicate Tables I and and II; see also Smith, Smith, 11980) indicate that the dynamics dynamics of of extinction extinction and recolonization recolonization of of patches patches at Bodie has not occurred occurred uniformly across the the landscape. landscape. In addition, addition, our model based based on stepping-stone stepping-stone metapopulation metapopulation dy dynamics namics failed failed to explain explain our observation observation that that the southern half half of of the Bodie Bodie meta metapopulation 99 1 census, population collapsed collapsed in the 11991 census, while the northern northern half half remained remained close to fully saturated. saturated. Thus, we examined examined this putative putative spatially nonrandom nonrandom pattern pattern by determining determining the extent to which there was spatial autocorrelation autocorrelation of of popula population-level tion-level events among patches. patches. Between-census Between-census correlated correlated changes changes in percentage percentage saturation were determined determined between focal patches patches and neighboring neighboring patches patches at three three effects effects radii for for each each of of the 972 - 1 977 and 1 989 - 1 99 1 interinter the three three census census intervals intervals (Table III). For For the the 11972-1977 and 1989-1991 vals, there patches that there were no significant differences differences between the number number of of patches exhibited exhibited positive positive spatial autocorrelation than negative spatial autocorrelation autocorrelation at the 2.0-km effects effects radii (Table III). Both intervals intervals showed showed significant significant differences differences (P . 1 ) at the 0.5-km radius; and for 1 989 - 1 99 1 interval, interval, there (P < < 00.1) for the 1989-1991 there was a .0-km radius. The highly significant significant difference difference at the 11.0-km The opposite result obtained obtained for 977 - 1 989 census interval: for the 11977-1989 interval: there there were significant significant differences differences at effects effects
17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulation Metapopula�on
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422 422
Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin Andrew
TABLE III III Between-Census Between-Census Correlated Correlated Changes Changes inin Percentage Percentage Saturation Saturation between between Focal Focal Mine Mine Tailing Tailing TABLE Patches and and Neighboring Neighboring Patches Patches at at Various Various Fixed Fixed Distances Distances ("Effects ("Effects Radii") Radii")a~ Patches Effects Effects radius (km) (km) radius
Census Census interval interval
Positive spatial spatial Positive autocorrelations autocorrelations
Negative spatial spatial Negative autocorrelations autocorrelations
G-test G-test value value
2.00 2.00
1 972- 1 977 1972-1977 1 977- 1 989 1977-1989 1989- 1 99 1 1989-1991 1972- 1 977 1972-1977 1 977 - 1 989 1977-1989 1 989- 1 99 1 1989-1991 1972- 1 977 1972-1977 1977- 1 989 1977-1989 1 989- 1 99 1 1989-1991
18 18 27 27 18 18 25 25 26 26 26 26 26 26 23 23 25 25
21 21 13 13 16 16 14 14 14 14 88 13 13 17 17 99
0.101 0.101 4.302** 4.302** 0.028 0.028 2.59 1 2.591 3.062* 3.062* 8.895*** 8.895*** 3.75 1 * 3.751" 0.627 0.627 6.849*** 6.849***
1 .00 1.00
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Positive spatial spatial autocorrelations autocorrelations occurred occurred when when the the focal focal patches patches increased increased or or decreased decreased in in perper a Positive centage saturation saturation and and the the average average change change in in percentage percentage saturation saturation in in neighboring neighboring patches patches (within (within centage respective radius) correspondingly increased increased or or decreased. decreased. Negative Negative spatial spatial autocorrelations autocorrelations ococ aa respective radius) correspondingly curred when when percentage percentage saturation saturation between focal patch patch and and its its neighboring neighboring patches patches were were of of difdif curred between aa focal ferent sign. sign. Only Only those those patches patches that that exhibited exhibited aa change change in in percentage percentage saturation saturation during during the the respective respective ferent census interval interval were were analyzed. analyzed. census p< 0. 1 0. ** P < 0.10. p< < 0.05. 0.05. ** P ** *** p < 0.0 1. *** P < 0.01.
radii ooff 2.0 and 11.0 .0 km, whereas difference between number of whereas there there was no difference between the number of positive positive and and negative negative spatially autocorrelated autocorrelated patches patches at the 0.5-km radius radius (Table III). III). The sensitivity of spatial autocorrelation autocorrelation at varying radii was was examined also using Moran's 972 - 1 977 census interval, Moran's I statistic (Fig. 5). During the 11972-1977 interval, there was positive spatial autocorrelation only for effects radii up to 0.3 km (Fig. 5), which is a distance that includes on on average very few nearest neighbors (Figs. 11,, 4). During this time overall saturation was high, there was an interspersion of occupied occupied and and unoccupied patches patches (see Fig. 2), and and population population extinctions and subsequent subsequent recolonizations on on patches appeared to to be in aa dynamic dynamic equilibrium (Smith, 11974a, 974a, 11980). 980). This result is consistent with the reported low vagility of of pikas at Bodie (Smith, 974b)- population sizes on focal patches were (Smith, 11974b)mpopulation were likely to be be influenced influenced only only by by nearby nearby patches patches (either (either growing growing in in part part due due to to immigration immigration of of surplus surplus animals from from nearby nearby patches, patches, or or declining declining and and concomitantly concomitantly not not re receiving ceiving immigrants immigrants from from nearby nearby patches patches that that were were also declining). declining). The 977 - 1 989 census The 11977-1989 census interval interval shows shows aa more more interesting interesting pattern. pattern. During During this 1 40 to 1 8; Table this interval interval there there was was aa net net loss loss of of animals animals ((140 to 1118; Table I), I), and and this this loss loss was was confined confined mostly mostly to to the the southern southern end end of of the the study study area area (Fig. (Fig. 2). 2). This This regional regional decline decline accounts accounts for for much much of of the the positive positive spatial spatial autocorrelation autocorrelation (largely (largely negative negative patch patch growth growth associated associated with with negative negative regional regional growth growth in in the the south) south) over over distances distances from .5 km from 0.3 0.3 to to 11.5 km (Fig. (Fig. 5). 5). ItIt is is also also possible possible that that the the longer longer time time interval interval between between censuses uenced the censuses infl influenced the autocorrelation autocorrelation statistic statistic by by allowing allowing more more "averaging" "averaging"
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over pikas at Bodie. Bodie. In over regions regions of of the cumulative cumulative small dispersal dispersal movements of of pikas this increased in population population size the census this way, many many patches patches in in the the north north increased size during during the census interval interval (Fig. (Fig. 4). 4). A A lack lack of of significant positive positive autocorrelation autocorrelation occurred occurred over over the the shortest shortest time interval 1 989- 1 99 1 ; Fig. 5). The remaining animals interval ((1989-1991; The few remaining animals in the southern southern portion portion high overall died out, and and the the northern northern region showed stability stability at at aa relatively high overall saturation. saturation.
VI. DISCUSSION DISCUSSION Our Our long-term long-term multigenerational multigenerational study study of of the the Bodie Bodie pika pika metapopulation metapopulation has has provided provided insight insight into into the the dynamics dynamics of of aa real real world world metapopulation. metapopulation. We We have have gained mechanisms of gained understanding understanding of of the the underlying underlying mechanisms of the the metapopulation metapopulation dy dyhave learned namics, and and we have learned something something about about the different different spatial spatial scales that that are are relevant these mechanisms relevant to to these mechanisms and and to to their their associated associated dynamics. dynamics. The which metapopulation dynamics are likely to has The regional regional scale scale at at which metapopulation dynamics are likely to occur occur has been that "at place been defined defined as as that "at which individuals individuals infrequently infrequently move from from one one place
424
Andrew and Michael Andrew T.T. Smith Smith and Michael Gilpin Gilpin
(population) (population) to another, typically across across habitat types which are not suitable for their feeding and breeding activities, activities, and often with substantial risk of failing to locate 99 1 ). locate another another suitable habitat habitat patch patch in which to settle" (Hanski and Gilpin, 11991). The American pika population at Bodie fits this definition closely. Pikas cannot and do not occupy the Great Basin sagebrush habitat that separates separates their obliga obligatory talus habitat -the mine tailing patches habitat~the patches at Bodie. Physiologically, pikas are at risk due to high temperatures temperatures and inability to find ameliorating microclimates (such as the cool interstices deep 974b). Pikas deep in talus) while dispersing (Smith, 11974b). are more vulnerable 974a; Ivins and Smith, vulnerable to predation predation when off off of of talus (Smith, I1974a; 1 983), and predators of 1983), and a high diversity of of potential predators of pikas pikas occupy occupy the the study area (Smith, 11979). 979). Juveniles, the primary dispersers in this system, normally settle Juveniles, dispersers on the closest unsaturated patch (1. A. Nagy, personal communication; Peacock, unsaturated (J. 1 995). 1995). Metapopulations are characterized characterized also by their between extinction Metapopulations their balance between of habitat of local popUlations populations and and establishment establishment of of new populations populations in vacant habitat patches (Hanski and Gilpin, 1991). 1 99 I ). Such extinction-colonization extinction-colonization dynamics, as well as a pattern of incomplete patch occupancy (in the range of 60-40%), 60-40%), have been a characteristic characteristic of of the Bodie pika population for several decades decades (Tables I, II; Severaid, 1955; 1 955; Smith, 11974a, 974a, b, 1980). 1 980). During the 1970s, 1 970s, extinction of pop of populations ulations on patches patches was a function of patch patch area, area, largely due to demographic stochasticity, and immigration, which was a function of nearest ' of the distance distance to the nearest' possible source patch (Table I; Smith, 11974a, 974a, 1980). 1 980). Indeed, the average small patch size at Bodie, thus the small carrying carrying capacity of pikas on these patches, appears appears to be a central central feature of this metapopulation. metapopulation. The strong effect of isolation of patches on occupancy rates of pikas in the of patches Bodie system indicates that that a basic assumption of of the spatially implicit Levins ((1970) 1 970) meta population model--that model- that dispersal occurs metapopulation occurs with equal equal probability be between any pair of patches, patches, independent independent of of their location~is tween location -is incorrect. incorrect. Smith ( 1 974a, 1980) 1 980) formulated his early analysis on a spatially (1974a, spatially explicit framework based 967). In the based on island biogeographic theory (MacArthur (MacArthur and Wilson, 11967). present analysis, we have extended parameterization of present extended this with the parameterization of a spatially explicit "stepping-stone" "stepping-stone" metapopulation model. There There are are many many forms, varying considerably in complexity, that that such a step steppingstone pingstone metapopulation model can take. We chose to work work with the simplest form of of model that that could include the the most important features of of the the Bodie pika metapopulation system. The of all patches. metapopulation The model includes the size and location location of We have have provided for distance-dependent distance-dependent rescue rescue and and recolonization through an "effects radius" that limits single time-step time-step dispersal to a maximal distance, which, "effects radius" distance, which, in our sensitivity analyses, we have kept to below I1 km. We modeled occupancy and al., this volume, for and extinction in a structured structured manner manner (see Gyllenberg Gyllenberg et et al., for a review) in which we distinguish the integer population size on a patch, most of of which fall in the 0 to 10 IO range. range. Transitions between states states on a patch patch are are governed ' by demographic demographic stochasticity, which is is" independent independent between between individuals. Based on data 1 970s, our analysis shows the metapopulation system data from the 1970s, to be stable, with continuing turnover and with patch patch occupancy occupancy in the range range of of
1177
Spatially Dynamics in SpatiallyCorrelated Correlated Dynamics in aa Pika Pika Metapopulation Metapopulation
425 425
60%. If prediction, then If these these results were were to be taken as a model model prediction, then the the actual system -the near sub behavior of of the systemwthe near extirpation of of populations in the southern southern subregionwould be quite surprising. surprising. Of region--would Of interest, then, is whether whether the model was misparameterized or incompletely structured. structured. have explored the possibility of of misparamterization. We have performed performed We have a wide sensitivity analysis by varying the effects radius and the expected expected growth rate per patch. We varied these parameters over a range that takes the metapop metapopulation from full saturation and occupancy to extirpation. What we found found is that it is highly improbable improbable (although not impossible) that the southern subregion should go extinct, while the northern northern subregion subregion should remain highly saturated. Thus, Thus, we conclude conclude that the the structure of of the the model is incomplete. Because of pre of the apparent inadequacy of of our stepping-stone stepping-stone model model for predicting the future of spa of the Bodie metapopulation, metapopulation, we analyzed the system for for spatially correlated patterns among neighborhoods neighborhoods (or clusters) of of patches. We did this in two ways: with analyses analyses of of correlated changes in population population growth within within neighborhoods, neighborhoods, and with the development development of of Moran' Moran'ss I statistic for our map mapbased data in which multiple effects radii were examined. The autocorrelation/neighborhood autocorrelation/neighborhood analysis indicated indicated that most patches patches were strongly influenced by the average level of patches of occupancy in surrounding patchesw thus thus that entire neighborhoods neighborhoods rather rather than distance to a single single potential source patch was important in determining the probability of of patch occupancy (Table III). These neighborhoods ned for neighborhoods were well-defi well-defined for the shortest effects radius, 0.5 km, for 1 972 - 1 977, 11989-1991). 989 - 1 99 1 ). The for the the two two censuses taken close together ((1972-1977, .0 km yielded the highest overall difference between positive and radius of of 11.0 negative autocorrelations autocorrelations of of patches for all censuses. These results highlight the influence of of neighboring patches at greater distances from target patches than is apparent from the spatially explicit metapopulation model model based based upon stepping steppingstone (or nearest-neighbor) nearest-neighbor) dispersal distances. The longest census interval was between 11977 977 and 11989, 989, and the strongest strongest effect (difference (difference between number number of of positive and negative autocorrelated patches) was found found at 2.0 km. Apparently, given more more time and the likelihood of of more more cumulative nearest-neighbor nearest-neighbor dispersal movements among patches, the radius affected by such movements increases. The results of of Moran' Moran' s I statistic also confirm the spatially correlated correlated pattern of occupancy of of pikas at Bodie and indicate that there may be distinct differences differences in the nature nature of of autocorrelation autocorrelation of of patch occupancy occupancy among census intervals (Fig. 5). Unfortunately, Unfortunately, because because of of the different intervals between censuses, censuses, we cannot cannot discriminate between between whether whether these different different responses responses were caused by time of of census interval or the actual spacing spacing of of occupied occupied and unoccupied unoccupied patches census patches during the census interval. Parsimoniously, the 1972-1978 1972 - 1 978 interval showed high levels of of autocorrelation autocorrelation at relatively relatively short effects radii radii distances, distances, indicative indicative of of the the ex extinction-colonization tinction-colonization dynamics that that characterized characterized this time frame. At the the other other 1 989 - 1 99 1 interval, and this extreme, no autocorrelation autocorrelation was evident for for the 1989-1991 result could be due to either (or both) the short time between censuses censuses or or the total half of popUlation. The 1 977 - 1 989 interval collapse of of the southern southern half of the study population. The 1977-1989 cant autocorrelations range of yielded signifi significant autocorrelations across across the full range of effects radii, again
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Andrew Andrew T.T. Smith and Michael Michael Gilpin Gilpin
indicating neighborhoods due to indicating that that there may have been been more more "averaging" of of neighborhoods the cumulative effects effects of of stepping-stone stepping-stone dispersal dispersal over this period. We We do do not not know know the the actual actual mechanism(s) mechanism(s) that that may may have have produced produced the the spatial spatial nonrandom ness of nonrandomness of extinction as demonstrated in this investigation. In his book book on 1 990) shows on spatial spatial statistics, statistics, Haining Haining ((1990) shows that that very very different different dynamics, dynamics, and and dif different ferent underlying mechanisms, mechanisms, can cause similar patterns of of spatial autocorrela autocorrelaThe tion. We We pose two general hypotheses about what occurred occurred in this system. The first is that the results could have been due to "position" effects. The second is results The of the southern half of the study that extinction events leading to the collapse of half of population were driven by spatial autocorrelation effects. effects. The of the The position position effects effects hypothesis hypothesis holds that the southern half half of the study study area is qualitatively different different from the northern half half or that external forces forces (such as climate, predation, competition) could affect affect pikas pikas in the southern and northern northern halves differentially. differentially. Although we have no direct measurements, available data data that we outline below indicate that there is no demonstrable impor demonstrable difference difference in imporparameters that may affect affect pika pika viability in the southern and northern halves tant parameters halves of of the study area. area. The The Great Basin sage plant community community is roughly roughly similar throughout the study area, area, and and each of of the habitat patches patches on which pikas pikas live was "thrown up" in the homogeneous habitat. There the middle middle of of this homogeneous There is very very little difference patch, and of the patches patches are difference in size of of rock rock in each patch, and the structure of are remarkably similar throughout the study area. Similarly, the area area is too small to have have been been affected affected differentially differentially by by climatic climatic patterns. patterns. All available observations on potential potential predators and competitors (Severaid, indicate that that they they occur throughout the study study area. area. The The most most 11955; 955; Smith, 11979) 979) indicate likely predators weasels (Mustela predators of of pikas pikas on their their preferred preferred talus habitat are are weasels -both of 979), as Jrenata frenata and M. erminea ermineamboth of which are common at Bodie; Smith, 11979), they can gain pikas (Ivins and Smith, 1983). 1 983). Weasels at gain entrance entrance to the dens of of pikas Bodie have the tendency to hunt repeatedly on the same patch (A. T. Smith, unpublished predation is the unpublished data), and and it is likely that that weasel predation the most common common cause cause of of extinction extinction of of a population of of pikas pikas on a patch. patch. If If a family of of weasels then moved to the next closest occupied patch (as they do when hunting Microtus; occupied patch 977), a cluster of How Fitzgerald, 11977), of patches patches could show "correlated extinctions." extinctions." However, as mentioned mentioned above, there there is no reason that such predator-caused predator-caused extinctions would occur occur only in one region of of the study area. Instead, such extinction extinction events could be said to operate as a form of of environmental stochasticity stochasticity that that would impact impact neighborhoods neighborhoods of of patches patches (see (see below). below). final position effect is that the the largest single single patch (High Peak; Peak; A fi nal potential position Fig. 11)) was found on the northern northern half half of of the study area area (although there there were were large patches-the l ] -found patches--the Noonday Noonday and the Red Red Cloud mine tailings [Fig. [Fig.1] - - found in the south). It could be that that the northern northern area area operates operates more more like a mainland mainland- island population, in which persistence depends depends on the existence of of one or more more extinction-resistent 99 1 ). Two extinction-resistent populations populations (Harrison, 11991). Two lines of of evidence ar argue against against this effect. First, the large patch patch (K = = 50) was included our spa spaincluded in our tially explicit model, and in spite of its size the northern northern population was more more
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extinction-prone than the 988 to 11991 99 1 all the southern population. Second, from 11988 pikas marked (Peacock, pikas in a portion of of the the High High Peak patch were individually marked (Peacock, 11995). 995). Although several patches patches are found in the neighborhood of of this site, none of them were colonized by any of 1 995). Thus, of the marked marked animals animals (Peacock, (Peacock, 1995). within within the time scale of of our censuses, the continued continued occupancy occupancy of a large large number number of patches patches in the northern half does not appear appear to be explained explained by mainland mainlandisland colonization colonization dynamics. The alternative alternative to the position effects effects hypothesis is that extinction extinction events leading half of leading to the collapse of of the southern southern half of the study population were were driven driven by spatial autocorrelation effects. In this case the southern and northern portions of of the study area are considered equivalent, equivalent, and a stochastic cause of of extinction may have impacted patch populations in the south rather than in the north. impacted rather than north. As fewer fewer and fewer fewer southern southern patches patches were occupied, occupied, then then there were fewer fewer source patches to produce number of produce propagules for for the increasing number of unoccupied patches. Ultimately, we believe a threshold may have been reached that the system itself was incapable -colonization equilibrium, incapable of of remaining remaining in extinction extinction-colonization equilibrium, and and the result was its total 989 census, which showed a greatly reduced number total collapse. The The 11989 of 99 1 . of occupied islands in the south, was a harbinger harbinger of the collapse observed in 11991. The 11989 989 census is important in that it shows that the southern half half did not collapse over a short period (such as might be expected expected should there have been an epidemic), epidemic), but rather rather that there there was a gradual gradual decline decline in the number number of of occupied islands. Effectively, what we observed in the southern half half of of the study area during the 11991 99 1 census was a regional extension of of the spatial autocorrelation among 1 99 1 census (Table patches seen throughout the Bodie landscape landscape leading up to the 1991 III; Fig. 5). We We have no direct evidence of of the stochastic event(s) that may have initiated half of of patch occupancy in the southern half of the Bodie study this downward spiral of area. Demographically, pikas are long-lived. It is possible that the age structure structure on some patches could have been skewed to older tum prohib older animals, animals, which in turn prohibited local settlement of 1 983a). A few of their offspring (see Smith and Ivins, 1983a). few key patches could have been structured in this way, followed by the death death of of all adults adults from "old age." Also possible is that that populations on individual patches patches became became genetically inbred, inbred, leading to inbreeding depression and local patch patch extinctions (Gilpin, 11991). 99 1 ). However, a detailed detailed study of of the genetics of of pikas in a subset of the Bodie popUlation patches population indicated indicated that enough movement took place among patches (with juveniles primarily colonizing the closest available territory territory as well as oc occasionally dispersing greater greater distances) that that habitat habitat fragmentation resulted in only limited genetic subdivision within this population (Peacock, 11995). 995). Another Another pos possibility, as mentioned above, is that that weasels cleared cleared out a few few key patches, patches, be beginning the decline. Naturally, there could also have been multiple mUltiple causes of of extinction of of patches patches in the south. The results of of this investigation show clearly the need for for metapopulation models to incorporate incorporate explicit spatial dimensions and to examine examine a range of of spatial and temporal scales in their analyses. Accordingly, efforts to understand understand and to
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T. Smith and Michael Gilpin Andrew T.
model further further the the Bodie Bodie pika pika metapopulation metapopulation continue. continue. First, First, we we desire desire to to underunder model stand dispersal dispersal as as aa function function of of source source patch patch density, density, degree degree of of patch patch isolation, isolation, stand intervening habitat habitat structure structure (Wiens, (Wiens, this this volume), volume), and and size size and and occupancy occupancy of of intervening target patches. patches. Second, Second, we we want want to to see see to to what what degree degree demographic demographic stochasticity stochasticity target plus rescue rescue m may explain local local patch patch fluctuations fluctuations in in population population size. size. Third, Third, we we plus a y explain want to explore alternative hypotheses for spatial autocorrelation of extinction. want to explore alternative hypotheses for spatial autocorrelation of extinction. For us, us, the the metapopulation metapopulation of of A American pikas at at the the ghost ghost town town of of Bodie Bodie remains remains For m e r i c a n pikas living laboratory. laboratory. aa living
ACKNOWLEDGMENTS ACKNOWLEDGMENTS Bodie California State Historical Historical Park for for their support and assistance We thank the staff staff of the Bodie California State Parks over the years. Permission for for access to our field site was kindly granted granted by the California of Land Management, and various mining companies holding patent rights rights at Association, Bureau of Peacock (1989 ( 1 989 and 1991), 1 99 1 ), and David Bodie. We appreciate the contributions of Chris Ray and Mary Peacock ( 1 991), who helped us conduct the censuses. Lyle Nichols assisted assisted in the statistical statistical analanal Quammen (1991), grateful. We thank Jan Bengtsson, Steve Dobson, Ilkka Hanski, John Nagy, yses, for for which we are grateful. Harriet Smith, and Chris Thomas for for their conscientious reviews of the manuscript.
18
A Case Case Study Study of Genetic Genetic Structure Structure in a
Plant Metapopulation Plant Barbara Barbara E.E. Giles Giles
Jerome J&~me Goudet Goudet
I. INTRODUGION INTRODUCTION Habitats that are suitable for for the establishment and maintenance of most across landscapes. Environ Environspecies of plants and animals are distributed patchily across mental patchiness populations patchiness forces species to be structured into systems of of local populations within which cs are more likely to interact which conspecifi conspecifics interact with each other other than than with conspecifi cs from other 995). Isolation, however, conspecifics other populations populations (McCauley, 11995). however, is not complete and since most real organisms have some power power of of dispersal, members of in of a local population have a low but positive probability of of interacting with individuals from other localities. Depending Depending on the rate rate of of migration, demographic demographic and genetic dynamics will be influenced by this migration as well as by local birth rates. birth and and death death rates. Local populations populations are seldom immortal. Demographic, environmental, and genetic stochasticities (Shaffer, 98 1 ; Lande, 11988b), 988b), and deterministic processes processes (Shaffer, 11981; such as succession (Olivieri et 1 990; 11995; 995; Harrison, 99 1 ) may cause the et al. al.,, 1990; Harrison, 11991) extinction of of local populations, populations, although some of of these habitats may be colonized colonized again again by dispersing dispersing propagules. As a consequence, consequence, local populations populations come and go on temporal scales that do not allow demographic demographic and/or and/or genetic equilibria to be attained, and the age structure structure of of the local populations populations reflects the time elapsed since 990). since they they were were formed formed (Whitlock (Whitlock and and McCauley, 11990).
Metapopulation Metapopulation Biology Biology Copyright © 997 by Academic Press. Inc. All rights of of reproduction 9 11997 Academic Press, reproduction in any form reserved.
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Barbara E.E. Giles Giles and and Jerome J6rSmeGaudet Goudet Barbaro
Patches Patches may also differ in quality, size, and and spatial spatial arrangement (Hanski, this volume). Since (re)colonization and migration depend on the distances between habitat patches relative to the dispersal range of a species, patch dispersion is for the dynamics in individual patches and for the maintenance maintenance of the critical for metapopulation as a whole (Harrison, 11991; metapopulation 99 1 ; Fahrig, 11992; 992; Slatkin, 11993; 993; Hastings Harrison, 11994; Where local popu popuand Harrison, 994; McCauley, 11995; 995; Hanski, this volume). Where connected by migration migration that is not extensive enough to entirely oblit oblitlations are connected erate 992), erate the local generation to generation dynamics (Gyllenberg and and Hanski, 11992), the system as if it were one one large homogeneous homogeneous population population at equilibrium equilibrium treating the is not appropriate 990). Instead, understanding appropriate (Olivieri et al. al.,, 11990). understanding population population and evolutionary processes in temporally and spatially structured populations populations requires requires evolutionary metapopulation level (Antonovics (Antonovics et al. al.,, 11994; work at a regional or metapopulation 994; Hanski, 11996a). 996a). metapopulations is now reasonably well developed developed While the theory of metapopulations (Hanski and and Simberloff, Simberloff, this volume) and many many ecological ecological and genetic genetic predic predic(Hanski tions have been been made, made, empirical empirical tests of theoretical predictions predictions lag behind. The tions of theoretical behind. The primary purpose purpose of of this chapter chapter is to present present a case study of of a plant metapopuprimary plant metapopu lation. The plant, plant, Si/ene Silene dioica, dioica, is a dioecious dioecious perennial perennial and a component component of of early of primary succession in northern northern Scandinavia. This study was carried out stages of islands in an area area of of the Baltic Sea subject subject to land uplift uplift so that that new on islands new islands islands continuously, though though slowly, being being formed. rate of of land land uplift allows the the are continuously, formed. The rate of the island island populations populations to be estimated; estimated; the successional successional processes processes together together ages of continual creation creation of of new new islands islands imply that that population population turnover turnover must with the continual occur. We We have have proceeded proceeded by constructing constructing groups groups of of islands differing differing in their spatial, temporal temporal or demographic demographic characteristics, characteristics, and compared compared the observed observed changes in genetic differentiation among groups with those changes genetic differentiation among these these groups those predicted predicted by metapopulation have been able to test test and, and, in many many cases, cases, metapopulation models. models. In this way, we have been able confirm effects of spatial and and temporal confirm the predicted predicted effects of spatial temporal heterogeneity on genetic genetic structuring resulting metapopulation dynamics. We short restructuring resulting from from metapopulation We begin begin with a short re view of of the genetic genetic theory theory of of metapopulations metapopulations to contrast contrast the differences the differences in the assumptions and and genetics genetics models. assumptions and questions questions of of the ecological ecological and models. After After presenting presenting the results other genetic genetic studies of metapopularesults of of our our study, we briefly briefly review other studies of metapopula tions. tions.
II. THEORY THEORY The consequences The consequences of of genetic genetic differentiation differentiation and and gene gene flow flow among among local local poppop ulations for the ulations for the rate rate and and pattern pattern of of evolutionary evolutionary change change has has been been studied studied for for a long long time time in in population population genetics. genetics. In In large large randomly randomly mating mating populations, populations, the the major major factor affecting affecting allele and and genotypic frequencies frequencies is selection. selection. In assemblages assemblages of of factor small small populations, populations, however, however, other other factors factors come come into into play, play, the the main main ones ones being being genetic drift (the genetic drift (the random random fluctuations fluctuations of of allele allele frequencies) frequencies) and and the the degree degree to to
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which which drift is counterbalanced counterbalanced by migration. migration. The evolutionary evolutionary role of of genetic drift long-standing debate in population drift is at the center center of of a long-standing population biology, biology, and and many theories 1 990). For For example, theories rely upon upon this force (Barton and Clark, 1990). example, in the shifting shifting balance evolution (Wright, 1977; 1 977; Barton Barton and Whitlock, this volume), volume), balance theory of of evolution the populations can landscape the general idea is that many many small small populations can explore explore the the fitness fitness landscape (where fitnesses, respectively) more (where valleys and peaks peaks represent represent low and and high fitnesses, more extensively than larger pressures need larger populations. populations. This This is because because the selective selective pressures to be much random genetic much stronger stronger in small populations populations to be effective effective against against random drift. consequences of drift. While the consequences of small population population sizes are much much debated debated in the literature in terms of ( 1 977, Chapter 1 3) showed showed of inbreeding inbreeding depression, depression, Wright Wright (1977, Chapter 13) that new combinations could arise. In this process, process, individuals individuals new favorable favorable gene gene combinations with gene combinations from large combinations that that selection selection would would eliminate eliminate from large populations populations may survive populations increases, ne survive to reproduce. reproduce. As the size of of these small populations increases, fi fine tuning population to reach tuning of of these gene gene combinations combinations may allow the population reach a higher higher adaptive adaptive peak. Migration Migration could could then spread spread these combinations combinations to other other localities (Barton Whitlock, this volume). however, poppop (Barton and Whitlock, volume). For shifting shifting balance balance to work, however, ulations need isolated from one another. would ulations need to be fairly isolated another. Too Too much much gene flow would prevent genetic prevent the spread genetic differentiation; differentiation; too little would would prevent spread of of favorable favorable combinations. combinations. Much effort effort has therefore therefore been been put into measuring the extent to which which populations populations are differentiated, differentiated, and and these measurements measurements have have been been used to infer populations. infer the level of of gene flow between between local populations. The The modeling modeling of of spatially structured structured populations populations was was pioneered pioneered by Sewall Wright 1 93 1 , 11943). 943). His first first model, model, the island island model 93 1 ), assumed Wright ((1931, model (Wright, (Wright, 11931), assumed a set of finite and equal sizes with equal probabilities migrant of populations populations of of finite and equal probabilities of of migrant exchange. exchange. At equilibrium equilibrium between the forces forces of of migration and genetic drift, drift, Wright ' s model showed that the degree of Wright's of genetic genetic differentiation differentiation among among local populations, variance of frequencies standardized populations, measured measured as FST (the variance of allele frequencies standardized by the maximum possible possible variance), variance), was a simple function numbers function of of the the effective effective numbers of populations (Wright, 11940, 940, 1943, 1 943, 1951): 1 95 1 ): of migrants migrants among among populations FST = FST ~
11/4Nm /4Nm + + 11..
((1) 1)
Further Further mathematical mathematical development development of of the model model allowed the effects effects of of spatial variance variance in migration migration rate rate to be incorporated incorporated and led Kimura Kimura to propose propose his stepping-stone 955; Kimura 964; Weiss stepping-stone model model (Kimura, (Kimura, 11955; Kimura and Weiss, 11964; and Kimura, 964), where adjacent populations migrants. Both Kimura, 11964), where only adjacent populations exchange exchange migrants. Both the island and the stepping-stone members stepping-stone models models assume assume that individuals individuals are members of populations within which For of discrete populations which mating mating occurs occurs at random random (panmixia). (panmixia). For many species, however, however, it is possible possible that truly random random mating units do not exist at all. To another set To account account for for this type of of spatial spatial popUlation population structure, structure, another of isolation by distance of models models without without panmictic panmictic units was developed. developed. In the isolation distance or neighborhood 1 943), the genes of individual disdis neighborhood model model of of Wright ((1943), of each individual perse as a decreasing function of of distance, neighborhood being ned decreasing function distance, with a neighborhood being defi defined for each each individual. individual. The size of of the neighborhood neighborhood corresponds for corresponds to the area from
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Barbaro E.E. Giles and Jerome Barbara Giles and J6rSme Goudet Goudet
which the parents parents of of the central central individual could have been drawn at random (Wright, 11943). 943). This area is defined defined as a circle of 20' around the central of radius radius 2o" o" is the standard standard deviation of of a Normal distribution of of parent parentindividual, where 0' offspring migration distances. Both isolation by distance and stepping stone modmod els of of migration, in which there there is spatial variance in migration rate, have been shown to increase the degree of of population population differentiation differentiation relative to the island 964; Crow and Aoki, 1984; 1 984; Whitlock, 11992a; 992a; GouGou model (Kimura and Weiss, 11964; det, 11993). 993). The island, stepping-stone, and isolation by distance models, however, however, still assume that that popUlations populations have have constant sizes and and are immortal. Although Although Wright ((1940) 1 940) was the fi rst to suggest that extinction could increase population popUlation differ first differentiation relative to an island model, the effects of of extinction and (re)colonization (re)colonization (population turnover) on among-population (population turnover) among-population variance variance in gene frequencies frequencies were not not studied before 1977, 1 977, when meta when Slatkin introduced introduced the ecological concept of of the metapopulation popUlation genetics. Population turnover implies that local pop population into population populations, ulations, or demes, vary in their their degree degree of of demographic maturity. Newly estab established populations are not demographically mature, and (re)colonization (re)colonization represents founder effects at the time represents an additional source of of genetic drift due to founder of introduction of of colonization. colonization. With the introduction of population population turnover turnover into models of of spa spatial population population structure, structure, spatial variance in migration rate rate is ignored ignored (i.e., migra migration patterns patterns are those assumed in the island model) and the focus lies entirely on the temporal temporal relationships among habitat habitat patches. Slatkin's 1 977) models are variations of Wright's island model and consist Slatkin's ((1977) of Wright's consist of of a collection collection of of local populations populations of of diploid monoecious monoecious individuals individuals of of identical m. Gen and fixed size N, which exchange exchange migrants migrants (gametes) at a common rate rate m. Generations represent the carrying capacity, or erations do not overlap overlap and and N is assumed to represent the number number of of organisms which the resources resources of of a habitat can support. Each generation, a proportion populations goes extinct, where the prob proportion e of of the local populations probability of recolo of extinction is equal for for all age classes. The extinct sites are then recolonized immediately by a fixed number number of of (diploid) colonizers, k, which in turn reproduce reproduce and give rise to N offspring offspring within one generation. This model differs from that of Levins ((1970) 1 970) in that: ((1) I ) patch size is a variable in the model (Levins was only concerned vol concerned with the proportion proportion of of occupied occupied patches) patches) (Hanski, this volume), (2) extant populations populations exchange migrants (they do not matter in the the Levins Levins model), and and (3) no site is empty at any time. Slatkin cast his model model in terms of of the variance in gene gene frequencies frequencies among populations populations and and considered two two forms of of colonization: the migrant migrant pool model, in which the k colonizers are drawn drawn as a random random sample from from the metapopulation; metapopulation; and the propagule pool model, where the k colonizers population chosen at random. His results colonizers are drawn drawn from a single population indicated that that under under the propagule propagule pool model, genetic genetic drift resulting from sampling effects during population differ population establishment would would increase the differentiation among local populations, populations, while under under the migrant pool model, mixing prior ow reducing reducing the prior to colonization resulted resulted in additional gene fl flow the degree degree of of dif-
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ferentiation. 1 985, 1987) 1 987) later concluded concluded that if if the ferentiation. Slatkin ((1985, the average average time (in generations) to extinction of populations was less than the ef generations) of local populations than or or equal to the effective size of of populations, populations, then, even even in the absence absence of of migration, migration, extinction extinction and recolonization populations by recolonization would would prohibit prohibit the genetic genetic differentiation differentiation of of local populations genetic genetic drift. drift. Wade and McCauley ((1988), 1 988), who This latter view was was challenged challenged by Wade and McCauley who recast FST> and the model in terms of standardized variance of frequencies, FST, of the standardized of allele frequencies, and asked: asked: "under what conditions conditions do do extinction, extinction, colonisation, colonisation, and and dispersion dispersion bind an array of of subdivided subdivided populations populations into a single evolutionary evolutionary unit, and and when do do they permit trajectories?" permit local populations populations to assume more more independent independent evolutionary evolutionary trajectories?" For ST would would be For the migrant pool pool model, model, these these authors authors found found that that F FST be increased increased compared compared to an island island model model if if the number number of of colonists was was less than than or equal equal to ST was was always increased propagule twice the number number of of migrants, migrants, whereas whereas F FST increased in the the propagule model. 1 990) generalized pro model. Whitlock Whitlock and and McCauley McCauley ((1990) generalized Slatkin's Slatkin's migrant migrant and and propagule pagule pool model by including including a new parameter, parameter,
(2) (2)
where where k is the the number number of of colonizers, colonizers, (1 -
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Barbara and Jerome Barbara E.E. Giles Giles and J6rOme Goudet Goudet
the values ST for nding the stated relationship values of of F Fsafor younger younger and and older older populations, populations, fi finding stated relationship between between the FST Fsv values does does not imply that that Eq. (2) has been been satisfied.
III. THE ON ISLANDS THE BAlTIC THE SILENE SILENEMETAPOPULATION METAPOPULATIONON ISLANDSIN THE BALTIC A. Environmental Setting A. The The Environmental Setting The Gulf of of Bothnia Bothnia at the The Skeppsvik Skeppsvik Archipelago Archipelago (Fig. 1l)) is located in the Gulf Savar River mouth of of the S~ivar River in the province province of of Vasterbotten, V~isterbotten, Sweden Sweden (63°44' (63o44 ' -48 - 4 8 ' 'N, N, 20°34' 1 00 islands within 20 20~ - 40 'E). The The archipelago archipelago contains contains about about 100 islands within 20 km km 22.• The The islands islands are composed composed of of morainic morainic deposits deposits initially left under under water when when the ice receded receded at the end end of of the last ice age 7700 7700 years ago. This This area has been, and and still is, subject subject to a rapid and and now now relatively relatively constant constant rate of of land land uplift of of ca 0.9 cm per 979). Because the glaciers per year year (Ericson (Ericson and Wallentinus, Wallentinus, 11979). glaciers did did not deposit deposit equal amounts raises the amounts of of material in all places, places, land land uplift naturally raises the highest highest points rst. This creates points above above the water water level fi first. creates islands, islands, which which will differ differ in age as the process continues. continues. After After an even longer longer period period of of land land uplift, the islands may eventually fuse as has occurred mainland in Fig. 11.. The occurred in what what is now now the mainland The rate of islands in the of land uplift also also provides provides means for for estimating the ages of of the islands Skeppsvik Skeppsvik Archipelago Archipelago from from their their heights above above sea level, corrected corrected relative relative to the theoretical mean sea level at the nearest mareographic station. theoretical mean nearest mareographic Once processes leading formation of Once the bare bare rocks rocks are are exposed, exposed, processes leading to the formation of soil begin, begin, and and the the islands islands gradually become become available available for for colonization colonization by plants. plants. The The Skeppsvik Archipelago undisturbed and and each each island through Skeppsvik Archipelago is relatively undisturbed island goes through autogenic however, autogenic primary succession. succession. The The age differences differences among among the islands, islands, however, mean that the plant populations on the islands differ in their ages and mean that the plant populations islands differ their and stages of of demographic development. The ages of the plant populations on these islands demographic development. The of plant populations islands can be estimated estimated from from Age Age ijo = = (( hhi/m m )} ) j -- titi,•
(3) (3)
where island}, him )} is the age of where Agei Age o} is the age of of population population i on on island j, ((h/m)j of island }j (i.e., (i.e., island island height, height, h, divided divided by by the rate of of land land uplift, m), and ti is the the average average time from from island exposure exposure to colonization colonization and establishment establishment of of species species i (Ericson (Ericson and and al., 11990). 990). ti Wallentinus, 979; Carlsson Wallentinus, 11979; Carlsson et et al., t~ has been been estimated estimated from from data data on new new colonizations, colonizations, changes changes in the species species composition, composition, and and changes changes in population population sizes sizes of 972, 11986, 986, and 1 990 (L. Ericson, unpub of target target species species occurring occurring between between 11972, and 1990 Ericson, unpublished). period, the lished). Since a number number of of islands islands were were colonized colonized during during this period, the time of of colonization known island island age. Time to colonization colonization could could be compared with known colonization also increases wind action, increases with exposure exposure to wave wave and and wind action, so ti was calculated calculated separately substituted for for different different parts of of the archipelago, archipelago, and and the appropriate appropriate tit; value was was substituted to obtain obtain the age of of each each population. population. Since successional successional processes, processes, i.e., "nonsea "nonseasonal, directional, and and continuous continuous patterns patterns of of colonization colonization and and extinction extinction on a site by species populations" popUlations" (Begon et 986), necessarily necessarily include turnover dydy et al., al., 11986), include turnover namics, we chose chose to work work with a plant species lying within the series of of successucces-
18 Genetic Structure Structure in in aa Plant Plant Metapopulation Metapopulation 1 8 Genetic
435 435
FIGURE 1i Skeppsvik Archipelago, in in the Gulf of Bothnia, Sweden (63°44' (63~ -48'N, 20°34' 20~ -40'E). FIGURE The The shaded shaded and and unshaded areas areas are are the exposed exposed and protected parts of the archipelago, respectively. As discussed in the test, A, B, and C are are the outer, middle, and inner chains, respectively; D and E are the north and south transects, respectively.
436 436
Barbara erome Goudet Goudet Barbara E.E. Giles Giles and and JJ6rOme
replacements occurring occurring on these islands. In this way, we are are able to work work sional replacements with a large populations at different stages of large number number of of populations of development to study the effects of of colonization and extinction on genetic differentiation differentiation among local populations. populations.
B. Life History of the Plant The species we have chosen chosen is the red bladder bladder campion, S. dioica dioica (L.) Clairv. (Caryophyllaceae), which requires requires fertile, disturbed habitats habitats and and is a member member of of the deciduous dis deciduous phase of of primary succession succession in the study area. area. This species disappears found in appears as later successional species close in the habitat habitat and and it is never never found evergreen forests (Baker, 11947), 947), which dominate the later stages of of primary suc succession in the region. region. Silene Silene dioica dioica is insect pollinated and and dioecious, and and the dynamics of of population population growth are dependent dependent on sexual reproduction. reproduction. The The bum bumblebees Bombus Bombus hortorum, hortorum, B. lucorum, lucorum, B. hypnorum, hypnorum, and B. pascuorum pascuorum ssp. sparreanus sparreanus are the the primary pollinators. This This herb herb has has a perennial life cycle; the average life span of 0 years, and of an individual surviving to adulthood adulthood is about 110 plants begin flowering in their second or third year. Flowering occurs occurs in June and July, and the seeds, which ripen in August, are dispersed dispersed by gravity and germinate following spring. Although Although seeds may remain viable for 2 --33 years (Baker, the following 11947), 947), a number number of of studies have shown that most viable seeds germinate within the first year (Baker, 11947 947 and 1 988), and references references therein; Elmqvist and and Gardfjell, Gardfjell, 1988), suggesting that seed bank dynamics may not be important for this species.
C. Population History Colonization and establishment of of S. dioica dioica populations in the Skeppsvik Archipelago Archipelago occur occur via individual diploid diploid seeds. Seeds are are transported transported between islands by water water in the drift material that is moved moved around around the archipelago archipelago with the prevailing winds, storms, and rising water levels each autumn. Colonizers Colonizers are found found growing growing singly in the decomposing decomposing shore deposits deposits (B. E. Giles, personal observation). observation). The advanced successional status of of the vegetation along the shores shores of of the mainland (Fig. 11)) suggests that the major contributors contributors of of pollen or seeds entering entering island populations are other islands in the archipelago. archipelago. Colonization and demographic histories of S. dioica dioica populations populations follow a repeatable pattern on 1 990; Giles and Goudet, 1 996a). The these islands (Carlsson (Carlsson et al. al.,, 1990; Goudet, 1996a). The range range of of ages given below refl ects the fact that reflects that colonization and successional processes processes are quicker quicker in the inner archipelago since the outer islands (shaded (shaded part in Fig. 11)) are subjected subjected to stronger wind, wave, and ice action. Seeds of of S. dioica dioica are not able to germinate and establish populations populations on islands that are less than 1 50 years old. This is the time required than 7070-150 required for for sufficient soil and nutrients nutrients to accumulate and for for an island to attain a height at which the risks of of being washed over by storm waves are small. Successful colonization by S. dioica dioica is observed when the central parts of of an island have been colonized by
1188
GeneticStructure Structure inin aa Plant Metapopulation Genefic Plant Metapopulation
437 437
the nitrogen-fi xing Alnus incana. The number nitrogen-fixing number of of individuals founding founding new island populations populations appears appears to be small. The The first evidence of of population population expansion expansion is seen as a dense circle of of even-aged seedlings about 0.5 m in diameter growing growing individuals on under under female female plants. We We have have observed observed this with with about about 5 --1100 individuals on an an island, as long as both sexes were present and flowering simultaneously (B. E. Giles, -6) individuals to arrive Giles, personal personal observation). The The first first few few (2 (2-6) arrive are are also observed observed in locations up to 1I or 2 m apart, apart, suggesting that it is unlikely that the seed capsule is the unit of popUlations of colonization. colonization. While it is clear that new populations expand expand from seeds produced produced by the original colonists and their offspring, offspring, migrants arriving as seeds and pollen will be able to establish within these expanding populations since space and nutrients are plentifully available. Germination Germination suc suc1 995). cess is is high high in in newly colonized habitats habitats (Carlsson, (Carlsson, 1995). den Silene dioica populations populations expand expand rapidly and attain large sizes and high den20 - 250 years old. Succession, Succession, however, however, continues continues as the S. sities on islands 1120-250 dioica populations populations expand. Sorbus Sorbus aucuparia succeeds succeeds A. incana incana in the middle of pendula, Picea of the islands, and as even later later successional successional species (Betula pendula, Picea abies, Juniperus Juniperus communis, communis, and Vaccinium myrtillus,) establish in the centers,A. centers,A, incana/ incana/ S. aucuparia aucuparia form form a border border which is forced forced toward the shores. These later arriving change the soil and and light conditions conditions and and germination success success of of S. dioica species change decreases dramatically in these darker darker habitats (B. E. Giles, personal personal observation). observation). At this stage, the S. dioica populations populations thin thin out and and disappear disappear from the the centers of of the islands, but form thick rings with high densities and numbers numbers in the A.. incana/S. incana/S, aucuparia aucuparia border. border. Populations at this stage are probably at their A largest. When When S. dioica populations populations reach this thick ring stage, they are often often invaded by Microbotryum Microbotryum violaceum, which is a systemic, perennial, sterilizing anther antherincidence of smut fungus and an obligate parasite parasite of of S. dioica. dioica. The The incidence of infection 1 990). varies from 5 to 60% in these middle-aged middle-aged populations populations (Carlsson (Carlsson et al., 1990). There are no differences differences in longevity between infected and healthy individuals and Elmqvist, 11992), and infected individuals continue continue to use space (Carlsson and 992), and and and other other resources. resources. Infected male and and female individuals no longer longer contribute contribute to the gene pool, pool, and M. violaceum may therefore therefore effectively reduce reduce the numbers of of reproducing reproducing individuals within an island population. population. On islands 200-400 popUlations rapidly 200-400 years old, P. abies abies populations rapidly expand expand out outward. The deciduous deciduous border border becomes increasingly restricted to the shores shores and the rings of of S. dioica become thinner. Deposition Deposition of of drift decreases decreases in regular regularity because islands continue to rise and increase in height, so that even the deciduous of later successional deciduous habitat becomes more stable allowing the invasion of populations decrease competitors. As these events occur, the Silene populations decrease in size and the ring of of S. dioica eventually breaks up into small discontinuous discontinuous patches patches or single individuals. On islands 250500 years old, the later successional species 250-500 have successfully occupied the islands to their shorelines and S. dioica populations populations go extinct. Recolonization Recolonization of of successionally advanced advanced patches does not occur.
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Barbara E. Giles Giles and and J6rSme Jerome Goudet Barbara E. Goudet
D. Skeppsvik SkeppsvikArchipelago Archipelagoand the Assumptions Assumptionsof the Metapopulotion Metapopulation Model Model The continual continual formation of of new islands and the patterns pattems of of population population de development velopment described described above above suggest that that the Skeppsvik Archipelago is a spatially and temporally structured metapopulation. metapopulation. Since models seldom fit the real world world exactly, but our inferences inferences about the world are are to be drawn relative relative to model predictions, predictions, it is important to be aware aware of of the differences differences between between the two and how they might affect these inferences. In our case, the deviations of inferences. of the system from the assumptions of the model (Section II) relax the stringency of the con of relax conditions required required for for increased increased differentiation. differentiation. gener Both ecological and genetic metapopulation models assume discrete discrete generations and growth to carrying capacity within a single generation. Silene dioica has overlapping generations and the 501 00 years taken by the island populations 50-100 to reach reach maximum size and density strongly suggests that that growth growth to carrying capacity takes much longer than the model stipulates. stipulates. Recent theoretical work (Ingvarsson, 11995), 995), however, suggests that delayed growth to carrying capacity is either without effect or it reduces reduces the stringency of of the conditions under which colonization increases the degree of of differentiation. differentiation. This occurs because because a period of of delayed popUlation population growth growth keeps the newly founded populations small for several generations. Since the accumulated accumulated effect of of genetic drift in finite popu popu989), the effective lations depends depends on the reciprocal of of population population size (Falconer, 11989), population population size is reduced and the levels of differentiation differentiation obtained during during the the prolonged growth period popUlations grow period will always be higher higher than than when when populations grow to generation. The presence of close relatives relatives from different genera generasize N in one generation. tions will also increase inbreeding during this initial phase (Crow and Kimura, 11970). 970). Thus, relative to a system with discrete generations generations started from the same number number but expanding to fill the niche in one generation, there there is a greater greater op opportunity for for differentiation among the young subpopulations in our system than than in the model. The model also specifies specifies monoecious organisms. Dioecy should have little effect on genetic genetic differentiation differentiation if sex ratio ratio is balanced balanced and migration the same for 99 1 ). While biased sex ratios have not been for both sexes (Chesser, 11991). 995), biases toward observed in established populations on the islands (Carlsson, 11995), one or another sex have been seen on five five islands colonized since this study began 99 1 (B. E. Giles, personal observation). Thus biased sex ratios, which reduce in 11991 effective 938), would effective population size (Wright, 11938), would also act to inflate differentiation differentiation arising from founder effects relative to the metapopulation model. We also believe that that the colonization and and migration dynamics dynamics in Skeppsvik Archipelago Archipelago satisfy Eq. (2). Our reasons reasons are are as follows. The The first seedlings may be produced III.C), produced in new populations with fewer fewer than than 10 individuals (Section Ill.C), suggesting that k is small. small. If If the the number number of of migrants (Nm) per per generation are of of the the same same magnitude as k, differentiation differentiation will be enhanced, enhanced, but but other evidence evidence suggests that that migration rates rates may actually be higher. higher. Only seeds can colonize, but seeds and pollen may contribute to migration. migration. If the pollinator-borne spores of of M. violaceum (Section III.C) Ill.C) are regarded as "natural" "natural" color markers for pollen,
18 Genetic Structure Structure inin aa Plant PlantMetapopulation Metapopulation 1 8 Genetic
439 439
the proportion proportion of offlowers flowers from from uninfected uninfected populations populations bearing bearing spores sporesprovides provides aa the roughestimate estimate of ofthe theextent extentof ofpollen pollencarryover carryoverbetween betweenpopulations. populations.Two Twostudies studies rough (P. K. K. Ingvarsson, Ingvarsson, unpublished; unpublished; B. B. E. E. Giles, Giles, unpublished) unpublished) have have shown shown that that 33-(P. 5% 5 % of of the the female female flowers flowers from from uninfected uninfected but but well well established established populations populations bore bore smut smut spores. spores. While While the the number number of offertilizations fertilizations will will be be lower lower than than the the number number of of transfers, transfers, the the magnitude magnitude of ofthese these numbers numbers plus plus continued continued seed seed flow flow suggests suggests that that the numbers numbers of of migrants migrants are are likely likely to to be be higher higher than than the the number number of of colonists. colonists. ItIt the is therefore therefore not not unreasonable unreasonable to to assume assume that that our our metapopulation metapopulation system system correcorre is sponds with the conditions stated in Eq. (2). sponds with the conditions stated in Eq. (2). One One final final difference difference between between our our system system and and the the model model can can be be dealt dealt with with in in the analyses. analyses. In In metapopulation metapopulation models, models, extinction extinction is is stochastic, stochastic, independent independent of of the population population age age and and aa requirement requirement for for recolonization recolonization (Levins, (Levins, 1970; 1 970; Slatkin, Slatkin, 1977; 1 977; Wade and and McCauley, McCauley, 1988; 1 988; Whitlock Whitlock and and McCauley, McCauley, 1990; 1 990; Whitlock, Whitlock, 1992a). 1 992a). Wade In In Skeppsvik Skeppsvik Archipelago, Archipelago, extinction extinction is is determined determined by by succession succession and and is is therefore therefore deterministic, dependent dependent on on population population age, age, and and not not aa requirement requirement for for colonizacoloniza deterministic, tion. Except Except in in terms terms of of the the rate rate of of local local population population extinction, extinction, the the metapopulation metapopulation tion. models models cited cited above above make make no no predictions predictions about about the the specific specific effects effects of of extinction extinction on on levels levels of of genetic genetic differentiation differentiation since since the the immediate immediate recolonization recolonization of of extinct extinct sites sites equates equates extinction extinction and and recolonization; recolonization; increased increased genetic genetic differentiation differentiation in metapopulation models arises metapopulation arises solely from from founder founder effects effects during during recolonization. recolonization. independent processes in the archipelago, archipelago, Since colonization and and extinction are independent expect that formation of of new new islands will enhance enhance levels of of we expect that the continual continual formation genetic differentiation differentiation as predicted predicted by the models and that that the drawn drawn out extinc extinction process could have an additional additional affect (see Section IV.C for details). details). We We have dealt with these differences differences between between our system and the model by removing those populations going toward extinction from the data set used to test the model predictions; the effects of successionally driven extinction on differentiation within the metapopulation are treated separately. Note also that the persistence dioica in Skeppsvik Archipelago depends on the continual formation of of S. dioica drawn from colonisation sites, which is consistent with one of the main inferences drawn the 969a, 11970; 970; Hanski and the ecological ecological metapopulation models models (e.g., Levins, Levins, 11969a, Simberloff, Simberloff, this this volume). volume).
IV. IV. PATTERNS PATTERNSOF OF GENETIC GENETICSTRUaURE STRUCTUREON ON THE THEISLANDS ISLANDS A. A. Methods Methods The The positions positions of of the the islands, islands, their their ages, ages, demographic demographic stages, stages, population population sizes sizes and and degree degree of of exposure exposure are are given given in in the the Appendix Appendix and and Fig. Fig. 11.. We We have have surveyed surveyed the the genetic genetic structures structures of of our our populations populations using using allozymes. allozymes. Six Six polymorphic polymorphic loci, loci, with with 20 20 alleles alleles in in total, total, were were presumed presumed to to represent represent neutral neutral characters characters (Giles (Giles and and Goudet, 996a). Individuals Goudet, 11996a). Individuals were were collected collected at at random random from from the the entire entire area area oc occupied cupied on on each each island island and and 4500 4500 individuals individuals were were screened screened from from 52 52 islands islands (Ap(Ap-
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Barbara and Jerome Barbara E.E. Giles Giles and J6rOme Goudet Goudet
pendix). program pendix). Genotypic Genotypic and and allelic compositions compositions were were analyzed using using the the program 1 .2 (Goudet, 995), which calculates 1 984) es FSTAT V 1.2 FSTAT (Goudet, 11995), calculates Weir and and Cockerham Cockerham ((1984) es's F from random random mating were timators timators of of Wright Wright's F statistics. Deviations Deviations from were assessed assessed heterozygote deficit, FST by means means of of F F statistics statistics (Fit (Fit estimates estimates the overall overall heterozygote Fsv esti estiFis estimates the heterozygote mates the degree of isolation of populations and mates the degree of isolation of populations and estimates the heterozygote cance of per deficit within within islands). islands). The The signifi significance of each each F F value value was tested using a permutation procedure. procedure. The The null hypothesis, hypothesis, Frv F,:~ = = 0 (where (where xy xy = = is, it, or or ST), ST), was was obtained obtained from from 5000 5000 permutations permutations and and the observed observed value value of of Fxy F~ was then then com compared with the null distribution distribution and and tested against against the alternative alternative hypothesis, hypothesis, pared Fxy Goudet, 11996a). 996a). For Fit ' alleles were were permuted Fxy > > 0 (Giles (Giles and Goudet, For Fis and and F~, permuted within within and individual may and among among populations, populations, respectively. Since the two two alleles in an individual may not ST were not be independent, independent, the permutation permutation units for for F FsT were the genotypes genotypes permuted permuted among populations. among populations.
among Islands B. Genetic Genetic Structure Structure among Islands While While it is essential essential that the the local populations populations composing composing a metapopulation metapopulation are connected If connected by migration, migration, migration migration must must also be shown shown to be restricted. restricted. If no assumed that that no evidence evidence of of population population structure structure can be be obtained, obtained, then then it must must be be assumed the local populations populations belong to the same same panmictic evolutionary evolutionary unit. Using Using the data rst asked data from from all study sites within within the archipelago, we fi first asked whether whether there there was population structure ow arising from was any evidence evidence of of population structure and and restricted restricted gene gene fl flow arising from habitat habitat subdivision subdivision within and among among islands. islands. As Table Table Ia shows, shows, the values values Fit loci, are Fit = = 0. 0 . 11 1100, , FST FST = = 0 0 ..00338 8 , and a n d Fis Fis = = 00.075, . 0 7 5 , calculated c a l c u l a t e d over o v e r all all loci, are signifi signifi- cantly 0.0002), providing restricted c a n t l y greater greater than than zero zero (P (P < < 0.0002), providing strong strong evidence evidence of of restricted gene ow among gene fl flow among (FST) (FsT) and and within within (Fis (Fis)) islands in in the archipelago archipelago (Fit (Fit).) . Note Note that that no no spatial or or temporal temporal dynamics have have been been taken taken into into account; account; this analysis re reveals only whether populations are structured structured where structured. whether the populations where the habitat is structured. ,
C. C. Genetic Genetic Differentiation Differentiation among among Cohorts Cohorts of Plant Plant Populations Populations Varying in Age The The geological geological and and successional successional processes processes occurring occurring in the Skeppsvik Skeppsvik Ar Archipelago chipelago clearly indicate indicate that that our our populations populations are are characterized characterized by colonization/ colonization/ F-Statistics F-StatisticsAveraged Averaged over over All All loci Loci and Populations Populations (a) (a) and over over All All loci Loci for for the the Three Three Age Age Classes Classesof of Populations Populations ((b)b)
TABLE I
(a) (b)
a
Class Class
N N
All Young Intermediate Old
52 13 13 30 9
Fit FIt 0. 1 10 0.110 0. 1 05 0.105 0. 1 07 0.107 0. 1 58 0.158
(.039) (.06 1) (.061) (.033) (.068)
Fsv FST
Fis Fis
0.038 (.0 1 0) (.010) 0.057 (.028) 0.030 0.030 (.006) 0.066 0.066 (.009)
0.075 (.035) 0.052 (.046) (.046) 0.080 0.080 (.032) 0.098 0.098 (.068)
a Standard Standard errors in parentheses. All F F values values are are significantly greater than than zero (P < < 0.0002). 0.0002).
18 Genetic Structure Structure inin aa Plant Plant Metapopulation Metapopulation 1 8 Genetic
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extinction dynamics dynamics and and differ differ in in terms terms of of their their demographic demographic maturity. maturity. The The quesques extinction tion then then is is whether whether these these particular particular dynamics dynamics increase increase genetic genetic differentiation differentiation tion among populations populations relative relative to to an an island island model model that that does does not not take take turnover turnover into into among account. account. The The predicted predicted increase increase in in differentiation differentiation in in the the model model arises arises solely solely from from founder effects effects during during recolonization recolonization since since the the effect effect of of extinction extinction is is seen seen as as founder recolonization (Section (Section III.D). III.D). This This also also partly partly explains explains why why the the FST FST values values of of recolonization only two two age age groups groups have have been been used used to to show show whether whether population population turnover turnover inin only creases differentiation differentiation (see (see Whitlock, Whitlock, 1992b; 1 992b; Dybdal, Dybdal, 1994; 1 994; McCauley McCauley et et al., ai., creases 1 995). In In these these comparisons, comparisons, the the younger younger group group contains contains the the newly newly founded founded poppop 1995). ulations, while all other populations are combined into a single older group since ulations, while all other populations are combined into a single older group since these populations populations are are assumed assumed to to be be approaching approaching equilibrium equilibrium between between migration migration these and genetic genetic drift. drift. Where Where extinction in in natural natural systems systems is stochastic, stochastic, dividing poppop and ulations into into two two groups groups may may be be appropriate. In contrast, contrast, our our populations popUlations decline decline ulations appropriate. In become extinct extinct as as succession succession proceeds proceeds (Section (Section III.C). III.C). The The slowly and and eventually eventually become decrease in size size is likely to be due due to a reduction reduction in germination success in in the the decrease germination success presence of of later later successional successional species. This reduction reduction in in germination germination appears to presence species. This appears to have two two effects on these these populations; populations; the the age age distribution distribution shifts shifts toward toward adult adult have effects on individuals (Carlsson, (Carlsson, 1995), 1 995), and be recruited individuals and since since migrants migrants can be recruited only by the cannot compensate the reduction reduction in N N which which growth of of seeds, seeds, migration (m) cannot compensate for for the is expected under model which which assumes assumes that is constant constant at at equilibrium. equilibrium. It It is expected under aa model that Nm N m is thus appears appears that that not only are the oldest populations moving away from demo demographic and genetic numbers and mi genetic equilibria in our system, the reductions reductions in numbers migration populations. Since gration are are likely likely to to increase increase the the F Fsv among these these populations. Since our our declining declining ST among populations do not fulfill the the criterion criterion for membership membership in a single group group called "older" populations 990; Whitlock, 11992b; 992b; Dybdal, populations (Whitlock (Whitlock and and McCauley, 11990; 11994; 994; McCauley et 995), we divided our populations into three groups, called et ai., al., 11995), young, intermediate, and old, where where our young and intermediate populations populations cor correspond to the younger and older populations in the papers cited above. To test whether colonization increases increases differentiation, we compared the F Fsv of ST values of intermediate populations; to test whether the population decline as asyoung and intermediate sociated with succession also infl ates levels of differentiation at the metapopu inflates metapopulation lation level, level, we we looked looked to to see see whether whether F FST values increased increased in in the the old old popUlations. populations. ST values We We proceeded proceeded as as follows. follows. The The young young group group contains recently founded populations. These These populations still still occupy occupy the the centers centers of of the the islands islands and and are are less less than than 30 30 years years old. old. The The inter intermediate group contains the populations which were well into the expansion expansion phase. These These had had the thick thick ring form and and were were between between 30 and and 250 years years old. The old populations populations were were decreasing in size, had had the the broken ring ring form, and and were were older older than than 250 years. The average average census sizes of of popUlations populations in the young, interme intermediate, and 3 000, and 300 individuals, respectively. The and old groups were 670, 670, 113000, and 11300 classifi cation of classification of each each population population into into an an age age group group is is given as as "stage class" class" in in the the Appendix. We We then then calculated the the F F statistics for for each each group group separately separately and and tested tested whether FST of whether the the Fsv of young young populations populations was was larger larger than than that that of of intermediate intermediate pop populations, ulations, and and whether whether the the F Fsv of intermediate intermediate populations populations was was smaller smaller than than that that ST of
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Barbaro and Jerome Barbara E.E. Giles Giles and J6rOme Goudet Goudet
of method, in of old populations. populations. We We tested tested our our hypotheses hypotheses using using a Monte Monte Carlo Carlo method, which partitions partitions of of the total total sample sample into into three three classes were generated at random. random. which were generated Each partition was made up 1 2, and Each three-class three-class partition was made up of of 30, 12, and 9 islands, islands, sampled sampled without without groups were were calculated repli replacement. replacement. Differences Differences in FST between between groups calculated for for 5000 5000 repliI ) FST,y FST,y = cates. cates. The The distributions distributions of of these these differences differences are are our our null null hypotheses; hypotheses; ((1) = FST,i (2) FST,i which we we tested FsT, i,, and and (2) FsT.i = = FST,o FsT,o,, which tested against against the the alternative alternative hypotheses; hypotheses; ((1) 1 ) FST FST,y > FST,i FST,i,, and and (2) (2) FST,i < < FST,o (y, (y, i,i, 0 o = = young, young, intermediate, intermediate, and and old, old, ,y > respectively). respectively). All F intermediate and F statistics estimated for for the the young, young, intermediate and old old age age classes classes are are among young significantly greater than than zero zero (Table (Table Ib). FST is high among young islands islands (0.057), decreases decreases to 0.030 0.030 in the intermediate intermediate age age class, and and reaches reaches its highest highest (0.057), value of of 0.066 0.066 among among the the old old islands. islands. The The FST of of the the young young populations populations was was (P < which significantly larger larger than than that that of of the intermediate intermediate populations populations (P < 0.05), 0.05), which suggests that that colonization colonization increases increases differentiation differentiation (Whitlock (Whitlock and McCauley, McCauley, 11990). 990). Continued migration among populations also appears to contribute Continued migration among populations also appears contribute to the the ST observed intermediate class. This This inference inference is supported decrease decrease in F FST observed in the intermediate supported by noting noting that that the means means and and standard standard deviations of of the numbers numbers of of alleles alleles ob observed served at all loci (maximum 20) in the the young, intermediate, intermediate, and and old old groups groups were were 114.33 4.33 ± 8.56 ± 5 .25 ± ___ 3.39, 3.39, 118.56 __ 0.73, 0.73, and and 115.25 ___ 2.05, 2.05, respectively. The The FST FsT for for the intermediate cantly greater intermediate class class remains remains signifi significantly greater than zero, suggesting that that gene fl ow is not popUlations into a single evolutionary evolutionary flow not high high enough enough to link these these populations unit. In the second model, the second test, which which is not not part of of the the model, the FST for for the old class class was significantly significantly higher higher than than that that for the the intermediate intermediate class (P (P < < 0.04). We was 0.04). We strongly suspect that suspect that that this increase increase is associated associated with extinction extinction processes, processes, and and that in our both founding our metapopulation metapopulation system, both founding events events and population population decay may island model at equilibrium. increase levels levels of of differentiation differentiation relative relative to an island equilibrium. We We know been reported, remains know of of no no other other study in which which such such an increase increase has has been reported, and and it remains to be seen whether metapopulation systems whether these results will be found found for for other other metapopulation with similar similar extinction extinction dynamics. However, However, in studies studies where where the the aim is to see whether which whether colonization colonization increases increases differentiation differentiation using the FST relationships relationships which follow - McCauley model, it may be wise to check follow from from the the Whitlock Whitlock-McCauley check whether whether FST values for for old or decaying decaying popUlations populations are are exceptionally exceptionally high. high. If If they are, group of assumed to these populations populations should should be removed from from the group of populations populations assumed be approaching their inclusion inclusion could could obscure approaching equilibrium equilibrium since since their obscure the the expected expected de decrease between between younger younger and and older older populations. populations.
D. Genetic Genetic Structure within Islands While While we have have demonstrated demonstrated that that extinction/colonization extinction/colonization processes processes increase increase the the genetic genetic variance variance among among island island populations, populations, this this is is not not the the entire entire story. story. The The within-island obtained in this study are large and within-island and and overall values values of of Fis Fis obtained and sig significantly greater implications of greater than than zero zero (Appendix (Appendix and and Table Table I). One One of of the implications of these these significant significant Fis values values is that that the the island populations populations themselves themselves consist consist of of more more than than one one random random mating mating unit. To To test this, a pilot study was was carried carried out out on
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Genetic GeneticStructure Structure inin ao Plant Plant Metapopulation Metapopulation
443 443
island 23 ((Fig. Fig. 11,, Appendix), in which all S. dioica dioica individuals within six 50 cm22 patches, separated by 11.5 .5 to 35 m, were surveyed electrophoretically. electrophoretically. Fis values FST values cal for all patches patches were were not statistically different from zero, zero, but all Fsacalculated culated for for different combinations of patches were were significantly greater greater than than zero. Converting the distance distance between between the two nearest patches patches in which FST was sig significant nificant into area revealed that that random mating mating units could be as small as 0.2 0.2-E. Lundqvist, J. Goudet, and B. E. Giles, unpublished). Some of of this within within6 m22 ((E. island differentiation differentiation probably results from restrictions to gene flow. Seed dis dispersal by gravity may lead to the establishment establishment of groups of of offspring near near the mother mother plant plant which are likely to be at least half-sibs. half-sibs. Nearest-neighbor Nearest-neighbor pollination has also been observed (Carlsson, 11995), 995), and the compounding of of these two processes over a number number of of generations could lead to an apparently continuous population consisting of of a patchwork patchwork of of differentiated differentiated family groups (see Turner Turner et al., 11982). 982). Spatial subdivision, however, does does not not appear to be the the only dynamic af affecting within-island within-island population structure. Because Because an island is rising all the time, the middle is always older than than the outer outer parts. parts. With With the exception of of shoreline species, each successional successional stage invades the middle first and moves outward. Close observation of of any single patch of of ground on an island reveals reveals that the plant species colonizing it go through a series of of successional successional and demographic demographic changes similar similar to those described at the island level (Section III.C). Ill.C). A patch is first colonized by S. dioica first dioica when conditions are right. fight. Initially, we observe that species may still share a few flowering adults, few if any seedlings, and shoreline species the habitat. When When seedlings begin to be produced within the patch, the numbers and densities of of the S. dioica dioica "population" "population" increase, plants plants at all life-history stages dioica is the most common plant. As 1. J. communis, communis, V. myrtillus, myrtillus, are present, and S. dioica B. pendula, pendula, and P. abies abies invade, fewer fewer and fewer S. dioica dioica seedlings are are observed, the number of of adults decreases, decreases, and S. dioica dioica eventually disappears disappears from the patch patch (Carlsson, 11995; changes associated with patch patch 995; B. E. Giles, unpublished). These changes colonization and extinction suggest that each island population is itself an age agestructured metapopulation. metapopulation. In a recent study of three islands (23, 35, and 39; Fig. dioica from patches of of different different ages, the FST values 11)) in which we collected S. dioica from young, intermediate, intermediate, and old patches patches within an island followed followed the same between young, intermediate, intermediate, and old islands (Giles and and Goudet, trend as those between 11996b). 996b). Clearly a new model will be needed to investigate how the dynamics occurring occurring at these two nested spatial and temporal temporal scales could affect affect each other other and the system as a whole. Nonetheless, fi nding that the degree of finding of demographic maturity of of mating units at two spatial and temporal scales has a consistent effect on the degree phe degree of genetic differentiation differentiation among those units suggests that that this phenomenon may be general.
E. E. The Effects Effects of Distance Distance on Genetic Genetic Differentiation among among Islands Islands If an organism's organism's potential for migration is limited limited relative to the the typical isolation of the habitat habitat patches patches it occupies, occupies, the arrangement arrangement and distances distances among
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Barbaro E.E. Giles and J6rSme Jerome Goudet Barbara Giles and Goudet
patches can affect population population differentiation differentiation even even in the absence absence of of turnover. turnover. We patches We next looked looked for evidence evidence of of isolation isolation by distance distance which has been shown to inin Whitlock, 11992a; 992a; crease the the degree of of differentiation differentiation among among populations populations (e.g., Whitlock, Goudet, 993; Slatkin, 993). Since Goudet, 11993; Slatkin, 11993). Since population population turnover turnover and and spatial spatial variance variance in migration migration have have not not been been jointly jointly investigated investigated in in aa single single model, model, it is difficult difficult to to know how how to disentangle disentangle their their effects. To To reduce reduce the confounding confounding effects of of age age know and possible, and concentrate concentrate on on the the effects effects of of spatial spatial variance variance in migration migration as as much much as as possible, we islands which, the we restricted restricted our our analyses analyses to to specific specific groups groups of of neighboring neighboring islands which, on on the basis of flow, we we believed believed could could not) basis of information information about about vectors vectors of of gene gene flow, could (or (or could not) exchange genes. exchange genes. Age Age was was ignored ignored in in these these analyses. analyses. From From the the time time of of seed seed release and and onward onward into into the the autumn, autumn, the the prevailing prevailing winds blow from the southwest, storms blowing from the southeast are winds blow from the southwest, storms blowing from the southeast are common, common, and cycle in and water water moves moves into into the the archipelago archipelago in in the the course course of of the the annual annual water water cycle in the 98 1 ). The forces will move seeds the Baltic Baltic (Ericson, (Ericson, 11981). The joint joint action action of of these these forces will be be to to move seeds from the south south to the north in the archipelago. archipelago. To To investigate the spatial patterns patterns of ow via seeds, we constructed of gene gene fl flow constructed three three "chains" "chains" of of islands, called called the outer, outer, middle, and -north direction middle, and inner inner chains, chains, which which are are oriented oriented in in aa south south-north direction (A,B,C, (A,B,C, ). Isolation three chains. respectively, respectively, Fig. Fig. 11). Isolation by by distance distance was was expected expected in in all all three chains. We We also also looked looked for for isolation isolation by by distance distance where where wind wind and and water water are are not likely agents east-west transects. Since bumblebee agents of of seed flow by setting setting up two two east-west bumblebee movement frequent in the inner inner movement occurs occurs among among islands islands and appears appears to be more more frequent (unshaded (unshaded part part in in Fig. Fig. 11)) than than in the outer outer archipelago archipelago (L. (L. Ericson, personal personal communication), isolation by unexpected in communication), the the occurrence occurrence of of isolation by distance distance is is not not unexpected in the the north D, Fig. 1). 1 ) . The second E, Fig. 1), 1 ), was set north transect transect ((D, second transect, transect, south south transect transect ((E, up in the most exposed exposed part part of of the archipelago archipelago as a contrast contrast to the other other four four linked by groups. Since Since we we suspect suspect that that south south transect transect islands islands are are not not directly directly linked by wind and expect to wind and water, water, we we do do not not expect to detect detect isolation isolation by by distance. distance. We We looked looked for for evidence evidence of of isolation by by distance distance within within these these chains chains and and transects using Mantel Mantel, 11967) 967) of transects using Mantel tests tests ((Mantel, of the the correlations correlations between between the the ma matrices physical distances. cant trices of of pairwise pairwise genetic genetic (based (based on on FST) FST) and and physical distances. A A signifi significant correlation correlation is taken taken as evidence of of isolation isolation by distance. distance. Since there there is little in information ST and is appropriate, formation as as to to what what type type of of transformation transformation of of F FST and distance distance is appropriate, we used three: 1 ) untransformed vs untransformed physical distances, we used three: ((1) untransformed FST FST vs untransformed physical distances, (2) (2) Slatkin' 1 993) In[ /4( l /FsT )] (the in an island Slatkin' ss ((1993) In[ l1/4(1/FsT -- 11)] (the log log of of the the number number of of migrants migrants in an island model) vs vs log log physical physical distances, distances, and vs ranked physical distances. model) and (3) (3) ranked ranked FST FST vs ranked physical distances. Table II shows cients (R) between two matrices matrices Table II shows that that the the correlation correlation coeffi coefficients between the the two were north transect were high, high, 0.40 0.40 or or greater, greater, for for the the outer outer and and inner inner chains chains and and the the north transect for cients for for for the the untransformed untransformed and and ranked ranked data. data. In In contrast, contrast, the the correlation correlation coeffi coefficients the middle the south were very our predictions the middle chain chain and and the south transect transect were very low. low. Thus, Thus, our predictions are are supported supported by these these tests for the outer outer chain, chain, the southern southern transect, transect, and marginally for for the the inner inner chain chain and and the the northern northern transect, transect, but but not not for for the the middle middle chain. chain. These These results results suggest suggest that that genetic genetic differentiation differentiation among among islands islands in in the the Skeppsvik Skeppsvik Ar Archipelago is inflated due to chipelago is inflated relative relative to to an an island island model model due to the the dispersion dispersion of of the the islands, islands, although although it it is as as yet yet impossible impossible to to estimate estimate the the relative relative contributions contributions of of
Genetic Plant Metapopulation GeneticStructure Structure inin aa Plant Metapopulation
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445 44S
Mantel in Groups Groups of Mantel Tests Tests of of Isolation Isolation by by Distance Distance in of Islands Islands Formed Formed on on the the Basis Basis of of Hypotheses Hypotheses about about Vectors Vectors of of Gene Gene Flowa Flow~
TABLE II II
Group Group
N N
Inside chain Middle chain Outer chain North transect South transect (b) Inside chain Middle chain Outer chain North transect South transect (c) Inside chain Middle chain Outer chain North transect South transect
7 9 I11I 6 5 7 9 1111 6 5 7 9 I11I 6 5
(a)
R
0.4669 0.0767 0.4296 0.5629 - 0.5034 0.3947 -0.3947 - 0.2548 0.3289 -0.3289 - 0.2275 0.2922 0.3961 0 . 1 68 1 0.1681 0.3980 0.5673 - 0.3697
P P
sdr
Threshold Threshold
0.01 0.0199 0.327 0.008 0.054 0.932 0.052 0. 1 28 0.128 0.021 0. 1 96 0.196 0.759 0.057 0.268 0.0 11 0.011 0.024 0.868
0.26 0.27 0 .16 0.16 0.29 0.36 0.24 0.21 0 .14 0.14 0.27 0.34 0.24 0.25 0 .15 0.15 0.29 0.33
0.01 25 0.0125 0.01
0.01 0.01
0.01
a
were used: ST against ~,Three transformations of pairwise genetic and physical distances were used: (a) F Fsx against distance, 1/4 FST (b) In ((1/4 Fsv - 1/4) 1/4) against against In distances, distances, (c) ranked FST Fsx against ranked distances. Threshold specifies sequential Bonferroni level of significance significance required for rejection of the null null hypothesis. specifies the sequential N, R, P, and sdr are sample size, size, correlation coefficient, coefficient, probability, and standard deviation, respectively. -
temporal levels of temporal and and spatial spatial heterogeneity heterogeneity to to the the total total levels of differentiation differentiation in in this this metapopulation. metapopulation. The The occurrence occurrence of of isolation isolation by by distance distance in in this this archipelago, archipelago, how however, to ever, also also allows allows us us to to examine examine another another factor factor which which is is critical critical for for the the degree degree to which which colonization colonization and and extinction extinction dynamics dynamics will will increase increase differentiation, differentiation, namely, namely, the Whitlock the parameter parameter � ~b and and the the degree degree of of propagule propagule mixing mixing at at colonization colonization (( W hitlock and McCauley, 990; McCauley, 99 1 ). and M c C a u l e y , 11990; McCauley, 11991).
F. Numbers Sources of Colonists Numbers and Sources Colonists The depend not not only typical The genetic genetic consequences consequences of of founding founding events events depend only on on the the typical numbers individuals involved involved in numbers of of individuals in a a colonization colonization (k), (k), but but also also on on the the nnumber u m b e r of of sources sources (�) (~b) from from which which the the individuals individuals entering entering empty empty patches patches are are drawn drawn (see (see Eq. 977; Wade 1 988; W Whitlock McCauley, Eq. (2); (2); Slatkin, Slatkin, 11977; W a d e and and McCauley, M c C a u l e y , 1988; h i t l o c k and and McCauley, 11990; 990; McCauley 995). The M c C a u l e y et et al., al., 11995). The observation observation that that groups groups of of seedlings seedlings are are found found on 10 individuals representing both sexes have have arrived on an an island island when when 5 5 to to 10 individuals representing both sexes arrived and and flowered first attempt attempt to to find find out out the the flowered simultaneously simultaneously suggests suggests that that k k is is small. small. As As aa first probability origin of the colonists, have looked looked for isolation by by probability of of common c o m m o n origin of the colonists, we we have for isolation distance groups. Isolation young popupopu distance within within age age groups. Isolation by by distance distance detected detected among a m o n g young lations, or populations, will be taken lations, or among a m o n g both both young young and and intermediate intermediate populations, will be taken as as evidence that propagules come come from small number of nearby islands evidence that propagules from one one or or a a small n u m b e r of nearby islands
Barbara E. Giles Giles and J6rOme Jerome Goudet Gaudet Barbara
446 446
(high ( high ~b) 4» and and aa propagule propagule pool pool model model will will best best describe describe the the archipelago. archipelago. If, If, howhow ever, isolation isolation by by distance distance is is detected detected among among intermediate intermediate but but not not among among young young ever, populations, it is is more more likely likely that that colonizers colonizers come come from from aa wide wide range range of of source source populations, populations rather rather than than neighboring neighboring ones, ones, and and aa migrant migrant pool pool model model will will be be inin populations ferred. We We carried out Mantel Mantel tests tests of of the the correlations correlations between between the the genetic genetic and and ferred. carried out distances of of the the young young and and intermediate intermediate island island populations. populations. As As above, above, aa physical distances significant correlation correlation between between the the two two matrices matrices was was taken taken as as evidence evidence of of isolation isolation significant by distance. distance. No No effect of distance distance was was detected any of of the the tests tests for the young young by effect of detected in any for the popUlations, but but all three Mantel tests tests were were highly highly significant significant for islands in the the populations, three Mantel for islands intermediate age class (Table (Table III). These These results suggest that that colonizers colonizers are are aa intermediate mixture from from several several locations in the the archipelago archipelago and and that that the the migrant migrant pool pool model model mixture an appropriate appropriate description description of of founding founding events. is an Finding isolation isolation by distance distance among islands in the the intermediate intermediate age age class class Finding among islands raises some some interesting interesting problems problems and questions. According According to existing existing genetic genetic raises and questions. metapopulation models, postcolonization migration among populations populations is supsup metapopulation posed to follow Finding isolation isolation by distance distance among interme posed follow an island model. Finding among intermemeans that their degree degree of diate-age islands in Skeppsvik Archipelago means of genetic differentiation these distance means that this this differentiation will be inflated by these distance effects. It also means metapopulation system pool colonization subsequent dis metapopulation system combines combines migrant pool colonization with subsequent distance-dependent differentiation is lower migrant tance-dependent migration. Since the the degree of of differentiation lower in migrant than in propagule propagule pool colonization, colonization, and and distance-dependent migration inin pool than distance-dependent migration differentiation relative to an island island model, the first problem problem is that that this flates differentiation of colonization and and migration modes modes is a "worst-case scenario" scenario" for for combination of detecting overall increases increases in differentiation differentiation arising arising from colonization. colonization. Too few few detecting studies of of metapopulation metapopulation systems have been carried carried out to know whether whether this combination FST of combination is a common common one. In our case, the FsT of intermediate populations populations was still significantly lower lower than than that that of of the young populations populations (Table Ib) as predicted by the model (which also strengthens strengthens our belief that the conditions conditions specified in Eq. (2) are are met in this metapopulation). metapopulation). It is, however, possible that these expected ST with age may not be detectable in other systems expected changes changes in F FsT showing showing a similar combination of of colonization colonization and migration migration patterns. Further Further
TABLE III III Mantel Mantel Tests Tests for for Isolation Isolation by by Distance Distance among among Islands Islands in in Young Young and and Intermediate Intermediate Age Age (Iassesa Classes"
(a) (b) (c)
a
Group
N
R
P P
sdr
Threshold
Young Intermediate Young Intermediate Intermediate Young Intermediate Intermediate
113 3 30 113 3 30 113 3 30
0.0561 -0.0561 0.3662 0.00 1 1 -0.0011 - 0.323 0.32311 0.0032 -0.0032 0.3298 0.3298
0.405 0.405 0.002 0.002 0.43 0.00 0.00 0.461 0.461 0.002 0.002
0.207 0.207 0. 1 18 0.118 0. 1 59 0.159 0.093 0.093 0. 161 0.161 0. 1 06 0.106
0.05 0.05 0.025 0.025 0.05 0.05 0.025 0.025 0.05 0.05 0.025 0.025
a See Table Table 2 legend for description of transformations (al, (a), (b), (c) and column column entries.
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Genetic Plant Metapopulotion GeneticStructure Structureinin aa Plant Metopopulation
447 447
development of of isolation by of the theoretical theoretical framework to include the effect of distance, distance, in addition to the effect of turnover turnover dynamics is clearly required. A second question is whether whether colonization and migration are really different different phenomena. In plants at least, the answer answer is probably yes, since migrants come in two forms, as diploid seeds and haploid pollen. Colonization can only occur through seed migration, migration, while migration into and and out of of established established populations may occur as seeds and pollen. While we do not know which form of of migration is more responsible for the gene flow which reduces the genetic differentiation differentiation among maturing popUlations, populations, we strongly suspect that gene flow via pollen be becomes more prevalent as populations age. As the islands age and rise, seed dep deposition even on the shores becomes more diffi cult. In addition to the large difficult. large pro proportion of flowers from uninfected M. violaceum uninfected populations which bore M. violaceum spore "pollen-markers" "pollen-markers" (see Section III.D), other evidence suggests that the potential potential Agren, 1996; 1 996; Handel, 11983; 983; for pollen carryover increases with population size ((,~gren, Fritz and Nilsson, 11994). 994). At At this stage of of the study, we can only speculate. speculate. The conclusive evidence will come from a study using markers markers which enable enable gene flow via pollen to be distinguished from that via seed. This can be done by comparing the patterns of gene flow inferred from maternally inherited organelle organelle markers markers (e.g., mtDNA mtDNA or cpDNA) with the patterns inferred from biparentally inherited nuclear nuclear markers will nuclear markers (e.g., allozymes), since only the nuclear be carried from one population to another Ennos, 1994; 1 994; McCauley, another as pollen pollen ((Ennos, 11994). 994). FST bladder campion, Fsv values of of newly founded populations populations of the white bladder Silene Silene alba, alba, have been found to be higher than than those of of older populations be belonging to the same metapopulation ((McCauley McCauley et 995), and a comparison et al. al.,, 11995), of estimates from allozymes and cpDNA suggested that the rate of of gene flow in nuclear nuclear genes was greater than the effective rate of organellar gene flow in this McCauley, 11994). 994). species ((McCauley,
G. Environmental Environmental Heterogeneity Heterogeneity Our studies have also revealed revealed that that the degree of of genetic differentiation is influenced by environmental factors. The values of Fjs , which describe of Fis, describe the degree degree of of structuring structuring within single islands (Appendix), (Appendix), were not only significantly greater greater than zero, they were also highly heterogeneous and ranged from from - 00. . 1188 to 0.38. It is therefore of of interest to know whether whether particular particular factors associated with high levels of within-island ed. Successional changes occur within-island structuring can be identifi identified. more more quickly in the protected protected than than in the exposed parts parts of of the archipelago. archipelago. Ex Exposure is likely to increase increase both the difficulty with which migrants (of any plant species) are deposited on islands and the degree of local patch destruction by salty waves and ice-blocks blown onto the islands during storms. We thus sussus pected that exposure could increase Fj" Fi~, but we could not ignore population age because of of the differences differences in rates of succession. We therefore therefore used age (stage class, Appendix) Appendix) and exposure exposure as factors in a two-way analysis of of variance on -
448 448
Barbaro and Jerome Barbara E.E. Giles Giles ond J6rOmeGoudet Goudet TABLE IV IV Two-Way Two-Way Analysis Analysis of of Variance Variance of of Ranked Ranked Within-Island Within-Island f;, F~ Values Values for for Exposure Exposure and and Stage Stage Classes Classes(Appendix) (Appendix) Source Source
DF DF
Exposure Exposure Age Age Interaction Interaction Residual Residual
2 2 46
1
MS MS
F F
P P
11819 819 284.3 284.3 2 [7.9 217.9 1175.5 75.5
110.36 0.36 11.62 .62 11.24 .24
0.002 0.002 0.209 0.299
the ranked ranked Fis Fis values values for the the islands. Islands Islands were grouped grouped into two two exposure classes, classes, protected protected and exposed, according to their position in the archipelago archipelago relative to the shaded and unshaded Appendix). unshaded sections of of Fig. 11 (see exposure, Appendix). The islands The shading shading marks marks the the boundary boundary between between those those islands islands in in the the lee lee of of other other islands (un shaded) and those that (unshaded) that are are directly exposed exposed to wind and and wave actions actions by autumn autumn storms (shaded) (shaded) and the area which is frozen solid during the winter (unshaded) (shaded) (Ericson, (unshaded) and and the the area area subject subject to to pack pack ice ice driven driven by by winds winds (shaded) (Ericson, 11981). 98 1 ). Table cant (P < Table IV indicates that exposure is signifi significant < 0.05), but age and and the interaction between age Fis values values interaction between age and and exposure exposure are are not. not. With With few few exceptions, exceptions, Fis for We have con for islands islands in the exposed exposed area area are are greater greater than than 0.09 (Appendix). (Appendix). We confi rmed this firmed this effect effect in in our our recent recent study study of of genetic genetic structure structure within within three three of of the the islands islands 996b); when (Giles and and Goudet, Goudet, 11996b); when islands islands were were ranked ranked according to to their their degree degree of exposure, within-island within-island FST Fsv values increased increased with exposure. We strongly strongly sus susof pect that local local patch destruction by wave and and ice action followed by recoloniza recolonization increases increases the genetic differentiation differentiation over and and above that induced by the suc successional dynamics within individual exposed islands, but this is currently currently a working hypothesis.
H. Conclusion Conclusion The populations of The results of of our study of of isozyme isozyme markers markers in 52 island populations of S. dioica popula dioica in Skeppsvik Archipelago Archipelago show show that migration cannot bind bind these these populations populations are tions into into aa single single evolutionary evolutionary unit and and that that each each of of the the island island populations are independent evolutionary likely to to have have independent evolutionary trajectories. trajectories. Furthermore, overall overall levels levels of inflated relative of genetic genetic differentiation differentiation in in this metapopulation will will be be inflated relative to an an island island model model at at equilibrium. equilibrium. In our our case, case, the the founder founder events events dealt dealt with with in in the the genetic 1 977; Wade 1 988; Whit genetic metapopulation metapopulation models models (Slatkin, (Slatkin, 1977; Wade and and McCauley, McCauley, 1988; Whitlock 1 990) do inflation lock and and McCauley, 1990) do not not appear appear to be the the only only cause cause of of this this inflation since we detected detected additional additional affects affects arising arising from other other temporal, spatial spatial and and en environmental rmed model predictions vironmental factors. factors. Our Our study study has has both confi confirmed predictions and and shed shed light on other areas areas which need further further theoretical theoretical and empirical development. We populations were more We found that that newly colonized populations more differentiated differentiated than than those nearer nearer their their demographic demographic and and genetic genetic equilibria, implying implying that that founder events events
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449 449
could increase differentiation at the metapopulation level as predicted predicted by the WhitlockMcCauley model. However, we also observed in Whitlock-McCauley observed that differentiation differentiation increased among the oldest populations undergoing demographic populations undergoing demographic and genetic changes as they approached approached extinction by succession. While these "old" popu populations will contribute contribute to the total increase increase in levels of of differentiation measured measured at a metapopulation metapopulation level, it is not yet known known what kind of of long-term influences these old populations populations could exert on the system (e.g., on the maintenance maintenance of of genetic genetic diversity) since the relative contributions contributions of of old, intermediate, and and young populations to migrant and colonizer colonizer pools are unknown. unknown. It will be interesting to see whether whether these results will be found found for for other other metapopulation metapopulation systems with similar successionally driven extinction dynamics. A spatial factor factor known to in increase differentiation, namely isolation by distance, a form of of spatial variance variance in migration rate 992a; Goudet, 1 993; Slatkin, 11993), 993), was also detected rate (Whitlock, 11992a; Goudet, 1993; in Skeppsvik Archipelago neighboring islands and Archipelago among several small groups groups of of neighboring and of the intermediate age group. For field among the islands of For the past 30 years, fi eld population consequences of subdi population geneticists have been focusing focusing on the consequences of spatial subdivision on popUlation population differentiation. differentiation. Our Our study indicates that spatial and and temporal temporal factors interact interact in a manner manner which may be complex, and the few few other other studies 992a,b; which account account for for temporal temporal as well as spatial effects (e.g., Whitlock, 11992a,b; 1 995) imply that McCauley McCauley et et al. al.,, 1995) that both are are crucial crucial for understanding understanding population population structure. structure. Clearly, the interplay of of these two factors needs to be more more thoroughly investigated in theoretical and empirical studies. Our results showing that the degree to which habitats are exposed to storm driven wind, water, and ice infl u influences the degree degree of of genetic differentiation differentiation also suggests that more attention should should be paid to environmental factors likely to affect the dynamics of ow (see of gene fl flow Whitlock, 11992a). 992a).
V. V. OTHER OTHERGENETIC GENETICMETAPOPULATION METAPOPULATIONSTUDIES STUDIES Studies of of single species metapopulation dynamics involving plants are rare rare (Silvertown, 1 99 1 ; Husband 995), as are empirical studies designed (Silvertown, 1991; Husband and Barrett, 11995), designed to test the predictions predictions of of genetic metapopulation models models regardless of of organism. To our knowledge, knowledge, there are are three such genetic studies studies in addition to our our own own (Giles and 1 996a). and Goudet, Goudet, 1996a). Bolitotherus cornutus, Whitlock l 992b) studied the Whitlock ((1992b) the forked-fungus forked-fungus beetle, beetle, Bolitotherus cornutus, which lives on patchily distributed fruiting fruiting bodies of of several species of of polypore bracket fungi growing on dead and dying trees. Whitlock was able to estimate migration and extinction rates, numbers numbers of of colonizers, and their probabilities probabilities of of common -recap common origin using a combination combination of of methods methods based based mainly on mark mark-recapture. He also created new habitat patches to estimate colonization rate and numnum bers and resurveyed habitat patches known to be occupied 10 1 0 years prior patches known prior to his specified by study. These These experiments enabled enabled him to show that the conditions specified Eq. (2) were met. To confirm confirm that these conditions conditions led to increased genetic dif-
450 450
Barbara Barbara E.E. Giles Giles and and Jerome J6rSrne Gaudet Goudet
ferentiation, Whitlock divided the populations populations of of forked forked fungus beetles into young and older groups using the size of of the fungal resource resource as an indicator indicator of of demo demographic maturity. An electrophoretic study confirmed that the young populations were more more differentiated than the older populations. populations. Dybdal ((1994) 1 994) studied populations populations of of the bottom-dwelling bottom-dwelling marine marine copepod, copepod, Tigriopus Tigriopus californicus, californicus, which inhabits inhabits intertidal splash-zone splash-zone tidepools on rocky shores shores of of the Pacifi Pacificc coast of of North North America. In contrast to the studies of of Whitlock ((1992b), 1 992b), Giles and 1 996a), and 1 995, see below), he and Goudet Goudet ((1996a), and McCauley et al. ((1995, found populations of found that that the older older populations of T. californicus californicus were were genetically more more differ differentiated than the younger populations populations and concluded concluded that extinction/recoloniza extinction/recolonization dynamics decreased decreased genetic differentiation in this metapopulation system. Dybdal ((1994), 1 994), suggested that this was due to violations of of two of of the assumptions assumptions of the metapopulation model, namely, not all extant populations populations were equally likely to be sources populations had prob sources of of colonists, and and not all extant extant populations had equal probof extinction. In the T. californicus californicus system, the extinction probabilities probabilities abilities of were were related related to the size and and location of of the the tidepools; tidepools; there was a tendency for the older older tidepools to be larger and more isolated than the younger younger pools, which reduce, according to Dybdal, the likelihood likelihood of of older older populations populations acting acting as sources sources of of colonists. In the light of of our our earlier earlier discussions discussions about about the importance of of migrant exchange among local populations, Iso populations, we suggest an alternative interpretation. Isolated populations populations are less likely to exchange migrants after colonization, colonization, and therefore some of populations may be more influenced of the older, more isolated populations influenced founder events that the genetic metapopulation metapopulation model suggests are are by the very founder responsible higher FST popula responsible for for the higher FsT values normally observed observed among among young young populations. tions. McCauley et al. ((1995) 1 995) have alba, a shorter have worked on S. alba, shorter lived but close Mountain relative to S. dioica, dioica, which which grows grows along along roadsides roadsides in the the vicinity of of Mountain Lake Biological Station in Giles County, Virginia. The dynamics of meta of this metapopulation system differ differ in several ways from from the S. dioica dioica system described unin above. The S. alba alba populations populations chosen chosen for for study are not separated separated by an uninhabitable matrix. As a consequence, the metapopulation ned as an area metapopulation was defi defined area (25 X • 25 km) and the local populations populations as those individuals inhabiting each 40-m segment of 50 km of of 1150 of roadside roadside contained contained in this area (Antonovics (Antonovics et et al. al.,, 11994). 994). These populations, followed since 11988, 988, are often much populations, which have been been followed 1 994), and the system smaller than ours (see Appendix and Antonovics et al. al.,, 1994), can can be be thought thought of of as as aa large large series series of of small small patches, patches, not not necessarily necessarily very very far far apart, apart, 994; D. E. McCauley, which show high rates of of turnover (Antonovics (Antonovics et al. al.,, 11994; personal personal communication). Extinction Extinction rates rates have been shown shown to be higher higher in small than in large large populations populations and and the extinction-colonization extinction-colonization dynamics dynamics of of the S. alba alba system, which are infl uenced by human activities (e.g., mowing, road mainte influenced maintenance), 994; D. E. McCauley, personal nance), are highly stochastic stochastic (Antonovics (Antonovics et al. al.,, 11994; personal 1 995) showed that communication). McCauley et al. ((1995) that the FST Fsv values calculated from both both isozyme and and cpDNA cpDNA data were higher higher for newly colonized populations populations than than those for for established populations. populations. Thus, Thus, even though though there are differences differences in
Genetic Structure Structure in i n aa Plant Plant Metapopulafion Metapopulation 18 1 8 Genetic
4S 1 451
the nature nature and and rates rates of of the the turnover turnover dynamics dynamics in in the the S. S. dioica dioica and and S. S. alba alba metameta the populations, the the two two systems show show qualitatively qualitatively similar similar results. results. populations, The degree degree of of genetic genetic differentiation differentiation between between 11 1 1 established established and and 77 newly newly The founded populations populations of of the the perennial perennial cowslip, cowslip, Primula Primula veris, veris, was was studied studied by by AnAn founded trobus trobus and and Lack Lack (1994). ( 1 994). While While this this study is concerned concerned with with the the effects effects of of subsub division per per se se and and whether whether founder founder effects effects associated associated with with colonization colonization may may lead lead division of genetic genetic variation, variation, the the metapopulation metapopulation concept concept is never never mentioned. mentioned. to a loss of These authors concluded that that there there were were no no differences differences in the the genetic genetic structure structure These authors concluded of new new and and established established populations popUlations because because the the genetic genetic distances distances and and degree degree of of of = differentiation among all the populations (F 0.039) were small and because T differentiation among all the populations (FsT S = 0.039) were small and because new populations populations did did not not possess possess fewer fewer rare or uncommon uncommon alleles alleles than than established established new rare or populations. However, However, Antrobus Antrobus and and Lack Lack (1994) ( 1 994) report FST values values of of 0.046 0.046 for for populations. report FST 0.033 for for established established populations. populations. While While the FST FST values values have have not been been and 0.033 new and tested for for differences differences from from zero zero or or from each each other, other, the the relative relative magnitudes magnitudes of of tested values are with the predictions predictions of of the metapopulation metapopulation model. model. these values are consistent consistent with Metapopulation dynamics dynamics have been been shown to account for changes Metapopulation account for changes in the the relative proportions of hermaphroditic plants plants in patchily distribdistrib relative of male sterile and and hermaphroditic of the gynodioecious, perennial perennial shrub, Thymus Thymus vulgaris uted populations of vulgaris (Couvet al. , 1986; 1 986; Belhassen et 989; Olivieri et al., aI., 1990; 1 990; see also Frank, Frank, this et al., et al., 11989; Male sterility is determined determined by cytoplasmic, maternally inherited inherited genes, volume). Male Thymus vulgaris vulgaris lives in but specific nuclear nuclear genes may restore male male fertility. Thymus but areas which are While fire leads leads to the extinction of of areas which are regularly destroyed by fire. While populations, the burned burned sites are recolonization. Newly colocolo local populations, are available available for for recolonization. nized and young populations populations contain of male male sterile individuals, individuals, nized contain high proportions of but as the popUlations hermaphrodites increase. but populations age, the the proportions of of hermaphrodites increase. New sites are colonized by seeds from different sources sources and and there is a strong probability that that the founding propagules will contain contain male steriles steriles and hermaphrodites which which contain the matching matching nuclear nuclear genes required to restore the fertility of of the do not contain male steriles in just that patch. Since male sterile plants produce more more seeds (Gouyon and Couvet, 11987), 987), they increase hermaphrodites, increase more more quickly than than hermaphrodites, leading to young populations populations with high proportions of females. With time and migration among populations populations in the form of of pollen, the restorer genes specific to the various male sterile forms eventually appear in the population. Since most pollination occurs within local populations, the matching restorer gene(s) spread(s) through the population and and the proportion of of hermaphrodites hermaphrodites increases as the local populations approach equilibrium. In a similar manner, recurrent extinction and recolonization dynamics have also been shown to affect sex ratio in local populations popUlations of the African butterfly, Acraea Heuch, 11978). 978). Two kinds of females occur in these populations, Acraea encedon encedon ((Heuch, normal females females which produce produce equal numbers numbers of sons and daughters daughters and abnor abnor-linked gene causing meiotic drive of the Y mal females (thought to contain a Y Y-linked chromosome) which only produce daughters. Since females are the heterogametic sex, daughters always have the same phenotype as their mothers. Population and eventually species extinction is expected since the numbers of females only pro-
452 452
Barbara E. E. Giles Giles and and J6rSme Jerome Goudet Gaudet Barbara
ducing daughters daughters increase increase in in aa population population with with time. time. Species Species extinction extinction is is not not ducing observed in in nature, nature, even even though though population population extinction extinction has has been been recorded. recorded. Heuch Heuch observed ( 1 978) showed showed that that turnover turnover within within the the metapopulation metapopulation could could explain explain the the perper (1978) sistence of of these these butterfly butterfly populations populations by by allowing allowing recolonization recolonization by by normal normal fefe sistence males. males. Selection associated associated with with high high rates rates of of population population turnover turnover may may also also account account Selection Carduus pycnocephalus pycnocephalus and and C. for the the maintenance maintenance of of aa seed seed polymorphism polymorphism in in Carduus for tenuifiorus (Olivieri (Olivieri et et al., al., 1983, 1 983, 1990, 1 990, 1995; 1 995; Olivieri Olivieri and and Gouyon, Gouyon, this this volume). volume). tenuiflorus These thistles thistles produce produce two two kinds kinds of of seeds: seeds: those those produced produced in in the the center center of of the the These capitulum, which which are are not not dormant dormant and and have have a pappus pappus allowing allowing wide dispersal, dispersal, capitulum, and the the outer outer seeds, seeds, which which have have no no dispersal dispersal mechanism mechanism but but show show dormancy. dormancy. and The proportion proportion of of seeds producing pappuses pappuses is under under partial partial genetic genetic control control (Oli(Oli The seeds producing 1 985). Older populations have been observed to produce higher vieri and Gouyon, vieri and Gouyon, 1985). Older populations have been observed to produce higher proportions of of nondispersed seed, and and it has has been been proposed that there there is selection selection proportions nondispersed seed, proposed that within populations populations against against interpopulation dispersal since since most most dispersing dispersing seeds seeds within interpopulation dispersal 1 985). However, However, their source source populations popUlations (Olivieri (Olivieri and Gouyon, Gouyon, 1985). will be lost to their popUlation extinction and occur frequently frequently and because all new new poppop population extinction and colonization colonization occur and because ulations must must be founded founded by migrants, aiding dispersal cannot ulations migrants, traits aiding cannot be lost at at a metapopulation level if the species is to persist. persist. This suggests that these these traits are metapopulation subject different ages. Olivieri subject to different selection pressures pressures in populations populations of of different et al. ((1995) 1 995) and Olivieri and Gouyon Gouyon (this volume) showed that between-deme selection for for high migration migration and and within-deme selection for for low migration could maintain maintain this this seed seed polymorphism. polymorphism. As evidenced by the metapop the papers papers written and cited in this volume, the the metapopulation concept is more than just a "rage." This concept concept recognizes and studies the the consequences consequences of of movement movement among among the the spatially spatially separated separated and and smaller-than smaller-thaninfi nite populations that typify nature. It also allows us to start to let go of our infinite comfortable view of of nature nature as an immortal, homogeneous, and constant constant entity and to examine the consequences of extinction, fl uctuation, and randomness fluctuation, randomness where it exists. At some level, metapopulation dynamics probably probably influence most species ((Husband Husband and Barrett, 11995). 995). In our case study and and several of the others that we have described, the focus has been on neutral genetic variation. These are important studies. That their results support the predictions of existing models suggest that the concept is not simply an empty fashion, but one that describes nature. For any particular particular species, studies involving neutral neutral characters are are an important fi rst step first step in establishing establishing the spatial and temporal scales at at which turn turnover over occurs occurs and and for for coupling the intimate details of of the biology of each species species to to the the magnitude of of the metapopulation effect. It is, however, however, too much to believe that ected in that the the effects of of metapopulation dynamics dynamics we have seen seen so so well refl reflected neutral neutral characters characters will not effect effect the rates rates and and patterns patterns of of evolution (see (see Barton and and Whitlock, Whitlock, this this volume). volume). The The difficult difficult but but necessary necessary theoretical theoretical and and empirical empirical next next step step demands demands that that we we learn learn to to understand understand and and measure measure the the consequences consequences of of this this dynamic dynamic population population structure structure for for adaptation and and speciation. speciation.
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Note. Exposure is the islands are subjected Note. Use Use Island Island No. for for location in Fig 11.. Exposure the degree to which islands subjected to wind, exposed islands in unshaded and wind, wave, and ice action; classes 11 and and 2 are protected and exposed and shaded shaded parts time since while Stages parts of of Fig Fig 11,, respectively. respectively. Age Age is is population population age age or or time since colonization, colonization, while Stages 11,, 2, 2, and and 33 divide divide the classes, respectively. respectively. Size Size is is census the islands islands into into young, young, intermediate, intermediate, and and old old age age classes, census size. size. Sample values. Sample describes describes the the numbers numbers studied studied electrophoretically. electrophoretically. F;, Fis gives gives within-island within-island values.
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Index
Age, Age, metapopulation effect effect on genetic variability, 440-442 estimation, 434 evolution of age structure at equilibrium, 3313 13
Basic reproduction number calculation for metapopulation, 1100-103, 00-1 03, 1121 21 mathematical derivation, 1113-114, 1 3- 1 14, 118-119 1 1 8- 1 19 steady state assumption, 100, 100, 1113 13 Butterfly metapopulation applications of metapopulation models, 385386 386 colonization, 367-368 conservation management, 359-360, 372-373 genetic structure, 1 -452 structure, 45 451-452 habitat change, population response change, meta metapopulation climatic variation, 379-380 heterogeneity of habitat, 379-38 379-381I loss of habitat, 377, 379
minimum habitat fragments for survival, 88-89, 3371-372 7 1 -372 vegetation alteration, 377 habitat identification, 361-362 incidence function modeling, 369-370 local extinction factors affecting extinction, 365-367, 385 rate, 365 time to extinction, 370, 379 migration area-dependent emigration, 375-376 capacity, 384 data collection, 373-374 estimation, 374-375 patch occupancy mapping, 362-364 population size in isolated patches, calculation, 375 rampant population turnover and and classical dynamics, 70-72 rescue effect, 280-282, 368 spatially explicit modeling, 368-369 spatial spatial synchrony of population fluctuations. fluctuations, 38 I -383 381-383
505 sos
506 506
Index Index
Butterfly metapopulation metapopulation (continued) (continued) Butterfly structure, 360-361 360-361 structure, usefulness in in metapopulation metapopulation studies, studies, 354354usefulness 355, 359 359 355,
Catastrophe Catastrophe examples in in nature, nature, 236 examples extinction scenarios, scenarios, 235 235 extinction genetic variability variability effects, effects, 236-237 236-237 genetic rate modeling, modeling, 235 235 rate Cinnabar moth Cinnabar 389 life history, 389 401-402 persistence mechanisms, 401 -402 ragwort-cinnabar moth-Cotesia moth-Cotesia popularis popularis ragwort-cinnabar metapopulation census data, 390 389-390 data collection, 389-390 extinction rates, 402-403 extinction population dynamics, 396-399 spatially extended dynamics and survival probability, 403-405 Climate, see Weather see Cytoplasmic male sterility CMS, see Colonization metapopulations, 367-368 butterfly metapopulations, capture- recapture estimation of immigration capture-recapture 260-261 interpopulation immigration, 260-261 259-260 single populations, 259-260 248-251 defined, 248-251 local extinction extinction and recolonization, recolonization, 33-34, 33-34, 57, 240 240 modeling bifurcation bifurcation parameter, 105-106 1 05-1 06 combining with local dynamics, 99, 99, 110 1 10 mathematical mathematical derivation, 108-109 1 08- 1 09 redistribution redistribution kernels, 97-98, 97-98, 108-109, 1 08- 1 09, 119 1 19 probability, probability, 81 81 rate rate estimation estimation difficulty, difficulty, 251 25 1 equation, equation, 74, 74, 105 1 05 experimental manipulation manipulation of of populations, populations, 263-264 263-264 gene gene flow estimation, estimation, 253-254 253-254 indirect indirect global global approach, approach, 251-253 25 1 -253 parameters, parameters, 252 252 patch-specific patch-specific estimation, estimation, 264-265 264-265 Community, Community, see see Metacommunity Metacommunity Conservation, Conservation, metapopulation metapopulation modeling modeling butterfly butterfly management, management, 359-360, 359-360, 372-373 372-373 ecosystem ecosystem management, management, 26 26 genetic genetic modeling, modeling, 25-26 25-26 island island biogeography biogeography theory, theory, 17-19 1 7- 1 9
landscape landscape ecology ecology impact, impact, 58-60 58-60 local 17 local extinction extinction parameters, parameters, 2217 minimum minimum viable viable metapopulation metapopulation size, size, 22, 22, 24, 24, 65 65 nonequilibrium nonequilibrium metapopulations metapopulations and and time time to to extinction, extinction, 85-86, 85-86, 88 88 parasitic parasitic impact, impact, 24-25 24-25 refuge refuge design, design, 22-23 rescue effect effect relevance, relevance, 27 2711 rise 9-2 1 rise in in popularity popularity of of models, models, 119-21 species 1 -22 species conforming conforming to Levins Levins model, 221-22 structured versus unstructured models, 1107071108 08 Cotesia Cotesia popularis popularis life history, 389 ragwort-cinnabar moth-Cotesia moth- Cotesia popularis popularis metapopulation census data, data, 390 data collection, 389-390 extinction extinction rates, 402-403 population dynamics, 399-401 spatially extended dynamics and survival probability, 403-405 Cowslip metapopulation, genetic structure anal analysis, 45 4511 Cytoplasmic male sterility colonization-extinction colonization-extinction dynamics of alleles, 343-347 cytotypes, cytotypes, 340 defined, 340 varia genotype dimensionality and and spatial variation, 341-343, 35 1 tion, 341-343, 351 343-344 natural history of of alleles, 343-344 nature, 340 prevalence in nature,
Demographic stochasticity, stochasticity, modeling modeling of of extincextinc Demographic tion tion demographic and and environmental stochasticity demographic stochasticity modeling, 228-231 228-23 1 modeling, diffusion approximation, approximation, 231 23 1 diffusion extinction, 232 time to extinction, Dispersal, see see Migration Migration Dispersal, Dormancy Dormancy evolution, 311-312 3 1 1-312 evolution, ragwort, 396 396 ragwort,
Effective population population size size Effective defined, 166 1 66 defined, estimation, 170, 1 70, 172, 1 72, 179 1 79 estimation, gene flow flow rate rate between between populations, populations, effect effect gene on, 178-180, 1 78-180, 188 188 on,
Index Index modeling of of genetic effects approaches, 167-169 167-169 colonization rate, 169, 172, 1 76, 1186 86 172, 174174-176, extinction rate, 169, 172, 172, 174-176, 86 rate, 169, 174-176, 1186 general pattern pattern of of results, 173 heterozygosity estimation, 170-172 170-172 patch carrying capacity, capacity, 169, 174 number of 80 of founders, effects on, 176, 176, 1180 simulation simulation of of effects on heterozygosity, 166167 subdivision, 87-1 88 subdivision, effects on, 1187-188 subpopulation subpopulation number, effects on, 177 variance, 1187-188 87- 188 Emigration butterfly metapopulations metapopulations area-dependent area-dependent emigration, 375-376 375-376 data data collection, 373-374 373-374 estimation, 374-375 374-375 defined, defined, 248-249 248-249 rate estimation capture-recapture capture-recapture estimation between populations, 260-261 260-261 experimental manipulation manipulation of populations, 262 patch-specific patch-specific estimation, 264-265 Environmental stochasticity defined, 2 1 8, 355 218, modeling of extinction autocorrelation, 232-233 232-233 ceiling adjustment, adjustment, 234-235 234-235 ceiling model, model, 220-223, 220-223, 225-227, 225-227, 242 demographic and environmental stochasticity modeling, 228-23 228-2311 diffusion approximation, approximation, 220, 228 extinction parameter estimation estimation from time series data, 223, 225 logistic density dependence, 233-234 233-234 time to extinction, 221-222, 221-222, 234 modes, 19 modes, 2219 regional stochasticity and correlated environ environmental fl uctuations, 238-240 fluctuations, species-area species-area curve dependence, 226-227 226-227 Evolution, metapopulations metapopulations biological traits, see see Metapopulation effect deleterious mutations fixation probability, 9 1 - 193 probability, 1191-193 manifestations, 191 manifestations, 191 spatial subdivision effects, 192-193 favorable allele establishment extinction effects, 190 fixation probability, 1189-190 89-190 mechanisms, 84 mechanisms, 1184 rate of 19 I of spread, spread, 191
S07 $01
stages, 1189 89 gene mutation 191 mutation rate, rate, 190190-191 genetic variation maintenance, maintenance, 208-209 208-209 limitations gene flow, 197-1 98, 200 197-198, population 99 population density, 198-1 198-199 local adaptation adaptation dispersal range of genes, 195 extinction 96 extinction effects, 195-I 195-196 gene flow rate, 193, 195 migration effects, 196 neutral variation effective population size variation, 1187871188 88 extinction/recolonization 187 extinction/recolonization models, 185185-187 migration effects, 186-187 186-187 stable local population 85 population models, 1185 shifting shifting balance balance model adaptive peaks, peaks, 201-202 201-202 extinction effects, 206-209 206-209 hybrid hybrid zone, 207-208 207-208 migration migration effects, 202-204, 202-204, 206-209 206-209 peak shift modeling, 202-204, 202-204, 206 phases, 201 speciation, 201 Extinction, see see Local extinction
Forked-fungus beetle metapopulation, metapopulation, genetic structure analysis, 449-450 449-450 F ST, see FST, see Genetic variability Genetic Host-parasite geGenetic variability, see see also also Host-parasite netics adequacy 10 adequacy in population modeling, 2210 calculation, 1171, 7 1 , 1185, 85, 4 3I 431 effects on F ST Fsx catastrophe, 236-237 colonization rate, 1169, 69, 172, 172, 174-176, 86 174-176, 1186 effective population population size, 166-167 166-167 extinction rate, 169, 172, 172, 1174-176, 74-176, 1186 86 rate, 169, gene flow rate, 178 habitat fragmentation, 65-66, 1181 81 number ooff founders, 176, 176, 180 patch 169, 174 patch carrying capacity, 169, subpopulation subpopulation number, 177 local extinction dependence, 169, 1172, 72, 1741176, 76, 1186 86 maintenance, maintenance, 208-209 208-209 measurement 165, 170measurement of heterozygosity, 165, 172 migration rate estimation, 28 1 , 287-288, 433 281,287-288,
508
Index Index
variability (continued) (continued) Genetic variability plant metapopulations effects on genetic differentiation, distance, effects 443-445 of environmental environmental heterogeneity, heterogeneity, effect of 447-448 effect of population age, 440-442 founder effects, 445-449 interisland genetic structure, 440 structure, 442-443 intraisland genetic structure, statistical analysis, 439-440
see also also Patch Habitat, see classification, 115 5 destruction 144, 146 effects on migration, 144, 143-145 multispecies models, 1126, 26, 1431 45 fragmentation effects genetic variability of population, 65-66 metapopulation dynamics, 296-297 minimum fragments for butterfly survival, 88-89 Host--parasite parasite genetics Host bacteria-phage bacteria -phage interactions, 348-349 cytoplasmic male sterility as host-parasite model colonization-extinction dynamics of alleles, 343-347 cytotypes, 340 defined, 340 defined, genotype dimensionality and spatial variavaria 3511 tion, 341-343, 341 -343, 35 natural history of of alleles, 343-344 prevalence in nature, 340 factors influencing spatial spatial variation, 350-351 350-35 1 genotype dimensionality and colonizationextinction dynamics complexity of natural dynamics, 330-332 four host-parasite pairs, pairs, 329-330 migration effects, 330 single-patch model, 327-330 two two host-parasite host-parasite pairs, pairs, 328-329 328-329 genotype variability, 325-327 insect insect herbivores, 349-350 349-350 major major histocompatibility complex, 349 349 plant-pathogen plant -pathogen interactions gene-for-gene systems, systems, 333-334 dimensionality of of host resistance, 334334336 flax and and flax rust, rust, 334-336, 338-339 338-339 crops, 335
colonization-extinction colonization-extinction dynamics dynamics of of al alleles, 336-339 legume legume and and fungus, 336-338
IF, see see Incidence function model Immigration, see see Colonization Incidence function model applications, applications, 83, 85 butterfly metapopulations, 369-370 characteristics, 14 colonization probability, 80-8 80-811 extinction probability, 80 incidence of species species in a patch, calculation, incidence 80-811 80-8 regression analysis for for parameter determina determination, 881-82 1 -82 testing, 82-83 Island biogeography theory comparison to metapopulation models, 116 6 conservation biology, 117-19 7-19 fall ooff popularity, 117-19 7-19 refuge design, 24 Isolation by distance model, 43 1 -432 431-432
Landscape defined, 45 heterogeneity, effects on local community composition colonization rate, 152-154 1 52-1 54 1 52 extinction rate, 152 growth rate of of rare species, species, 153-154 1 53- 1 54 of species, 1 52- 1 55, habitat specialization of species, 152-155, 1 64 164 1 5 1 - 1 52 metapopulation modeling, 151-152 species distribution, 150-151 1 50- 1 5 1 spillover between habitats, 154 1 54 high coverage landscapes and metapopulametapopula tion applicability, 4-5 low coverage landscapes and metapopulation applicability, 4-5 Landscape Landscape ecology 1 5, comparison with metapopulation ecology, 15, 48, 50 48, defined, 45-48 defined, development, 45-47 fusion with metapopulation dynamics conservation impact, 58-60 58-60 conservation impact, 61 importance, 58, 61 local extinction and recolonization, 57 effects, 51-57 5 1 -57 migration effects,
Index Index patch characteristics, 50-5 50-511 success, 43 patch theory, 44-45, 47, 58 Levins metapopulation assumptions of model, 9, 113, 3, 73-74, 94-95 basic reproduction number, calculation, 1101, 01, 1114-115 14-1 15 colonization, rate equation, 74 1 -72 conditions for application, 771-72 conservation biology, 221-22 1 -22 definition and synonyms, I11I fraction of occupied patches, 93 homology with susceptible- infected-suscepinfected- susceptible model, 94 migration and local dynamics, 95-96 patch area and extinction risk, 74-75 species candidates, 75-76 stochastic version, calculation of time to extinction, 76-78 Levitt-Boorman metapopulation, see see Main Mainland land--island island metapopulation Life history, evolution of traits age structure at equilibrium, 3313 13 annuals versus perennials, 3313-317 1 3-3 1 7 migration effects, 320-321 Local extinction butterfly metapopulations factors affecting extinction, 365-367, 385 rate, 365 time to extinction, 370, 379 catastrophe and genetic impoverishment, 235-237 causes, 30-3 1 , 429 30-31,429 defined, 17 defined, 30, 2217 effects favorable allele establishment, 1190 90 genetic variability, 1169, 69, 1172, 72, 1174-176, 74-1 76, 1186 86 local adaptation, 1195-196 95- 1 96 mUltispecies 57-1 58, multispecies metapopulation, 36, 1157-158, 1161-163 6 1 - 1 63 nonequilibrium metapopulations, 85-86, 88 shifting balance model, 206-209 size of population, 2 1 6, 237-238 216, landscape mixtures mixtures and metapopulation ex extinction, 308-3 11 308-311 migration and persistence, 32-33, 40, 57, 272, 275, 290-291 09- 1 1 0, 2 1 5-21 6 modeling, 98-99, 1109-110, 215-216 conservation parameters, 2 17 217 features of a good model, 2 16-21 7 216-217 metacommunities, 240-241 population growth, 2217, 1 7, 2 19 219
509 509
stochastic modeling, see see Demographic sto stochasticity; Environmental stochastic stochasticity time to extinction, 2218, 221-222, 1 8, 221 -222, 229, 234 recolonization, 33-34, 57, 240 Local population definition and synonyms, 1111 migration and metapopulation approaches, 9110 0 modeling of growth, 98-99, 1115-118, 15 - 1 1 8, 2 1 7, 217, 2 19 219
Mainland-island metapopulation, definition and synonyms, 1111 Metacommunity, see see also also Multispecies metapopulation apparent competition modeling, 1161-164 6 1 - 1 64 50 defined, 1150 extinction modeling, 240-24 240-2411 food web complexity, 1161 61 landscape heterogeneity, effects oonn local community composition colonization rate, 1152-154 52- 1 54 extinction rate, 1152 52 growth rate of rare species, 1153-154 53 - 1 54 habitat specialization of species, 1152-155, 52-1 55, 1164 64 metapopulation modeling, 1151-152 5 1 - 1 52 50- 1 5 1 species distribution, 1150-151 spillover between habitats, 1154 54 Metapopulation, see see also also specific specific models m o d e l s and and populations populations conservation biology ecosystem management, 26 genetic modeling, 25-26 minimum viable metapopulation determi determination, 22, 24, 65 non equilibrium metapopulations and time nonequilibrium to extinction, 85-86, 88 parasitic impact, impact, 24-25 refuge design, 22-23 rise in popularity of models, 119-21 9-21 species conforming to Levins model, 221-22 1 -22 structured versus unstructured models, 107-1 08 107-108 definition and synonyms, 9-1 1 , 48, 50 9-11, density-dependent regulation, 297 growth in literature, 5-6, 2 211 history of studies, 9 studies, 5, 7-8, 119 homology with epidemiology, 94, 1124-126 24-126
5510 10
Index Index
Metapopulation Metapopulation (continued) (continued) mathematical derivation, 1111-113 1 1- 1 1 3 rampant rampant population population turnover, butterfly exam example of of classical dynamics, 70-72 steps in modeling, 96-100 96-100 Metapopulation Metapopulation effect defined, 294 donnancy, 1 1-312 dormancy, 3311-312 interacting genomes and reproductive sys systerns, 1 7-3 1 9 tems, 3317-319 interacting species, 3317 17 life history age structure at equilibrium, 3313 13 annuals versus perennials, 3313-317 1 3-3 1 7 migration effects, 320-321 migration rate adaptation adaptation to landscape heterogeneity, 308-3 11 308-311 conditional conditional migration, 303-304 evolutionarily stable migration migration rate balancing with optimal migration rate, 305-308 balancing with polymorphism, 300-301 calculation, 302 landscape influence, 301 survival rate effects, 302-303 302-303 metapopulation metapopulation effect, 295 polymorphism, 298-301 294-295 rationale for selection, 294-295 1 -322 selection levels, 32 321-322 thyme, 3317-319 1 7-3 1 9 Metapopulation structure, definition and syno synoI , 267 nyms, I11,267 Migration, see see also also Rescue effect bifurcation 04- 1 05 bifurcation parameter, 1104-105 butterfly metapopulations metapopulations area-dependent area-dependent emigration, 375-376 capacity, capacity, 384 data data collection, 373-374 estimation, estimation, 374-375 374-375 computer computer simulation, simulation, 257 connectivity of habitats, habitats, 55 defined, 248-250 248-250 effects genetic variability, 28 1 , 287-288 281,287-288 habitat 46 habitat destruction, 144, 144, 1146 life history trait evolution, 320-321 local adaptation, adaptation, 196, 196, 268 local extinction, 32-33, 32-33, 40, 57, 272, 275, 290-291 , 304 290-291,304 neutral variation models, 1186-187 86- 1 87 shifting balance model, 202-204, 206-209 206-209 38- 1 4 1 spatially explicit models, 1138-141
evolution adaptation to landscape heterogeneity, adaptation 308-3 11 308-311 conditional migration, 303-304 evolutionarily evolutionarily stable migration migration rate balancing with optimal migration rate, 305-308 balancing with polymorphism, 300-301 calculation, 302 landscape influence, influence, 301 survival rate effects, 302-303 metapopulation effect, 295 polymorphism, 298-301 rationale for selection, 294-295 selection levels, 321-322 frequency, 268, 28 2811 gene selection, 2213 13 ground insects, methods of study, 257-258 257-258 insect model, 53-54 landscape ecology effects, 54-55, 6 611 mating probability, 290 modeling in metapopulations, 91 0, 551-57, 1 -57, 9-10, 1104-105, 04-105, 272, 274 mortality during migration, 258-259 258-259 optimization of of rate, 305-308 percolation theory, 55-56, 61 plant studies, 255 rate estimation experimental manipulation manipulation of of populations, 262-263 genetic variability, 433 patch-specific patch-specific estimation, 264-265 264-265 timing, 268-269, 275 Minimum Minimum viable metapopulation, detennina determination, 22, 24, 65, 76-78 Moran Moran statistic, statistic, computation for pika metapop metapopulation, 4 1 4-41 5 , 425 414-415,425 Multispecies metapopulation, metapopulation, see see also also Host Hostparasite genetics; Metacommunity; Spa Spatially explicit metapopulation model classical dynamics, 38-39 competition modeling conditions conditions for for inferior species existence, 1128 28 habitat destruction effects, 1128-130 28- 1 30 Levins model, 1127-130 27-1 30 rate equations for population growth, 1271 271128 28 mutualism modeling eradication threshold, 1135 35 habitat destruction effects, 135-136 1 35- 1 36 rate 1 34rate equations for population growth, 1341135 35
Index Index persistent refuges, refuges, 37 37 persistent predation modeling modeling predation apparent competition competition modeling, modeling, 161-164 1 6 1 - 1 64 apparent biogeographic donor donor control, control, 157 157 biogeographic colonization-extinction dynamics, dynamics, 157, 1 57, colonization-extinction 1 6 1 - 1 63 161-163 habitat destruction destruction effects, effects, 131-133 1 3 1 - 1 33 habitat laissez-faire predators, predators, 133 1 33 laissez-faire prey colonization colonization scenarios, scenarios, 158-159 1 58- 1 59 prey prey extinction extinction scenarios, scenarios, 157-158, 1 57-1 58, 163 1 63 prey rate equations equations for for population population growth, growth, 131 131 rate two-prey modeling, modeling, 163 1 63 two-prey ragwort-cinnabar moth -Cotesia Cotesia popularis popularis ragwortcinnabar mothsystem system census data, data, 390 390 census data collection, collection, 389-390 389-390 data extinction rates, rates, 402-403 402-403 extinction persistence mechanisms mechanisms persistence cinnabar moth, moth, 401-402 40 1 -402 cinnabar ragwort, 395-396 395-396 ragwort, population dynamics dynamics population cinnabar moth, moth, 396-399 396-399 cinnabar Cotesia popularis, popuiaris, 399-401 399-40 1 Cotesia ragwort, 39 1 , 393, 395 ragwort, 391,393, properties of of individual individual species, species, 389 properties spatially dynamics and and survival survival spatially extended extended dynamics probability, probability, 403-405 stability stability of of populations populations predator, predator, 37 37 prey, prey, 35-37 35-37 subdivision subdivision of of populations, populations, 38 38 theory, theory, 29 29 three-level three-level food food chain chain colonization55colonization- extinction extinction dynamics, dynamics, 11551156, 56, 1160 60 occupancy 60 occupancy of of intermediate intermediate competitor, competitor, 1160 trophic 1 , 1155, 55 , 1161 61 trophic complexity, complexity, 441,
Patch, Patch, see see also also Habitat; Habitat; Landscape Landscape approaches approaches to to patchiness, patchiness, 44-45 44-45 definition definition and and synonyms, synonyms, 1111 metapopulation metapopulation effects effects configuration configuration of of edges, edges, 50-5 50-511 linking linking elements. elements, 5511 number 25 number of of patches, patches, 1125 size. size, 50 50 spacing, 1 , 89-90, spacing, 50-5 50-51, 89-90, 430 430 weather weather effects, effects, 90-91 90-91 Persistence I Persistence time, time, definition definition and and synonyms, synonyms, I11 Pika Pika metapopulation metapopulation Bodie 1 1 -4 1 2 Bodie site site properties, properties, 407, 407, 4411-412 census 1 2-4 1 3 census data data collection, collection, 408, 408, 4412-413
51 Sl !1
competitors, competitors, 426 426 natural natural history history of of pikas, pikas, 409-411 409-4 1 1 patch patch area effects, effects, 415-417 4 1 5-4 1 7 area isolation isolation effects, effects, 417-418, 4 1 7-4 1 8, 424 424 occupancy, occupancy, 415, 4 1 5, 424 424 size size assessment, assessment, 413 413 predators, predators, 424, 424, 426 426 rescue rescue effect, effect, 283,408 283, 408 spatial spatial autocorrelation autocorrelation of of population population events, events, 414-415,420, 414-4 1 5, 420, 422-423,425-428 422-423, 425-428 spatially spatially explicit explicit modeling, modeling, 413-414, 4 1 3-414, 419-420 4 19-420 Plant Plant metapopulation, metapopulation, see see also also Ragwort; Ragwort; Red Red bladder bladder campion campion metapopulation Carduus, 452 452 Carduus, metapopulation metapopulation effect effect in in thyme thyme populations, populations, 3 1 7-3 1 9 317-319 Primula veris, veris, 451 Primula Thymus Thymus vulgaris, vulgaris, 451 45 1
Ragwort life history, 388-389 persistence mechanisms, 395-396 ragwort-cinnabar mothmoth- Cotesia Cotesia popularis popularis metapopulation census data, 390 data collection, 389-390 extinction rates, 402-403 population dynamics, 39 391,393, 1 , 393, 395 spatially extended dynamics and survival probability, 403-405 Red bladder campion metapopulation age estimation, 434 colonization, 436-439. 436-439, 445-447 445-447 colonization, genetic genetic variability variability distance, effects effects on genetic differentiation, distance, 443 -445 443-445 effect of of environmental environmental heterogeneity, heterogeneity, effect 447-448 447-448 effect of of population population age, age, 440-442 440-442 effect founder effects, effects, 445-449 founder interisland genetic genetic structure, structure, 440 440 interisland intraisland genetic genetic structure. structure, 442-443 442-443 intraisland statistical analysis, analysis, 439-440 439-440 statistical life history history of of plant, plant, 436 436 life migration, 438-439, 438-439, 442 442 migration, patch carrying carrying capacity. capacity, 438 438 patch population population growth, growth, 437 437 Silene alba alba metapopulation, metapopulation, genetic genetic structure structure Silene analysis, 450-451 450-451 analysis, Skeppsvik Archipelago Archipelago characteristics, characteristics, 434, 434, Skeppsvik 438 438
5512 12
Index Index
see also also Conservation Refuge, see 22-24 design, 22-24 and management, 59-60, landscape ecology and 1123-24 23-24 effect Rescue effect 271I conservation relevance, 27 defined, 2213 13 effect ooff population growth rate, 278 evidence within metapopulations butterflies, 280-282 chipmunks, 285 chipmunks, mice, 284-285 natterjack toads, 287-288 pikas, 283 pool frogs, 288-289 prairie dogs, 284, 286 proof requirements, requirements, 279 proof red-spotted newts, 287 285-286 squirrels, 285-286 283-284 voles, 283-284
Shifting balance model 201-202, 4311 adaptive peaks, 201 -202, 43 206-209 extinction effects, 206-209 hybrid zone, 207-208 4311 migration effects, 202-204, 206-209, 43 peak shift modeling, 202-204, 206 phases, 201 speciation, 20 201I Single species metapopulation metapopulation local extinction causes, 30-31 30-3 I defined, 30 migration migration and prevention, 32-33, 40 recolonization, recolonization, 33-34 low turnover, 35 mixed metapopulation metapopulation structures, 34-35 34-35 persistence, 331-32 persistence, I -32 theory, 28-29 theory, Source-sink metapopulation Source-sink metapopulation definition and synonyms, 10-11 1 0- I I genetic genetic drift, drift, 207
migration, 269-270 pseudosinks, 380-38 380-3811 explicit metapopulation model Spatially explicit advantages and disadvantages, 14 14 advantages butterfly metapopulations, 368-369 definition and synonyms, 112-13 2- 1 3 migration effects, 1138-141 38-141 muitispecies multispecies modeling complex spatial dynamics, 140-143, 146147 habitat destruction and and spatial dynamics, 143-145 host-parasitoid host-parasitoid modeling, 140-143 local stability, 1137-140 37-140 three-species systems, 145I 46 145-146 pika metapopulation, 4 1 3-414, 4 19-420 413-414, 419-420 population size, calculation, 1138, 38, 140 Spatially implicit metapopulation model advantages, 113 3 definition and synonyms, 112 2 see Spatially realistic metapopulation model, see also Incidence function model; State trantran also sition model applications, 78-79 2, 114 4 definition and synonyms, 112, State transition model applications, 79 characteristics, 79 Stochastic modeling, see see Demographic stochas stochasStochastic ticity; Environmental stochasticity
Tigriopus Tigriopus calif californicus amicus metapopulation, metapopulation, genetic
structure analysis, 450 Transfer, see see Colonization; Emigration; MigraMigra tion Turnover, II Turnover, definition and synonyms, 11
Weather, variation within habitats, habitats, 90-91 90-91,379, 379380
