Metapopulation Biology: Ecology, Genetics, and Evolution

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

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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

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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

IIkka Ilkka Hanski Hanski and and Daniel Daniel Simberloff Simberloff a

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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

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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

21 21

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-

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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

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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-

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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

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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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4S 45

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.

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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

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Individual Individual

Metapopulation MetapopulationDynamics Dynamicsand ond Landscape LandscapeEcology Ecology

53 53

B ' Population Population B

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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

SS 55

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.

33

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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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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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

6611

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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4 II

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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to km --r-

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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78 78

IIkko Ilkka Honski Honski

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

I 0.0 I� ~ 0.0 '-J-jLJ 0. 0 0.2 0.0 0.2

0

FIGURE FIGURE77

• 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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450 450

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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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IIkko Hanski Honski Ilkka

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

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Metapopuiation MetapopulationDynamics Dynamics

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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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9

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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.

93 93

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Mols Mats Gyllenberg Gyllenberget et 01. al.

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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Mats Gyllenberg Gyllenberg et et al. al. Mats

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-

98 98

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

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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-

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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.

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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"

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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-

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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).]

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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)

|

~� 9

0; c: 0~,

'

�

0.2

00

(b) (b)

a) ((a)

0.2

,

SPIRALS SPIRALS

',I,;,,,

,

o.s I

CHAOS CHAOS

I,

~,,

0

Oo -0.2 -~

8' ...J

0.5

'::

'5 0.

0 ~ a.

,,

i~ !

,,,

,,,

~

,,

-o.s ,,

0 50 . . . . 1160 -"0"40 .40 . . . . 50 00 . . . . 1150 50

r fI) Q) N '0; c: r 0B .0

�

'5 0. ~-

0 0 n a. r 0> 0 0 ...J

--J

1143 43

" " 200 260

9

-I0

,

,

,.'I'',i' .

.

t .

.

.

.

.

.

.

50

.

.

100

.

.

.

.

.

.

.

150

200

(c) (e)

22 1

00 -1 -1

"

00

I'

,

' CRYSTAL LATIICE ICE

1100 00

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

0.4 io. 2 ,12 HH Il 0.1

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.

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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.

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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

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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

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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.

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Consequences of Heterogeneily Consequences of Spano SpatialI Heterogeneity

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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

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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

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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.

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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.

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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.

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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

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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.

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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.

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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

,

,

,

1

Als Z,/s

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

-

]

-

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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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.

99

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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Patrick PatrickFoley Foley

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

1100

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