Metapopulation Biology: Ecology, Genetics, and Evolution

Metapopulation

BIOLOGY Ecology, Genetics, and Evolution

Meta population Ecology, Genetics, and Evolution Edited by

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llkka Hanski Department of Ecology and Systematics Univer.;ity of Helsinki FIN-00014 Helsinki. Finland

Michael E. Gilpin Department of Biology University of California, San Diego La Jolla, Califomia

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An Elsevier Science Imprint

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Front cover photograph: The landscape of forests, fields, and lakes In Aland, Finland, with the network of small meadows used by the Glanville fritillary butterfly (Melitaea cinxia) shown as red and White dots. The red dots represent meadows with a local population in 1993 (see Chapter 4 by Hanski, this volume). Reproduced with permission from the National Land Survey of Finland.

This book is printed on acid-free paper.

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Copyright €>1997, Elsevier Science (USA). All righlS reserved. No pan of this publication may be reproduced or ttansmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system. without permission in writing from the publisher. RequeslS for permission to make copies of any pan of the work should be mailed to: Permissions Depanmcnt, Academic Press, 6277 Sea Harbor Drive, Orlando, Florida 32887-6777

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Academic Press 32 Jamestown Road, London NWI 7BY, UK http:l/www.academicpress.com Library of Congress Cataloging-in-Publication Data Metapopulation biology : ecology, genetics, and evolution I edited by llkka Hansk.i, Michael E. Gilpin. p. em. tncludes index. ISBN 0-12-323446-8 I. Population biology. I. Hanski, lllka. II. Gilpin, Michael E., date. QH3S2.M47 1997 S74.S'248~c20 96-2842 CIP

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Contents

Contributors Preface xv

P ART

xi

CONCEPTUAL FOUNDATIONS

1 The Metapopulation Approach, Its History, Conceptual Domain, and Application to Conservation 1/kka Hanski and Daniel SimberloH I. II. Ill. IV. V.

Introduction 5 Brief History of Metapopulation Studies 7 Conceptual Domain and Metapopulation Approaches Metapopulations and Conservation Biology 16 Conclusions 26

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Contents

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Empirical Evidence for Metapopulation Dynamics Susan Hoflison and Andrew 0. Toylot I. II. Ill. IV.

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Introduction 27 Single-Species Metapopulations 30 Multispecies Metapopulations 35 Discussion 39

Metapopulation Dynamics and Landscape Ecology John A. Wiem I. II. Ill. IV.

Introduction 43 Approaches to Patchiness 44 What Is Landscape Ecology? 45 How Is Landscape Ecology Relevant to Mctapopulation Dynamics? 48 V. Metapopulations. Landscapes, and Conservation 58 VI. Conclusions 60

P ART

METAPOPULATION THEORY

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Metapopulation Dynamics: From Concepts and Observations to Predictive Models 1/kko Honski I. Introduction 69 II. An Example of Classical Metapopulation Dynamics with Rampant Population Turnover 70 Ill. Classical Metapopulation Dynamics: The Levins Model 73 IV. Minimum Viable Metapopulation Size 76 V. Predictive Models of Metapopulation Dynamics 78 VI. Nonequilibrium Metapopulations 85 VII. Four Conservation Messages 88 VIII. Concluding Remarks 91

5 Structured Metapopulation Modek Mots Gyl/enbelg, 1/kko Honski, and Alan Hastings I. Why Use Structured Models'? 93 II. Modeling Structured Metapopulations

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Contents III. IV. V. VI.

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Steady States and Metapopulation Extinction Examples 10 I Discussion 107 The Mathematical Fonnalism 108

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Two-Spec:ies Metopopulation Models Sean Nee, Robed M. May, ond Michael P. Hassell I. II. III. IV.

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Introduction 123 Two-Species Patch Models 127 Spatially Explicit Metapopulations Conclusion 146

137

From Metopopulotion Oynomics to Community Structure: Some Consequences of Spotiol Heterogeneity Robert 0. Holt I. Introduction 149 II. Effects of Landscape Heterogeneity on Local Community Composition 150 III. Metapopulation Dynamics of Food Chains 155 IV. Apparent Competition in Metacommunities 161 V. Conclusions 164

8 Genetic Effective Size of o Metopopulotion Philip W. Hedrick ond Michael E. Gilpin I. II. III. IV.

An Example 166 Methods 167 Results 171 Conclusions 179

9 The Evolution of Metopopulotions N. H. Bmton and Michoel ( Whitlock I. II. III. IV. V. VI.

Introduction 183 Neutml Variation in Metapopulations 185 Selection in Metapopulations 189 Adaptation to Local Conditions 193 Speciation and the "Shifting Balance" 200 Discussion 209

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Contents

METAPOPULATION PROCESSES

10 Extinction Models for Local Populations Patrick Foley I. Introduction 215 II. Local Populations Are Vulnerable to Extinction due to Demographic and Environmental Stochasticity 217 Ill. Environmental Stochasticity 220 IV. Models with Demographic and Environmental Stochasticity 228 V. Robustness of the Model 232 VI. Catastrophics and Genetic Impoverishment 235 VII. Implications for Metapopulation Dynamics 237 VIII. Conclusion 242

11 Studying Transfer Processes in Metapopulations: Emigration, Migration, and Colonization Rolf A. lms ond Nigel G. Yo((oz I. Introduction 247 II. Transfer of Individuals in Metapopulations: Defining the Components 248 III. Indirect, Global Approaches to Parameter Estimation 251 IV. Direct. Patch-Specific Approaches 254 V. Experimental Approaches to Studying Transfer Rates in Metapopulations 262 VI. Conclusion 264

12 Migration within Metapopulations: The Impact upon Local Population Dynamics Peter B. SIDcey, Veronica A Johnson, and Marl: L Toper I. Introduction 267 II. The Effect of Stochastic Variation in Migration Rate on Metapopulation Dynamics 271 Ill. Empirical Evidence 279 IV. Conclusions 289

Contents

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13 Evolution of Migrotion Rote ond Other Troits: The Metopopulotion Effect Isabelle Olivieri and Pie11e-Henri Gouyon I. Introduction 293 II. How Does a Metapopulation Persist in a Landscape? An Example of a Model with Local Disturbances and Successional Processes 295 Ill. Within-Population Selection versus Colonization Selection: An Insight into the Metapopulation Effect 298 IV. Does Selection Adjust Migration Rate at the Metapopulation Level'? 301 V. The Metapopulation Effect: Other Traits 311 VI. Influence of Migration on the Evolution of Life-History Traits 320 VII. Conclusion 321

14 Spotiol Processes in Host-Porosite Genetics Steven A. Fronk I. Introduction 325 II. Dimensionality and Colonization-Extinction Dynamics 326 III. Introduction to the Examples 332 IV. Plant-Pathogen Interactions 333 V. Cytoplasmic Male Sterility 339 VI. Other Systems of High Dimension 347 VII. Theories and Test'\ 350

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CASE STUDIES

15 Butterfly Metopopulotions Chris D. Thomas and 1/kka Honski I. Introduction 359 II. Consequences of Landscape Structure on Population Structure 360 Ill. Population Tumover in Butterflies 364 IV. Theoretical Predictions Tested 368

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Contents V. Adding Realism to the Metapopulation Concept VI. Conclusions 383

373

16 Tritrophic Meta population Dynamics: ACase Study of Ragwort, the Cinnabar Moth, and the Parasitoid (otesia popularis Ed van der Meifr/en and Cothorino A. M. van der Veen-wn Wiik I. II. Ill. IV. V.

Introduction 387 Marerials and Methods 389 Population Dynamics of Ragwort 391 Mechanisms of Persistence in the Ragwort 395 Population Dynamics of the Cinnabar Moth and the Parasitoid Cotesia popu/aris 396 VI. Mechanisms of Persistence in the Cinnabar Moth 401 VII. Discussion 402

17 Spatially Correlated Dynamics in a Pika Metapopulation Andrew T. Smith and Michael E. Gilpin I. II. Ill. IV. V. VI.

Introduction 407 Relevant Pika Natural History The Bodie Site 411 Methods 412 Results 415 Discussion 423

409

18 ACase Study of Genetic Structure in a Plant Metapopulation Barbaro E. Giles and Jerome Goudet I. II. Ill. IV. V.

Introduction 429 Theory 430 The Silene Metapopulation on Islands in the Baltic 434 Patterns of Genetic Structure on the Islands 439 Other Genetic Metapopulation Studies 449 Appendix: Infonnation About the Studied Islands 453

Bibliography Index

505

455

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

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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Year

FIGURE 1I Numbers Numbersof citations citations to the key key words words "island "islandbiogeography" biogeography"and "metapopulation" "metapopulation"in FIGURE the BIOSIS 970-1995, standardized BIOSIS data data base base in 11970-1995, standardized by by the respective respective total total number number of papers papers in the data base. base.

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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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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9

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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IIkka Hanski Hanski and and Daniel Daniel SJmberloff Simberloff Ilkka

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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1133

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

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IIkka Ilkka Hanski Hanski and and Daniel Daniel Simberloff Simberloff a

FIGURE FIGURE 22

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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,

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the the main main salvation salvation of o f small small sites sites was was the the shift shift by b y conservationists conservationists to to the the metameta­ population paradigm. paradigm. In In the the Levins Levins model, model, at at least, least, small small sites sites containing containing small small population populations were were the the only only homes homes of of aa species species and and thus thus the the proper proper locus locus of of concon­ populations servation concern. concern. The The model model even even suggests suggests that that aa certain certain number number of of unoccupied unoccupied servation required for metapopulation persistence persistence (Lande, (Lande, 1988a; 1 9 88a; Hanski, this volvol­ sites is required for metapopulation thus relieving relieving beleaguered beleaguered conservation biologists from having to to justify justify aa ume), thus conservation biologists from having refuge for for a given species by by confirmed confirmed residence. A A famous famous example example in which which refuge local extinction extinction rates high and and aa supply supply of of suitable suitable empty empty sites sites is necessary necessary local rates are high is Pedicularisfurbishae, Pedicularis furhishae, the Furbish Furbish lousewort lousewort (Menges, ( Menges, 1990). 1 990). In sum, sum, from from a conservation conservation standpoint, standpoint, it is not not surprising surprising that that citations citations of of metapopulation studies increase exactly when those of island biogeography de­ metapopulation studies increase exactly when those of island biogeography decline (Fig. (Fig. 1). 1 ) . These trends represent represent aa paradigm paradigm shift. shift. cline These trends

Misuse of the Metapopulation Concept Concept in Conservation Conservation Biology C. Use and Misuse Hanski and 1 99 1 ) observed have recently Hanski and Gilpin ((1991) observed that that "metapopulation "metapopulation ideas have become become the vogue in conservation biology," and and numerous numerous conservation strategies strategies are explicitly explicitly based based on metapopulation models (references metapopulation models (references in Harrison, Harrison, 1994b). 1 994b). The general effect been to draw draw attention to landscapes and networks, The effect has has been landscapes and networks, as opposed individual reserves reserves in isolation, isolation, for metaphor of of island opposed to individual for which the island metaphor biogeographic theory is appropriate. This is a salutary development. Even Even if if there biogeographic appropriate. This were no different refuges, no significant significant interactions interactions among populations populations in different refuges, it would would be good good to have have multiple refuges refuges simply as insurance insurance against against local catastrophes catastrophes (Soule and 1 986). Doak Doak and and Mills (1994) ( 1 994) and and Harrison Harrison (1994b) ( 1 994b) argue (Soul6 and Simberloff, Simberloff, 1986). argue not structured that, even if most species species are are not structured as Levins-type metapopulations in nature, the rise of of the metapopulation paradigm has served and continues continues to serve function by forcing conservation conservation biologists to gather gather data that are imim­ a useful function portant to effective conservation strategies of of individual speciesspecies--movement portant movement rates at different different sites, rates from site to site, relative reproduction reproduction and mortality rates and 1 982) for and the like. This is precisely the the view of of Haila and and Jarvinen J~irvinen ((1982) for island biogeographic biogeographic theory. theory. The problems arise, for for metapopulation models as for for island biogeographic theory, when it is assumed without empirical evidence that all species, or all species in a large class, conform to some particular model ((Doak Doak and Mills, 11994; 994; Harrison, 11994b). 994b). If a conservation conservation strategy is based based on metapopulation dynamics that do not exist, it can misfire. Thus, for example, Murphy et al. ((1990) 1 990) suggested that small-bodied, short-lived species with high reproductive rates and high hab­ habitat specificity typically constitute Levins metapopulations, but there there are are simply insuffi cient data to evaluate this claim ((Harrison, Harrison, 11991, 99 1 , 11994b). 994b). To focus auto­ insufficient automatically on metapopulation dynamics for such species would not constitute ef­ effective science. No wide-ranging generalizations are yet possible, because few data really demonstrate the existence of classical metapopulations. Harrison ((1991) 1 99 1 ) could cite only pool frogs (Rana lessoniae) in Sweden and waterfties waterflies (Daphnia) in rock pools as unequivocal cases (for some new examples, see Har-

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Ilkka Hanski HanskJand and Daniel DanielSimberloff Simberloff IIkka

rison and Taylor, this volume). The endangered endangered Glanville fritillary butterfly (Mel­ (Melitaea Hanski et al., 11995a,b) 995a,b) and itaea cinxia) cinxia) in Finland is another another good example ((Hanski may represent many other butterfly species ((Hanski Hanski and Thomas, 11994; 994; Hanski and Kuussaari, 11995; Thomas and Hanski, this volume). In such instances, an 995; understanding of metapopulation dynamics is crucial to effective conservation. understanding metapopulation Research is required required on local extinction and migration rates rates and how these are Research affected by patch size and isolation (c. (C. D. Thomas et al., al., 11993; et al., al., 993; Hanski et affected D. Thomas 11995b). 995b). Harrison ((1994b) l 994b) suggests that the the species most most convincingly conforming conforming to the Levins model occupy habitats that inevitably change because of of succession succession the (see also Thomas Thomas and and Hanski, Hanski, this volume). In many such species, the extinction populations is deterministic deterministic rather rather than stochastic, stochastic, but this fact fact does not of local populations conception of metapopulation metapopulation dynamics. The fundamentally undercut the Levins conception lousewort, for example, requires a riverside habitat that is endangered Furbish lousewort, neither too little nor too heavily disturbed disturbed (Menges, (Menges, 11990). However, any local 990). However, population is ultimately destroyed destroyed by ice scour and and bank bank slumping, slumping, so the metapopulation meta­ population requires a supply of of temporarily suitable suitable sites that are not not too isolated isolated population colonized. A metapopulation to be colonized. metapopulation analysis ((Menges, Menges, 11990) 990) including including observaobserva­ and recolonization suggests that the species is in decline decline tions on local extinction and rather than than at equilibrium equilibrium and and that that tempering of the disturbance disturbance (flow) regime rather tempering of exacerbate the situation. situation. Further, Further, in this species species as in the Levins scewill likely exacerbate Levins sce­ nario (Lande, 1988a; 1 988a; Hanski, this volume; Nee et al., this volume), volume), nario in general general (Lande, et al., restriction of of conservation conservation measures measures to occupied occupied sites only would would be fatal. A restriction of nature nature would not have have led to the recognition recognition static, nonmetapopulation nonmetapopulation view of of the importance importance of of currently currently unoccupied habitat. Much Much of of the history of of refuge refuge of unoccupied habitat. establishment simply of apparently healthy populations and preestablishment consists consists simply of locating locating apparently populations and pre­ serving their sites sites (Simberloff, (Simberloff, 1988). 1 988). Metapopulation Metapopulation models models have have been been used used to to deduce deduce the the minimum minimum viable viable metameta­ population population (MVM) ( MVM) size under under certain certain assumptions assumptions (Hanski ( Hanski et et al., al., 1996b). 1 996b). This This concept analogous to the concept is analogous the minimum minimum viable viable population population (MVP) ( MVP) size (Shaffer, (Shaffer, 1981), 1 98 1 ), but with the the critical difference difference that MVM MVM involves involves both the the minimum minimum viable number number of of populations of suitable suitable habitat habitat patches ( Han­ viable populations and and the availability availability of patches (Hanski et et al., al., 1996b). 1 996b). In practice, practice, use use of of these these concepts concepts may degenerate degenerate into specious specious "magic more constructive constructive approach approach is to use "magic numbers." numbers." A more use metapopulation metapopulation models models to rank scenarios of of landscape landscape change persistence of of a focal focal to rank alternative alternative scenarios change in terms terms of of persistence species. entire removal species. One One may may ask, ask, for for instance, instance, whether whether the the entire removal of of one one large large habitat habitat patch is more more detrimental detrimental to to aa metapopulation metapopulation than than reducing reducing the the areas areas of of several several patch patches patches (Hanski, (Hanski, 1994a,b; 1 994a,b; Hanski Hanski et et al., al., 1996c; 1 996c; Wahlberg Wahlberg et et al., al., 1996; 1 996; note note the the connection to to the the SLOSS SLOSS rule, Fig. 2b). The The theory theory of of island island biogeography biogeography inin­ connection spired spired the the rules rules of of refuge refuge design design discussed discussed above above (Fig. ( Fig. 2). The The analogous analogous contricontri­ bution from from metapopulation metapopulation theory theory is predictions predictions about about the the relative relative performance performance bution of particular particular species species in particular particular fragmented fragmented landscapes landscapes based based on on relatively simsim­ of ple ple but but spatially realistic realistic models models (Fig. ( Fig. 3, 3 , Hanski, Hanski, 1996b). 1 996b). There There are are two two reasons reasons to to expect expect the the latter latter sorts sorts of of predictions predictions to to be be more more helpful helpful than than the the island island biogebioge-

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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

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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.

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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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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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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

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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-

40 40

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)

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lation terms. tenns. We fi nd only a few find few examples of single species existing in a balance balance between the extinction and colonization colonization of populations populations and almost none none of of sys­ systems in which multiple species coexist through through tightly coupled coupled metapopulation metapopulation dynamics at comparable comparable spatial scales. Is this apparent apparent scarcity real? Clearly, the great difficulty of among-population processes of studying among-population processes in nature nature is aa major major ob­ obstacle to reaching reaching finn firm conclusions. conclusions. However, However, it may also be that systems in which migration too low, or which some populations migration among among patches patches is too high high or or too or in which some populations are much more more common approxi­ are highly persistent persistent (Fig. 2), are are truly much common than than ones ones approxistructure. mating the classical structure. If metapopulations sufficiently to include include the nonnon­ If we we broaden broaden our our view of of metapopulations classical how does perception of prevalence and imporimpor­ classical kinds, kinds, how does this affect affect our our perception of the prevalence tance of metapopulations? This This is clearly an an area area of of active work, work, as this volume volume tance of metapopulations? illustrates, but but we we will advance some of of our our own own speculations. speculations. Regional illustrates, advance some Regional persist­ persistence may may be be an overemphasized concern, concern, arising from models models that were based based ence an overemphasized arising from that were on a simple analogy and and that that overlooked overlooked variation patch size, on simple birth-and-death birth-and-death analogy variation in patch detailed local local population dynamics, explicit explicit spatial patches, and and detailed population dynamics, spatial relations relations among among patches, spatio-temporal correlation correlation in in the the environment. environment. Adding Adding such such real-world real-world refinerefine­ spatio-temporal ments to to models models may may have have the the general general effect effect of of reducing reducing the the relative relative importance importance ments of migration migration and and recolonization, recolonization, and and increasing increasing that that of of local population population processes, processes, of for regional regional persistence. persistence. If If this this is true, true, then then what what other other types of "metapopulation "metapopulation for types of effect" effect" may may we we seek seek in in nature? nature? When levels levels of of migration migration among among patches patches are are moderate, moderate, and and patches patches vary vary in in When their degree degree of of spatial spatial isolation, isolation, aa system system may may be be demographically demographically unified unified in in their central patches patches and and exhibit exhibit rescue rescue effects effects or or extinction extinction and and recolonization recolonization on on central increasingly marginal marginal ones. ones. The The defining defining feature feature of of such such aa metapopulation metapopulation is is not not increasingly the dependence dependence of of regional regional persistence persistence upon upon local local extinction extinction and and recolonization, recolonization, the but the the strong strong effect effect of of patch patch structure structure and and dispersal dispersal on on local local population population persistpersist­ but ence and/or and/or regional regional distribution. distribution. This This type type of of metapopulation metapopulation structure structure seems seems to to ence us us aa highly highly plausible plausible one. one. Consideration of of variation variation in in patch patch size size or or quality quality leads leads into into the the realm realm of of Consideration mainland - island and and source-sink source- sink dynamics, dynamics, where where again again the the appropriate appropriate quesquesmainland-island

22

Empirical EmpiricalEvidence Evidencefor Metapopulation MetapopulationDynamics Dynamics

41

tions are not about regional persistence, but about regional distribution, and about local persistence in habitats too small (islands) or poor in quality (sinks) to support Mainland-island are well documented, but long-lived populations. Mainland -island dynamics are source-sink source - sink dynamics are virtually untested; how frequently species are found in habitats where they are unable to replace themselves without immigration, and how much this affects their overall demography, remains a very open area for empirical research. The related refuge models of predation and competition, in which coexistence is made possible by the net dispersal of the victim species from habitats of low to those of high predation or competition (e.g., Hochberg and Holt, 11995), 995), also deserve more empirical study. another important area for for exploration is the For multispecies systems, another effect of trophic complexity. Both theory and empirical work have emphasized two or three tightly coupled species, but real food webs are nearly always more M. cinxia cinxia system comprises four important species on complex than this. The M. three three trophic trophic levels, plus some eight minor species of which some are gener­ generalists (G. Lei and Hanski, unpublished manuscript). manuscript). Four spider and three lizard Bahamanian islands (Schoener and species and their shared prey interact on Bahamanian sevSpiller, 11987a,b). 987a,b). Even relatively simple biocontrol systems typically include sev­ enemies (e.g., Quezada, Quezada, 1969; Murdoch et et at. al.,, 1996). eral important natural enemies 1 969; Murdoch 1 996). have yet to address systems of many species at multiple Metapopulation models have charactertrophic levels, each with different population dynamics and dispersal character­ istics, each coupled to other species to varying degrees (but see Holt, this volume). sitWhether spatial subdivision retains its potentially stabilizing effect in these sit­ other important important consequences, is an open area area for theoretical and uations, or has other empirical work. empirical In all of of these these extensions of of metapopulation metapopulation dynamics, an approach that comcom­ bines empirical general empirical work and modeling will be very helpful. helpful. For For example, example, no general guidelines are are available available for for empiricists empiricists to decide decide how much much migration migration is enough, too little or too much for metapopulation occur. Merely observing some metapopulation effects effects to occur. asynchrony in local population adequate, since this will be population fluctuations fluctuations is not adequate, shaped not only by migration, but by patterns patterns of of environmental variability and by the sampling regime. Moreover, the critical critical level of of migration will depend depend greatly on the exact hypothesis, or type of of metapopulation behavior, behavior, that is of of interest. Thus, combining combining field moderately detailed detailed system-specific interest. field work with moderately system-specific models models will be valuable valuable in many cases. In conclusion, this review review of of empirical empirical studies studies makes makes it clear clear that that a great great of spatial population structures exists exists in nature, nature, and and recognition of of this this variety of diversity suggests and theoretical suggests changes changes in how how both both empirical empirical and theoretical metapopumetapopu­ lation lation research research are are approached. approached. Whether Whether experimental experimental or or observational, singlesingle­ or or multispecies, empirical empirical studies studies need need to to take take fully into account account the the different different types of of metapopulation metapopulation structure structure that that are are possible, possible, perhaps perhaps treating them them as alternative alternative hypotheses to test. While empiricists empiricists attempt attempt to better better characterize characterize metapopulations for theorists metapopulations in nature, nature, an important important task for theorists is to to continue exploring exploring the ways ways in which which metapopulation metapopulation behavior behavior changes changes as as patch patch configuration, configuration, disdis­ persal, and local population dynamics are altered persal, and population dynamics altered in realistic realistic ways. Through Through this this

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SusanHarrison Harrisonand and Andrew AndrewD.D. Taylor Taylor Susan

combined c o m b i n e d effort, effort, we we will will be be much m u c h better better able able to to identify identify the the ways ways that that within­ withinand and between-population b e t w e e n - p o p u l a t i o n processes p r o c e s s e s interact interact to to determine d e t e r m i n e the the behavior b e h a v i o r of of natural natural systems. systems.

ACKNOWLEDGMENTS ACKNOWLEDGMENTS We We thank thank Ilkka Ilkka Hanski. Hanski, Daniel Daniel Simberloff, Simberloff, Chris Chris Thomas, and and an an anonymous reviewer reviewer for helpful comments comments on on an an earlier earlier draft. draft. helpful

II

Metapopulation Dynamics and Landscape LandscapeEcology Ecology John A. A. Wiens Wiens John

INTRODUCTION II.. INTRODUGION The fusion of metapopulation metapopulation studies studies and and landscape ecology should should make make for for an exciting exciting fusion of landscape ecology scientific and Gilpin, Gilpin, 1991) 1 99 1 ) scientific synthesis synthesis (( HHanski a n s k i and

The synthesis o metapopulation studies and landscape landscape ecology anticipated anticipated The off metapopulation Hanski and and Gilpin Gilpin has has barely barely yet yet begun. There There are are at least two reasons reasons for for this by by Hanski ( Wiens, 1995a). 1 995a). First, First, as many many of of the the chapters chapters in this this volume volume illustrate, illustrate, metameta­ (Wiens, population theory theory continues continues to to be be tied tied to to a view of of spatial spatial patterning patterning of of environenviron­ population ments in which which patches patches are are embedded embedded in in a featureless featureless background background matrix. matrix. Second, Second, ments landscape ecology seems seems still to to be be in the the process process of of defining defining what what it is about and landscape ecology about and describing complex complex spatial spatial patterns, patterns, but but it it has has not not developed developed much much theory theory to to deal deal describing with spatial spatial patterning. patterning. By By focusing focusing on on some some shared shared areas areas of of interest, interest, perhaps perhaps the the with synthesis synthesis of of these these disciplines disciplines can can be be accelerated. accelerated. In this this chapter, chapter, I consider consider the the relationship relationship between between the the emerging emerging (but (but yet yet In immature) discipline discipline of of landscape landscape ecology ecology and and the the emerged emerged (but (but perhaps perhaps adolesadoles­ immature) cent) discipline discipline of of metapopulation metapopulation dynamics. dynamics. I will will argue argue that that considerations considerations of of cent) metapopulation structure structure may may often often be be incomplete incomplete unless unless they they are are framed framed in in the the metapopulation context context of of the the underlying underlying landscape landscape mosaic. mosaic. Metapopulation Metapopulation Biology Biology

Copyright Copyright 9 © 1997 1997 by by Academic Academic Press, Press, Inc. Inc. All All rights rights of ofreproduction reproduction in in any any form form reserved. reserved.

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John A. John A. Wiens Wiens

II. II. APPROACHES APPROACHESTO TO PATCHINESS PATCHINESS Ecologists have always known known that nature nature is patchy and heterogeneous, heterogeneous, even Ecologists if much much of of their their theory has not not treated it so. Habitats Habitats in areas used by humans if humans fragments, and the patchwork occur as sharply defined defined blocks blocks or fragments, patchwork nature nature of of the landscape landscape mosaic mosaic is especially evident in such environments. environments. Even in more more natural natural settings, habitats are heterogeneous heterogeneous at virtually any scale of settings, however, however, habitats of resolution. resolution. Although indistinct gragra­ Although patch patch boundaries boundaries in such situations situations may sometimes sometimes be indistinct dients 992b), the spatially variable dients rather than sharp discontinuities discontinuities (Wiens, (Wiens, 11992b), variable character follow the convenconven­ character of of environments environments still remains. remains. In this chapter chapter I will follow tion that has become become widespread under widespread in ecology of of considering considering such variation variation under the rubric though "patches" "patches" are not always evident rubric of of "patchiness," "patchiness," even though evident in nature. nature. Dealing Dealing with such spatial heterogeneity heterogeneity has been been a major major challenge challenge in both empirical and theoretical ecology. Faced with the daunting daunting complexity of of spatial patterns patterns in the real world, world, field ecologists ecologists historically tended tended to focus focus on patterns and dynamics dynamics of of ecological systems within relatively homogeneous homogeneous habitat habitat types (e.g., watersheds, watersheds, woodlots) woodlots) or aggregated spatial spatial variation into into dimensionless dimensionless indices indices of of heterogeneity heterogeneity or dispersion. dispersion. More More recently, recently, it has become become fashionable fashionable to map map spatial patterns patterns at broad broad scales using geographic geographic information information systems systems and impor­ spatial statistics, statistics, but the link between between such technologies technologies and ecologically important questions questions is not always apparent. apparent. Spatial variance also strains the capacities of of analytical models models and theory if it is viewed viewed explicitly (i.e., by location) location) rather than averaged as "noise." "noise." As a consequence, consequence, many theoreticians theoreticians concerned concerned with heterogeneity have contented themselves themselves with simple models in which which spatial patterning is collapsed into patches Kareiva, 11990b; 990b; Wiens, 1995a). 1 995a). patches and an ecologically neutral neutral "matrix" "matrix" ((Kareiva, Such patchmatrix theory is usually spatially implicit ( Hanski, 1994c), 1 994c), in that patch-matrix implicit (Hanski, the locations 996a). The inin­ locations of of patches patches in the matrix are not specified (Wiens, (Wiens, 11996a). teresting in­ teresting dynamics occur in the patches, patches, which are usually considered considered to be internally homogeneous; homogeneous; the matrix is viewed viewed as inhibiting inhibiting interactions interactions among among predators). patches (e.g., migration, migration, colonization, colonization, gene flow, prey discovery by predators). Traditional -matrix Traditional metapopulation metapopulation theory is an elaboration elaboration on this patch patch-matrix theme. 1 970; Hanski, theme. Levins' Levins' metapopulation model ((1970; Hanski, this volume) volume) considered the patches habitat of of a population population to be subdivided subdivided into an infinite number of of similar similar patches undefined locations mod­ occupying undefined locations in a background background matrix. As metapopulation metapopulation modeling has progressed, progressed, however, details about patch sizes, patch clumping, indi­ individual movement movement capacities, capacities, local patch dynamics, and explicit patch locations locations have been Hanski, 11994a,c; 994a,c; see Hanski, been incorporated incorporated ((Hanski, Hanski, this volume; Gyllenberg Gyllenberg et et al. al.,, this volume). volume). Most Most patch theory deals deals with with the the dynamics dynamics of of populations populations occupying occupying a patchy environment (Wiens, 976; Levin, 11976; 976; Kareiva, 11990b; 990b; Shorrocks Swing­ environment (Wiens, 11976; Shorrocks and Swingland, 990). Another heterogeneity has focused land, 11990). Another approach approach to heterogeneity focused on the the dynamics dynamics of of the patches patches themselves. themselves. Although Although the spatial pattern pattern of of some patches, patches, such as the islands islands considered considered in island biogeography biogeography theory, may be relatively static in ec-

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Metopopulation Dynamics and and Landscape MetapopulationDynamics LandscapeEcology Ecology

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

46 46

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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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metapopulation metapopulation dynamics dynamics has been been done, done, a discussion discussion of of how how the major major themes themes of of landscape landscape ecology m spatial and temporal temporal variations variations in patch patch quality, boundary boundary effects, landscape connectivity, patch context-affect connectivity, and and patch context--affect the three components components of local extinction, of metapopulation metapopulation dynamics dynamics ((local extinction, interpatch interpatch movement, movement, and recolo­ recolonization) nization) must must necessarily be somewhat somewhat abstract abstract and and conceptual. conceptual. It may be useful, therefore, therefore, to preface preface this discussion discussion with a few few examples of of the effects effects of of land­ landscape structure provided by Angelstam Angelstam structure in the real world. Additional Additional examples examples are are provided ((1992), 1 992), Fahrig and Freemark 1 993), and Hobbs ((1995). 1 995). Freemark ((1993), and Hobbs

A. Some Some Examples Examples of landscape LandscapeEffects Effects Some Some of of the effects effects of of landscape landscape structure are related related to patch characteristics characteristics such as patch patch size or spacing. spacing. For For example, the size of of habitat habitat patches patches has been been related Verboom et related to the persistence persistence of of local populations populations of of forest forest birds birds ((Verboom et al. al.,, 11991a; 99 1 a; Villard 992), and the degree habitat patches patches has been Villard et et al., al., 11992), degree of of spacing of of habitat has been shown shown to affect affect the likelihood likelihood of of recolonization recolonization of of vacant vacant patches patches by the Glanville Glanville fritillary (Melitaea Hanski et 995a). Both Both patch (Melitaea cinxia) cinxia) in Finland ((Hanski et al. al.,, 11995a). patch size and remnant forest spacing spacing influenced influenced the use by brown brown kiwis (Apteryx (Apteryx australis) australis) of of remnant forest fragments New Zealand Zealand (Potter, 990). Kiwis fragments in an agricultural matrix in New (Potter, 11990). Kiwis are flightless, isolated remnants. remnants. All fragments flightless, so they must must walk between between isolated fragments less than regardless of 80 m from from other other forest forest remnants remnants were used by the birds, regardless of their size. Movements Movements of of more more than than a kilometer kilometer from the reserve, reserve, however, however, were were accom­ accom"stepping stones." situation, the spatial plished by using small fragments fragments as "stepping stones." In this situation, interspersion of of habitat habitat patches patches was a critical factor factor in determining determining the effects effects of of patch isolation and, consequently, consequently, the potential for for metapopulation metapopulation dynamics. Patch guration may also be important. important. The emigration Patch edges edges and and their their confi configuration emigration of from patches of GIanviIIe Glanville fritillaries from patches of of suitable suitable habitat, habitat, for for example, increases increases with Kuus­ with the proportion proportion of of the patch patch boundary boundary that is bordered bordered by open open fields ((Kuussaari et 996). Gates 1 978) found passerine et al. al.,, 11996). Gates and and Gysel ((1978) found that that the abundance abundance of of passerine birds elds and forests, birds increased increased at the boundary boundary between between fi fields forests, and they suggested suggested that that individuals individuals might be drawn drawn to the edge edge as nesting nesting habitat habitat because because of of greater greater food 992; Andren, 992, food availability there. Numerous Numerous studies (e.g., Angelstam, Angelstam, 11992; Andr6n, 11992, 11995), 995), however, however, have have documented documented that predation predation rates may may be greater at such ecotones, ecotones, presumably presumably due to predators predators living in adjacent adjacent areas. For For some some species, species, edges individuals to areas in edges may function function as an "ecological "ecological trap" trap" by attracting attracting individuals which 978). Predation hab­ which predation predation losses are great (Gates and and Gysel, 11978). Predation risks at habitat edges 985; Angelstam, edges vary as a function function of of the surroundings (Wi1cove, (Wilcove, 11985; Angelstam, 11992; 992; Wiens, 11995b), 995b), so the landscape Pear­ landscape context context of of patches patches is also important. important. Pear' s ((1993) son 1 993) work on habitat occupancy also son's occupancy by birds in the Georgia Piedmont also illustrates illustrates the effects effects of of landscape landscape context. context. There, There, the composition composition of of the sur­ surrounding 74% of of the variance rounding matrix matrix explained explained as much much as 74% variance in habitat habitat occupancy occupancy by some but was unimportant unimportant for some species but for other other species. The The demographic demographic con­ consequences of sequences of such edge- and context-related context-related effects have have received received very little at­ attention, tention, but but they may have have important important effects effects on metapopulation metapopulation dynamics, es-

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pecially where where populations populations are subdivided subdivided among among many small habitat habitat patches patches and cant. predation risk is signifi significant. The effects landscape mosaic have effects of of corridors corridors linking linking elements in a landscape have also been eld studies been documented documented by by fi field studies (although (although not not to the degree degree that that the the widespread widespread adoption management option adoption of of corridors corridors as a management option would lead one to believe; Bennett, Bennett, ' s cock­ 11990, 990, 11991; 99 1 ; Hobbs, 992). In Western Hobbs, 11992). Western Australia, Australia, for for example, example, Carnaby Carnaby's cockfor­ atoos (Calyptorhynchus (Calyptorhynchus funereus) funereus) use roadside roadside vegetation vegetation as a pathway pathway for for foraging movements movements among among woodland woodland patches patches in their their large home home ranges ranges (Saunders, (Saunders, 11990). 990). Where linked or are Where woodland woodland patches patches are not linked are not visually apparent apparent to the cockatoos, they are not used, even though though food food may be available available there. On the other other hand, hand, singing honeyeaters honeyeaters (Lichenostomus (Lichenostomus virescens), virescens), which are habitat habitat generalists, y across farmland Merriam and Saun­ generalists, readily fl fly farmland with little vegetation ((Merriam Saunders, 11993) 993) and 1 984) found and apparently make make little use of of corridors. Osborne ((1984) found that richness in an area that hedgerow hedgerow area area was the best predictor predictor of of bird bird species species richness area of of Great Great Britain, and the presence presence of of red squirrels squirrels (Sciuris (Sciuris vulgaris) vulgaris) in wooded wooded fragments fragments in The Netherlands Netherlands was positively related related to the the amount amount of of hedgerow hedgerow surrounding Verboom and van Apeldoorn, 1990). 1 990). In Australia, surrounding the fragments fragments ((Verboom van Apeldoorn, the occupancy occupancy of of corridors corridors by arboreal arboreal marsupials could not be predicted predicted by habitat habitat features features within the corridor corridor but required required additional information information on the composition Lindenmayer and 1 993). composition of of the surrounding surrounding landscape landscape ((Lindenmayer and Nix, 1993).

B. B. Movement Movement and and Migration Migration Individual meta­ Individual movement movement is the most most important important unifying unifying element element in both both metapopUlation 1 99 1 ; Wiens, 1 992b, population dynamics dynamics and landscape ecology (Saunders (Saunders et al. al.,, 1991; Wiens, 1992b, 11995a; 995a; Wiens 993; Ims, 11995). 995). Moreover, Wiens et al. al.,, 11993; Moreover, how fast fast and and how how far organisms organisms move het­ move imposes imposes a scale on the the environment: environment: highly highly vagile animals animals integrate integrate heterogeneity over therefore perceive over broader broader scales than do sessile sessile individuals individuals and therefore the environment lter or "grain" Wiens, 11985; 985; Fahrig environment with a coarser coarser fi filter "grain" ((Wiens, Fahrig and and Palo­ Paloheimo, 988; Kotliar and Wiens, 990; De 99 1 ; With, 994). At the heimo, 11988; Wiens, 11990; De Roos Roos et aI., al., 11991; With, 11994). outset of of any field study or modeling modeling exercise, then, the mean mean and shape of of a species migration function responses species'' migration function determine determine the the scale(s) at which population responses to environmental environmental patchiness patchiness must must be investigated. investigated. In the tradition most metapopulation tradition of of island island biogeography biogeography theory, most metapopulation models models use interpatch migration rates major determinants patch­ interpatch distance distance and and migration rates as the major determinants of of patch1 994a). The 1 988) colonization probabilities (e.g., Hanski, 1994a). The Fahrig and and Paloheimo Paloheimo ((1988) simulation guration of popu­ simulation studies studies of of the effects effects of of the the spatial confi configuration of patches patches on population indicated that migration lation abundances abundances in a metapopulation, metapopulation, for for example, indicated migration distance, distance, rather rather than migration migration rates alone alone (or demographic demographic features features such as birth birth rate), was critically important, when interpatch important, especially especially when interpatch distances distances were great. 1 995) modeled Bachman's Bachman ' s sparrow In contrast, contrast, when Liu et al., ((1995) sparrow (Aimophila (Aimophila aestivalis) aestivalis) population population dynamics, dynamics, they found found that demographic demographic parameters parameters were more important important than mortality during during dispersal dispersal (although (although not not necessarily necessarily dispersal dispersal

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John JohnA. A. Wiens Wiens

rate rate or distance). These These differences differences may stem from differences differences in model structure, structure, but they may also refl ect basic reflect basic differences differences in the life histories of of the organisms organisms modeled. Traditional Traditional metapopulation metapopulation models usually usually do do not consider consider the the details details of of movement movement in in even even an an abstract abstract sense. sense. Movement Movement is is modeled modeled as as transition proba­ probabilities among cells in a grid ((Liu Liu et ai., 1995) 1 995) or et al., or movement rates rates and distances distances are ed or are distributions. Whether move­ are simply specifi specified are drawn drawn from frequency distributions. Whether movement ment through through the matrix matrix between between patches patches is directional directional (e.g., Fig. 2A) or follows aa diffusion, 980; diffusion, correlated correlated random random walk, walk, or or some some other other algorithm algorithm (e.g., Okubo, Okubo, 11980; Turchin, 1 989; Johnson et ai., 1 992a) is not considered, even though the differ­ Turchin, 1989; Johnson et al., 1992a) is not considered, even though the differences ences among among these these movement movement patterns patterns can can produce produce substantial substantial differences differences in in the the mi­ probability of of encountering encountering a patch in the matrix. This is especially true if if migration gration rates are low or if the number number of of individuals individuals available available to migrate migrate is quite limited (as may occur when local populations populations are are small). Movement Movement patterns patterns such such as as diffusion diffusion or or random random walks walks are are handy handy modeling modeling devices that that may have some relevance relevance to how real real organisms organisms move through through a featureless featureless matrix, but they are of of limited value (other than as neutral neutral models) in specifying specifying how individuals might respond respond to a complex landscape landscape mosaic (e.g., Fig. 2B). Conceptually, the movements movements of of individuals individuals through a landscape landscape may may be viewed as a consequence consequence of of their movements movements within individual patches patches and and 993). Within-patch their movements between patches Fig. 3A; Wiens patches ((Fig. Wiens et et ai. al.,, 11993). Within-patch movement patterns patterns vary vary among among different different patch patch types. The The probability that that an an ed time interval is a individual will encounter encounter a patch patch boundary boundary during during a specifi specified function function of of these patch-specific patch-specific movements and of of patch size and shape shape (perimeter Whether or not an individual will cross a patch (perimeter:: area area ratio). Whether patch boundary boundary upon bound­ upon encountering encountering it is a function both of of features features of of the boundary boundary itself ((boundet al., and of of the characteristics characteristics ai., 1987; 1 987; Wiens, 11992b) 992b) and ary "permeability"; Stamps et of of the adjoining adjoining patch patch (patch context). [This is where where another another behavior, behavior, patch patch or habitat habitat choice, choice, becomes important.] important.] Both Both costs (e.g., (e.g., predation predation risk, risk, physiological physiological stress) and benefi ts (e.g., shelter, benefits shelter, food availability, mating mating opportunities) may differ among among elements elements in a mosaic, and and movement patterns patterns within and and between patches ect these relative costs and benefits (i.e., patch quality), at least patches may refl reflect 993; Wiens, 11996a). 996a). Some simulation models of of metapopmetapop­ in part ((Wiens Wiens et et ai., al., 11993; 1 992; Adler ulation migration migration in in patchy patchy environments environments (e.g., (e.g., Pulliam et et ai., al., 1992; Adler and and Nuernberger, 994) vary migration costs incorporate Nuernberger, 11994) costs as a function of of distance distance or incorporate differences differences in in patch patch quality. quality. To make such an individual-based conceptualization movements conceptualization of of mosaic movements relevant to metapopulation popu­ metapopulation dynamics, it must be extended extended to the scale of of population Fig. 3B). matter lation rather rather than than individual individual patches patches ((Fig. 3B). In In simple simple terms, terms, this this is is aa matter of of movements and patches of shifting shifting the scale from that of patches defined by individual home home ranges ranges to to the the broader-scale broader-scale movements movements of of populations populations (i.e., (i.e., migration) migration) and and the the scale scale of of patchiness patchiness represented represented by by interactions interactions within within aa local local population population (i.e., (i.e., nodes nodes in a metapopulation). Exactly how how the translation translation from individual move­ movements ments to to population population distribution distribution and and interactions interactions should should be be accomplished accomplished is is one one

33

A A

(

Individual Individual

Metapopulation MetapopulationDynamics Dynamicsand ond Landscape LandscapeEcology Ecology

53 53

B ' Population Population B

p

FIGURE33 (A) (A) Patterns Patterns of movement movement of an individual individual among among elements elements of a landscape landscape in its home FIGURE range. The movement movement pathway pathway consists consists of within-patch within-patch and between-patch between-patch components; components; both both may range. affected by the characteristics characteristics of patches patches and by their their spatial spatial configuration be affected configuration ((patch patch context). context). (B) Extension of individual individual movements movements to the the population population level. level. A local local population population may may occupy occupy patch patch i,i, Extension which individuals individuals move according according to the local local habitat heterogeneity heterogeneity within within that patch. These within which movements (characterized probability that individuals individuals will will encounter movements (characterized by a function, function, IJ,) 0;) determine the probability boundary between between patch i and patchj patch j during a given given time time interval. interval. The probability that individuals the boundary encountering the boundary will cross into into patchj, j, ' ~b0, boundary 1 ' is a function of the permeability permeability of the boundary encountering (e.g., patch patch choice). choice). Within and the behavior of the organisms organisms (e.g., Within patch j, a proportion of the dispersing individuals ((l-p) residency in the patch. Movements (0j) determine individuals l -p) may die or establish establish residency Movements within within patch patchjj (IJ ) determine probability that that the boundary boundary between between patch jj and another element the probability element in the landscape landscape (patch k) will encountered; lk ~bjkdetermines p, p, the proportion of dispersers dispersers from from patch i that move move into patch be encountered; patch k. 0 and ~bare patch-specifi patch-specific which may have density-dependent density-dependent effects Values of IJ c (as is patch density, which on movement movement and migration). migration). Developed 1 993). Developed from Wiens Wiens et al. ((1993).

of the most vexing vexing problems problems confronting confronting aa metapopulation-Iandscape of the most metapopulation-landscape synthesis. synthesis. It more general general problem problem of of translating across scales It is is part part of of the the more translating across scales in in ecology ecology (Wiens, et al., al., 1992). 1 992). (Wiens, 1989a; 1 989a; King, King, 1991; 1 99 1 ; Rastetter Rastetter et My colleagues and and II have have used used systems systems of small animals My colleagues of small animals (insects) (insects) moving moving through grassland "microlandscapes" as through grassland "microlandscapes" as experimental experimental model model systems systems (Ims ( lms et et al., al. , 1 993; Ims, Ims, 1995) 1 995) to to investigate investigate how how movements movements are affected by by mosaic mosaic structure, structure, 1993; are affected following following the the framework framework of of the the model model of of Wiens Wiens et et al. al. (1993). ( 1 993). Initial Initial studies studies of of tenebrionid beetles E l e o d e s spp.) that individuals o v e d differently tenebrionid beetles ((Eleodes spp.) indicated indicated that individuals m moved differently in in microlandscapes a few square meters differed in microlandscapes of of a few square meters that that differed in internal internal heterogeneity, heterogeneity, as as measured measured by by the the fractal fractal dimension dimension of of the the landscape landscape pattern pattern (Wiens (Wiens and and Milne, Milne, 1989). o v e m e n t alternated 1 989). M Movement alternated between between matching matching the the predictions predictions of of an an ordinary ordinary diffusion ovement diffusion model model and and those those of of anomalous anomalous diffusion diffusion depending depending on on m movement "rules," "rules," landscape landscape pattern, pattern, and and spatial spatial and and temporal temporal scales scales (Johnson (Johnson et et al., at. , 1992a). 1 992a). In particular, particular, diffusion diffusion exponents exponents changed changed significantly significantly at at spatial spatial scales scales correcorre­ In sponding 42 cm), cm), suggesting suggesting that that sponding to to the the size size of of vegetation vegetation patches patches (a (a radius radius of of ~= 42 the effects effects of of spatial spatial heterogeneity heterogeneity on on beetle beetle movements movements at at finer finer scales scales differed differed the fundamentally et al., al. , 1992) 1 992) demdem­ fundamentally from from those those at at broader broader scales. scales. Other Other work work (Crist (Crist et onstrated that structure within onstrated that variations variations in in vegetation vegetation structure within 25 25 m: m2 areas areas had had significant significant

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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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boundary another will be al­ boundary itself, the likelihood of of moving from one patch patch to another altered. particular characteristics of tered. If If patch patch context context is important, then then the particular of what what is beyond beyond a given patch boundary boundary will further further modify transition transition probabilities. Landscape Landscape controls controls over over movement patterns have have yet to receive receive detailed at­ attention in either field studies. Moreover, Moreover, all of either models models or field of these approaches approaches con­ consider the structure structure of of the landscape landscape mosaic to be fixed; patch dynamics in time would add another another level of of realism (and further further computational complications) to the research program. One aspect of of landscape structure that is implicit in the spatial arrangement of of mosaic elements and the transition probabilities probabilities among among them is connectivity. Landscape Landscape connectivity refers to the degree degree to which the landscape facilitates or 993). Corridors impedes movement among patches patches (Taylor (Taylor et et al. al.,, 11993). Corridors of of similar habitat 990; habitat linked together together are are thought thought to enhance enhance connectivity (Bennett, (Bennett, 11990; Hobbs, 1992), patches among among which transition probabilities Hobbs, 1 992), but dissimilar habitat patches probabilities are Through the patterns are high may also result result in high connectivity. Through patterns of of connectivity that characterize a landscape, landscape, movement pathways pathways are directed in spatially non­ nonrandom random manners manners (Fig. 2B), which can either increase increase or or decrease decrease the likelihood that movement among specifi specificc patches in the landscape landscape (e.g., subpopulations subpopulations in a metapopulation) metapopulation) will occur. Connectivity is related related to the coverage coverage of of a given habitat habitat type in the land­ landIf a continuous scape, but the relationship is strongly nonlinear. If continuous habitat is broken broken into fragments by habitat conversion, conversion, the initial effects are due primarily to the below some threshold threshold value, loss of of habitat coverage alone. As coverage coverage drops drops below however, landscapes however, the effects of of patch patch isolation begin to be more more important. In landscapes with a low proportion further decreases proportion of of suitable habitat, further decreases in coverage coverage result in a rapidly increasing distance between between habitat habitat patches patches and even even greater isolation effects (Fig. 4). For example, Andren 1 994) found that habitat loss was a good Andr6n ((1994) predictor landscapes with predictor of of fragmentation effects on birds birds and mammals in landscapes > > 30% coverage coverage of of suitable habitats, but in more highly fragmented fragmented landscapes the effects of of patch isolation and size also became important. threshold effects also emerge in simulation studies based based on percolation Such threshold theory. In simple percolation percolation models, a landscape landscape mosaic is divided divided into suitable and unsuitable habitat patches (cells) that are distributed distributed over the landscape at random, random, with a specified coverage or proportion, proportion, p, p, of of the suitable patches (Gard­ (Gard987, 11989). 989). Above Above some ner ner et et al. al.,, 11987, some critical critical threshold, threshold, Peril' Pcrit, cells cells of of the the suitable suitable habitat An or­ habitat are likely to form a continuous continuous cluster that that spans the landscape. An organism in this "percolating cluster" will be able to move or "percolate" "percolate" across ' Neill et 1 988). For the landscape; landscape; connectivity is high (O (O'Neill et ai. al.,, 1988). For a random landscape has aa in Perit has in which which organisms organisms move only to to adjacent adjacent (but not not diagonal) cells, cells, Pcrit nomandom algo­ value of of 0.5928. 0.5928. If If the landscape pattern is generated generated using a nonrandom algo993; With et press), the value rithm (e.g., fractal curdling; Lavorel et et ai. al.,, 11993; et ai. al.,, in press), in the the of - 0.50). Similar in Pcrit Perit occur of P Pc,.it is lower lower (0.29 (0.29-0.50). Similar reductions reductions in occur with with changes changes in crit is movement patterns patterns to allow individuals individuals to move to any adjacent cell or to cross cross

S6 56

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0% 0% Proportion of of Suitable Suitable Habitat Habitat Proportion FIGURE 44 A hypothesized hypothesized relationship between the the proportion of suitable habitat in a landscape FIGURE importance of habitat loss and patch isolation to individual individual movement or population and the relative importance availability of suitable habitat decreases, decreases. the importance of habitat loss increases dynamics. As the availability monotonically. The effects effects of patch isolation (the (the inverse of landscape connectivity) connectivity) are relatively monotonically. suitable habitat is high, but increase sharply when a connectivity threshold slight when coverage coverage of suitable percolation theory parlance). Increases Increases in individual individual vagility vagility will will move this threshold is passed (P,.,.i, (Pent in percolation values of the suitable habitat. to lower coverage values

gaps where where suitable suitable cells cells are are not not immediately adjacent ((Dale et al. al.,, 1994; Pearson gaps immediately adjacent Dale et 1 994; Pearson et 996). Field beetles moving moving through et al. al.,, 11996). Field experiments experiments with with Eleodes Eleodes beetles through random random landscapes landscapes (Wiens (Wiens et et al., al., in in press) press) indicated indicated aa threshold threshold change change in in movement movement patterns patterns when when coverage coverage of of grass grass in in aa bare-ground bare-ground matrix matrix increased increased from from 0 0 to to 20%. 20%. Changes Changes in in either either the the spatial spatial pattern pattern of of the the landscape landscape or or the the scale scale over over which which individual individual organisms organisms "perceive" "perceive" landscape landscape patterns patterns (as (as judged judged by by their their move­ movements) ments) can can therefore therefore produce produce high high connectivity connectivity in in aa mosaic mosaic even even when when the the favored favored habitat habitat type type occupies occupies aa relatively relatively small small proportion proportion of of the the landscape. landscape. Differences Differences in in vagility vagility among among organisms organisms (e.g., (e.g., the the grasshoppers grasshoppers studied studied by by With With and and Crist, Crist, 11995) 995) may may also also affect affect the the location location of of aa percolation percolation threshold threshold (Fig. (Fig. 4), 4), as as Fahrig Fahrig and 1 988) also and Paloheimo Paloheimo ((1988) also suggested suggested in in aa somewhat somewhat different different context. context. Details Details of of the the spatial spatial arrangement arrangement of of habitat habitat patches patches in in the the mosaic, mosaic, such such as as those those modeled modeled by 1 985) or 1 994), are by Lefkovitch Lefkovitch and and Fahrig Fahrig ((1985) or Adler Adler and and Nuernberger Nuernberger ((1994), are likely likely to to become become important important only only around around or or below below this this threshold. threshold. Most Most models models that that link link animal animal movements movements to to landscape landscape structure structure assume assume that that movement xed species movement parameters parameters are are fi fixed species traits traits and and that that migration migration can can adequately adequately be be represented represented using using average average values. values. Individuals Individuals do do vary vary in in movement movement charac­ characteristics, teristics, of of course, course, and and the the effects effects of of this this variation variation may may be be profound. profound. For For example, example, 1 994) found Lens Lens and and Dhondt Dhondt ((1994) found that that crested crested tit tit (Parus (Parus cristatus) cristatus) young young dispersed dispersed 11 week week later later from from small, small, isolated isolated pine pine stands stands than than did did those those in in large large pine pine forests. forests.

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Metapopulation and Landscape landscape Ecology MetapopulationDynamics Dynamicsand Ecology

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Chicks Chicks from second second broods were were also more more likely to disperse disperse into less suitable habitat fragments than were young from rst broods. Collectively, these move­ from fi first movement ment characteristics characteristics reduced reduced the probability that second-brood young would be integrated into winter flocks, which would affect affect their overwinter survival prob­ prob( 1 994) sug­ abilities. In another vein, the simulation studies of of Goldwasser et et al. al. (1994) suggested that that variability among among individuals could markedly increase the rate of of spread spread of of a population, even if if only a few few individuals in the population population migrated rapidly. The prospect that individual movement behavior may be facultatively adjusted to landscape landscape patterns patterns such as the interspersion or isolation of of suitable suitable 995; Fahrig and Merriam, 1 994) may further habitat patches ((Matthysen Matthysen et et al. al.,, 11995; Merriam, 1994) further complicate attempts to model migration dynamics in heterogeneous heterogeneous landscapes. Nonetheless, it is apparent between fine-scale movemove­ apparent that the complex interplay interplay between ment patterns, broad-scale nonlinear effects of broad-scale migration dynamics, and the nonlinear of land­ landscape-mosaic structure may have fundamentally important effects on the inter­ interpatch movements that lie at the heart of of metapopulation dynamics.

C. C. Locol Local Extinction Extinction and Recolonization Recolonization In addition addition to interpatch interpatch movement, the extinction of local populations populations in habitat patches and the subsequent recolonization of of those patches are are what drive metapopulation dynamics. Local popUlation population extinctions are often associated associated with the stochastic stochastic dynamics that that characterize characterize small populations. Deterministic Deterministic local habitat changes, however, can produce produce patch patch dynamics in the landscape landscape that that also result in the extinction of 994c). If of local populations (Thomas, 11994c). If this is the case, the local patch environment may remain unsuitable unsuitable for for some time after extinction occurs. Under these conditions, the persistence of the metapopulation depends Under of depends on how well the organisms can track the shifting spatial locations of of suitable habitat unpredictable in time patches. Because Because the location of suitable patches may be unpredictable as well as in space, how organisms move through the landscape mosaic and the scales on which they perceive environmental patchiness become all the more important. The pattern of interspersion of of suitable habitat patches through a landscape mosaic also influences extinction and colonization probabilities. The degree to which a patch is connected connected to other suitable areas or is isolated may have little direct effect on extinction, uence the immigration flow and extinction, although it may infl influence therefore determine Brown and Kodric­ determine the magnitude magnitude of the "rescue "rescue effect" effect" ((Brown KodricBrown, 11977). 977). Colonization, on the other hand, is clearly related related to the interplay between individual migration abilities and both both the distribution (i.e., isolation) and the connectivity of habitats in the landscape. landscape. If If fragmentation alters the land­ landscape so that the interspersion of of habitat patches patches no longer coincides with the migration patterns patterns of of a species, metapopulation dynamics may be disrupted. To some degree, this situation characterizes Hanski characterizes the Glanville fritillary in Finland ((Hanski et al. al.,, 11995a). et 995a).

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JohnA.A. Wiens Wiens John

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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59

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.

60 60

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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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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.

P

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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.

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

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FIGURE land islands in southwestern habitat FIGURE 1! Map of of A ]kland southwestem Finland, showing the locations of of the habitat patches Melitaea cinxia cinxia (dots). Patches Patches that that were patches (dry meadows) suitable for for the Glanville fritillary Melitaea occupied 00 km2 occupied in late summer summer 1993 are shown by black dots. The size of the grid is 1100 km 2 (modified from 995a). from Hanski Hanski et et al al.,.. 11995a). mean, median, and maximum distances moved by migrating butterflies among habitat habitat patches in one 50-patch network were 590, 330, and 3050 m respectively 994). ((Hanski Hanski et et at. al.,, 11994).

Condition Condition44 Local populations do do not have completely synchronous dynamics. If they have, the the metapopulation metapopulation would would not persist persist for for much much longer than than the the local local pop­ population with the M. cinxia, cinxia, we we have have demonstrated demonstrated the smallest risk risk of of extinction. extinction. In In M. substantial asynchrony asynchrony in in the the dynamics of of populations within within an an area area of of 55 by by 55 km2 Hanski et 995a). The et al., al., 11995a). The most most recent recent results results suggest dynamics dynamics that that may may km 2 ((Hanski be be somewhat somewhat correlated correlated across across areas areas up up to to some some tens tens of of square square kilometers, kilometers, but but at at the the scale scale of of the the entire entire metapopulation metapopulation changes changes in in population population size size occur occur in in opposite opposite directions Fig. 2). directions ((Fig. 2). The The question question about about spatial spatial synchrony synchrony and and its its causes causes is is aa com­ complex plex one one (Thomas (Thomas and and Hanski, Hanski, this this volume), volume), but but the the point point which which II wish wish to to make make here here is is that that in in our our butterfly butterfly metapopulation metapopulation there there is is certainly certainly enough enough asynchrony asynchrony to to make make simultaneous simultaneous extinction extinction of of all all local local populations populations aa very very unlikely unlikely event event under under the the prevailing prevailing environmental environmental conditions. conditions.

44 Metapopulation Metopopuiotion Dynamics Dynamics

73 73

Aland islands from 11993 993 till FIGURE FIGURE22 Observed Observedchanges changes in the population population sizes sizes of M. M. cinxia cinxia on Aland islands from 11994. 994. The study study area area was divided divided into 2 by 2 km2 km2squares squares for the purpose purpose of this this analysis. analysis.The symbol symbol indicates indicates the the sign sign and the magnitude magnitudeof the change change in the number number of larval larval groups groups per 4 km2 km2 square square between between the 22 years years (stippled (stippled triangles, triangles, decrease; decrease; black black triangles, triangles, increase; increase; logarithmic logarithmic scale) scale) (data (data from from I.I. Hanski, Hanski, J. Pogry, P6gry, and T. Pakkala, Pakkala, unpublished). unpublished).

III. III. CLASSICAL CLASSICALMETAPOPULATION METAPOPULATIONDYNAMICS: DYNAMICS:THE THE lEVINS LEVINSMODEL MODEL The purpose and and "validity" of simple simple models in population ecology is often The misunderstood. misunderstood. Their Their purpose purpose is is not not to to replicate replicate in in the the model model as as many many details details of of real real populations populations as as possible. possible. Models Models which which do do that that are are not not simple simple and and their their pur­ purpose ). The pose is is different different (Section (Section V V). The purpose purpose of of simple simple models models is is to to isolate, isolate, for for aa theoretical theoretical study, study, some some feature feature of of real real populations populations that that happens happens to to be be of of interest. interest. A A simple simple model model is is not not invalid invalid just just because because all all known known real real examples examples deviate deviate in in some some respect respect from from model model assumptions; assumptions; these these differences differences may may be be immaterial immaterial for for the the purpose purpose that that the the model model was was constructed. constructed. A A simple simple model model is is defective defective if if itit fails fails to to incorporate incorporate the the critical critical variables variables and and processes processes affecting affecting the the phenomenon phenomenon under under scrutiny scrutiny or or if if itit makes makes some some critically critically unrealistic unrealistic assumptions. assumptions. 1 969a, 11970), 970), the In In this this spirit, spirit, II suggest suggest that that the the well-known well-known Levins Levins model model ((1969a, the mother mother of of all all metapopulation metapopulation models models with with population population turnover, turnover, provides provides aa valu­ valuable able theoretical theoretical framework framework for for studying studying systems systems such such as as shown shown in in Fig. Fig. 11 and and satisfying satisfying the the four four conditions conditions detailed detailed in in the the previous previous section. section. The The Levins Levins model model assumes assumes aa large large number number of of discrete discrete habitat habitat patches, patches, ideally ideally of of the the same same size, size, and and

74 14

IIkka Ilkka Hanski Hanski

all connected connected to each other other via migration. migration. In reality, not all populations populations are are directly directly connected restricted, but but this makes connected to each other, other, because because migration migration distances distances are are restricted, makes no behavior of unless the no important important difference difference to to the the steady-state steady-state behavior of the the model model unless the net­ network heterogeneous. In the Levins model, model, work of of habitat habitat patches patches is strongly strongly spatially heterogeneous. the Levins habitat habitat patches patches are are scored scored only as occupied occupied or or not, as shown shown in Fig. 11,, and and the the actual best actual sizes of of the the local populations populations are are ignored. ignored. The The model model therefore therefore applies applies best to situations situations in which which local dynamics dynamics occur occur at a fast fast time scale compared compared with with metapopulation metapopulation dynamics, either either because because the the habitat habitat patches patches are are relatively small and and hence hence local populations populations quickly reach reach the local "carrying capacity" capacity" or or because because colonization colonization rate rate is low. All extant extant popUlations populations are are assumed assumed to have have a constant constant risk risk of of extinction. extinction. The The rate rate of of colonization colonization is assumed assumed to be proportional proportional to the fraction P, and fraction of of currently occupied patches patches (sources of of colonists), colonists), denoted denoted by P, and With to the fraction fraction of of currently currently empty patches patches (targets of of colonization), colonization), 11 - P. With continuous time is given by these these assumptions, assumptions, the rate rate of of change change in P P in continuous -

dP

dP dt == cP( cP(1 l dt

- P) P) - eP eP, ,

((1) 1)

parameters, respectively. The where where c and and e are the colonization colonization and and extinction extinction parameters, The equilibrium equilibrium value value of of P P is given by

P = = 11 f>

e e. cC

(2) (2)

The fraction of habitat at equilibrium The Levins Levins model model thus thus predicts predicts that the the fraction of occupied occupied habitat equilibrium metapopulation is prepre­ the ratio e/c. increases with with decreasing decreasing value of of the e/c, and and the metapopulation < 1. 1 . In spite of dicted to persist persist (P (P is positive) positive) as long long as e/c e/c < of its simplicity, the Levins feature of metapopulation Levins model is most useful useful in highlighting highlighting a key feature of metapopulation dynamics: for for the metapopulation metapopulation to persist, recolonization recolonization must must occur occur at a suf­ sufdynamics: ficiently high increase from from high rate rate to compensate compensate for extinctions and and to allow an increase small metapopulation im­ metapopulation size. More More specifically, condition condition e/c < < 11,, or 11 < < cle. c/e, implies that that a local local population population surrounded surrounded by empty patches patches must must cause cause the estab­ estabfor the metapopmetapop­ lishment lishment of of at least one new new population population during during its lifetime lifetime (lIe) (l/e) for ulation to persist. persist. Equation important predictions Equation (2) leads leads to some some straightforward straightforward but but important predictions when when we we recognize recognize that, very generally generally and and not surprisingly, the risk of of population population extinction extinction decreases decreases with increasing increasing patch patch area, area, and and the probability probability of of coloniza­ colonization decreases Hanski, 1991, 1 99 1 , decreases with increasing increasing distance distance from from the extant extant popUlations populations ((Hanski, I1994b). 994b) . The predicts that the fraction habitat at equi­ The Levins Levins model model predicts fraction of of occupied occupied habitat equilibrium librium (P) (P) decreases decreases with decreasing decreasing average average size and and decreasing decreasing density of of habitat these predictions habitat patches patches in a patch patch network. network. The The results in Fig. 3 support support these predictions for 970s for M. cinxia. cinxia. This This species species went went extinct on the Finnish Finnish mainland mainland in the late 11970s and Hanski and from from many many other other regions regions in northern northern Europe Europe during during the the past past decades decades ((Hanski and Kuussaari, 11995). 995). The and Kuussaari, The most most probable probable reason reason for for these metapopulation-Ievel metapopulation-level

Metapopulation MetapopulationDynamics Dynamics

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FIGURE FIGURE33 Effects Effects of of average patch patch size and and regional density on the the fraction of of occupied patches A land islands (Fig. 11). ). The was divided into into 2 by by 2 km km22 patches in M. M. cinxia cinxia on on/~land The study area was patches (as in Fig. 2). (a) Squares patch area in the square; Squares are divided into into four four classes based on the average patch square; (b) squares number of per square (patch Note squares are divided into four four classes based based on the number of patches patches per (patch density). Note that the fraction of patches are large of occupied occupied patches patches is high in the squares squares where where patches large and where where patch patch 995a). density is high (both Hanski et (both effects are are highly significant; statistical analysis in Hanski et al., al., 11995a).

extinctions is decreased decreased density of of suitable habitat habitat patches, patches, forcing the equilib­ equilibrium metapopulation size to zero ). > 11). zero (e/c > The large M. cinxia cinxia metapopulation metapopulation shown in Fig. 11 is a good example of of Levins-type metapopulations. However, how representative representative is this example? example? Han­ Hanski and Kuussaari ((1995) 1 995) attempted to answer this question for Finnish butterflies. butterflies. By our count, 57 of of the the 94 resident Finnish species may may belong to this this category. This fi gure includes much uncertainty, though, because because the spatial population figure structures cinxia are not well known. Collecting structures of of all the 93 species apart apart from from M. cinxia the kind of of large-scale information we have collected collected for for M. cinxia cinxia (Fig. 11)) is expensive, obtaining obtaining funding for for this kind of of work is difficult, and the life cycles and and larval biologies of of most species make them much harder harder to study than than M. large spatial scale. These These are are some of the reasons why we do not cinxia on a large know how common common Levins-type metapopulations metapopulations are are in nature. Promising candidates candidates for species with Levins-type metapopulations can be found in forest insects living in small and patchy microhabitats such as dead tree trunks. trunks. One One such such example involving involving beetles specializing specializing on on dead dead aspen aspen trees trees in boreal forests is described 1994; see also described in detail by Siitonen and Martikainen ((1994; Hanski and Hammond, 995). As most insects live in forests, and as most forest­ Hammond, 11995). forestliving insects are more or less specialized specialized on discrete discrete microhabitats, Levins-type metapopulation metapopulation structures structures may be common in insects (see also van der Meijden Meijden and van der Veen-van Wijk, this volume). Other Daphnia water Other examples examples include include Daphnia fl eas in rock pools ((Hanski Hanski and Ranta, 11983), 983), frogs in ponds (Sj6gren, (Sjogren, 11991; 99 1 ; fleas

76 76

IIkka Ilkka Hanski HanskJ

Sjogren 994), passerine Sj6gren Gulve, 11994), passerine birds in small woodlots (nuthatch; (nuthatch; Verboom Verboom et et aI. 99 lb), 1 b), and small mammals al.,, 1199 mammals in small patches patches of of suitable habitat (pika; Smith, 11980; 980; Smith and Gilpin, this volume).

IV. SIZE IV. MINIMUM MINIMUM VIABLE VIABLEMETAPOPULATION METAPOPULATIONSIZE The minimum MVP) size has become a well-established well-established minimum viable population ((MVP) concept in population and conservation biology. MVP esti­ MVP is intended intended to be an estimate of of the minimum number number of individuals in a population population which has a good chance of of surviving for for some relatively long period of of time, for for instance, 95% chance of 1 00 years (Soule, 980). Though MVP of surviving for at least 100 (Soul6, 11980). MVP is difficult to apply in practice 987; Lande , 1 988b), it is a useful concept high­ practice (Soule, (Soul6, 11987; Lande,1988b), concept in highlighting the need for for a quantitative analysis of of the risk of of population extinction. extinction. extinction-prone In the case of of Levins-type metapopulations, metapopulations, consisting of of extinction-prone local populations, an analogous concept concept of of minimum viable metapopulation ((MVM) MVM) size may be defi ned as the minimum number popu­ defined number of of interacting interacting local populations necessary for Hanski et 996b). Apart for long-term persistence ((Hanski et af. al.,, 11996b). Apart from MVM, MASH) MVM, one also has to consider the minimum amount of of suitable habitat ((MASH) necessary for metapopulation persistence, persistence, because because not not all suitable habitat may may be simultaneously occupied by a metapopulation persisting in a balance between between local extinctions and recolonizations ). recolonizations (that is, P P is generally less than than 11). The original Levins model cannot be used to answer questions about MVM, MVM, because Eq. ((1) deterministic model and only applicable applicable to large networks because 1 ) is a deterministic of of habitat patches in which the stochasticity involved in local extinctions and metapop­ recolonizations becomes drowned by large numbers. In reality, many metapopmetapopulations may go extinct ulations live in small patch networks. Such metapopulations when all local populations happen to go extinct at the same time, even if the expected colonization and extinction rates would allow long-term persistence by Eq. ((1) l ) or by some other deterministic model. Gurney and Nisbet ((1978; 1 978; summarized summarized in Nisbet and Gurney, 1982) have analyzed analyzed a stochastic version of of the Levins model. Their Their analysis yielded the following approximation for the expected M, expected time to metapopulation extinction, extinction, T TM, (Hh/(2( l -P)) TM = T Tce M = T , Le

(HP2)/(2(1-/5)),

(3) (3)

where TL TL is the expected time to local extinction, H H is the number number of of suitable habitat patches, and P P is the fraction of of occupied patches at a stochastic steady state. nes long-term TM > 1 00 TL state. If If one one defi defines long-term metapopulation metapopulation persistence persistence as as TM > 100 TL,, Eq. (3) leads to the following condition for for reasonably large H H (Gurney and Nisbet, 11978): 978):

pJH P ~ 2: -> 3.

(4) (4)

44

Metapopulation Dynamics Metapopulation Dynamics

17 77

For example, if there there are 50 habitat habitat patches, patches, Eq. (4) says that that the colonization and extinction rates must be such that P P > > 0.42 for for the metapopulation metapopulation to persist for longer than 00 TL• P is large, than roughly 1100 TL. Assuming a good colonizer, for for which P the critical 1 0 (however, critical minimum patch number number is of the order order of of 10 (however, the approxi­ approximation becomes less satisfactory for H ). Empirical results for for small H). for M. M. cinxia cinxia and for other butterfl butterflyy species (Thomas and Hanski, this volume) are in broad agreement 0 - 20 small agreement with these predictions, suggesting that a minimum of of 110-20 and persistence. and well-connected well-connected habitat habitat patches patches are are needed needed for long-term long-term persistence. 1996b) have studied the stochastic Hanski et et al. al. ((1996b) stochastic Levins model numerically, numerically, incorporating such realistic realistic features as variation in patch areas areas and and the rescue rescue effect (decreased risk of extinction due to immigration; Brown and Kodric­ Kodric977; Hanski, 11991). 99 1 ). Figure 4 gives the predicted Brown, 11977; predicted time to metapopulation extinction in the model parameterized M. cinxia Hanski et parameterized with data on M. cinxia ((Hanski et al. al.,, 11996b). 996b). These results are in good agreement with the analytical results of of Gumey Gurney and Nisbet ((1978) 1 978) and strengthen the conclusions about about the minimum numbers of habitat patches and local populations necessary for for long-term metapopulation metapopulation lifetime combines persistence. Notice that the condition about metapopulation characteristics P, with the properties characteristics of of the species, as reflected reflected in the value of of P, properties of the landscape ((patch patch number MASH number H H).). Hence Hence the concepts concepts of of MVM and MASH cannot be applied independently. independently. Equation (3) is not very sensitive to varying assumptions about metapopulation dynamics, because the effects of these as­ assumptions are refl ected in the value of in­ reflected of P, itself a part of of the condition. For instance, making migration more restricted in space will lower the colonization rate 11 000000 cc

0 O

�

9 _~

u 0 cc ._ � x x ill G) 0 0

� .4-,

ill (1)

FIGURE FIGURE44

•

6 6 0000

• • •

4 0000 4

c c 0 -0 ill G) L

2 2 0000

9

• •

• ••

•

8 8 0000

E E

�

!

•

g,

•

• •

• •

• • •

',~..o . •o

•

. 0 w!'. 9 0 0 0

2 2 i

4 4 1

-V H pp~/H

6 6 I

8

The relationship between the expected time to metapopulation extinction TM TM and the product P..{H Px/-H in simulations of an incidence function model parameterized parameterized with data from real metapopulations of of the butterfly M. M. cinxia cinxia (the median time to local extinction TL = = 3.3 in these at., I1996b). 996b). examples; details in Hanski et et al.,

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

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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-

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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

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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

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Metopopulotion MetapopulationDynomics Dynamics

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(A) Metapopulation size of ass measured measured by the the fraction fraction of of (A) Metapopulation size of the butterfly Melitaea Melitaea cinxia cinxia a occupied P in the landscape shown in Fig. 8. The occupied patches patches P landscape shown The results results were were obtained obtained by iterating iterating the the incidence parameterized with started by assuming incidence function function model model parameterized with field data. data. The The model model iteration iteration was was started assuming the (from year the patch patch network network 20 20 years years ago ago (Fig. (Fig. 8). During During a period period of of 20 20 years years (from year 300 300 to to 320 320 in in the the figure), this this network network was was reduced reduced to to its its present present size size (Fig. (Fig. 8) as as described described in in detail detail by by Hanski Hanski et et al. al. (1996b). ( 1996b). The The broken broken line gives the the expected expected (equilibrium) (equilibrium) metapopulation metapopulation size, size, whereas whereas the the contincontin­ uous uous line gives gives the the actual actual metapopulation metapopulation size size in the the declining declining network network (the (the lines lines give the the average average PP value value in in 200 200 replicate replicate simulations). simulations). (Middle) ( Middle) Difference Difference between between the the actual actual and and equilibrium equilibrium metameta­ population population sizes; sizes; (bottom) (bottom) numbers numbers of of metapopulation metapopulation extinctions extinctions in in the the 200 200 simulations simulations (no extincextinc­ tions for A, current patch (8) As As for A, but but now now starting starting with with the the current patch network network (Fig. (Fig. 8) and and halving halving tions in in this this case). case). (B) the area area of of each each patch patch in 20 20 years. years. Note Note that that the the equilibrium equilibrium metapopulation size drops drops to to zero, zero, but but the metapopulation size decades for most metapopulations metapopulations to to reach reach the the equilibrium equilibrium (extinction). (extinction). (Top) (Top) The The P it takes decades for most P value value in the the beginning beginning of of simulation simulation is is higher higher than than the the final value value in A A because because the the number number of of habitat habitat patches patches is et al., al., 1996b). J 996b). now smaller (reprinted with with permission permission of of University University of of Chicago from Hanski now smaller (reprinted Chicago Press Press from Hanski et

88 88

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

44

Metapopuiation MetapopulationDynamics Dynamics

89

may advance advance so slowly that that there is time to act if there is wish to act. In the case of managed of metapopulations on the brink brink of of extinction, intervention in the form of of managed recolonizations recolonizations is likely to become an increasingly necessary, and accepted, form of of management.

C. Ideal Spacing Spacing of Habitat Fragments Is a Compromise Compromise Even Even a large large number number of of small habitat fragments is no guarantee of of metapop­ metapoprecoloni­ ulation survival if the patches are located so far from each other other that recolonization and population popUlation rescue from extinction by immigration are are unlikely. A tentative practical answer answer to the question of of minimum density of of suitable habitat patches patches necessary for for long-term survival has been sought from the Levins model, Eq. ((1). 1 ). To model habitat loss, assume that fraction 11 - h of of the patches patches is per­ permanently because the density manently destroyed. destroyed. The The colonization rate rate becomes becomes lowered lowered because of P, and the model of empty but suitable patches patches is decreased decreased from 11 - P P to h -- P, 994; becomes ( May, 1 99 1 ; Nee, 1 994; Nee and May, 1 992; Lawton becomes (May, 1991; Nee, 1994; Nee and May, 1992; Lawton et et aI., al., 11994; Moilanen 995) Moilanen and and Hanski, 11995)

dP dP dt.· dt

- = P)) = cP(h ce(h - P - eP. ee.

(( 1111) )

At equilibrium, the fraction of of empty patches (out of of all patches, including the destroyed ones) is given by

e6' h --P *p* = - . cC =

- .

((12) 1 2)

Thus of all patches Thus the fraction of of empty patches patches out of patches remains constant as long as the metapopulation does not go extinct, which happens happens when h < < e/c. e/c. This is a seemingly very useful result, because it gives an estimate of of the critical minimum patch density from the very limited information of of the number number of of empty patches patches in still survives; no detailed knowledge in aa landscape landscape in in which which the the metapopulation metapopulation still survives; no detailed knowledge of 994). In practice, of metapopulation metapopulation dynamics dynamics is is required required (Nee, (Nee, 11994). practice, though, though, this this rule of of thumb is liable to yield an underestimate, and possibly a severe underestimate, of Hanski et al., 1996b): 1 996b): the of the critical patch density, because because of of three reasons reasons ((Hanski et al., extinction stochasticity in small patch networks (Sec­ rescue effect, colonizationcolonization-extinction (Section tion IV), IV), and and nonequilibrium nonequilibrium metapopulation dynamics, dynamics, when when a metapopulation metapopulation is approaching approaching the equilibrium from above (Section VI). Increased Increased patch density facilitates colonization and and is hence hence helpful, but if if habitat fragments are located located close to each each other other the the degree degree of of spatial synchrony in local dynamics may become elevated (Fig. 2), which has a negative effect on long-term survival. In theory, a row of of well-connected habitat fragments fragments might often of long-term survival than often provide provide a better better chance of than a tight cluster, but such considerations considerations are seldom practical. The The main recommendation recommendation is simply to pro­ provide sufficient connections their density connections among among habitat habitat fragments by maintaining maintaining their

90 90

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

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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).

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

94 94

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

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desirable to to be be able able to to relax relax this this assumption. assumption. Finally, Finally, being being aa deterministic deterministic patch patch desirable model (Gilpin and Hanski, this volume), the Levins model tacitly assumes a very model (Gilpin and Hanski, this volume), the Levins model tacitly assumes a very large (effectively infi n ite) number of patches. large (effectively infinite) number of patches. There is is aa clear clear analogy analogy between between the the simplifying simplifying assumptions assumptions on on which which the the There Levins model model is is based based and and the the corresponding corresponding assumptions assumptions of of classical classical models models in in Levins popUlation ecology, ecology, such such as as the the logistic logistic equation. equation. Classical Classical population population models models population are concerned concerned with with the the total total number number (or (or density) density) of of individuals individuals in in aa population population are but neglect any differences among individuals (age, size, sex, etc.). To take these these but neglect any differences among individuals (age, size, sex, etc.). To take differences into into account account one one has has to to turn tum to to structured structured population population models, models, which which differences allow one one to to use information information about about individual individual behavior behavior to to draw draw conclusions conclusions about about allow the dynamics dynamics of of a population. population. The The book book by by Metz Metz and and Diekmann Diekmann (1986) ( 1 986) presents presents the comprehensive introduction introduction to to the the philosophy philosophy of of using using structured structured population population a comprehensive models as as well as a wealth wealth of of examples. examples. More More recently, recently, Diekmann Diekmann et et al. al. (1993a,b, ( 1 993a,b, models 1 995a,b) have have developed developed a slightly slightly different different approach approach to to structured structured population population 1995a,b) models, which we we apply apply in in this this chapter. chapter. models, which Our concept of population of of populations popUlations (for (for alteralter­ Our of a metapopulation is a population Hanski and Simberloff, this volume; Harrison and and Taylor, native approaches, approaches, see Hanski and Simberloff, Hanski, 1996c; 1 996c; Hastings and Harrison, 1994). 1 994). As As pointed pointed out out by this volume; Hanski, and Harrison, et al. al. ((1988, 1 988, 1989), 1 989), the theory theory of of structured structured populations populations can can be be applied Diekmann et to straightforward manner manner if if one one makes makes the analanal­ to metapopulations metapopulations in a relatively straightforward ogy ogy between between local popUlations populations and and individuals individuals and and between local populations and and metapopulations, metapopulations, respectively. In more detailed metapopulation metapopulation models, where for for instance dynamical dynamical changes in in patch patch quality quality are are included, included, one has has to to replace, replace, in this analogy, local popUlations populations by some other other kind of of local entities. One of of the first structured metapopulation models was presented presented by Levin and Paine ((1974, 1 974, 11975). 975). Their model was structured by the age and the size of a patch. Extinction was assumed to be age-dependent and size-dependent, but col­ colonization (establishment of new populations) populations) was not modeled explicitly and the effect of of migration on local dynamics was not considered. considered. Hastings and Wolin ((1989) 1 989) used a McKendrick-type model in which local populations populations are structured by age (time since colonization). They assumed that the size of a local population is a function of its age, and they could thus predict the size distribution of local populations. In this framework, it was not convenient to model the effect of migration on local dynamics, since migration does not affect the age of a popu­ population. Gyllenberg and Hanski ((1992) 1 992) chose local population size as the structur­ structuring ing variable variable and and could could thereby thereby model model explicitly explicitly the the within-patch within-patch consequences consequences of migration. Later they extended their their model to account for for variation variation in patch 1 995) have quality ((Hanski Hanski and Gyllenberg, 993). More recently, Val et Gyllenberg, 11993). et al. al. ((1995) made aa detailed analysis analysis of of the effect of migration upon local dynamics using similar similar structured structured models. models. In In this this chapter chapter we we present present aa unified unified treatment treatment of of aa large large class class of of deterministic deterministic structured structured metapopulation metapopulation models and and illustrate illustrate the the mathematical mathematical framework framework with with several nite several examples. examples. Being deterministic, deterministic, the the models models continue continue to to assume an an infi infinite number number of of patches patches and and local local populations, populations, and and the the results are are hence hence applicable to to

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large metapopulations. metapopulations. Deterministic Deterministic metapopulation metapopulation models models with with aa finite finite number number large of patches patches have have been been investigated investigated by by Levin Levin (1974), ( 1 974), Holt Holt (1985), ( 1 985), Davis Davis and and Howe Howe of et al. al. (1993, ( 1 993, 1996), 1 996), Doebeli Doebeli (1995), ( 1 995), and and ( 1 992), Hastings Hastings (1993), ( 1 993), Gyllenberg Gyllenberg et (1992), many others. others. All All these these papers papers are are concerned concerned with with the the effect effect of of migration migration on on local local many dynamics, with with a special special focus focus on how how migration migration may may synchronize and stabilize dynamics, and stabilize dynamics. On On the the other hand, these these models models ignore ignore local local extinctions extinctions and and local other hand, local dynamics. recolonizations (for (for stochastic stochastic models models of of finite metapopulations, metapopulations, see Gyllenberg Gyllenberg recolonizations and and Silvestrov, Silvestrov, 1994; 1 994; Hanski, Hanski, 1994a). 1 994a). It is practically practically impossible to to incorporate incorporate the spatial arrangement arrangement of of patches patches into into deterministic deterministic structured structured models of of the the type the treated in this this chapter. chapter. Perhaps Perhaps the the most that that can can be be done done is to to analyze analyze the the qualqual­ treated itative effects effects of of spatial aggregation of of habitat habitat patches (Adler and and Nfirnberger, Nurnberger, itative spatial aggregation patches (Adler 1 994). 1994). This chapter has two parts. parts. The first part part (Sections (Sections I --V) gives a nonmathenonmathe­ This chapter has two The first V ) gives matical description of the basic principles of modelling structured populations matical of the basic principles of modelling structured populations and shows by examples examples the the kind kind of results that that can can be be obtained such models. and of results obtained by such An empirical empirical example example illustrates the relevance of structured models. illustrates the relevance of models. The The mathmath­ ematical outlined in Section VI. This approach was first first used by ematical formalism is outlined al. (1993b, ( 1 993b, 1995b; 1 995b; see also Diekmann et et al., al., 1993a,c, 1 993a,c, 1995a) 1 995a) for for Diekmann et Diekmann et al. "ordinary" populations. "ordinary" structured structured populations.

II. MODELING MODELINGSTRUaURED STRUCTUREDMETAPOPULATIONS METAPOPULATIONS We consider structured We consider structured metapopulation metapopulation dynamics as the study of of the interinter­ processes at the local level and on the metapopulation metapopulation level level under under relation between processes the influence of the environment. We shall interpret interpret "environment" "environment" in a wide instance the fraction of of empty sense and it will often be convenient to include for instance patches among the environmental variables. Recall that in Levins' Levins'ss model the rate at which a local population gives rise to new local populations depends on the fraction of empty patches. The crucial point is that that all (nonlinear) (nonlinear) feedback takes place through the environment. By a "virgin" environment we understand understand an environment with no local populations and where the patch quality distribution distribution has has settled settled down down to to an an equilibrium. equilibrium. The most essential features of of classical classical metapopulation dynamics dynamics are are recur­ recurrent local extinctions and colonizations of empty patches. The Levins model is concerned with only these two processes and treats them directly at the level of the metapopulation, thus entirely ignoring local dynamics. Modeling structured metapopulations and analyzing the models take place in three steps. First one has to model mechanisms at the local level, that is, at the level of patches and local populations. In the second step one lifts the model to the level of the metapopulation by simple book-keeping, and fi nally one studies finally population dynamical phenomena phenomena at this level. Local dynamics may include both the dynamics of local popUlations populations and the dynamics of patches. Local populations grow or decline as a consequence of

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reproduction, reproduction, death, emigration, and immigration. Patches Patches may change change in size and quality, be destroyed, and and new patches patches may may be created. created. In order to model these local processes processes one has to start by specifying the basic basic local local entity corre­ corresponding to an individual in ordinary population var­ population dynamics and by choosing variables that adequately describe the local states. Here "adequate," "adequate," of of course, refers coloni­ to quantities that affect processes like growth, migration, extinction, extinction, and colonization. If If we, for for instance, instance, consider consider a metapopulation in a set of of patches patches of of dy­ dynamically changing changing quality a relevant relevant choice of of basic local entity would be a Xl ' X patch and its state would be described by the vector ((xl, x2) where Xl Xl denotes the 2 ) where quality (e.g., resource density) of of the patch and X x22 denotes the size of of the local population inhabiting it. We call the set of all conceivable conceivable local states the local local state space space and denote it by il. 11. In the example example above il f~ is (a subset of) the positive R2 . quadrant R� The following processes have to be modeled:

(i) patch quality dynamics, for for instance how do resource resource density and patch patch area change with time; change (ii) local population growth; (iii) extinction, extinction, patch destruction; destruction; (iv) colonization, patch patch formation, and production production of of dispersers, that that is, how many new basic local entities and with what what local state state at "birth" "birth" a given local entity will give rise to. When modeling these four processes one has to describe describe how they depend depend on the local state and on the environmental state. Some of of the processes, for instance the formation of new patches, may be the result of of processes independent independent of the metapopulation. If that is the case, time enters the description of patch of independent variable. formation as an independent As the metapopulation metapopulation affects its environment one has to close the loop by modeling modeling the

(v) feedback mechanism. mechanism. Next we describe in some detail how the processes (i-v) (i-v) should be modeled.

A. Colonization Without recolonization recolonization of of empty patches a metapopulation metapopulation consisting of of ex­ extinction-prone local populations certainly goes extinct. The foremost modeling task task is is therefore therefore to to prescribe prescribe the the colonization colonization process. process. To To answer answer some some simple simple but important questions like "will the metapopulation metapopulation persist or go extinct? extinct?"" a precise characterization characterization of of this process is sufficent. The basic idea in the present present approach is to model colonization by describing mathematically the expected cumulative number number and structure structure of of new local en­ entities produced in the future by a given local entity whose present produced present state state is known and when the course of of the environment is known. All this information is con-

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Mots Mats Gyllenberg Gyllenberget et 01. al.

densed rigorously densed into into aa mathematical mathematical object object called called the the colonization kernel and and rigorously defi n ed in Section VI. To give an example, suppose that colonization is modeled defined To an suppose that modeled as a two-step process: dispersers which may colonize process: local populations populations produce produce dispersers empty empty patches. patches. Then Then the the colonization colonization kernel kernel should should contain contain at at least least the the following following information: Given a local population present population with a given state (e.g., size) at present produce so and so many dispersers time, it will on average average produce dispersers in the future. Given Given a a disperser disperser having having a given given state state at at present it it will on on average colonize colonize so so and and so so many future. The future devel­ many empty empty patches patches in in the the future. The "so "so and and so" so" depend depend on on the the future development opment of of the environmental state, state, which for the the time time being being is considered considered to be known. known. The The populations populations on on the the patches patches colonized colonized by by aa given given local local population population will in in tum populations that regarded as "second turn give rise to new local populations that can be regarded "second generation generation offspring" offspring" of of the given ancestral popUlation. population. By applying the colonization kernel to itself itself in way to to be precise in Section VI, VI, we mathematical to in aa way be made made precise in Section we obtain obtain aa mathematical description of of this this second second generation generation or or "grandchildren" "grandchildren" to use an an analogy with ordinary popUlations. This ordinary populations. This procedure procedure can can be repeated repeated ad infinitum to to obtain obtain "great "great grandchildren," grandchildren," etc. Summing up over all generations generations we obtain what we shall At; contains call the clan kernel. The The clan kernel A~ contains information about all "descend­ "descendants" ants" of of a given ancestor ancestor population population with respect to the time of of colonization and and state state at colonization.

B. Patch B. Patch Dynamics, Dynamics, Population Population Growth, Growth, and Extinction Extinction The state of of the basic local entity (e.g., local population, patch) patch) changes changes The because of of popUlation population growth, changes changes in patch quality, and during its lifetime because so on. We We refer refer to all these processes processes by the common common term local state develop­ develop-

ment. Our Our framework allows for for stochastic stochastic local state state development. development. As our second model model ingredient ingredient we therefore choose the probability distribution of of future local states of of a local entity whose present present state is known. Here we assume assume again that that the future course of of the environment environment is known. From From this model model ingredient many many important important quantities can be computed, for for instance instance the survival probability, probability, that that is, the probability that a given local entity still exists at a given instant of of time in the future. The future state The state of of a local entity can move from its present present state to a future along local state space. Assume Assume that local entity entity has has a along different different routes routes in in the the local state space. that the the local a certain nal state. certain state state before before it it reaches reaches the fi final state. Using the the aforementioned aforementioned probability probability distribution one can compute the probability distribution of of the final final state, state, given that had this time. Summing up over interme­ that it it had this intermediate intermediate state state at at aa given given time. Summing up over all all intermefinal state, given the diate states states one should get the probability distribution of of the final initial state. This leads leads to a consistency relation that the second second model ingredient has has to to satisfy. The The same same argument argument also applies to to colonization; colonization; hence hence we get aa consistency relation combining the colonization kernel and the transition consistency relation combining the colonization kernel and the transition proba­ probabilities.

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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

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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

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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

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)
,,

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

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9

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,

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' 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

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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.

From From Metopopulotion Metapopulation DDynamics ynomics to Communit Community Structure y Structure Some Some Consequences Consequencesof Spatial Spatial Heterogeneity Heterogeneity Robert D. Holt

NTRODUCTION I. IINTRODUCTION The most fundamental fundamental structural properties of of a local community are the member species and the pattern of of their number and relative relative abundances of of its member dynamical interactions ((Roughgarden Roughgarden and Diamond, 11986). 986). The history of of com­ community ecology largely revolves around variations on a small number number of of perennial perennial 1 ) the relationship between species diversity and environenviron­ themes, including: ((1) mental heterogeneity (e.g., resource resource diversity or disturbance disturbance regimes; Chesson, 11986; 986; Huston, 11994), 994), (2) the implications of of direct and and indirect indirect interactions interactions for community structure (e.g., dynamical constraints on food chain length; Pimm, 11982; 982; Schoener, 993; Wootton, 11994), 994), and Schoener, 11993; and (3) historical contingency, such as multiple stable states (e.g., priority effects effects in competition). A consideration consideration of of spatial spatial dynamics can enrich enrich all these traditional themes in community ecology. The insight that local colonizations and extinctions de­ determine rst articulated 960s in the theory termine local community structure structure was fi first articulated in the 11960s of MacArthur and Wilson, 11967). 967). This seminal work was of island biogeography ((MacArthur soon complemented interactions of complemented by analyses of of the effects on species species interactions of patch 1 97 1 ; dynamics and spatial fluxes in mosaic landscapes (e.g., Levins Levins and Culver, 1971; Horn and MacArthur, 972; Levin, 11974; 974; Whittaker and Levin, 1977; 1 977; Holt, 1985), 1 985), MacArthur, 11972; Metapopulalion Metapopulation Biology Biology Copyright © 997 by Academic Press, Press. Inc. All rights of 9 11997 of reproduction in any form reserved.

1149 49

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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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Consequences Consequencesof Spatial Spatial Heterogeneity Heterogeneity

1! S511

years of of the study. For For some of of these these species, species, "extinctions" "extinctions" may be be recorded recorded experi­ because the site provided only a small sample drawn drawn from populations populations experiencing the landscape at a coarser coarser spatial scale (1. (J. Bengtsson, Bengtsson, personal commu­ communication). nication). For For other other species, the Eastern Eastern Wood Wood population population may may be part part of of a clas­ classical metapopulation, metapopulation, in which which a balance balance between between colonization colonization and and extinction extinction across across the landscape landscape permits permits regional regional persistence, persistence, despite despite the ephemeral ephemeral occur­ occurrence Hanski, this volume). rence of of populations populations in local communities communities ((Hanski, volume). However, 1 994b; see also Harrison However, Harrison Harrison ((1994b; Harrison and and Taylor, this this volume; volume; Schoener, 11991) argued that species species which which in a particular particular patch patch network network show show Schoener, 99 1 ) has argued frequent frequent extinctions extinctions and and colonizations, colonizations, may may actually have have a few few persistent persistent pop­ poppersistence. Moreover, ulations, which which permit permit overall overall persistence. Moreover, a local population that never never goes extinct extinct may nonetheless nonetheless prove prove to be a sink population, population, maintained maintained by a regular ow of regular fl flow of individuals individuals from from self-sustaining self-sustaining source populations populations (Shmida (Shmida and Ellner, 984; Holt, 985, 1993; 1 993; Pulliam, 988; S. Hubbell, Hubbell, personal Ellner, 11984; Holt, 11985, Pulliam, 11988; personal communi­ communication). cation). Local species species richness richness thus thus may be enhanced enhanced if, at the landscape landscape scale, habitat habitat long­ heterogeneity provides provides each each species species with some some habitat habitat patches permitting longterm persistence. persistence. Guaranteed Guaranteed local survival survival in some some habitats habitats allows allows a species species to be present range present (e.g. (e.g.,, as rare transients transients or sink populations) populations) over over a much much broader broader range of useful to consider meta­ of habitats. To examine examine this effect effect in more more detail, detail, it is useful consider a metapopUlation model that incorporates population incorporates habitat habitat heterogeneity in colonization colonization and and ex­ extinction rates. rates.

A. A Metapopulation Metapopulation Model Model for a Heterogeneous Heterogeneouslandscape Landscape Assume patches, of Assume that that the landscape landscape consists of of a large large number number of of habitat habitat patches, of For simplicity, consider which which a fraction fraction H H are are suitable suitable for for the community. community. For simplicity, I will consider that that just just three three habitat habitat types are are present: present: patches patches of of two two distinct distinct habitat habitat types, potentially potentially occupied occupied by species in the community, community, embedded embedded in a third third matrix matrix habitat, unsuitable unsuitable for for any of of them. Let Let hi h/be of habitat habitat patches of be the fraction fraction of patches of 1 . Some species hab­ type i. Necessarily, h h i1 + -k- h h 2 2 = -- H H � --< 1. species in in the community may be habitat specialists on just just habitat I1,, others specialists on habitat 2, and yet others may be habitat generalists, able to use both habitat habitat types (possibly to different degrees). patches of p;i denote denote the fraction f r a c t i o n of of habitat habitat patches of type i occupied occupied by a focal focal Let P species. species. The The total total occupancy occupancy of of this species species over over the entire entire landscape landscape is P p = = Let ei be the rate of extinction of the focal species in patches of habitat · PlI + + P P2. P 2 Let eg be the rate of extinction of the focal species in patches of habitat type i and patches due to emigration from and ci cijj the rate rate of of colonization colonization of of type i patches from patches following model dynamics of patches of of type type jj (i, jj = = 11,, 2). The The following model describes describes dynamics of the the total metapopulation: metapopulation:

dp dp~i

=- - (( CCIl l PIPI l dt dt d dp~ P2

=- - (( cC221 PI PI l dt dt

+ P I ) - ee~IPl I -k- C1 Clzpz)(hl 2P2 )(hl --Pl) -

+ -k-

-

cc22P2)(h - e e2P2 22P2 )(h22 --PP22)) 2 P2

((1) 1)

1152 52

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;

1154 S4

Robert Robert D. D. Holt Holt

it cannot 1 ), for for cannot persist in any any single single habitat habitat alone. At At equilibrium equilibrium in model model ((1), habitat habitat i 0 = [ciiP~ ( h i -

p?

-

e;)] + cijp* (hi - Pi* ).

(4) (4)

If O. The The bracketed bracketed If the cross-habitat cross-habitat parameters parameters cij are positive, positive, both both p p*i > > 0. larger P term term on on the the left left of of (4) (4) is is zero zero at at P Pii = = hi h i -- eJcii ei/cii and and is is negative negative for for larger Pi.i' Because negative; Because the right-hand right-hand term term is positive, positive, the bracketed bracketed term term must must be be negative; because hence hence p p*i > > P P ii.' Thus, Thus, aa species species'' incidence in in one one habitat habitat type type is is enhanced enhanced because permits a of spatial spatial coupling coupling with the second second habitat habitat type. Habitat Habitat generalization generalization permits of species habitat, given species to be present present with a higher higher than expected incidence in each each habitat, given colonization colonization and and extinction extinction rates rates for for each each habitat habitat in isolation. isolation. Model 1 ) illustrates Model ((1) illustrates how how "spillover" "spillover" between between habitats habitats can can enrich enrich local com­ comHolt and 992), munities. Assume Assume that that habitat habitat 2 is a "black-hole" "black-hole" sink ((Holt and Gaines, Gaines, 11992), which which can can be colonized colonized but does does not not provide provide colonists colonists for for either habitat habitat type (for (for concreteness, concreteness, one one can imagine imagine that that popUlation population densities densities are very low in habitat 2, so these popUlations populations provide negligible negligible sources for for colonists). colonists). Hence, Hence, Cc~1 I and and C2 CI2 O. The 12 = c2~1 > > 0, but but C22 c22 = = C12 = - - 0. The incidence incidence of of the the species species in in habitat 2 is 12 i i i = pi Ih2 = which increases with pt. If H is then p , + c2I l(e2 p*/h2 = c2 c2~*/(e2 + c2,p*), which increases with p*. If H is fixed, then p* = IP P ) H , which decreases linearly with h2 • Hence, the incidence ( prob­ eJc H h 2 e l / C l l , I I which decreases h2. (probh2 ability of increases in habitat 2, of occurrence, occurrence, per per patch) of of this spillover spillover species species increases the less habitat type, relative frequency of habitat type less frequent frequent is this habitat relative to the frequency of the habitat that metapopulation. that actually sustains a viable metapopulation. Species rates in their preferred hab­ Species with high high colonization colonization or or low extinction extinction rates their preferred habitat should should exhibit a high high occupancy occupancy in this habitat habitat and and can can secondarily secondarily have have a high high incidence incidence in habitats habitats where where they cannot cannot persist. persist. Such Such spillover spillover effects effects should should be most involving in particular particular those those species most noticeable noticeable in rare rare habitats, habitats, involving species with high high occupancy occupancy in frequent frequent habitat habitat types. In some some circumstances, circumstances, utilizing a second second habitat habitat may may permit permit a species species to persist persist in a landscape landscape even even if if there there is no no colonization colonization among among patches patches of of the second second habitat habitat type. For For instance, instance, imagine that that patches patches of of habitat habitat 2 are are overdispersed, overdispersed, sufficiently far the species cannot far apart apart that that C22 c22 = - - 00, , and and furthermore furthermore that that the cannot persist in habitat habitat 11 alone. The The condition condition for for such such a species to persist persist in the entire land­ landscape scape is is

Cllhl Cllhl hlhzClzC21 < 1< + ~ . el el ele2

(5) (5)

The The right-hand right-hand inequality is always always met met if if e2 e2 is sufficiently small. Sparse Sparse habitats habitats with with low local extinction extinction rates rates can have have a large large effect on the the overall overall persistence persistence of of a species, even even if if the geometry geometry of of the landscape landscape does does not permit permit such such habitats habitats to sustain the species on their their own. In effect, colonization colonization of of sparse sparse but provides a kind "spatial storage but low-extinction low-extinction habitat habitat patches patches provides kind of of"spatial storage effect" effect" (Holt, (Holt, 11992), 992), amplifying colonization colonization rates overall in the the more more widespread widespread habitat.

77

Consequences of Heterogeneily Consequences of Spano SpatialI Heterogeneity

1155 55

This test­ This two-habitat metapopulation model model leads to several interesting and testable conclusions. In a heterogeneous landscape: conclusions. landscape: 11.. Habitat Habitat specialists will be disproportionately disproportionately common common in those habitats habitats that are most common in the landscape. landscape. because they can 2. Some generalists generalists may persist persist in the the landscape landscape precisely precisely because can exploit a range range of of habitat habitat types. 3. Species which which can persist persist in one habitat habitat can thereby thereby incidentally incidentally occupy other communities. This spillover effect other habitats, habitats, enriching enriching those those local communities. effect should be particularly defining the community membership in sparser particularly important important in defining sparser habitats habitats and be characterized commoner habitats. characterized by species species with with high occupancies occupancies in commoner habitats. rates, or 4. Specialists on on rare rare habitats habitats should should have have unusually low low extinction extinction rates, or high landscape high colonization colonization rates, rates, relative relative to the entire entire ensemble ensemble of of species in the landscape (including habitat generalists). im­ (including both specialists specialists on common common habitats habitats and habitat generalists). This This imfactors plies a systematic bias bias at the community community level in entire entire suites suites of of ecological ecological factors correlated correlated with local extinction or colonization rates. The The above model model deliberately deliberately ignored ignored species species interactions. interactions. Yet, Yet, in practice, practice, habitat habitat suitability for for a given species and its local colonization colonization and and extinction extinction rates rates may be largely largely determined determined by interactions interactions with other other species. Several Several au­ authors homo­ thors have have considered considered metapopulation metapopulation models for for species interactions interactions in homoexamined com­ geneous geneous landscapes landscapes (e.g., see Nee Nee et et al. al.,, this volume) and and have have examined competitive interactions interactions in heterogeneous heterogeneous metapopulations metapopulations (Hom (Horn and MacArthur, MacArthur, 11972; 972; Hanski, 11992b). 992b). In the consider some implications the remainder remainder of of this paper, paper, I consider implications of metapopulation, using of trophic interactions interactions in a heterogeneous heterogeneous metapopulation, using natural natural exten­ extensions of of the above model. model.

III. FOOD CHAINS III. METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICSOF OF FOOD CHAINS The simplest specialist predator simplest trophic interaction interaction is the the one one between between a specialist predator and its prey, and the the simplest simplest food food web web is an unbranched unbranched chain of of trophic specialists. specialists. Here I first first consider consider a metapopulation metapopulation model model for for a three-level three-level food chain. chain. A food food Here chain describes describes a set of of tight sequential sequential dependencies dependencies among among species. species. In many many chain circumstances, circumstances, it is reasonable reasonable to expect expect that that such sequential sequential trophic trophic dependency dependency will lead lead to nested nested distributional distributional patterns, in which which a given species will be nec­ necessarily absent 1 993, absent in a patch patch if its required required prey population population is absent absent (Holt, (Holt, 1993, 11995). 995). Let the state of ed by the food chain of a patch patch be identifi identified the length length of of the the food chain it contains, contains, such that 1 " a patch that "0" "0" denotes denotes an empty patch, patch, ""1" patch with with just the basal basal prey species, species, "2" patch "2" a patch patch with both both the basal basal prey prey and and an intermediate intermediate predator, predator, and and "3" a patch with both both these plus plus a top predator. predator. The The fraction fraction of of patches found found in state i is the food chain denoted by Pi p;.' We We assume that the basal species species in the chain is a habitat habitat specialist specialist and that its required habitat habitat occupies occupies a fraction fraction hh < < 11 of of available available

1156 S6

Robert Robert D. D. Holt Holt

patches landscape. The following model -extinction patches in the landscape. model describes describes colonization colonization-extinction dynamics metapopulation: dynamics in this metapopulation: Npl

dt

-- (co~p~ + co~p2'

+

ColP3")(h - P~

-- (cl2P2 4- c ' ( 2 P 3 ) P l + c~ p2

dt p3

dt

-- P2

-- P3)

+ e21P2 -

elopl,

(6)

-- (cl2P2 + C'[2P3)Pl -- (e20 + e 2 1 ) P 2 + e32P3 -- r

= c23P3P2 -- (e30 + e31 + e32)P3.

For For clarity, the order order of of the subscripts subscripts for for the the colonization colonization and extinction extinction coef­ coefficients indicates the direction ow among ficients direction of of fl flow among states states (read them from from left to right). right). Thus, Thus, the ccij's denote the rate rate at which which colonization colonization transforms transforms patches patches from state u 's denote i to state j; j; the e;/s eij's likewise likewise set the rates rates of of extinction, extinction, changing changing patches from state i to state state j. j. In the basal parameter C� basal prey equation, equation, the parameter c~I arises arises because because empty patches patches can be colonized colonized by prey prey originating from patches patches with both both the basal prey prey and and the intermediate intermediate predator. predator. Likewise, C� c0'I~denotes denotes colonization colonization of of empty patches patches by basal prey emigrating predators emigrating from patches patches with both both the intermediate intermediate and and top predators (as well as the basal prey), and C'{ c'~2 describes colonization colonization of of prey patches by 2 describes intermediate intermediate predators predators dispersing dispersing from patches patches with the full food food chain. chain. If If these these parameters parameters are are positive, positive, colonization colonization dynamics dynamics at lower lower trophic levels levels involves habitat 1 ) (although such heterohetero­ habitat heterogeneity, heterogeneity, comparable comparable in spirit to model model ((1) geneity is not a fixed landscape feature, feature, but but instead instead emerges as a dynamical dynamical feature feature of interactions). of the trophic trophic interactions). The most important important assumption assumption made made in the the above above model model is that the the food chain 982), and that if chain builds builds up via sequential sequential colonization colonization (see, e.g., Glasser, Glasser, 11982), a prey population population goes extinct extinct in a patch, patch, so does any predator directly or indi­ indirectly supported nes of premises, a wide supported by that that prey. Within Within the confi confines of these key premises, wide range range of of assumptions assumptions about about local local dynamics dynamics can be embodied embodied in the model. model. It is useful useful to examine examine the the properties properties of of this model by building building it up up from its base. The The basal species, species, on its own, satisfies the standard standard metapopulation metapopulation model model dpl

dt h

= Colpj(h - P l )

- elOPl.

p; = ok'o l ' The At equilibrium, equilibrium, p'~ = h - ee r~o/Co~. The basal prey species species persists persists provided -

h > elo

e ro . CCO1 Ol

h>~.

This This inequality also ensures ensures that the basal basal species increases increases when when rare.

(7)

Consequences Spotial Heterogeneily Consequencesof Spatial Heterogeneity

77

1157 57

A. Two Trophic Trophic Levels Levels When When the intermediate intermediate predator predator is also present, present, the model model takes the the form form

dpl == ((co~p~ CO I PI + P2 ) + c C oIP ' o l P2z ))((hh - P P lI - P2) dt dt

dp,

I

--

Cl2P2Pl

+

2 dt

=

Cl2P2Pl

--

e21P2 -- e loPl

(8)

(e20 + e21)P2.

The The model model resembles resembles a standard standard predator-model, predator-model, but but with with the crucial dif­ difference ference that that the predator predator patches patches also contain contain prey and and can can therefore therefore contribute contribute rate of of generation generation of of new new prey prey patches, either by prey colonization colonization of of empty to the rate patches, either patches, patches, or by by predator predator extinctions extinctions unaccompanied unaccompanied by by prey extinctions. extinctions. For For sim­ simplicity, we will assume only if assume that that the predator predator goes goes extinct extinct locally only if the prey prey also goes the former former effect. goes extinct extinct (i.e., (i.e., ee21 = 0), so here here I consider consider only the 21 = There predator may have on its prey There are are two two kinds kinds of of effects effects a specialist specialist predator may have prey in this model: it may may alter the prey prey extinction extinction rate, rate, or or it may change change the rate rate of of prey colonization colonization of of empty patches. patches. In general, these effects effects could could be either positive positive or negative: negative: 1. Biogeographic Biogeographic "Donor "Donor Control" Control"

Some Some predators predators may may have have negligible negligible effects effects on local prey prey dynamics dynamics and and so = ee:0 are are unlikely to alter prey prey colonization colonization or or extinction extinction rates, rates, i.e., eel0 and 10 = 20 and 'I This (DeAngelis, 11992) 992) in a spatial = C� C~l. This is "donor "donor control control"" (DeAngelis, spatial context: context: prey Cc01 Ol = dynamics may constrain predator, without constrain the the distribution distribution of of the predator, without reciprocal effects predator on its prey. by the predator 2. Increased Increased Prey Prey Extinction Extinction

The the literature The scenario scenario that that has has received received by far far the most most attention attention in the literature on prey interactions predators reduce prey abunabun­ local predatorpredator-prey interactions is the one one in which predators dances dances so greatly that both both populations populations face face enhanced enhanced extinction extinction risks (i.e., eel0 < lO < ee20) 975; Taylor, 11991; 99 1 ; Hassell eett al., 1 992). Even Gilpin, 11975; a l . , 1992). Even in the absence absence 20 ) (e.g., Gilpin, of of any effect effect of of the predator predator on prey abundance abundance in "typical "typical"" years, the predator predator may during episodes disturbance, reduced reduced may heighten the risk of of prey prey extinction extinction during episodes of of disturbance, prey prey resources, resources, or or extreme climatic events. events. Even in predator-prey predator-prey models with with stable equilibia bounded bounded well away from from zero, following following large perturbations perturbations there there can can be transient phases phases at low densities, densities, greatly increasing increasing the likelihood likelihood of of local extinction unpublished results). extinction for for both both species species (R. D. Holt, unpublished results). 3. 3. Decreased Decreased Prey Prey Extinction Extinction In a wide wide range range of of circumstances, circumstances, predators predators can reduce reduce the magnitude magnitude of of fluctuations 973b; Rosenzweig, Rosenzweig, 11973) 973) or fluctuations in prey abundances abundances (May, 11973b; or even even in-

1 58 158

Robert D. D. Holt Holt Robert

crease average average prey prey abundances abundances (Abrams, (Abrams, 1992). 1 992). For For instance, instance, if if prey prey respond respond crease behaviorally to to predators predators by by reduced reduced exploitation exploitation of of their their own own resources, resources, overover­ behaviorally et al. al. (1985) ( 1985) exploitation may may be be less less likely likely in in the the presence presence of of aa predator. predator. Sih Sih et exploitation reported aa surprising surprising number number of of cases cases in in which which removal removal of of aa predator predator led led to to aa reported decrease in in the the abundance abundance of of the the focal focal prey. prey. Many Many of of these these cases cases seem seem to to involve involve decrease indirect interactions interactions in in multispecies multispecies assemblages (e.g., competitive competitive interactions interactions indirect assemblages (e.g., among prey, prey, held held in check check by by generalist generalist predators), predators), but but it it is is not not clear clear that that all all do. do. among In cases where aa predator predator enhances enhances the the mean mean abundance abundance or or reduces reduces the temporal In cases where the temporal variability of of its its prey, prey, it it is is conceivable conceivable that that ee 10 e 20 ' variability J O >> e2o.

4. Decreased Decreased Prey Prey Colonization Colonization 4. If local local prey prey densities densities are reduced by by predation, predation, the the flux flux of of dispersers dispersers If are greatly greatly reduced available for colonizing empty empty patches patches is is likely to to be be reduced reduced and, and, hence, hence, we we available for colonizing COl > C� could expect that could expect that c01 > c~.I '

5. Increased Prey Prey Colonization Colonization 5. Increased If predators predators increase above, then predators may may If increase local prey density, as noted noted above, then predators also indirectly facilitate facilitate prey if prey differentially also indirectly prey colonization. Alternatively, if prey differentially disperse in response response to perceived perceived increases increases in the local risk risk of of predation, of disperse predation, rates rates of prey emigration may be higher higher from from patches patches with predators predators than from patches patches prey emigration may than from without predators. In such cases, one might expect expect that C� I 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 < > 0, 0, then then the positive branch in the above solution leads leads to a unique positive equilibrium. The condition that that B > > 0 0 is equivalent equivalent to the condition for for invasion by the predator, predator, when the prey prey is is at at aa predator-free predator-free equilibrium. equilibrium. The The condition for for B B > > 0 0 can can be be expressed expressed invades, the system settles into a unique unique p*� < < p; P'I-. Thus, Thus, when the predator predator invades, as p equilibrium in which the prey is reduced reduced to a lower occupancy than than when alone. If when rare. However, a joint equilib­ If B < < 0, 0, the predator predator cannot cannot increase increase when equilibrium rium may nonetheless nonetheless exist if A > > 0, 0, (the larger branch branch in the above above solution). When When this occurs, occurs, the system exhibits exhibits alternative, alternative, locally stable stable states, one with, and pred­ and one without, the specialist predator. Moreover, Moreover, the equilibrium with the the predator present present present has the prey at a higher occupancy than when the prey is present alone. However, bl < I O • For However, this this outcome outcome is is impossible if if Cc~l < CO c0~, and ee20 > eelo. For alter­ alterl ' and 20 > prey system, the predator native equilibria equilibria to exist in this predatorpredator-prey predator must either either enhance As enhance the prey colonization rate, or reduce reduce the prey extinction rate, or both. As noted noted above, above, there there are are reasonable reasonable circumstances circumstances leading leading to to such such counterintuitive counterintuitive effects effects of of predation predation on prey dynamics. When When such effects effects are are present, present, it is feasible feasible for the metacommunity metacommunity to exist in alternative stable states.

B. Three Three Trophic Trophic Levels Levels Let Let us us then then return return to to the the full full food food chain chain model, Eq. (6). (6). Rather Rather than than attempt attempt aa full full analysis analysis of of this model here, here, II will will simply touch touch on on some some interesting limiting limiting

1160 60

RobertD.D. Holt Holt Robert

donor-controlled at each level and where pred­ predcases. Consider a system which is donor-controlled face extinction only when its prey goes extinct, but predators do ators in a patch face not c, e30 e3 11 =not affect affect prey prey extinction extinction rates rates (i.e., (i.e., CO! Col = = C� Col e30 = e20 == ' I == -=c, = e20 = ee, , and ande3 ezl1 = = 0). Without the top predator, the intermediate predator has an equilibrial e2 occupancy of of occupancy

ee ee Clll cC el The top predator predator invades if p� p* > > e/ e/c23, c23 , or p� p * ==h h -

-

- .

ee ee ee hh >>- +-m++ - + ~ CC Cl2 C23 C 12 C23

((11) 1 1)

predator shuts down prey emigration Alternatively, if the intermediate predator (i.e., (i.e., C� C~lI = = 0), 0), and and e2l e2~ = = e31 e3~ = = e32 e32 = = 0, 0 , the the top top predator predator increases increases when when rare rare provided provided h > el___~0+ C01

e2_.__0.0-4-

C12

+ C01

.

((12) 1 2)

\ C23 fl'

For both both special cases, the condition for for invasion by the top predator For predator is more more stringent than than the condition condition for for invasion predator (compare (compare invasion by the intermediate predator conditions ((11) or ((12) (4)).. A habitat rare to sustain the intermediate conditions 1 1 ) or 1 2) to (4» habitat that is too too rare intermediate predator will not contain contain the top top predator, predator, either. However, However, more more common common habitats predator habitats able to sustain a specialist intermediate predator, predator, but not may be able not a similarly specialized top predator. predator. Thus, Thus, if if there are constraints constraints on species species'' colonization colonization specialized abilities, food chains trophic specialists are are not abilities, long long food chains composed composed of of trophic not likely to to charchar­ acterize rare rare habitats (see also Schoener, Schoener, 1989). 1 989). The emerges from The basic basic conclusion conclusion that that emerges from this this model model is that that metapopulation metapopulation dynamics can can constrain of specialist chains, particularly particularly in hetconstrain the length length of specialist food food chains, het­ erogeneous where the erogeneous landscapes landscapes where the basal basal species is specialized specialized to a rare habitat. habitat. Trophic specialization specialization on such automatically forces habitat specialization Trophic such species automatically forces habitat specialization on species of of higher trophic rank, thereby experience experience all the on higher trophic rank, which which thereby the spatial spatial concon­ straints on on the distribution of compounded by additional straints the distribution of the the lower-ranked lower-ranked species, species, compounded additional limitations of of their their own own (Holt, 1993). 1 993). The full model alternative stable The model can can admit admit alternative stable landscape landscape states. For For instance, instance, a top top predator predator may may be be able able to to stabilize stabilize an an intrinsically intrinsically unstable unstable interaction interaction between between an an intermediate intermediate predator predator and and its its own own prey prey (May, (May, 1973b; 1 973b; Rosenzweig, Rosenzweig, 1973), 1 973), thus thus extinction for patches extinction rates rates may may be be low low for patches with with the the full full chain. chain. If If the the landscape landscape initially has has all all species in in all patches, patches, then then it it may may persist persist in in this this state state because because of of low extinction extinction rates. rates. However, However, if if the the system system starts with just just the the intermediate intermediate low starts with predator predator and and its own own prey, prey, the the intermediate intermediate predator predator may may go go extinct extinct because because of of highly cannot highly unstable unstable local local dynamics. dynamics. In In this this case, case, obviously obviously the the top top predator predator cannot invade, prey is absent. landscape may invade, because because its its own own prey absent. Thus, Thus, the the landscape may either either have have just just the prey the entire the prey alone alone or or the entire food food chain. chain.

77

Consequences Consequencesof Spotiol Spatial Heterogeneity Heterogeneity

1161 61

IV. APPARENT APPARENTCOMPETITION COMPETITIONIN METACOMMUNITIES METACOMMUNITIES Food Food chains are are useful useful starting points for examining examining the implications implications of of meta­ metapopulation dynamics for for community structure, structure, but most natural food webs are are much more complex, because because there are typically multiple species species on each each trophic 99 1 ). I will next level and complex linkage patterns patterns across levels (e.g., Polis, 11991). examine the potential for for strong indirect indirect interactions interactions arising in metapopulations, metapopulations, constraining constraining species membership membership in local communities. In standard standard food web models, species richness richness at intermediate intermediate trophic trophic levels is often limited by a com­ combination of of two mechanisms: mechanisms: exploitative compettion (via effects effects of of species at these levels on abundances abundances of of lower trophic trophic levels) and apparent apparent competition (via effects 1 982). effects on the abundance abundance of of higher trophic trophic levels) (Pimm, 1982). Consider a landscape of patches of two habitat types, each containing a single landscape of patches of two habitat habitat specialist. A generalist generalist predator predator which can exploit both prey species in the two habitats patches of habitats is present. If If predators predators can colonize patches of both habitat habitat types from patches patches of of either either type, the dynamics of of the two prey species species are are indirectly linked. If If predators predators increase increase prey extinction rates or depress prey colonization rates, it may be possible for other for one prey species to exclude indirectly the other species, in effect by providing a reservoir popu­ reservoir maintaining a resident predator population (Holt and Lawton, 11994) 994) To explore the potential for apparent landscape context apparent competition competition in a landscape consider the following model, which splices the forms of of model ((1) 1 ) (metapopu­ (metapopulation dynamics in a heterogeneous heterogeneous landscape) landscape) and model (7) (predator-prey (predator-prey metapopulation dynamics):

dPp lr

dt dt

== cClPl(hl - P ~P r -- qr ql)) - ee lrPr pl - p lP( cr (lcl qr rqr l r P r (h r -

+ +

Cc12q2) r2q2 )

dP2

p2 = Pic2rqr + -- c C2Pi z p z (hh 22 -- P2 P 2 - - q2 q2)) - e e 22P2 P 2 - - P2(czlql + C c22q2) 22q2 ) dt dt

((13) 1 3)

dql dt

= p l ( c l l q l + c12q2) -- e lqql

dq2 dt

- - P 2 ( c 2 1 q l q- r

-- e2qq2-

patches of Here, the P Pii are the fraction fraction of of patches of type i occupied occupied by by prey i, and the are the fraction fraction of of such patches occupied by both this prey and the generalist generalist qqii are

predator. standard metapopu­ predator. In the absence absence of of the predator, predator, each prey obeys a standard metapopulation model, in which Ccii and ei e i are respectively the colonization and extinction parameters parameters of the prey species specialized specialized to habitat i (which occupies a fraction hi of patches of the landscape). landscape). The The quantities quantities cij cij scale the rate rate of of colonization of of prey patches 1 ). of of type i by generalist generalist predators predators dispersing from type jj patches patches (as in model 1). The predator habitat-specific rate predator and prey go extinct in each each habitat at a habitat-specific rate eiq• eiq.

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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 ci (i = = 1, 2), then one of of the two inequalities will not hold. In this case, cq 1 , 2), the can increase the prey prey species species with with higher higher cihi cihican increase when when rare rare and and the the other other prey prey species species is common, whereas whereas the alternative alternative prey cannot reciprocally increase when rare. The model shows that given a predator which is both a habitat habitat generalist and a trophic generalist, habitats may generalist, alternative prey species specialized specialized to different habitats apparent compe­ indirectly interact interact via predator predator colonization of prey patchespatches wapparent competition (Holt, 11977, 977, 11984; 984; Holt and Lawton, 11994) 994) at the landscape If such landscape level. If predators are effective colonizers and can induce local prey extinctions, one prey species restricted restricted to the community in one habitat habitat can indirectly exclude another another prey species in a different local community. The potential for prey exclusion via metacommunity dynamics raises an in­ interesting methodological dilemma. Given such exclusion, a survey of of seemingly suitable but empty habitat - gener­habitat patches patches will not reveal reveal the cause of of absence absencemgener alist predators, which which can colonize only after the missing prey has invaded. The usual sort of descriptive surveys may completely miss the dynamical cause for species exclusion landscape. exclusion from a heterogenous landscape. A criterion for for dominance in apparent competition is given by the compound parameter parameter cihi• cihi. Prey species with a low value of of this quantity are particularly vulnerable to exclusion by shared predation. Prey specialized specialized to rare rare habitats (low h) hi) are are more more likely to be excluded excluded by predators sustained by prey prey inhabiting more widespread widespread habitats. habitats. Likewise, prey species which are poor colonists (low c) c;) are more prone to exclusion exclusion by apparent apparent competition. A low ci ci may reflect either poor individual dispersal abilities or low local prey population sizes. I have 984) analyzed a one-predator, have previously (Holt, 11984) one-predator, two-prey species model in which each prey was specialized specialized to a different different habitat. habitat. This model explicitly tracks abundances 1 3» and assumes density­ abundances in each habitat (unlike ((13)) densityindependent sink population structure). independent predator predator dispersal (leading to a sourcesource-sink Such dispersal permits prey to experience experience apparent competition, competition, despite habitat habitat segregation. The prey species with lower intrinsic growth rate is vulnerable vulnerable to exclusion by the alternative prey, and the likelihood of of such exclusion increases with increasing predator predator dispersal.

1! 64 64

Robert Robert o. D. Holt Holt

These These earlier earlier results are consistent consistent with the conclusions conclusions drawn drawn above above for for shared shared predation predation in a metacommunity. Given Given low low inherent inherent extinction extinction rates, the "intrinsic metapopulation is its rate "intrinsic growth growth rate" rate" of of a prey prey metapopulation rate of of colonization, colonization, which is cihi. c i h i . This This compound compound parameter parameter determines determines prey community composition, composition, just just as the usual within-patch usual intrinsic growth growth rate does does in determining determining dominance dominance in within-patch apparent 984; Holt 993). apparent competition competition (Holt, 11984; Holt and and Lawton, Lawton, 11993).

CONClUSIONS CONCLUSIONS Classical metapopulation metapopulation theory theory assumes assumes that landscapes landscapes are comprised comprised of of a large Most models large number number of of patches patches available for for colonization. colonization. Most models assume that the patches are physically homogeneous. homogeneous. Yet in natural natural landscapes, metapopulations metapopulations are are likely to span span a wide wide range range of of local environmental environmental conditions. conditions. In this chapter, chapter, I have po­ have used used variants of of the Levins Levins metapopulation metapopulation model model to examine examine some some potential consequences consequences for for community community structure structure of of habitat habitat heterogeneity. heterogeneity. These These theoretical theoretical results suggest suggest that sparse sparse habitats habitats in a heterogeneous heterogeneous land­ landscape scape are likely to sustain sustain a biased biased array of of species, including including habitat habitat specialists specialists with unusually unusually high high colonization colonization or low extinction extinction rates and and habitat habitat generalists generalists sustained sustained via spillover from more more abundant abundant habitats. Trophic cation of Trophic specialization specialization leads to a kind kind of of magnifi magnification of these these effects, so that each persist­ each additional additional level must must satisfy increasingly stringent stringent criteria for for persistence. One One broad broad implication implication of of this result is that that metacommunity metacommunity dynamics au­ automatically tends tends to constrain constrain food food chain chain length. length. Trophic Trophic generalization generalization leads to an avenue avenue for indirect interactions interactions among among alternative alternative prey species. If If alternative prey species are are habitat habitat specialists, but a predator predator is a habitat habitat generalist, predator predator colonization colonization can can couple the dynamics dynamics of of these these prey prey species. This gives rise to apparent apparent competition competition at the metacommunity level, which which in some some circumstances circumstances can can lead to the exclusion exclusion of of prey species that that are poor poor colonists, colonists, or are are specialized specialized to rare rare habitat habitat types. The The ideas ideas presented presented here here provide provide a first pass pass through through the the potential potential implica­ implications tions of of habitat heterogeneity heterogeneity for for metacommunity metacommunity dynamics dynamics and and structure. structure. One One promising promising direction direction for for future future work work will be be in developing developing spatially spatially explicit explicit models models (Kareiva and and Wennergren Wennergren 11995; Nee et et al. al.,, this volume) volume) with limited dispersal dispersal (Kareiva 995; Nee and and various various patterns patterns of of spatial heterogeneity. heterogeneity. My My expectation expectation though, though, is that that the the general conclusions reached general conclusions reached here here will prove prove robust. robust.

ACKNOWLEDGMENTS ACKNOWLEDGMENTS I thank reviews of the manuscript thank Ilkka Ilkka Hanski Hanski and Jan Bengtsson Bengtssonfor very very thoughtful thoughtful reviews manuscript and the National National Science Science Foundation Foundation for financial financial support. support.

8

Genetic Effective Size of a Metapopulation Philip W. Hedrick

Michael f.E. Gilpin

Population structure has long been been recognized recognized as having a major influence influence Population on the of genetic genetic variation has been been the topic topic of of exten­ the maintenance maintenance and and loss of variation and and has extensive research in population genetics (e.g., Wright, 11978; 978; Slatkin, 1985, 1 985, 1987). 1 987). of the impact of population structure on genetic variation Generally, investigation of subpopuhas assumed that subpopulation sizes remain constant over time, i.e., subpopu­ lations do not go extinct. It has been shown that if a population exhibits meta­ metapopulation dynamics, i.e., patches in which subpopulations exist become unoc­ unoccupied because of local extinction, that many of of the generalizations generalizations of of earlier studies of population structure 1 977; Maruyama and Ki­ structure may not hold (Slatkin, 1977; Kimura, 980; Wade mura, 11980; Wade and and McCauley, 1988; Gilpin, 1991). 1 99 1 ). The amount of genetic variation in a population is generally determined using the measure of both measure heterozygosity because because of both its biological importance (individuals are are either heterozygotes or homozygotes) and the extensive theory that predicts heterozygosity levels due to various evolutionary factors. In the present context, we are concerned with two different aspects of heterozygosity: the average average level distriof heterozygosity in a subpopulation or a metapopulation and the spatial distri­ bution of heterozygosity due to the structure of the population. Generally, the steady-state values of these heterozygosity values are of interest to evolutionary genetics, while changes, particularly losses, in heterozygosity are of particular importance to conservation biology. Metapopulation Metapopulation Biology Biology Copyright © All rights orm reserved. Copyright 9 1997 1997 by Academic Academic Press, Inc. All rights of of reproduction reproduction in in any any fform reserved.

1165 65

1166 66

PhilipW. W. Hedrick Hedrickand and Michael MichaelE.E. Gilpin Gilpin Philip

Both Both the the steady-state steady-state levels levels and and changes changes in in heterozygosity heterozygosity are are governed governed by by the the effective effective size size of of the the population. population. In In general, general, the the effective effective population population size size cor­ corrects rects census (or (or breeding) breeding) population population number number to to account account for for aa variety variety of of (mainly (mainly demographic) demographic) real real world world considerations considerations such such as as the the sex sex ratio ratio of of breeding breeding individ­ individand life history characteristics (e.g., Lande and and BarrowcIough, Barrowclough, 11987; Cabauals and 987; Caba­ I1ero, 994). The llero, 11994). The effective population size is usually calculated for for a group of individuals with given particular particular demographic demographic properties properties in which there there is random random mating (although other mating structures have also been examined, CabaIlero, Caballero, 11994). 994). The The effective population size size is generally defined as the size of of an ideal population that results in a given variance in allele frequency or amount of in­ inbreeding. However, because of our interest in the level of genetic variation, we will estimate the effective population size in an ideal population that results in a given loss of heterozygosity (this has been termed the eigenvalue effective pop­ population size by Ewens, 11989). 989). This effective size can be estimated either for subpopulations of a metapopulation or for total metapopulation composed of a group of subpopulations. Population dynamics similar to that in a theoretical metapopulation in natural populations are not uncommon (Harrison and Taylor, this volume) and there do appear to be particular instances in which habitats are fragmented that metapopmetapop­ ulation dynamics is an appropriate description of the population structure structure at a 995b; Thomas and Hanski, this volume). As regional level (e.g., Hanski et et al. al.,, 11995b; a result, there has been increasing interest in the impact impact of of metapopulation struc­ structure on genetic variation in endangered species and other organisms that exist in either of of natural or human causation (Hastings extremely fragmented habitats, either and Harrison, 11994). and 994).

I. AN AN EXAMPLE EXAMPLE Before examining examining the the specific effects effects of of metapopulation metapopulation dynamics dynamics on on effeceffec­ Before tive population population size, size, it is useful to to give give an an heuristic heuristic example example to demonstrate demonstrate how how metapopulation dynamics dynamics can can influence influence the the maintenance maintenance of of genetic variation. variation. GilGil­ metapopulation pin (1991) ( 1 99 1 ) gave simulation example in which which he he assumed assumed that that there there are are pin gave a simple simple simulation example in three subpopulations subpopulations or or patches patches in the metapopulation, metapopulation, each with with an effective effective three (and census) census) population population size size of of 500. 500. All AIl the the subpopulations subpopulations were were initiated initiated with with aa (and high high level level of of heterozygosity. The important important sequence sequence of of events events in in this this simulation simulation starts starts in in generation generation 48 48 The (see Fig. Fig. 1) 1 ) when when patch patch 2 goes goes extinct extinct and and is is recolonized recolonized from from patch patch 3 with with aa (see consequent reduction reduction in in heterozygosity. This This loss loss occurs occurs because because it it is is assumed assumed consequent that recolonization recolonization is is by by only only two two individuals, individuals, e.g., e.g., aa fertilized fertilized female. female. The The next next that significant event event is is when when empty empty patch patch 11 is is recolonized recolonized from from patch patch 22 with with aa founder founder significant population having having no no genetic genetic variation. variation. Finally, FinaIly, when when patch patch 22 goes goes extinct extinct in in population

Genetic Genetic Effective Effective Size Size Of of aa Metapopulafion Metapopulation

88

167 1 67

PATCH PATCH 1

HIGH H H HIGH

aI

2

HIGH H H HIGH

3

HIGH H H HIGH

00 I

FIGURE ]1 FIGURE

++ ,

LOW H H LOW

++

J

,4I

+

,~l . . ,

H == 0O H

i i '

: LOW LOW H H

,

II

20 20

40 40 I

60 60 I

GENERATION GENERATION

.

II

. ' '

,

. , . .

,t;' . H=0 H=O I , , .

, .

H= O ,t+ H=0 80 80 I

_

100 1 00 I

The level level of of heterozygosity heterozygosity (H) (H) over over time time in in aa simulation simulation of of aa population population existing existing in in The three patches patches (after (after Gilpin, 1 99 1 ). The short vertical vertical bars bars on on the the right-hand right-hand end end of of horizontal horizontal lines lines three Gilpin, 1991). The short indicate extinctions in aa patch and the the arrows arrows indicate indicate recolonization. recolonization. indicate extinctions in patch and

generation 71, 7 1 , the metapopulation metapopu1ation has has no variation although although there are still 500 500 generation no variation individuals remaining remaining in patch patch 1. 1 . All All of of these individuals individuals can be traced traced back back to to individuals can be some 1 . Gilpin 199 1 ) termed this some individuals individuals in patch 3 before before generation generation 551. Gilpin ((1991) this pro­ procoal­ cess through which metapopulation dynamics reduces reduces genetic variation the coalesence of metapopu1ation, i.e., the loss of of the metapopulation, of genetic variation being traced traced back to a few individuals that are the ancestors present ancestors of of all the individuals in the present metapopulation. metapopulation. This example illustrates an extreme case in which the loss of of genetic variation in the metapopu1ation metapopulation can be dramatically lower than that expected from a pop­ population the size of the average census number in the system. Gilpin ((1991) 199 1 ) found that in general the most dramatic lowering of of genetic variation occurred for for ex­ extinction and and recolonization values at which the average number number of occupied patches was low enough that that the metapopu1ation metapopulation itself was in danger danger of extinction. However, genetic variation may be of secondary interest in metapopulations with high extinction expectation so we will examine metapopulations that include more patches and with a balance of local extinction and recolonization rates that makes the the likelihood likelihood of of extinction extinction low. low.

II. II. METHODS METHODS One One methodical approach used used to examine examine metapopulations in a population genetics 1 977), Maruyama 1 980), Wade genetics context context is is that that of of Slatkin Slatkin ((1977), Maruyama and and Kimura Kimura ((1980), Wade and 1 988), Ewens 1 989), and and McCauley ((1988), Ewens ((1989), and Barton Barton and and Whitlock (this volume). volume). These 1 970) These authors authors use use an an infinite infinite (or (or finite) finite) patch, patch, spatially spatially implicit, implicit, Levins Levins ((1970)

1168 68

Philip Hedrick and and Michael Michael E. E. Gilpin Philip W. Hedrick Gilpin

metapopulation structure in which there is instant recolonization of of empty patches. Thus, Thus, some constant constant fraction of of the local populations go extinct each each generation, all of which are immediately recolonized by some number of colonists of are recolonized number of which then, during during the time step (or over time, Barton and and Whitlock, Whitlock, this volume), grow grow up to the the local carrying capacity. Our Our model, on the other other hand, is expanded expanded from the earlier earlier approach approach of of Gilpin ((1991) 1 99 1 ) as introduced above and 1 977) and and differs differs from the the approaches approaches of of Slatkin ((1977) patches, each of of others in several ways. First, our model has a finite number of of patches, which can support a local population, but which but which can be empty for a number of of time steps (see examples examples in Figs. 11 and and 2). Second, we decouple decouple gene flow from the the number 1 988). Third, number of of colonists, a possibility suggested by Wade Wade and and McCauley McCauley ((1988). the time that is governed by the the colonization the time that aa local local population population remains remains extinct extinct is governed both both by colonization probability and also by the number of extant source patches. Finally, we examine probability and also by the number of extant source patches. Finally, we examine the influence influence of of metapopulation dynamics on genetic genetic variation separate separate from ge­ genetic drift within patches infinite population size within a patch. patches by assuming assuming an infinite patch. While the previous approaches approaches can be approximated analytically, our our approach approach appears to be tractable tractable only using computer simulation (however, see Whitlock appears and 996). Further, and Barton, 11996). Further, while while the the general general behavior behavior of of the the two two approaches approaches are are similar, in some cases appear to yield quantitatively different cases they appear different answers. Because of parameters parameters that Because there there are are a number number of that can influence influence the effective effective population metapopulation, and nature of population size size of of aa metapopulation, and because because of of the the complicated complicated nature of the interaction model, the interaction of of stochastic stochastic processes processes within within and and between between patches patches in in our our model, understand the process of of heterozygosity loss we will use computer computer simulation to understand and estimate estimate the effective effective size in a metapopulation. We We will check check the simulations through the use of analytical approximations for the behavior of through of approximations for of single patches. patches. Our approach will be to assume Our approach assume some standard standard conditions conditions and and then then sequentially sequentially alter these these various parameters parameters and and assumptions assumptions to determine their effects. effects.

Description of Parameters Parameters A. Description First, let us assume that the metapopulation is divided up into Np Np patches, patches, each of patch, random assumed. of local population population size K. Within Within each each patch, random mating mating is assumed. We We will examine examine the the changes changes in in heterozygosity heterozygosity for for aa single single locus locus with with two alleles, both of of which have an initial frequency frequency of of 0.5 in all patches. patches. both 1 970) we assume Following Levins ((1970) assume a colonization rate rate (probability) of of c and extinction rate e and based on occupancy of all other patches in the meta­ and extinction rate and based on occupancy of all other patches in the metapopUlation. patches in the meta­ population. The The variable p p** gives the observed fraction of of patches metapopUlation that We assume population that are are occupied at at any any one one time. We assume that the the actual actual proba­ probac* = cp* so bility of of colonization colonization to to an an unoccupied unoccupied patch patch is c* = cp* so that that if if some some of of the the patches are not occupied, occupied, the rate of of colonization is lowered because because the pool of of potential potential colonizers colonizers is is reduced. reduced. For For cc** < < e, the the metapopulation metapopulation will go go extinct. extinct. For nite number patches, extinction is possible even even For a metapopulation metapopulation with a fi finite number of of patches, for for cc** > > e, much in the the same way that that a small population population can go extinct from demographic demographic stochasticity even with the individual birth rate greater greater than the

88

Genetic Metapopulafion GeneticEffective Effective Size Size of of aa Metapopulation

1169 69

individual death rate. Note that this is a spatially unstructured unstructured model, essentially equivalent to the original model of 1 970). of Levins ((1970). NtI founders randomly chosen from a A patch is assumed to be colonized by N given occupied patch, termed the propagule-pool model or randomly chosen from all occupied patches, patches, called the migrant-pool model (Slatkin, 1977). After reco­ reco1 977). After lonization, Ionization, it is assumed that the subpopulation expands expands in one generation to its generation are randomly carrying capacity, K. The individuals in the following generation drawn from the parental allele frequency pool to simulate genetic drift within a patch. After uence of After evaluating evaluating the infl influence of genetic drift in local populations, populations, to de­ determine the impact impact of of metapopulation dynamics independent independent of of genetic drift within a patch, we will assume that the number number of of individuals within a patch is infi nite. When the infinite. the population size size is assumed to be infinite within the the patch, then there is no change in allele frequency from from genetic drift and the only change change When there is gene flow, flow, within a patch occurs from gene flow when it is present. When each generation a proportion m of the individuals in a given patch patch comes from another given occupied patch, making the total amount of gene flow into a patch patches. per generation, mN;, mN*, where N; N* is the number number of of occupied occupied patches.

B. Estimation Estimation of Effective Effective Metapopulation Size Size and Other Values To estimate estimate the the effective effective population size, Ne, the the relationship relationship which which gives the the change in heterozygosity between consecutive generations,

-( ( -1))

1 H , , + !1 = H Ht+ H ,t 11 - 22~V~ Ne ' -

-

"

((la) 1 a)

is used whereH where HtI andH andH,+l I+ ! are the mean heterozygosities over all occupied patches and over replicate computer simulations in two consecutive generations t and ft + 985). Ne + 11,, respectively (e.g., Hedrick, 11985). N e is is the effective population size that results in the given amount of loss of heterozygosity between the two generations. Therefore, an estimate of the effective population size is Therefore,

=

H H,

, N Nee = 2(H 2(H,, - H O , +, +l ) !" )

(( l1 bb))

If � wee assume assume that the average heterozygosity within a subpopulation (or patch) patch) is H Hs, of the average effective subpopulation size is s , then the estimate of H ,(s) Ht(s~ = 2(H,~s)Ne(s~ = 2(H H, + ! (s) ,(s) - H,+l~s~)"

Ne(S)

(2a)

(2a)

Likewise if H metapopulation (calcu­ HvT is the average heterozygosity in the total total metapopulation (calculated from the global allele frequency in the metapopulation), metapopulation), then the estimated estimated for the metapopulation is effective population size for

Ht~ Ne~ = 2(H,~ - H,+~)"

(2b) (2b)

1170 70

Philip W. Hedrick Hedrick and and Michael Michael E. E. Gilpin Philip Gilpin

To for To estimate estimate heterozygosity, heterozygosity, aa given given metapopulation metapopulation simulation simulation was was run run for 21e The first 2/e + + 25 25 generations. generations. The first 21e 2/e generations generations were used used to allow the metapop­ metapopulation dynamics dynamics to to become become stabilized stabilized (the (the expectation expectation is is that that approximately approximately 90% more extinctions during this period, period, 90% of of the patches patches would would have have had had one one or or more extinctions during of the last 25 pairs of - ((1I - ee)(2/e)), and the heterozygosities heterozygosities of of consecutive consecutive gen­ gen) > 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 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

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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

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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

207 207'

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

208 208

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

99

209 209

b

a

0.2

G' 0.7 '"
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)

2211 9?

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 > 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 ._

= '1:1

;Z

.S

800

600

400

200 0

f

J

2

3

4

5

6

7

8

9

10

Patch age

FIGURE population age FIGURE 77 Age structure structure of individuals individuals in each each population age class class at metapopulation metapopulationequilibrium. equilibrium. The model model was was iterated iterated with with disturbance disturbance rates rates of landscape landscape L2 (Table (Table I) and population population lifespan lifespan of :z = 0, migration 0.5, constant rate of 0.9, fecundity of 2. = 110, migration rate of 0.5, constant adult adult survival survival rate 0.9, and annual annual fecundity

3314 14

Isabelle Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-Henri Gouyon Gouyon

and and adult adult survival rate rate of of 0.45. 0.45. Finally, let genotype genotype A be annual annual (semelparous), (semelparous), with net 2. The net fecundity fecundity of of 112. The results results are given Table Table II. The lifespan, the more peren­ The results results show show that the longer longer the population population lifespan, more the perennial genotypes genotypes are selected for. /, 4; A, 6), interesting By dividing dividing all fecundities fecundities by half half (P, 2; I, interesting differences differences emerge perennials emerge (Table (Table II). Notice Notice that that a polymorphism polymorphism between between annuals annuals and and perennials is possible 1 ), indicating indicating that possible (here (here for for zz = = 111), that disruptive disruptive selection selection on life histories histories high, the may occur. When When zz is small, the annual annual genotype genotype is favored. favored. When When zz is high, the may perennial perennial type is favored. favored. When When zz is intermediate, intermediate, the intermediate intermediate I/ may be be able to replace pairwise contests intermediate replace one or the other other genotype genotype in pairwise contests (here (here the intermediate 1 ) and natural selection wins against the annual annual for for zz = = 111) and still be be eliminated eliminated by natural selection when when both both other other types are are present. present. This This example example illustrates illustrates that that dynamics dynamics of of game game theory models with more Maynard-Smith, more than than two two strategies strategies may may be complex complex ((Maynard-Smith, 11982). 982). Such Such disruptive disruptive selection occurs occurs here here because because the empty empty patches patches are faster faster recolonized by the perennial type for recolonized the annual annual type, whereas whereas the the more more perennial type is selected for niches, the intermediate intermediate type does than the in old populations; populations; in both both niches, does less less welI well than best adapted adapted genotype. In this particular particular case, the dynamics dynamics are highly dependent dependent conditions, even though though the the only evolutionarily evolutionarily stable state is the oc­ ocon initial conditions, currence perennial types in frequencies frequencies of 2 : 3. This currence of of both both annual annual and and perennial of ca 2:3. This is illustrated 1 00 illustrated in Fig. 8, which which shows shows that when when the simulation simulation is initiated with with 100 seeds of of each each genotype genotype in each each empty patch, patch, the annual annual genotype genotype almost goes extinct because of of competition with the intermediate intermediate genotype genotype /, I, before before /I starts starts to decline because of alIowing the annual of competition with the perennial, perennial, allowing annual type to increase increase again. again. The The metapopulation metapopulation processes processes may thus thus lead lead to particular particular and and

Invading P, or or Pairs Invading Genolype Genotype in in Contests Contests Involving Involving Either Either the the Three Three Types, Types, A, A, I,I, P, Pairs of of Them, Them, as as aa Function Function of of Population Population lifespan, Lifespan, zZo~

TABLE TABLE IIII

Genotypes Genotypes intially intiaily present

A, I, I, P P A,

z

77 8 9 9 1100 I11I 1122

A, A, II

A, A, P P

I, I, P P

P 1"

1/2 c 1I2c

11

1/2 112

11

1/2 112

A A

A A A A II

I I I P P PP P P

I I I II P P P P

A A A A A A P P P P P P

A A A A A A A A A A ++PP P P

I I P P P P P P

I

A+P A+P P P

"" A A is is intermediate. is annual, annual, P P is is a a perennial, perennial, I is intermediate. h Fec Fec = = II ;; see see text text for for fecundity fecundity and and survival survival of of each each genotype. genotype. h '' Fec 1/2; same I, but all fecundities fecundities were were divided divided by 2. Fec = = 1/2; same as as Fec Fec = 1, but all by 2. =

1

A A I I I I I

112 1/2 A A A A II I I I

1133

EvoluTIon Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits

3315 15

600 600 SOO 500

A A

r~ '"

-; 400 400

= "C

.> . .

"C

Perennial Perennial

s 300 .. 300

.

Q

'"'

�

a i.

200 200 Annual

100 100

+ Intennediate Intermediate

0

o 0

~ I : Q I I I I I I I I I : : : : ,-v-y~,, o 0 o 0 on

o 0 o 0 o -

0 0

0 0

Generations Generations FIGURE F[GURE 8 8 Evolution of the number of of individuals of of three genotypes with variable life-histories. The y-axis gives the numbers per patch of age class I1 (newly founded populations). Genotype A is of 6; genotype P P is perennial with fecundity of of 2 and adult survival rate of of 0.9; annual with fecundity of genotype I is intennediate intermediate with fecundity of 4 and adult adult survival rate of of 0.45. All three genotypes have the same migration rate of I I was assumed. In this of 0.5. Landscape L2 (Table I) with zz = = 11 example, the simulation was started with 1100 O0 seeds of of each genotype in each patch. patch. The evolutionarily stable state (A (A + + P) was insensitive to initial conditions.

unexpected metapopulation, it is likely u n e x p e c t e d ddynamics y n a m i c s of of gene g e n e frequencies. f r e q u e n c i e s . In In a a finite finite m e t a p o p u l a t i o n , it is likely that increase. that the the annual a n n u a l genotype g e n o t y p e would w o u l d go go extinct extinct bbefore e f o r e its its frequency f r e q u e n c y starts starts to to increase. In selects for In this this eexample, x a m p l e , decreasing d e c r e a s i n g fecundity f e c u n d i t y selects for increasing i n c r e a s i n g reproductive r e p r o d u c t i v e effort effort (annual is apparently the colonization of nnew (annual life life cycles). cycles). This This is a p p a r e n t l y bbecause e c a u s e the c o l o n i z a t i o n of e w patches patches limiting factor. bbecomes e c o m e s the the limiting factor. When When a a stable stable ppolymorphism o l y m o r p h i s m bbetween e t w e e n annual a n n u a l and and pperennial e r e n n i a l types types can can be be maintained, in yyoung m a i n t a i n e d , the the frequency f r e q u e n c y of of the the annual a n n u a l type type is is hhigh i g h in o u n g ppopulations, opulations, whereas Fig. 9). w h e r e a s the the frequency f r e q u e n c y of of the the perennial p e r e n n i a l type type increases i n c r e a s e s after after colonization c o l o n i z a t i o n ((Fig. 9). Within W i t h i n populations, p o p u l a t i o n s , selection s e l e c t i o n favors favors perennials, p e r e n n i a l s , while w h i l e between b e t w e e n ppopUlations o p u l a t i o n s an­ annuals are is exactly in nuals are selected s e l e c t e d for. for. The T h e two-level t w o - l e v e l selection s e l e c t i o n pprocess r o c e s s is e x a c t l y the the ssame a m e as as in

316 31 6

Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-HenriGouyon Gouyon Isabelle - - ~>C - - Annual Annual genotype genotype

-

-

--D-

Perennial Perennial genotype genotype B

A

0.9 0.9

0.8 >. 0.8

0.7 = 0.7 � 0.6 0.6 "

;.

t 00.5 .5 &

......

0.4 � 0.4 = 0.3 � 0.3

� 0.2 0.2 0.11 0. 0

-�

FIGURE99 FIGURE

�.......

I

N

I

M

t

' 22 22

Migration rate Migration d d 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.70 0.70 0.20 0.20 0.50 0.50 0.50 0.50

Winners Winners of of pairwise pairwise contests between between annuals annuals and perennials perennials

Annual Annual Annual Annual Annual Annual Annual Annual Annual Annual Polymorphism Polymorphism Annual Annual Annual Annual Perennial Perennial Annual Annual Polymorphism Polymorphism Perennial Perennial Annual Annual Perennial Perennial Perennial Perennial Polymorphism Polymorphism Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial Perennial

"An annual with and aa perennial annual fe" An annual with fecundity fecundity of of 66 and perennial with with constant constant annual fe­ cundity and survival rate rate of of 0.9 0.9 were were assumed assumed in landscape landscape L2 L2 cundity of of 2 and (A" = = 0.4; 0.4; A A II == 0.9, 0.9, A_ k= = 0.95). 0.95). (Ao

1133

Evolution Evolutionof of Migration Migration Rate Rate and and Other Other Traits Traits

321 321

which they compared populations of compared the equilibrium density of of monomorphic monomorphic populations of either annuals annuals or or perennials, they suggested that "long-distance clonal spreading" spreading" of of herbaceous herbaceous perennials perennials might explain their finding, especially if seeds were were more sensitive to disturbances disturbances than stems. Our Our own own simulation results point to the same direction, though we provide a different different explanation. In highly disturbed disturbed habitats, high migration rates are expected to evolve. In such habitats, only species with high fecundity can sustain sustain a viable metapopulation. We showed showed in Section Section III that perennial life histories histories are favored when fecundity and and migration migration rates rates are high. For certain landscapes, landscapes, migration rates and fecundities may be such that perennials perennials are favored over annuals when disturbance disturbance rates increase, as found found by 1 994). Fahrig et al. ((1994). 1 99 1 ) conducted Ben-Shlomo Ben-Shlomo et al. ((1991) conducted a selection experiment on the the migration ability of the flour beetle Tribolium found that correlated Tribolium castaneum, castaneum, and they found correlated responses to selection included shorter generation times. Roff ( 1 986) has shown responses generation Roff (1986) that there are fitness costs associated with the ability to disperse. These correla­ These correlations are likely to affect the evolution evolution of of migration behavior, as formally shown shown by Cohen and Motro 1 989). Den 1 990), in contrast, found Motro ((1989). Den Boer Boer ((1990), found that dispersing production than nondis­ individuals of of some arthropod arthropod species had a higher higher egg production than nondispersing persing individuals. A review of studies of of genetic correlations correlations among migration characters tness components characters and other fi fitness components in insects may be found found in Roderick Roderick and Caldwell ((1992; 1 992; see also Roff, 990). The idea that life-history characters Roff, 11990). characters do not of course not new. evolve independently is of

VII. CONCLUSION CONCLUSION We have shown in this this chapter chapter that the processes processes determining the migration rate in a metapopulation specific to the very functioning metapopulation are specific functioning of of the metapopulation metapopulation (the metapopulation effect) and interacting with the processes determining the evolution cant life-history traits. These evolution of of most signifi significant These processes result in a partial adaptation of of the metapopulation to its landscape. landscape. This adaptation is incomplete because the processes processes involved act between between genes at both both the population population and and the metapopulation metapopulation levels. They thus necessarily involve frequency dependence dependence and ' s fundamental theorem of Fisher's of natural selection does not apply at the therefore Fisher metapopulation metapopulation level. One One of of the difficulties involved in the study study of of metapopulation metapopulation evolutionary of processes is the general confusion that characterizes the questions questions of of levels of selection. The controversy between group group selectionists (sensu (sensu Wynne-Edwards, Wynne-Edwards, 11971) 97 1 ) and individual selectionists (e.g., Williams, 11971) 97 1 ) has led to radical radical posi­ positions and cult to sort of view and made made the whole whole point diffi difficult sort out. Roughly, one point point of is the one developed 970s, which states developed during during the 11970s, states that group group selection acts necessarily against individual selection. A more recent point of of view defends defends the idea that all levels can be taken into account, whether they act in different or similar directions. This latter point of of view seems much more promising and

322

Isabelle Isabelle Olivieri Olivieri and and Pierre-Henri Pierre-Henri Gouyon Gouyon

tractable than Lloyd, 11994). 994). It can be formalized in diverse ways, the than any other other ((Lloyd, best known 1 970), extended by Wade ((1985), 1 985), known being the one developed by Price ((1970), 1 992) to, and applied by Frank ((1986, 1 986, 11987, 987, 11994b) 994b) and Goodnight Goodnight et et at. al. ((1992) to, for for instance, migration, sex-ratio, and altruism [see also Heisler and Damuth 1 987) Damuth ((1987) and Damuth 1 988), for Damuth and Heisler Heisler ((1988), for extensions extensions to multiple characters characters evolution evolution and discussion). It consists of and further further discussion]. of a nested nested decomposition decomposition of of forces acting acting on allelic frequencies, assigned to each frequencies, with one component component being assigned each level at which which supposed to act. The levels may be within individual individual (e.g., repeated selection is supposed repeated sequences), populations, etc. sequences), within colony (or deme), within population, population, among populations, If If aa DNA DNA sequence sequence that that can can be be repeated repeated in in the the genome genome of of an an individual individual acts acts positively or negatively on its fitness, nobody nobody will deny that that the study of of the evolution evolution of of this sequence sequence has to take take into account account both both the the within- and and the among-individuals component of understand its fate. Oddly of selection to understand Oddly enough, enough, it seems much cult for evolutionary biologists to jump much more more diffi difficult jump one more more level. If a gene can be selected selected within a population population and a n d influence influence the probability that that a new patch is colonized, then, its fate se­ fate can be analyzed in terms of of two-level selection. Whether Whether this is done explicitly or not in the models is just a question of of words. Unfortunately, Unfortunately, the confusion confusion between this hierarchical approach approach to levels of of selection selection and and the the naive naive Panglossian Panglossian group group selection postulates postulates still survives. In their very clear review about about the evolution of of migration, Johnson Johnson and Gaines Gaines ((1990) 1 990) still mix in their "group "group selection selection models" section section a paper paper by Wynne­ WynneEdwards 1 962) with a model 1 97 1 ). This last model explicitly Edwards ((1962) model by Van Valen ((1971). distinguished distinguished the individual and and demic levels of of selection, while others others (e.g., Comins et 980; Levin et 984), who et al., al., 11980; et at. al.,, 11984), who have treated these levels implicitly, are classified in the "individual "individual selection" selection" section. The The fact that these these models models do do not not differ differ in their assumptions but in their wording is made clear by the the fact that they produce the same results! (Assuming survival rate during migration is very low, all these models find find that the ES migration rate is equal to the local extinction rate.) The same confusion confusion can be found found in a recent recent review about about metapopulation metapopulation ( Hastings and Harrison, Harrison, 1994), 1 994), where consequences of dynamics and genetics (Hastings of 1 985), Olivieri et 1 990), the metapopulation metapopulation effect described described by Rice and and Jain Jain ((1985), et al. al. ((1990), 1 992) are treated as interdemic selection, leading the and and Manicacci et et al. al. ((1992) the authors authors to conclude that "however, this is open open to explanation explanation in terms of of individual selection." The confusion, confusion, once again, comes from the lack of of distinction between between the entity that 976; or the information, that is selected selected (the replicator, Dawkins, 11976; Gouyon 988; Gliddon 989) and the level of Gouyon and Gliddon, Gliddon, 11988; Gliddon and Gouyon, 11989) of inte­ integration 976; gration at which differences differences in reproduction reproduction exist (the interactor, Dawkins, 11976; or 988; Gliddon 1 989). All formal­ or avatar, Gouyon and Gliddon, Gliddon, 11988; Gliddon and Gouyon, 1989). All formalized models of of migration assume that the selected entity is the genetic information. ex­ information. Sometimes the two levels (within and among populations) populations) are are explicitly treated treated and sometimes they are not. Both levels are nonetheless nonetheless always included in the analyses, leading leading to convergent convergent analytical and simulation results.

1133

EvoluTIon Evolutionof of MigraTIon Migration Rote Rate ond and Other Other Traits Traits

323 323

ACKNOWLEDGMENTS ACKNOWLEDGMENTS Sandrine Bernard Godelle, Sandrine Maurice, Yvain Dubois, Dubois, Stuart Baird, Baird, Bob Bob Holt, Bernard Godelle, and Stephanie St6phanie Brachet made of this chapter. made useful comments comments on a final version of chapter. Susan Mazer, Mazer, Yannis Yannis Michalakis, Michalakis, Ilkka Hanski, Simon Levin, and an anonymous anonymous reviewer raised raised very interesting interesting points points on the sub­ submitted version. Anne-Marie Duffour helped with the documentation, documentation, and and Stephanie St6phanie Brachet Brachet was very helpful in drawing 3-D figures and Fig. I1 was drawn by Jean-Yves Pontallies. The present present version benefited from the considerable considerable help of of Ilkka Hanski, who made extensive (useful! (useful!)) changes changes on the paper. This is publication publication ISEM95-097 ISEM95-097 of the Institut des Sciences Sciences de l'Evolution, Montpellier.

14

Spatial Processes Processes in Host - Parasite Genetics Host-Parasite Genetics Steven Steven A. Frank

I. INTRODUCTION INTRODUCTION Host - parasite diversity can be described first is Host-parasite described in two different different ways. The The first simply the observed parasites in a particular particular observed variability among among the hosts hosts and and parasites population. For 1 99 1 ) classifi ed 67 wild flax plants plants population. For example, example, Burdon Burdon and Jarosz Jarosz ((1991) classified into 110 0 distinct distinct resistance genotypes when when tested against six races races of of flax rust. One One host genotype genotype was was completely resistant resistant to all six pathogen pathogen races, races, whereas whereas another ve of another genotype was susceptible susceptible to fi five of six races. races. The The second second type type of of variability is the range range of of potential potential genotypes genotypes that can can occur over 1 985) used field transplant ex­ occur over space space and time. For For example, example, Parker Parker ((1985) transplant experiments fungal pathogen periments to study the legume Amphicarpaea bracteata and its fungal pathogen Synchytrium decipiens. heavy in each of three locations. decipiens. Fungal Fungal infection infection was heavy each of three locations. However, a plant However, plant moved moved to a new new location location developed developed little or or no no infection, infection, sug­ suggesting among sites. In a second gesting that that the pathogen pathogen populations populations differ differ among second experiment, experiment, host their ability to resist host lines derived derived from from different different locations locations varied varied in their resist a single single pathogen host populations. populations. pathogen isolate, indicating indicating spatial spatial differentiation differentiation among among the the host Parker's Parker's study suggests suggests that that the potential potential range range of of diversity over over space space and and time is often observed in a single location. The The poten­ often greater greater than than the variability observed single location. potential diversity is limited by the biochemistry morphology of parasite biochemistry and and morphology of hosthost-parasite traits, whereas controlled by the local dynamics of whereas the observed observed diversity is controlled of disMl'tap(} Metapopulation l'u/alioll Biology

Copyright «) 1997 by Academic Press, Press. Inc. All rights of reproduction in any form reserved. reserved. 9 1997

325 325

326 326

Steven A. A. Frank Frank Steven

ease and and the the global global processes processes of of extinction extinction and and colonization colonization in in the the metapopulametapopula­ ease tion. tion. The first first goal goal of of this this paper paper is is to to suggest suggest that that increasing increasing potential potential diversity diversity The causes aa qualitative qualitative shift shift in in metapopulation metapopulation dynamics. dynamics. Local Local processes processes dominate dominate causes when potential potential diversity diversity is is low. low. Colonization-extinction Colonization-extinction dynamics dynamics of of alleles alleles in in when the metapopulation metapopulation become become more more important important with with an an increase increase in in the the potential potential numnum­ the ber of of distinct distinct genotypes. genotypes. In In the the next next section section I present present aa simple simple model model to to illustrate illustrate ber the importance importance of of potential potential diversity. diversity. the After briefly briefly discussing discussing the the model, model, I review review evidence evidence that that many many host-parasite host- parasite After systems do do in fact fact have have high potential potential diversity. diversity. Examples Examples include include plant-pathplant- path­ systems ogen genetics genetics and and bacterial bacterial defense defense systems against viral parasites parasites and and conspecific conspecific ogen competitors. I also discuss the antagonistic interaction between cytoplasmic and competitors. also discuss the antagonistic interaction between cytoplasmic and nuclear genes in cytoplasmic male sterility. nuclear genes in cytoplasmic male sterility. Data from from these these studies studies suggest suggest that that spatial spatial variation and colonization-excolonization - ex­ Data variation and tinction important in the the observed patterns of of diversity. However, tinction dynamics dynamics are important observed patterns diversity. However, data are interpret because because of of limited limited sampling sampling over space the data are difficult difficult to interpret space and and time. This difficulty leads leads to my second goal: the emphasis of space-time space-time scaling scaling emphasis of interpreting host-parasite host-parasite diversity. Spatial when interpreting Spatial scales that that are are small relative distance have well-mixed populations populations dominated inter­ to migration distance dominated by local interactions. Local processes also dominate on temporal scales that are short relative to the expected times to extinction and recolonization of of genotypes. By contrast, observations aggregated over long spatial and temporal scales may ob­ obscure colonizations, extinctions, extinctions, and rapid rapid changes in genetic composition that ner scales. Thus the patterns of observed variability are strongly infl u­ occur on fi finer influenced by the spacetime scaling of colonizations and extinctions in the meta­ space-time metapopulation. population.

II. DIMENSIONALITY AND COLONIZATION - EXTINGION DYNAMICS DIMENSIONALITYAND COLONIZATION-EXTINCTION DYNAMICS In this section I describe more precisely the relationship between between potential variation and observed diversity. I defi n e the potential number of genotypes as define the dimensionality of the system. I begin with a verbal illustration of the link dimensionality verbal between dimensionality and colonization -extinction dynamics. I then tum and colonization-extinction turn to a simple simple model. model.

A. A. Verbal Verbal Description Description The The observed diversity of host-parasite host-parasite genetics depends on the range of of possible variants and and the the processes that that govern local extinction extinction or success success of each each genotype. genotype. For For example, example, suppose suppose that that the the host host has has just just two two alternative alternative geno­ genotypes, and the parasite has two genotypes, PI and P ' The host h I can types, hh Il and and hh2, and the parasite has two genotypes, pl and P2. The host h~ can 2' 2 recognize I but ' Likerecognize and and resist resist the the matching matching parasite, parasite, P p~, but hh~I is is susceptible susceptible to to P P2. Like2 '

1144

Spatial Processes Processesinin Host-Parasite Genetics Spatial Host- Parasite Genetics

327 327

to P wise, h h 22 can c a n resist resist P P22 but but is is susceptible susceptible to p~. In this this case case strong strong frequency frequency de­ deI ' In pendence pendence will will favor favor rare rare genotypes, genotypes, and and genotype genotype frequencies frequencies will will fluctuate fluctuate around around 0.5. 0.5. Thus Thus diversity diversity is is controlled controlled by by the the local local dynamics dynamics of of frequency frequency de­ dependence. pendence. Now host-parasite interaction interaction but with more Now consider consider the same same pattern pattern of of host-parasite but with more genotypes. host genotypes genotypes. In In particular, particular, each each of of the the n n host genotypes h il . . . h h,n matches matches the the single single corresponding corresponding parasite parasite genotype genotype from from the the set set of of PI pl '9 9. 9. P P,. Thus Thus h Il resists resists h 2 resists pPIl but but is is susceptible susceptible to to all all other other parasite parasite genotypes, genotypes, h2 resists P2, so on. on. The The same same P2 , so frequency frequency dependence dependence occurs, occurs, favoring favoring equal equal abundance abundance of of all all genotypes. genotypes. How­ However, in. As ever, the the frequencies frequencies now now fluctuate fluctuate about about l1In. As the dimensionality dimensionality n n increases, increases, the declines, and to cause the average average frequency frequency declines, and small small fluctuations fluctuations are are more more likely likely to cause local extinction of a genotype. local extinction of a genotype. An An extinction extinction leads leads to to aa sequence sequence of of events events that that changes changes the the local local dynamics. dynamics. hi is locally extinct. Then matching For example, suppose that host genotype For example, suppose that host genotype is locally extinct. Then the the matching because it it can attack all parasite parasite P Pii has has an an advantage advantage over over other other parasite parasite types types because can attack all hosts by their matching hosts in in the the local local popUlation. population. The The other other parasites parasites are are resisted resisted by their matching host decline toward toward local host genotypes. genotypes. Thus Thus Pi Pi increases increases and and the the other other parasites parasites decline local extinction. recolonization and by hi, hi ' extinction. The The patch patch is is now now ripe ripe for for recolonization and rapid rapid increase increase by which point which would drive drive Pi pi and the other other host host types toward toward local extinction. extinction. The The point is that dynamics are now now controlled by the times to extinction extinction and and recolonization. recolonization. The population at will be The observed observed variation variation in in aa particular particular population at aa particular particular time time will be much much lower lower than than the the potential potential diversity. diversity. n '

B. B. The Model Model II now the same in the now tum turn to to the the formal formal model. model. The The ideas ideas are are the same as as in the verbal verbal model model Some readers readers may may prefer just given, but but the points points are are made made more more precisely. Some prefer to skip later to the details skip ahead ahead to to the the sections sections on on natural natural history history and and return return later to the details of of the the model. model. I focus focus on a single-patch single-patch model model with with extrinsic extrinsic colonizations colonizations rather rather than an explicit, explicit, multipatch multipatch metapopulation metapopulation analysis. analysis. In In the the next next section section II discuss discuss single­ singlepatch models. patch and and multipatch multipatch models. The locus. Each The model model has has aa single single haploid haploid locus. Each of of the the n n host host alleles alleles causes causes recognition recognition and and resistance resistance to only one one of of the n parasite parasite alleles. Thus Thus each host host is )In resistant and each resistant to to lin 1/n of of the the parasite parasite genotypes, genotypes, and each parasite parasite can can attack attack (n (n -- 11)/n of the host genotypes (Frank, 1 99 1 a, 1 993a). I call this the "matching-allele" of the host genotypes (Frank, 1991a, 1993a). I call this the "matching-allele" model. model. In In aa popUlation-genetic population-genetic context context the the different different alleles constitute constitute aa polymor­ polymorphic phic locus locus of of a single single species. In an ecological ecological context context each allele is associated associated with species. II will will use the population-genetics popUlation-genetics language language of with aa different different species. use the of allelic allelic polymorphism, ecological interpretation polymorphism, but but an an ecological interpretation of of species species diversity diversity is is equivalent equivalent for for these these assumptions. assumptions. II use -Volterra equations to describe system. These use Lotka Lotka-Volterra equations to describe the the system. These equations equations show show the rather than just the relative genotype the dynamics dynamics of of genotype genotype abundances abundances rather than just the relative genotype frefre-

328 328

StevenA. A. Frank Frank Steven

population sizes and quencies. Thus the model tracks epidemic fluctuations in population disease intensity in addition to changes in genotype frequency. The model is Ah; = = hh;[r(1 - H/K) H/K) - m(P m(P - p ;pJ] ) ] At Ahi ; lr( l Apj = p j [ - s

+ b(H-

((1) 1)

hj)] At.

The values of hhii and P pjj are the abundances abundances of hosts of of genotype i and parasites parasites of genotype j. The total abundance abundance of hosts is H H = = sLk� 1 h kk,> and the total abun­ abunof of parasites is P P = = Lk� s 1P Pkk' dance of host'' s intrinsic rate ooff increase, H/K H / K iiss the strength ooff density The term r iiss the host dependent competition competition among among hosts with carrying capacity capacity of K, m is the mor­ mordependent and mortality per parasite attack, attack, s is the parasite death death rate, and b is the bidity and parasite'ss intrinsic intrinsic birth birth rate rate per hosthost-parasite contact. The The At term term is the size of of parasite' parasite contact. which the interactions occur. occur. For For example, example, At may be the the time step over which of one host generation or one season season in a discrete-time discrete-time model. model. When When birth, birth, length of death and and disease cause continuous continuous change change of of the the abundances abundances of of hosts hosts and and par­ pardeath asites, At � ~ 0. O. The system in Eq. ((1) easier to analyze when when rewritten in nondimensional The 1 ) is easier nondimensional form (Segel, (Segel, 11972; Murray, 11989). Nondimensional analysis focuses focuses attention attention on form 972; Murray, 989). Nondimensional of parameters parameters and and highlights highlights relative magnitudes magnitudes (scaling relations) relations) a minimal set of among the processes processes that that drive the dynamics. This This is accomplished accomplished without without al­ alamong interpretation because because one can translate translate freely between between the tering the dynamics or interpretation biologically motivated motivated formulation formulation and the nondimensional nondimensional quantities. quantities. biologically The system can be rewritten rewritten with the following following substitutions substitutions The

= hi~K, h;!K,

hhii =

s/r, s = = sir,

mp/r, fPjj= = mpi/r,

7~'== rr At At

=

(2) (2)

l) Kb/r. b = Kb/r.

Dropping Dropping the hats yields the the nondimensional nondimensional system system

Ahi Ahi Apj Apj

p;)]7 = h i [ 1 -- HH -- (P (P -- Pi)]7 = hi[1

= pj - ss + + bb(H ( H -- hj hj )]'1". )]7. Pj [[-

=

(3)

The dynamics the system are are controlled by the the equilibrium all hosts The dynamics of of the controlled by equilibrium with with all hosts and parasites and p* / and parasites present, present, which which occurs occurs at h* = = s/[b(n s/[b(n -- 1)] 1 )] and = (1 ( 1 -- H H**))/ P* = * * 01 nh and, and, by by the the symmetry symmetry of of the the system, system, h* ht = = h* and and 1 ), where where H H* -= nh* (n -- 1), p* for all i and pj == p* and j. j. This This equilibrium equilibrium point point is unstable unstable when when there there are are discrete discrete P * for time lags lags in in the the competitive effects effects among among hosts hosts and and in in the the interactions interactions between between time host host and and parasite. parasite. This This equilibrium equilibrium is neutrally neutrally stable stable when when interactions interactions occur occur in in continuous + 0). A A detailed detailed analysis is given given in in the the Appendix Appendix of of Frank Frank continuous time time ((7r -� (1993a). ( l 993a). Figure shows the the dynamics Figure 1I shows dynamics of of this this system system with with two two hosts hosts and and two two parasites parasites (n = = 2). Each Each panel panel shows shows how how one one of of the the two two host-parasite host- parasite pairs pairs changes changes from from an an initial initial condition. condition. In In each each case the the abundances abundances follow follow aa stable stable limit limit cycle cycle that that

1144 II) u c: cu "0 c:

a

Spatial Processes Parasite Genetics Spatial Processesinin HostHost-Parasite Genetics

329

c

.E
> 50 50 8

1100 00 1110 10 27

Kheyr-Pour ((1980) 1980) Gouyon and Couvet ((1985) 1 985) Van Damme Damme and Van Delden ((1982) 1982)

Inferred Inferred Direct Direct

115 5 113 3 31 31

27 27 88 88 22

Koelewijn ((1993) 1 993)

Inferred

Boutin-Stadler et al. ((1989) 1 989)

Direct

Range

Median

Origanum vulgare Thymus vulgaris vulgaris Plantago lanceolata lanceolata Plantago

11-62 - 62 5-95 5-95 11-23 -23

With With IN IN Plantago coronopus coronopus With IN Beta maritima

11-34 - 34 0-35 0-35 113-61 3-61 119-62 9-62

Species

a

The second and third columns "The columns show the range and median median in percentage of females per per population population for for samples samples from from N popUlations. populations.

14 1 4 Spatial Spatial Processes Processes in Host-Parasite Host - Parasite Genetics

343 343

crosses will will expose expose cytoplasmic cytoplasmic genotypes genotypes from from the the female female parent parent to to nuclear nuclear crosses backgrounds of of the the male male parent parent that that have have aa low low frequency frequency of of matching matching restorers. restorers. backgrounds species, the the "IN" "IN" rows rows show show the the frequency frequency of ofpartially partially malemale­ For the the Plantago species, For sterile (IN) (IN) plants. plants. Partial Partial male male sterility sterility also also depends depends on on an an interaction interaction of of cytocyto­ sterile plasmic and and nuclear nuclear genes. As As noted noted by by Koelewijn Koelewijn (1993), ( 1 993), partial partial male male sterility sterility plasmic common phenomenon phenomenon in in CMS, CMS, but but phenotypes phenotypes are are often often reported reported with with didi­ is aa common chotomous classification. classification. This This is is similar similar to to the the partial partial resistance resistance that that is common common chotomous in plant-pathogen plant-pathogen interactions, interactions, as as noted noted in in the the previous previous section. section. In In both both CMS CMS in and plant-pathogen plant -pathogen interactions, interactions, the the intermediate intermediate phenotypes phenotypes often often depend depend on on and specific interactions interactions between between genetic genetic polymorphisms polymorphisms of of the the host host and and parasite. parasite. specific

Colonization - Extinction Dynamics Dynamics B. Colonization-Extinction The frequency frequency of of females females in in a population population is the the frequency frequency of of unrestored unrestored malemale­ The sterile cytoplasms. The data suggest that the frequency of females varies among sterile The data that the of females varies among variation appears be associated associated with widespread widespread genogeno­ populations. Phenotypic variation appears to be restorer frequencies. frequencies. typic variation in cytoplasmic types and restorer Two related related metapopulation metapopulation scenarios scenarios have have been been proposed to explain phe­ Two explain phevariation. The The first theory concerns concerns the the colonization-extinccolonization -extinc­ notypic and and genetic variation. tion dynamics of 985; of alleles among among existing popUlations populations (Gouyon and Couvet, 11985; Frank, 11989). 989). The second theory focuses on the colonizationextinction colonization-extinction dynamics of populations 985). I will briefl populations (Gouyon and Couvet, 11985). brieflyy outline the eld study that hints at how natural populations may allelic theory, along with a fi field be influenced by these processes. At the end of this section I mention the popupopu­ lation -extinction theory. lation colonization colonization-extinction theory. To understand the colonization - extinction dynamics of alleles one must colonization-extinction imagine imagine aa sequence sequence of of events. events. ((i) i ) Initially, one of the cytoplasmic type is lost from a local population. Loss may occur by drift or because the alternative types have higher fi tness. fitness. Increasing dimensionality (more types) raises the probability that one or more cytoplasms will be absent locally. al((ii) ii ) When a cytoplasmic type is absent, the associated nuclear restorer al­ leles do not have any benefi cial effects. These specifi beneficial specificc restorer alleles may be lost from the the local population by a variety variety of processes. If there are are no fitness differ­ differences between restorer and alternative nonrestorer alleles, then the restorers may ences restorer and alternative nonrestorer alleles, be lost by drift. If the restorers, which must in some way infl u ence pollen devel­ be lost drift. restorers, influence development, ciency when opment, reduce reduce effi efficiency when their their matching matching cytoplasm cytoplasm is absent, absent, then then the specifi c restorers will specific will be lost lost by selection. selection. ((iii) iii) After steps i ) and steps ((i) and (ii), (ii), aa cytoplasmic cytoplasmic genotype genotype and and its specific specific re­ restorers storers are are absent locally. locally. If an an unrestored unrestored cytoplasm cytoplasm arrives arrives by by immigration, immigration, it tness it will have have aa fitness fitness advantage advantage and and spread spread quickly quickly in in the the population. population. The The fifitness advantage advantage occurs occurs because because an an unrestored unrestored cytoplasm cytoplasm causes causes aa male-sterile male-sterile pheno­ phenotype. type. Male-sterile Male-sterile plants plants typically typically produce produce more more seeds seeds than than hermaphrodites hermaphrodites ((Lloyd, Lloyd, 11976; 976; Van 984; Van 984). Because Van Damme, Damme, 11984; Van Damme Damme and and Van Van Delden, Delden, 11984). Because

344 344

Steven Fronk Steven A. A. Frank

cytoplasmic fitness depends depends only on success through through the maternal line (seeds) and not on pollen pollen success, the male-sterile male-sterile plants plants have greater cytoplasmic fitness than than hermaphrodite hermaphrodite plants. Thus Thus the cytoplasms cytoplasms that that cause male sterility spread in the local population, population, causing an increase in the frequency of of females. ((iv) iv) Cytoplasmic genotypes are essentially alternative alleles at a hap­ hapgeno­ loid locus. When When one genotype increases in frequency, then the other genotypes necessarily decline in frequency. In the case of of mitochondria, an increase in the frequency of mito­ of one mitochondrial type will cause a decline in other other mitochondrial chondrial types. Thus the selective spread of of an unrestored unrestored cytoplasm may cause the local extinction of of alternative cytoplasmic genotypes. Loss of of cyto­ cytoplasmic genotypes may be associated with loss of of matching restorers, restorers, as in step step (ii). (ii). (v) The The population now has a high frequency of of females and a dominant cytoplasmic genotype. The restorers restorers matching the dominant cytoplasm are locally extinct. If a matching matching restorer restorer arrives by immigration, it will combine with the dominant dominant cytoplasmic type to produce produce hermaphrodites. hermaphrodites. The restorer allele spreads spreads rapidly because pollen is rare rare locally, thus the few few hermaphrodites hermaphrodites are the source of paternal alleles for for all members of of the population. population. As the restorer restorer spreads, spreads, the frequency of of females declines. The frequency of of cytoplasmic genotypes may be unaffected by the initial spread of restorers. unaffected (vi (vi)) A locally absent cytoplasmic genotype can invade and spread if its spe­ specific restorers are absent. The cycle then repeats, with a genotypic turnover in the local population. -the number population. The greater the dimensionality dimensionality-the number of of cytoplasmic genotypes and the more and matching matching specific restorersrestorers-the more likely an immigrant cyto­ cytoplasmic type will be locally absent absent and can start a new round round of genotypic turn­ turnover. Van Damme 's study of P. lanceolala Damme's of P. lanceolata provides just enough enough detail to show show how parts of population. Van Van Damme of the above scenario may work in a natural population. Damme and Van 1 982) distinguished two cytoplasmic genotypes in P. lanceo/ala Van Delden Delden ((1982) P. lanceolata each with its own set of of nuclear restorers. restorers. Table IV shows phenotypic frequencies frequencies in 12 populations pop­ populations in two habitat groups; the original paper paper lists data data for for 27 poppopUlation are abbreviations ulations in five categories. The labels for for each population abbreviations for for locations. locations. The cytoplasmic genotype MS I when when genotype R causes the male-sterile phenotype MS1 unrestored unrestored and IN IN1I when partially restored. The cytoplasm P P causes MS2 when unrestored four types are unrestored and and IN2 when partially restored. All four are morphologically distinct and can be scored scored by direct direct examination. examination. Restored Restored cytoplasms of of either either type are hermaphroditic, H. The cytoplasmic type of of a hermaphrodite hermaphrodite can be determined only by crossing crossing until the cytoplasm is exposed exposed in an unrestored unrestored nuclear background. background. The two population population groups shown in Table IV are the most differentiated of of the five groups populations either groups listed in the original paper. Five of of the hayfield populations lacked the R cytoplasm or were fixed for for the R restorers. In the pasture pasture popula-

1144 TABLE TABLE IV IV

Population Population Hayfield Hayfield Dr Ze An Re Me Me l1 Ve Ve Br Br Pasture Pasture Wd Wd Bm2 Bm2 Pa Pa Ac2 Ac2 Ju

Spatial - Parasite Genetics Spatial Processes Processesinin Host Host-Parasite Genetics

345 345

Phenotype PhenotypePercentages Percentagesinin Natural Natural Populations Populationsof of Plantago Plantago lanceolataa lanceolata ~ MSI MS 1

IN! IN 1

MS2 MS2

IN2 IN2

H H

Sample size

0 0 0 0 0 0 0 0 0 112.2 2.2 23.0 23.0

0 0 0 0 0 0 0 0 2.2 2.2 7.0 7.0

0.2 0.2 5.0 5.0 8.2 8.2 5.0 5.0 3.9 3.9 0 0 0.3 0.3

0.2 0.2 0.8 0.8 11.3 .3 3.6 3.6 5.5 5.5 0.6 0.6 0.2 0.2

99.6 99.6 94. 94.11 90.5 90.5 9 1 .4 91.4 90.6 90.6 85.0 85.0 69.5 69.5

8811 11 742 742 754 695 695 688 688 623 623 601 601

4.6 4.6 7.3 7.6 7.6 111.8 1 .8 2 1 .5 21.5

0.6 0.6 3.9 3.9 7.8 7.8 110.8 0.8 7.0

0.5 0.5 0 0 0.5 0.5 0 0 0

0.9 0.9 11.3 .3 0.9 0.9 11.0 .0 0.5 0.5

93.4 87.5 87.5 83.2 83.2 76.4 76.4 7 1 .0 71.0

6902 6902 386 386 437 437 305 305 4 14 414

""From From Van 1982). Van Oamme Damme and and Van Van Oelden Delden ((1982).

tions, rare or -specific restorers tions, either either the the P P cytoplasm cytoplasm was was very very rare or the the P P-specific restorers were were com­ common. population groups groups were for MS mon. The The other other three three population were relatively relatively more more mixed mixed for MS 11 and and MS2 phenotypes. phenotypes. Van Damme 1 986) made variation within within the Van Damme ((1986) made an intensive study of of spatial variation the Westduinen (Wd) population listed listed in picture of the fi eld at West­ Westduinen (Wd) population in Table Table IV. IV. A A picture of the field at Westduinen Fig. 6, 6, with listed in V. Females were duinen is is shown shown in in Fig. with some some of of the the data data listed in Table Table V. Females were rare whole population, MS 11 more common than than MS2. rare over over the the whole population, with with MS more common MS2. However, However, larger in a few few locations locations the the frequency frequency of of MS MS1I was high high (Fig. 6). Within Within the the larger clusters -p4, the zero at clusters of of MS MS I1,, pJ pl-p4, the frequency frequency of of MS MS I1 phenotypes was was close close to zero at the borders borders and and rose to 60% 60% near the center. The eld as a whole MS2) cytoplasm, with The fi field whole was dominated dominated by the P P ((MS2) with an overall frequency frequency of of 0.94. The The frequencies frequencies of of the P-specific P-specific restorer restorer alleles alleles were also high. Thus also high. Thus most most plants plants were were hermaphrodites hermaphrodites with with aa P P cytoplasm cytoplasm and and P P restorers. restorers. The overall frequency frequency of of the R cytoplasm was 0.06, 0.06, and the R-specific restorers 0.02 and restorers at at the the two two restorer restorer loci loci had had frequencies frequencies of of 0.02 and 0.08. 0.08. Genotypic Genotypic composition composition was was very very different different in in those those few few areas areas that that had had high high frequencies V). The MS 1) I ) cytoplasm, frequencies of of the the MS MS 1I phenotype phenotype (Fig. (Fig. 6 6 and and Table Table V). The R R ((MS cytoplasm, rare in the population population as a whole, had had frequencies frequencies ranging ranging between between 26 26 and and 39% 39% in -p4. The restorers, also in populations populations pJ pl-p4. The R-specifi R-specificc restorers, also rare rare in in the the whole whole field, field, were were more cult more frequent frequent in the MS I1 clusters, clusters, although although the exact frequencies frequencies were diffi difficult to estimate. ' s interpretation Van Van Damme Damme's interpretation agrees agrees with with the the scenario scenario outlined outlined above. above. Initially Initially most eld was most of of the the fi field was dominated dominated by by P P cytoplasms cytoplasms and and P-specific P-specific restorers. restorers. R­ Rbearing MS I1 spots -specific restorers bearing colonists colonists founded founded the the MS spots and, and, since since the the R R-specific restorers were were initially rare, the MS I1 females spread from a central focus. MS plants produce produce MS 11 plants

346 346

Steven Steven A. A. Fronk Frank

�

II

�....... // 0 •

@ p3

p •

•

JL .,0 @ p4

• •

•

0

• •

>=

• •

.0 > 11.0

Occupancy

n n

P P

No. of patches per 4 4 km2 km 2 per

23 23 1 38 138 88 88 6 6

0.24 0.24 0.24 0.24 0.40 0.40 0.56 0.56

I1 2-3 2-3 4-7 4-7 > >7 7

Occupancy n n

P P

6 611 70 70 58 58 66 66

0.21 0.21 0.32 0.32 0.25 0.25 0.41 0.41

a

a Effects Effects of of both both average average patch patch area area and and regional regional density density on on occupancy occupancy are are highly (from Hanski highly significant significant (from Hanski et et al., al., 1995). 1995).

editha editha bayensis bayensis is dominated dominated by one very large and and apparently persistent "main­ "mainland" population with transient "island" "island" populations found nearby in smaller smaller hab­ hab1 988). At the other extreme, the metapopu­ itat patches patches (Fig. 2; Harrison et et at., al., 1988). metapopuA land islands in the Baltic is an extensive lation of of M. M. cinxia cinxia on the the/~land extensive system of of hundreds of of small local populations, each of which is potentially susceptible susceptible to at., 11994, 994, 11995a,b). 995a,b). Important extinction (Fig. 11 in Hanski, this volume; Hanski et et al., differences differences between these two metapopulations metapopulations appear appear to be due to differences differences of habitat patch areas in the landscape, landscape, rather rather than to a fun­ funin the distribution of damental difference Hanski et difference in the biology of these closely related species ((Hanski et at., al., 11994). 994). Indeed, different Ptebejus argus different metapopulations of of the the blue butterfl butterflyy Plebejus argus show nearly as much variation in their spatial structure as do all comparisons of metapopulation structure across all species studied to date. Differences Differences in the rates and patterns patterns of of local extinction and colonization in different metapopulations of P. P. argus argus are apparently largely due to differences in the distributions of of patch sizes and vegetation dynamics, not to differences in the butterfly (c. D. Thomas differences (C. and Harrison, 1992). 1 992). Most metapopulations occupy an intermediate intermediate position along this continuum, with some relatively large, but not necessarily permanently pop­ populated patches, and other small and/or higher turnover and and/or isolated patches with higher probabilities of being occupied. lower probabilities One diffi culty in making difficulty making deductions deductions about about the persistence persistence and and dynamics of of metapopulations hab­ metapopulations from a "snapshot" "snapshot" distribution of of "occupied" "occupied" and "empty" "empty" habitat is that empty habitat might not be suitable of some subtle suitable after all, because because of unrecognised attribute attribute of particular particular habitat patches. patches. Although this potential prob­ problem should be considered carefully in every empirical study, we believe that it is empirical issue in most most butterfly studies. A few misclassified patches are a relatively minor issue unlikely to change the overall conclusions in any study of which we are aware. Introductions of of butterflies to empty habitats have succeeded succeeded in establishing new local populations on numerous occasions (Oates and Warren, 990; C. D. Warren, 11990; Thomas, 11992; 992; C. D. Thomas and Harrison, 11992; 992; Neve 1 995), and other N~ve et et at., al., 1995), studies have reported natural colonization of patches which had previously been

Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski

364, 364

�" 'm"

Extinct m in mo, 1976, ,"w > 20% in several 996c; H. H. comma, comma, Hill et 1 996; species species (e.g., A A.. hyperantus, hyperantus, Sutcliffe Sutcliffe et et al., al., 11996c; et al., al., 1996; see Warren, 987a; C. D. Thomas, 994a; Hanski and Kuussaari, 1 995, for Warren, 11987a; Thomas, 11994a; Kuussaari, 1995, for other exchange exchange rates). Overall, high exchange exchange rates among patches patches and and rapid colonization butterfly meta­ colonization suggest that the rescue effect is common in butterfly metapopulations populations and and that that isolation isolation may not be a major major constraint constraint on colonization colonization of of empty habitat within empty habitat within many many existing existing metapopulations. metapopulations. Often Often only a small small fraction fraction of unless patch of habitat habitat patches patches is empty within within a connected connected patch patch network, network, unless patch sizes are (c. D. Thomas 992). High migration are extremely extremely small small (C. Thomas and and Harrison, Harrison, 11992). migration rates and consequent consequent strong rescue effects effects may generate generate alternative stable equilibria equilibria in metapopulation metapopulation metapopulation dynamics; i.e., almost all patches patches occupied occupied or metapopulation extinction. 1 995b; extinction. A putative example is described for for M. M. cinxia cinxia by Hanski et et al. al. ((1995b; see Fig. 4 in Gyllenberg Gyllenberg et et al., al., this this volume). In contrast contrast to the the situation situation within within patch patch networks, networks, isolation isolation is an extremely important important reason why empty habitat habitat does does not not become become colonized colonized beyond beyond the rec­ rec(c. D. Thomas Harrison, ognized boundaries boundaries of of existing metapopulations metapopulations (C. Thomas and Harrison, 1992). Groups of of habitat patches patches are are often separated separated by distances distances which 1 992). Groups which will prevent colonization colonization from another another patch patch network network ((1I to > > 20 km separation separation may may provide provide an effective effective barrier, depending depending on the the species). species). Once Once such metapopula­ metapopulations become become extinct, reestablishment reestablishment can be very slow, which highlights the importance importance of of obtaining obtaining an empirical and theoretical theoretical understanding understanding of of the factors factors that contribute to the persistence of populations. of entire meta metapopulations.

IV. IV. THEORETICAL THEORETICALPREDIGIONS PREDICTIONSTESTED TESTED Despite metapopulations, empirical Despite many complications complications in specifi specificc butterfly metapopulations, empirical data lend considerable considerable support support to the central central tenets of of metapopulation metapopulation theory, namely that extinction /population size and extinction is related related to patch patch area area/population and colonization colonization is distance-dependent distance-dependent and that these these relationships contribute contribute to the observation that populations populations are most most likely to be present present in habitat habitat patches patches which are large 994b ). These general confidence that and close Hanski, I1994b). close together ((Hanski, general results results give us confidence the approach approach is valuable, valuable, even if if more more specifi specificc and complex models may be required required to predict predict the dynamics of of a particular particular system. The potential potential ability to test model assumptions assumptions and and predictions predictions is one of of the attractions c, spatially attractions of of using using specifi specific, spatially explicit explicit models, models, of of the the type outlined outlined in Hanski l 994a, this volume; 994; Sjogren Hanski ((1994a, volume; Hanski Hanski and and Thomas, Thomas, 11994; Sj6gren Gulve and and Ray, 11996). 996). In these models, habitat patch is specifi ed, models, the location location and and area area of of each habitat specified, with rules governing governing the probability of of local extinction extinction (mainly dependent dependent on population isola­ population size) and the probability of of colonization colonization (mainly dependent dependent on isolation and predictions of and source source popUlation population sizes). In this section, we deal deal with the the predictions of existing following existing models models and and consider consider some some of of the model model assumptions assumptions in the following section. section.

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able to test the predictions of a spatially explicit metapopu­ metapopuWe have been able H. comma c o m m a ((Hanski lation model with data for H. Hanski and Thomas, 11994). 994). This model explicitly iterates local dynamics in each occupied patch, which are connected to each other via distance-dependent migration (modeled at the population rather than individual level). Colonization is a mechanistic consequence of immigration to an empty patch. Altogether, the model has nine parameters. Where possible, parameter values should be estimated independently, but some parameters are pavery difficult or extremely time-consuming to estimate reliably. In our case, pa­ rameter values which could not be measured independently were estimated by simulation from a metapopulation which was assumed to be at equilibrium; this was done by choosing parameter parameter values which generated model predictions that matched empirical patterns of patch occupancy. The model was then applied, using these parameter parameter values, to a different different metapopulation, which was clearly not at equilibrium, to test whether the model could successfully predict dynamics metapopulation. in a nonequilibrium metapopulation. H. comma c o m m a occupies short, sparse, dry grasslands. Myxomatosis In England, H. rabbit grazing grazing in the midmid-1950s habitat became became over­ overremoved rabbit 1 950s and the grassland habitat butterflyy became very localized between grown, with the result that the skipper butterfl localized between (J. A. Thomas et et al. al.,, 11986). c o m m a became 986). In East Sussex, H. H. comma became 11960 960 and 11975 975 (1. restricted to one large. large population. In 1982, restricted 1 982, the skipper was still thriving in this refuge population and had colonized two small nearby habitat patches. patches. By large refuge was clear clear that substantial areas of of habitat were again suitable for breeding 11982, 982, it was by H. H. comma, comma, as rabbits had partially recovered from from myxomatosis and conser­ consermanagement on some vation organizations had begun to undertake active grazing management of the previously overgrown grasslands. This represented represented an ideal situation in of which to test the predictions predictions of of the model; model; a network of of empty patches which which could be mapped could mapped and a known known distribution distribution of of the skipper skipper in 1982. 1 982. The The predictions predictions of the models of models could then be compared compared with the observed observed skipper skipper distribution after after of colonization colonization (C. (c. D. D. Thomas Thomas and and Jones, Jones, 1993). 1 993). The The results were a 9 years of qualified qualified success. The model model predicted predicted that the skipper skipper would spread spread in this region and that it would not spread spread in other regions regions where where it failed, failed, in reality, to expand expand its distribution, distribution, but but the model underestimated underestimated the real rate of of expansion (Hanski ( Hanski and Thomas, 1994). H e s p e r i a ccomma o m m a actually occupied occupied 21 after 9 years 2 1 patches patches after 1 994). Hesperia ((Fig. Fig. 7 in Hanski and Thomas, 1994), 1 994), which which was more than than the model model predicted predicted patches occupied 1 1 , in 100 1 00 replicate replicate simusimu­ (mean 8.6 patches occupied after after 9 years, maximum 11, lations). quantitative mismatch prediction and observation prompts lations). The The quantitative mismatch between between prediction and observation prompts a series of of new new questions. questions. Was Was the the region region used to parameterize parameterize the the model model really equilibrium (not quite; quite; C. D. D. Thomas Thomas and and Jones, Jones, 1993)? 1 993)? Was Was the the negative at equilibrium exponential exponential distribution distribution used for for migration/colonization migration/colonization appropriate appropriate (new markmark­ release-recapture data were migration disrelease-recapture data were collected collected to resolve resolve this issue, and and migration dis­ tances found to tances were were found to fit a negative negative power power function function better, better, implying more more longlong­ distance distance migrants migrants than than in the the original simulations; simulations; Hill et et al., al., 1996)? 1 996)? et al. al. (1996c) (1 996c) have have attempted attempted to to predict predict the the distribution distribution of of M. M. cinxia cinxia Hanski et Hanski Aland in the over the Baltic, Baltic, using using an an incidence incidence function function model model over some 1000 1 000 km km22 on ,~land

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Chris and IIkko Chris D. D. Thomas Thomas and Ilkka Hanski Hanski

((Hanski, Hanski, 11994a). 994a). Again Again the the results results are are encouraging encouraging but but mixed. mixed. The The model model was was parameterized 1 99 1 (Hanski parameterized using using data data collected collected from from aa small small part part of of Aland Aland in in 1991 (Hanski et aI., 11994) 994) and et al., and model model predictions predictions were were tested tested with with independent independent data data collected collected in 993. Over fractions in 11993. Over large large parts parts of of Aland, Aland, the the model-predicted model-predicted and and observed observed fractions of Fig. 8 in in Hanski, Hanski, this volume), but of occupied occupied habitat habitat were were in in good good agreement agreement ((Fig. this volume), but this level this began began to to break break down down in in drier drier areas areas of of the the island island and and in in areas areas where where the the level of y ' s distribution. distribution. As As with with of grazing grazing was was an an important important determinant determinant of of the the butterfl butterfly's H. comma, testing the model revealed that there were further aspects of the bi­ H. c o m m a , testing the model revealed that there were further aspects of the biology of species that ology of the the species that needed needed to to be be understood. understood. Another fragmented Another kind kind of of problem problem for for predicting predicting species' species' distribution distribution in in fragmented landscapes is which there is landscapes is posed posed by by the the possibility possibility of of mUltiple multiple equilibria, equilibria, for for which there is M. cinxia Fig. 4 this empirical case of empirical evidence evidence in in the the case of M. cinxia ((Fig. 4 in in Gyllenberg Gyllenberg et et al., al., this 995b). Multiple impose inherent inherent uncertainties volume; volume; Hanski Hanski et et al., al., 11995b). Multiple equilibria equilibria impose uncertainties on particular netnet­ on our our ability ability to to predict, predict, accurately, accurately, the the distribution distribution of of species species in in particular works. role when when habitat is itself itself dynamic. dynamic. Al­ works. History History can can also also play play aa crucial crucial role habitat is Although meta­ though aa habitat habitat network network may may be be extensive extensive enough enough to to support support aa persistent persistent metapopulation now, now, the be too been colonized. population the whole whole network network may may be too isolated isolated to to have have been colonized. In some some areas areas where where H. H. comma c o m m a used to occur, occur, and and where where it became extinct extinct when when used to it became In the the habitat habitat was was overgrown, overgrown, the the habitat habitat has has now now recovered. recovered. Simulations Simulations suggest suggest be reestablished least one that that aa substantial substantial H. H. comma c o m m a metapopulation metapopulation could could be reestablished in in at at least one such recolonize this this area such area, area, but but the the butterfl butterflyy has has failed failed to to recolonize area naturally naturally and and simu­ simu(c. D. D. Thomas lations 1 00 years lations indicate indicate that that it it is is likely likely to to take take more more than than 100 years to to do do so so (C. Thomas and Jones, 11993; 993; Hanski Thomas, 11994). 994). and Jones, Hanski and and Thomas,

Metapopulation Persistence Persistenceand Establishment Establishment A. Metapopulation One One of of the the major major potential potential uses uses of of models models is is to to predict predict whether whether metapopu­ metapopulations are in specific specific networks habitat patches. lations are likely likely to to persist persist in networks of of habitat patches. The The models models referred Hanski, 11994a; 994a; Hanski 1 994) can be used referred to to above above ((Hanski, Hanski and and Thomas, Thomas, 1994) can be used for for this that one one has has been been able them. this purpose, purpose, assuming assuming of of course course that able to to parameterize parameterize them. One models to size and persistence of One may may use use the the models to explore explore how how the the size and persistence of aa particular particular metapopulation loss of is likely likely to to metapopulation is is affected affected by by further further loss of suitable suitable habitat, habitat, which which is increase the risk of local extinction and to decrease the probability of recoloni­ increase the risk of local extinction and to decrease the probability of recolonization. 1 994a,b) and 1 994) give zation. Hanski Hanski ((1994a,b) and Hanski Hanski and and Thomas Thomas ((1994) give examples. examples. More More generally, generally, one one may may ask ask how how the the expected expected metapopulation metapopulation lifetime lifetime depends depends on on the the number local populations populations that connected to to each number of of extinction-prone extinction-prone local that are are connected each other. other. 1 996c) have explored this question with with the the incidence incidence function function Hanski Hanski et et al. al. ((1996c) have explored this question islands. The model using the model using the observed observed patch patch networks networks for for M. M. cinxia cinxia on on the the Aland ,~land islands. The expected to the expected time time to to metapopulation metapopulation extinction extinction is is closely closely related related to the product product px/~, where where p p is is the the fraction fraction of of occupied occupied habitat habitat patches patches at at stochastic stochastic steady steady state state pJH, patches ((Hanski, Hanski, this volume). A A rough of and and H H is is the the number number of of suitable suitable patches this volume). rough rule rule of thumb metapopulation is is likely likely to thumb is is that that if if this this product product exceeds exceeds 33 the the metapopulation to persist persist much of a the much longer longer than than the the expected expected lifetime lifetime of a local local population. population. Assuming Assuming that that the

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species patch network which it species occupies occupies most most patches patches in in the the patch network in in which it is is present, present, a a minimum of some 1 5 20 well-connected patches are required for long-term minimum of some 1 5 - 2 0 well-connected patches are required for long-term per­ persistence. in local local dynamics dynamics and sistence. With With spatial spatial correlation correlation in and much much environmental environmental sto­ stoal., 1996b). 1 996b). Hanski et chasticity, even this this would would not not be enough ((Hanski et al., Empirical tests tests of of the the persistence persistence of of entire entire metapopulations metapopulations are are very very limited Empirical limited and, and, understandably, understandably, difficult difficult to to accomplish. accomplish. The The extinction extinction of of an an entire entire meta­ metapopulation is is much scale than population much rarer rarer and and takes takes place place on on a a longer longer time time scale than does does the the extinction populations. As As Harrison Harrison ((1991, 1 99 1 , 11994b) 994b) and and C. C. D. D. extinction of of individual individual local local populations. Thomas 1 994a,b) have noted, metapopulation metapopulation extinction is generally generally related Thomas ((1994a,b) have noted, extinction is related to to an the amount to widespread an overall overall decline decline in in the amount of of suitable suitable habitat habitat due due to widespread human­ humancaused in the landscape. Here, Here, we caused changes changes in the landscape. we are are concerned concerned whether whether a a metapopu­ metapopulation is likely to persist in a network of habitat patches which is not becoming lation is likely to persist in a network of habitat patches which is not becoming further In studies studies of of P. D. Thomas 1 994b) further degraded. degraded. In P. argus argus and and H. H. comma, comma, C. C. D. Thomas ((1994b) found 5 to tended to populated, but but found that that regions regions with with over over 115 to 20 20 habitat habitat patches patches tended to be be populated, that 1 0 patches populated, suggesting the latter latter that regions regions with with < < 10 patches were were rarely rarely populated, suggesting that that the may shows just just may be be inadequate inadequate for for long-term long-term metapopulation metapopulation persistence. persistence. Figure Figure 4 4 shows the M. cinxia. cinxia. T. T. Ebenhard Ebenhard (personal (personal communication) communication) was was able able the same same pattern pattern for for M. to extinctions directly, the cranberry to examine examine metapopulation metapopulation extinctions directly, when when working working on on the cranberry fritillary B. aquilonaris wet year. year. Boloria Boloria aquilonaris aquilonaris in in an an extremely extremely cloudy cloudy and and wet aquilonaris fritillary B. occurs in within remnant with occurs in bogs bogs within remnant forest forest patches patches in in southern-central southern-central Sweden, Sweden, with of cranberry-containing cranberry-containing bogs per per forest fragment fragment representing representing the the number number of

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Aland as a FIGURE occupancy of habitat habitat patch patch networks networks by M. cinxia FIGURE 4 The frequency frequency of occupancy cinxia on on/~land fraction number of patches this analysis, analysis, the habitat patches (shown Fig. fraction of the number patches in the network. network. For this habitat patches (shown in Fig. 11 in Hanski, Hanski, this this volume) volume) were 1 27 semi-isolated networks, isolated isolated by a physical physical were divided divided into into 127 semi-isolated patch patch networks, barrier to dispersal Numbers of networks networks in each dispersal or by ca. ca. I1 km from from the nearest nearest other other patches. patches. Numbers each class given by the number number in the fi figure. is given gure.

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Chris Chris D. D. Thomas Thomas and and IIkka Ilkka Hanski Hanski

number meta­ number of of habitat habitat patches patches per per network. network. Ebenhard Ebenhard found found that that many many of of the the metapopulations fewer than than 20 patches patches became populations with fewer became extinct while those those with > > 20 patches population collapse. patches survived a massive population collapse. At accord. At this stage, theory and and empirical empirical data data appear appear to be in approximate approximate accord. A metapopulation metapopulation rarely persists for for very long at < < 1100 local populations populations and and usually does so for populations. There for extended extended periods periods at > > 20 local populations. There is obviously obviously a trade-off between the number patches and patch, with fewer trade-off between number of of patches and area area per per patch, fewer patches patches needed xed length length of needed for for some some fi fixed of metapopulation metapopulation survival when when each patch patch is large; one persistence if if it is vast and heterogeneous (the theory one patch patch alone alone may ensure ensure persistence and heterogeneous theory described above metapopulation lifetime in relation to the lifetime of described above predicts predicts metapopulation of local interpreted). When habitat local populations, populations, and and the the results results should should be thus thus interpreted). When the habitat consists consists of of transient transient or successional successional vegetation, much much larger areas areas may be required required ensure that appropriate appropriate habitat continuously available, available, the amount amount of of extra extra to ensure habitat is continuously vegetation required the natural vegetation and vegetation required depending depending on the natural dynamics of of the vegetation and on patterns of of human management. Finally, if there regional stochasticity stochasticity patterns human management. there is much regional metapopula­ (spatially correlated correlated environmental environmental stochasticity), stochasticity), even even very large large metapopulations may go extinct, extinct, the extreme case case being a catastrophe catastrophe of of some some kind kind sweeping sweeping away species from Given away the species from a large large area area (see below below and and Hanski, Hanski, this volume). Given these and other other uncertainties, uncertainties, we must that the patch numbers numbers mentioned mentioned these and must stress that the patch above rules or planning. Much above should should not not be taken taken as rules or guidelines in conservation conservation planning. Much larger required in some larger numbers numbers of of patches patches will be required some circumstances, circumstances, but but one one vast vast patch ce in some conservation, each must patch may suffi suffice some cases. In the context context of of conservation, each case case must be considered considered individually. One most valuable uses of One of of the most valuable uses of specific specific models models is in predicting predicting the con­ consequences management options. Predicting the actual proba­ sequences of of possible possible future future management options. Predicting probability of bound to be fraught uncertainties, but but prepre­ of long-term persistence persistence is bound fraught with uncertainties, dicting response to changes changes in the distribution dicting the likely direction of of response distribution of of habitat periods of few decades is potentially potentially of over periods of a few of immense immense value to managers, and and may be be possible explicit metapossible to achieve. We We have already seen that that a spatially explicit meta­ population model predicted population predicted correctly, in qualitative qualitative terms, where where H. H. comma comma would would expand its distribution distribution in response response to increased increased habitat quality and and where it would nements, such models would not. Even without without any further further refi refinements, models could be of of use to conservation been possible habitat conservation managers. managers. It has has been possible to draw draw maps maps of of existing existing habitat around around surviving surviving H. H. comma c o m m a populations populations and to assess assess the potential for for further further spread. present spread. In East Sussex, Sussex, managers managers can can relax in the sense that that continuing continuing the present management is likely to permit permit continued continued expansion of of H. H. comma. c o m m a . In three other other expansion is not not predicted (c. D. regions in southeast southeast England, England, further further expansion predicted (C. D. Thomas Thomas and 993; Hanski, 11994a; 994a; Hanski and 994). When and Jones, 11993; and Thomas, 11994). When the species species is unable unable to spread spread within within the existing patch patch network, network, the consequences consequences of of many many different different management management options options can be be explored explored by changing changing the distribution distribution and sizes of of specifi specificc habitat patches in the model. model. For For example, example, would enhanced enhanced management management to increase increase existing existing population population sizes facilitate further further spread, spread, and and would management would management of of the surrounding surrounding areas areas to increase increase target target patch patch areas or or to decrease spread? In a model, decrease distances distances between between patches facilitate spread? model, many different different

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options options may may be be compared compared quickly, quickly, before before embarking embarking on on time-consuming time-consuming and and expensive expensive management management work. work. Even Even if if the the predictions predictions are are not not quantitatively quite quite correct, correct, the the relative relative merits merits of of different different practical practical options options may may be be robustly robustly assessed. assessed. An An additional additional important important application application is is likely likely to to be be in in predicting predicting the the potential potential of species translocation translocation projects. Hanski and and Thomas ((1994) that success of 1 994) found that one one set of parameter values correctly correctly predicted the existing existing distribution of of P. argus argus on limestone grassland, and its invasion of two new networks of habitat habitat introduced in the the past. For H. comma, patches to which it had been successfully introduced the the model has been used to identify identify at least one network which which could potentially support support a viable metapopulation metapopulation of the skipper, but which is too isolated for colonization to occur naturally. This approach may help to eliminate some of the effort wasted on releasing rare butterflies in single or very small groups of of habitat patches where long-term persistence is improbable 990; improbable (Oates and Warren, 11990; explicit models, the the target species species can can be C. D. Thomas, 11992). 992). Using spatially explicit introduced to the the same same patch network repeatedly, and and the the value value of releasing a fixed number of individuals at one versus many sites can be explored. There are number of individuals several lessons to learn. For example, managers should not necessarily give up even if the fi rst attempt fails; the introduced first introduced population may have to exceed some threshold threshold before before establishment is likely. This is especially true if there are multiple equilibria ((Hanski Hanski et ai., 995b; Gyllenberg al., 11995b; Gyllenberg et ai., al., this volume). volume). In most cases, the largest, largest, high quality and and least isolated habitat patches patches in the network should should the isolated habitat the network be targeted Hanski, 11994b), 994b), even targeted for for releases releases ((Hanski, even if if these these patches patches fall outside outside existing existing nature reserves reserves in the network. network. nature

V. ADDING CONCEPT ADDING REALISM REALISM TO TO THE THE METAPOPULATION METAPOPUlATION CONCEPT At this relatively relatively early stage stage in the the development development of of specific specific and and predictive predictive At metapopulation models, field field studies studies are needed to test model model assumptions assumptions and metapopulation are needed and to identify additional additional behavioral behavioral and and ecological factors factors which which may may need need to to be be to incorporated into into the the next next generation generation of of models. models. Having Having identified identified existing existing shortshort­ incorporated comings, comings, it it is important important to to find find out out whether whether the the predictions predictions of of refined refined models models differ adding many differ substantially substantially from from the the predictions predictions of of simpler simpler models, models, because because adding many more parameters parameters to to models models creates creates new new problems. problems. In In this this section section we we address address more model assumptions assumptions and and complications complications that that we we believe believe will will have have important important imim­ model plications for for metapopulation metapopulation persistence, persistence, patterns patterns of of distribution, distribution, and and rates rates of of plications colonization. colonization.

A. Migration Migration In Hanski Hanski and and Thomas Thomas (1994, ( 1 994, p. p. 170), 1 70), we we highlighted highlighted "the "the need need for for good good In In the the model, model, empiricai data data on on emigration emigration and and immigration immigration rates rates in in butterflies." butterflies." In empirical we assumed assumed that that the the distribution distribution of of dispersing dispersing individuals individuals could could be be fitted fitted to to aa we negative exponential exponential function, function, which which is is typical typical of of metapopulation metapopulation models models (Har( Harnegative

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rison et et al., al., 1988; 1 988; Hanski, Hanski, 1994a; 1 994a; Hanski Hanski and and Thomas, Thomas, 1994; 1 994; Akqakaya, Ak�akaya, 1994; 1 994; rison Sjogren Gulve Gu1ve and and Ray, Ray, 1996). 1 996). This This distribution distribution clearly clearly gives gives the the right right general general Sjrgren pattern, of of many many short-distance short-distance movements movements and and aa few few long-distance long-distance movements, movements, pattern, but needs needs to to be be tested tested more more rigorously with with empirical empirical data. data. but When P. argus argus were were released released into an extensive extensive area area of of empty empty habitat, habitat, When adult adult P. it was found found that that the the distribution distribution of of migration distances distances was was a close close fit fit to to a exponential (O. (0. T. Lewis, Lewis, C. D. Thomas Thomas and and J. K. Hill, Hill, unpublished). unpublished). negative exponential However, individuals individuals that that moved moved the longest distances distances may have left left the whole However, the longest may have the whole area, and and hence hence the the tail tail of of the the distribution distribution is likely likely to to have have been been underesunderes­ study area, timated. Mark-release-recapture Mark- release-recapture work work has has shown shown that that the the between-patch between-patch distridistri­ timated. H. ccomma ( Hill et et al., al., 1996) 1 996) fits a negative negative power power bution of of distances distances moved moved by H. bution o m m a (Hill function well, function M = zD -~,

(1) (1)

M iiss the the fraction fraction of of individuals individuals reaching reaching distance distance D and zz and and k are are concon­ where M where D,, and For H. H. ccomma, negative exponential stants. For o m m a , the negative exponential distribution distribution gives a slightly poorer poorer fit to the the data ( 1 ); in particular, particular, the negative negative exponential exponential underesunderes­ fit data than does Eq. (1); timates the proportion of of butterflies butterflies that fly relatively long distances. We We surmise timates that this tail of of the distribution distribution is generated generated either either by individuals that change change habitat, behavior during migration, after after they have initially failed to locate locate new habitat, or by individuals that are are inherently dispersive. When appropriate, this alternative alternative individuals that When appropriate, assumption about about migration could easily be incorporated into spatially explicit explicit models, and would presumably generate generate more long-distance colonizations than and would negative exponential exponential distribution. the negative distribution. 1 994) assumed, with In the absence absence of of good field data, Hanski and and Thomas Thomas ((1994) proportion of of individuals emigrates from each local misgivings, that a constant proportion population. Empirical data now show that the fraction of individuals emigrating are relatively high when patches are small and have high perimeter-to-area perimeter-to-area ratios 996; Sutcliffe et 996c; Kuussaari et et al., al., 1996; 1 996; M. Baguette and ((Hill Hill et et al. al.,, 11996; et aI. al.,, 11996c; G. Neve, N~ve, personal personal communication; see Kareiva, 1985). 1 985). Therefore, it would be desirable to incorporate area-dependent area-dependent emigration rates in spatially explicit meta­ metacult to calculate calculate the actual population models. Unfortunately, it is very diffi difficult actual (as opposed to relative) rate of emigration in relation to patch area, because emigrants are rarely detected unless they immigrate into another habitat patch; substantial numbers of of emigrants may fail to arrive in any patch. Since the spread spread of of mi­ miindividuals is neither entirely random (migrants may be attracted to new grating individuals patches from some distance, and may then then stay stay there) nor nor entirely directed, it may not be feasible to estimate the fraction lost accurately. C. D. Thomas, O. T. Lewis, and and 1. J. K. Hill (unpublished) have used a sim­ simulation approach in order to estimate emigration rates (see also Buechner, 11987; 987; 987). We took imaginary habitat patches with a range of areas Stamps et et al. al.,, 11987). and placed butterflies in random locations within those patches. Butterflies were then allowed to migrate, by making each move away from its origin at a random = per generation) angle, and for a distance chosen at random from the empirical ((~ distribution of migration distances that we had already recorded recorded for P. P. argus a r g u s and

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H. H. comma. comma. The The fraction of individuals leaving the the patch was then then recorded. recorded. These estimates i ) In reality, individuals may estimates of emigration emigration rate rate have two main biases: ((i) perceive perceive patch patch boundaries boundaries and be reluctant to leave (we assumed that boundaries were fully permeable), ii ) permeable), leading to an overestimate of of the emigration rate, and ((ii) the proportion of of individuals moving long distances may bbee under-represented under-represented in the empirical empirical distributions of of migration distances, distances, leading to an underestimate underestimate of of the the emigration fraction. These two biases biases act act in opposing directions. In any case, case, this exercise produced a strong and and negative relationship between between patch area and the fraction of of individuals emigrating, with most individuals individuals emigrating from from small patches with high perimeter-to-area perimeter-to-area ratios. For a given area, the emigration rate differed considerably between ecting differences between the two species, refl reflecting differences in dis­ dispersiveness. persiveness. The implications -area relationships implications of of such emigration emigration-area relationships have not yet been explored explored in the context context of metapopulation models, but they are likely to be im­ important. Within Within central parts parts of metapopulations, high perimeter-to-area perimeter-to-area ratios ratios also result in high per-unit-area per-unit-area rates of of immigration into small patches and potentially 994; H. H. comma, 1 996; to high local densities ((Hanski Hanski and Thomas, 11994; comma, Hill et et at. al.,, 1996; 994). However, the consequences M. M. cinxia, cinxia, Hanski et et al. al.,, 11994). consequences are are much more more sig­ signifi cant when nificant when habitat patches are are relatively isolated. Changes in population size in isolated patches (c. D. Thomas, patches with no immigration can be described described by (C. O. T. Lewis, and J. K. Hill (unpublished) O. T. Lewis, and J. K. Hill (unpublished) N,+ 1 = e N t e'(1-RN'/K),

(2)

where where Nt N, is the number number of of individuals in generation t, rr is the intrinsic rate of of population population increase, R is the proportion of of individuals which are resident in the local population (proportion 11 - R emigrates), and K is the local carrying ca­ ca+ In R/r) pacity. The equilibrium population size in this model is (K/R)( (K/R)(11 + R/r) and extinction takes place reproduction fails place when In R/r R/r < < - 11,, that is, when local reproduction to replace replace losses due to emigration. Furthermore, Furthermore, even when isolated populations can sustain emigration losses, depression of of local population size by emigration would make the populations more susceptible to other causes of of extinction. Incorporating the emigration -area relationship emigration-area relationship described described above in the model, independent estimates of rr and K, allows us to predict the expected expected and using independent population population size of of H. H. comma comma in habitat habitat patches patches which are isolated enough for immigration to have little effect on local population size, but not so isolated that colonization is unlikely ((Fig. Fig. 5). In Fig. 5, we assume that emigrants for that generation have already left at the time of of census, but this assumption makes little difference difference to the overall pattern. pattern. The good match match between between the model predictions of apparently apparently suitable and empirical data suggests that many isolated patches of habitat are not populated because the losses due to emigration are too high for 985). The match is equally good for local reproduction to match (see Kareiva, 11985). P. argus, P. argus, though much smaller smaller patches can be populated populated in isolation by this less dispersive dispersive insect. These These results are are encouraging and and suggest that that the immigration immigration-­ emigration balance balance may determine patch occupancy patterns to a previously un­ unexpected extent. At the same time, as these patterns can also be explained explained by -

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Predicted logj() patch area (ha) Cha) for Hesperia Predicted and actual population population sizes in relation to log~o for Hesperia comma. Thin line, predicted predicted population population size in the absence absence of emigration emigration and stochastic stochastic extinction. extinction. Thick line, predicted predicted patch patch occupancy and population sizes, with carrying capacity reduced reduced by area­ areadependent emigration, emigration, according to Eg. Eq. (2) (adult butterflies were assumed to have emigrated at the popUlation sizes to allow zero values time of of census). census). One was added to all measured measured and predicted population values - ) and 982 (A). (A). Patches plotted to be plotted. Measured population sizes in 1991 plotted. Measured 1991 C(11) and 11982 plotted were were > > 0.6 to 5 km from the nearest Cother) (other) population, population, which are regarded regarded as sufficiently isolated that immigration is likely to have negligible effects on local density, but not so isolated isolated that sites could not be colonized migrants (from C. D. Thomas, O. T. Lewis, and J. K. Hill, unpublished). by occasional migrants unpublished).

alternative hypotheses (area-dependent extinction rate for reasons other other than em­ emigration), critical field experiments would be welcome. Area-dependent Area-dependent emigration rates have important dynamic implications. For example, successful invasion of empty patch patch networks is expected expected to proceed proceed by the colonization of large patches first first (as observed in H. H. comma; c o m m a ; C. D. Thomas and Jones, 11993), 993), or by stepwise colonization of small patches, gradually eroding isolation before before popUlation population establishment is possible. Loss of of a part of an isolated patch may result in rapid population extinction. Finally, the area of habitat needed to support a single isolated population is much larger than the minimum area of of habitat habitat that that can be populated within a metapopulation where immigration roughly equals emigration in each patch.

B. Deterministic Deterministic Population Population Responses Responsesto Habitat Change Change Perhaps the most serious shortcoming of metapopulation theory has been been the general assumption that the distribution of suitable habitat remains constant through time. Many butterfl butterflyy species occupy successional vegetation. Even when habitats are potentially permanent, landscape changes brought about by human activities may drive signifi cant changes in species distributions (Arnold, 11983; 983; significant J. A. Thomas, 11991; 99 1 ; C. D. Thomas, 11994a,b,c; 994a,b,c; New 1 995). For example, New et et al., al., 1995). many local populations of fritillary butterflies inhabiting woodland clearings clearings in

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the United United Kingdom Kingdom have have gone gone extinct extinct because because of of successional successional changes changes in in the the the vegetation, not not because because of of stochastic stochastic fluctuations fluctuations in in local local population population size size (Fig. ( Fig. 6; vegetation, Warren, 1991; 1 99 1 ; Warren Warren and and Thomas, Thomas, 1992). 1 992). Reviews Reviews of of local local and and regional regional exex­ Warren, tinctions have have argued argued that that successional successional changes changes in in vegetation, vegetation, changes changes in in human human tinctions management of of surviving surviving habitat habitat fragments, fragments, and and outright outright habitat habitat loss loss are are principrinci­ management pally responsible responsible for for most most extinctions extinctions of of substantial substantial butterfly butterfly populations populations in in modmod­ pally em landscapes; landscapes; extinctions extinctions are are frequently frequently aa deterministic deterministic consequence consequence of of the the dede­ em terioration of of local local breeding breeding conditions conditions (Harrison, ( Harrison, 1991, 1 99 1 , 1994b; 1 994b; Warren, Warren, 1993; 1 993; terioration 1. A. A. Thomas, Thomas, 1991; 1 99 1 ; J. A. Thomas Thomas and and Morris, Morris, 1994; 1 994; C. C. D. D. Thomas, Thomas, 1994a,b,c, 1 994a,b,c, J. 1 996). Similarly, most most colonizations colonizations appear appear to take take place place when when environmental environmental 1996). improve locally (C. (c. D. D. Thomas, Thomas, 1994a,b,c, 1 994a,b,c, 1996). 1 996). The The spatial spatial dynamdynam­ conditions improve of many species appear appear to be driven driven by the the changing changing distribution distribution of of their their ics of habitats. These These insects insects are habitat mosaic. mosaic. habitats. are tracking a shifting shifting habitat Examples described this chapter chapter clearly illustrate the the importance importance of of stosto­ Examples described in this clearly illustrate chastic extinctions "traditional" metapopulation dynamics are extinctions too, but these "traditional" are su­ superimposed on a dynamic habitat habitat mosaic. mosaic. When When the the dynamics of of the the butterfly are fast relative to vegetation dynamics, ignoring the latter may still leave a good fit between between observed and predicted predicted species distributions distributions in the short short term. This may insects, with their fast dynamics, have become popular popular subjects be one reason why insects, for for metapopulation studies. However, these models may not be particularly useful for predicting long-term trends or long-term probabilities of persistence if longlong­ term changes are determined by underlying vegetation dynamics. The task of of superimposing stochastic dynamics of a butterfly on top of vegetation dynamics has has hardly hardly begun. begun. We give two examples examples to show how butterfly metapopulation dynamics dynamics may lag behind behind changes changes in the spatial distribution of suitable suitable habitat/vegetation. After earlier declines, H. H. comma c o m m a has recently enjoyed an increase in the amount and extent of suitable habitat in southern England, particularly in the late 11970s 970s and early 11980s 980s (described above). After After 9 years of documented colonizations, this butterfl y had still not colonized all of the "new" habitat available to it, and it is butterfly likely that the distribution will take several more decades to reach an equilibrium (C. D. Thomas Thomas and and Jones, 11993; and Thomas, 11994). (c. 993; Hanski and 994). Over the past 1100+ 00 + years, there has been no period of 330 0 years or more when the distribution of H. comma c o m m a habitat has remained even approximately stable, stable, and and it is hard to imagine that it will remain stable over the next 30 years. At least in modem landscapes, specialized specialized species may be continuously continuously chasing after after their their habitats. habitats. Another Another common common scenario scenario may be continuing continuing loss of of habitat. In parts parts of of A land, M. Aland, M. cinxia cinxia habitat has been lost in recent decades, decades, and and predictions of spa­ spatially explicit explicit models suggest suggest that the dynamics may not be at equilibrium equilibrium in all areas. O in Hanski areas. Figure Figure l10 Hanski (this volume) illustrates such such nonequilibrium dynamics with land. Within with an an example example from from M. M. cinxia cinxia on on A Alan& Within an an area area of of ca ca 25 25 km km22,, the the total total area area of of suitable suitable habitat habitat has has declined declined to to one-third one-third and and the the number number of of habitat habitat patches patches has 5 - 20 years. has decreased decreased from from 55 55 to to 42 42 over over the the past past 115-20 years. The The metapopulation metapopulation of of M. M. cinxia cinxia is is predicted predicted to to have have followed this this decline rather rather rapidly, apparently apparently

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the distribution of Mellicta athalia in response to rotational cUlling cutting (cop­ (copChanges in the pieing) of deciduous woodland in Blean Woods, Kent, England. Numbers show year of cUlling; picing) cutting; shaded areas areas indicate indicate the distribution distribution of adult adult bUllerfties. butterflies. Glades (GL) and wide rides (WR) were sporadically used for for breeding. Reprinted from from Warren, 11991, from Elsevier only sporadically 99 1 , with kind permission from Kidlington OX5 11GB, Science Ltd, The Boulevard, Langford Lane, KidlinglOn GB, UK.

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because because it occupied most of of the habitat habitat before patch patch reduction, reduction, and most of of the dynamics has occurred in small patches patches with fast turnover. turnover. However, an entirely different picture is predicted predicted if if the area area of of each patch patch is halved again over the next 20 years. In this case, further loss of habitat is further of habitat predicted predicted to lead to a network Fig. 10 1 0 in Hanski, this vol­ smaller than that required required for for long-term persistence ((Fig. volume). However, However, the actual extinction of the metapopulation metapopulation is predicted predicted to take a long time, and for for tens of of years we would see a metapopulation metapopulation slowly but inevitably oscillating to extinction. The final decline to extinction extinction. extinction is slow because the last populations to go are typically the largest ones with the smallest risk of of local extinction. This latter latter finding is especially worrying, in that that the status of of many or even most species in landscapes which are are gradually being degraded may presently presently be "better" Hanski, 11996b; 996b; Hanski et "better" than than the expected expected status status at equilibrium equilibrium ((Hanski, et al., 11996b,c). 996b,c). Nonequilibrium systems of of this type may lead us to conclude conclude that that po­ potential tential colonization distances distances are are greater than than they really are are and to overestimate local population persistence persistence in small patches. These biases result in overestimates overestimates of of metapopulation metapopulation lifetimes. Some currently surviving metapopulations metapopulations may be doomed doomed even if all further habitat loss is prevented.

C. Habitat Heterogeneity Habitat heterogeneity is probably crucial to the persistence persistence of of many local butterfly populations populations and metapopulations. metapopulations. For convenience, convenience, there there is a strong tendency in metapopulation metapopulation ecology to define some parts of of the landscape as "habitat" "matrix" - and to ignore the latter. "habitat" and the remainder as "nonhabitat" "nonhabitat" or "matrix"--and There are several problems with this (see also Wiens, this volume). ((i) i ) The en­ environment may exist as a series of of habitats which vary in suitability, and it may be difficult to distinguish between Rodriguez et al., 11994). 994). between habitat habitat and nonhabitat nonhabitat ((Rodrfguez Typically, not Hochberg et al. not all suitable habitat will be of of equivalent quality ((Hochberg al.,, 11992, 992, 11994; 994; C. D. Thomas, 11996). 996). (ii) (ii) Vegetation dynamics and human human activities change the suitability of of a given habitat patch; some of of these changes changes will be can change predictable (succession in a woodland clearing clearing resulting in a gradual decline decline in patch quality for for clearing clearing butterflies), butterflies), but other other changes changes may be unpredictable unpredictable and reversible. iii) Temporal environmental variability may affect an entire en­ reversible. ((iii) environmental gradient. For example, example, the warmest and driest parts of of a habitat habitat patch may represent represent the environmental optimum in normal years, but be inhospitable in a drought year, requiring the drought-affected veg­ drought-affected species to move into taller vegetation (H. comma, 994a) or to more mesic slopes (E. editha comma, C. D. Thomas, 11994a) editha bayensis, 988; the large blue Maculinea Maculinea arion, bayensis, Weiss et aI. al.,, 11988; arion, J. A. Thomas, 1 988) found that personal communication). Singer ((1972) 1 972) and Weiss et et al. ((1988) that com­ complex spatial variation in the microdistribution of of disturbance disturbance and of two host plant species, and the aspect of of the slope, interacted interacted with climatic variation to affect population persistence editha bayensis pop­ persistence and changes in population size in E. editha bayensis populations on serpentine serpentine grassland. Since a greater range of of microhabitats microhabitats is more

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Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski

likely to be present present in large large than than in small areas, areas, this may be an extremely im­ important habitat fragfrag­ portant reason reason why why populations populations are are often often most most persistent persistent in in large large habitat ments. ((iv) iv) Some occupied habitat patches patches may not be suitable for for population because of persistence; populations populations may be present present in these habitats habitats (sinks) only because of 994; immigration from source populations (Pulliam, 988; Rodriguez (Pulliam, 11988; Rodrfguez et et al., 11994; Warren, 994). Warren, 11994). Adding Adding realistic variation in habitat quality is likely to be one one of of the the key key issues facing empirical and theoretical metapopulation biologists in coming facing coming years. The 996) is almost certainly unusual, unusual, The following example (C. D. Thomas Thomas et et al., 11996) but nonetheless nonetheless highlights the the potential potential complexity of of metapopulation dynamics when more than than one habitat habitat type is present. A metapopulation of of the checker­ checkerspot E. editha editha (an unnamed race which differs differs from bayensis) bayensis) occurs occurs at 2000 2000 to 3000 m elevation in openings in coniferous coniferous forest in Sequoia National Forest and Sequoia 967, the butterflies butterflies were restricted Sequoia National National Park Park in California. Before 11967, to natural of their eggs on Pedicularis Pedicularis semibarbata natural rocky outcrops and laid most of semibarbata (Scrophulariaceae). 967, clearings (Scrophulariaceae). Around Around 11967, clearings in the forest were made made by logging. Pedicularis in­ Pedicularis semibarbata semibarbata disappeared disappeared from clear-cut clear-cut areas, areas, but the the butterfly invaded this habitat and colonized Collinsia Collinsia torreyi, torreyi, which is also in the Scrophu­ Scrophulariaceae, 1 983; Singer 1 993, lariaceae, but which is not used on outcrops outcrops (Singer, 1983; Singer et et al. al.,, 1993, 11994). 994), By 11985, 985, a patchwork of 00 + km km22,, of host use use had been been established established over over 1100+ with P. semibarbata semibarbata as the principal host on unlogged outcrops and C. torreyi torreyi as the principal host in clear-cuts. Clear-cuts 980s. Although Clear-cuts acted acted as as population population sources sources during during the the 11980s. Although the the clear­ clearcut habitat received received fewer eggs, it generated generated more adults due to higher higher survival there 983; Moore, 11989). 989). The there than than on outcrops outcrops (Singer, 11983; The butterflies moved from clear-cut to outcrop about twice as frequently as they moved in the opposite direction. Biased Biased movement generated generated a gradient in insect density, such that em­ emigration from clear-cuts (c. D. Thomas clear-cuts raised insect densities on nearby nearby outcrops outcrops (C. Thomas et 996). et al. al.,, 11996). Then, Then, a severe summer summer frost killed virtually all of of the C. torreyi torreyi in the clear­ clear992 (Singer 994). Although E. editha cut habitat habitat in 11992 (Singer et al. al.,, 11994). Although E. editha eggs and larvae larvae were were not damaged damaged by the cold, the larvae starved. The populations populations in this habitat de­ de1 992 to two in 1993, 1 993, even though C. torreyi clined from - - - 11 04 0 4 egg batches batches in 1992 torreyi regenerated 1 993. Pedicularis Pedicularis semibarbata semibarbata was regenerated in abundance abundance in all clear-cuts clear-cuts in 1993. unaffected 992 frost, unaffected by by the the 11992 frost, E. editha editha survival survival was was apparently normal normal on on out­ outcrops, and there there was no mass mass extinction in this habitat. habitat. Extinction of of the source populations populations set up a fascinating fascinating natural experiment. In the sourcesink theory, sources are areas source-sink areas of of habitat which generate generate individuals and sinks consume consume them; sinks are are areas which are are populated populated because because there is a net influx of of migrants migrants into the habitat, and they are predicted to become extinct in the absence 988). "Pseudosinks" absence of of immigration (Pulliam, 11988). "Pseudosinks" are are areas areas which can support support a population population without immigration, but where where immigration immigration increases increases population population density density above above the the local local equilibrium; equilibrium; removal removal of of immigration immigration should should result in a decline in density to the local carrying capacity rather rather than than in extinction

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(Holt, 985; Watkinson 1 995). Following sudden extinction extinction (Holt, 11985; Watkinson and Sutherland, Sutherland, 1995). Following the sudden of overall egg egg densities of the population population sources, sources, overall densities fell on P. semibarbata, semibarbata, the decline decline was greatest 993 densities greatest close close to former former population population sources, sources, and and the 11993 densities on out­ outcrops crops were were no no longer longer correlated correlated with with isolation isolation from former former sources. sources. In this case, we know pseudosinks, and know that that natural natural outcrops outcrops were were pseudosinks, and not not true true sinks, because because E. editha editha populations populations occurred occurred on outcrops outcrops before before the the clear-cut clear-cut habitat habitat was was created, created, they survived on on outcrops outcrops after the frost, frost, and and they persisted persisted throughout throughout the study period period at moderate moderate abundance abundance on undisturbed undisturbed outcrops outcrops in Sequoia National National Park, Park, et al., 1 996). to the south of our disturbed study sites (c. D. Thomas south of our disturbed (C. Thomas 1996). Source Source populations populations are often considered considered especially important important for for metapop­ metapopulation ulation persistence persistence -they m they clearly are are when when long-term long-term survival is impossible impossible in sink habitats, as has has been been shown shown for for the blue blue butterfly Cyanaris Cyanaris semiargus semiargus (Rod­ (Rodriguez 994). However, matters to rfguez et al. al.,, 11994). However, it is resistance resistance to extinction extinction that that really really matters persistence, persistence, not not the balance balance of of birth birth and and death death in "normal" "normal" years. In this particular particular pseudo sinks were example, pseudosinks were more more resilient resilient to a particular particular type of of extreme extreme envi­ environmental ronmental event, although although it may well be that that in most most metapopulations metapopulations the the sources sources are are usually the more more resilient resilient to environmental environmental extremes. extremes. Empirical Empirical evidence is lacking. for lacking. Both Both source source and and pseudosink pseudosink populations populations may may be prone prone to extinction extinction for all of of the reasons reasons given in Section Section lILA. III.A. Unfortunately, Unfortunately, there is almost almost no no em­ empirical information between local population population information with which which to assess assess the relationship relationship between productivity in a "normal" "normal" year, and ability to survive environmental environmental extremes. extremes.

D. Spatial Spatial Synchrony Synchrony in Population Population Dynamics Dynamics When uctuate in synchrony, When populations populations fl fluctuate synchrony, and and particularly particularly when when they they become become extinct much extinct in synchrony, synchrony, the probability probability of of metapopulation metapopulation persistence persistence may be be much lower Hanski, 1991). 1 99 1 ). When When the chance lower than than predicted predicted by standard standard models models ((Hanski, chance of of extinction the probability extinction is completely independent independent in each each local population, population, the probability of of metapopulation metapopulation extinction extinction rapidly rapidly declines declines with increasing increasing number number of of local pop­ populations Fig. 4; Hanski, 11991, 99 1 , this volume), ulations ((Fig. volume), but but if if extinction probabilities probabilities are correlated, correlated, for for instance instance because because local popUlations populations are responding responding in a similar similar way to climatic variability, metapopulations popu­ metapopulations with even even large large numbers numbers of of local populations may Hanski, 11991, 99 1 , this may be susceptible susceptible to extinction extinction ((Hanski, this volume). volume). There There are examples examples of of synchronous synchronous butterfly extinctions extinctions in response response to single climatic Ehrlich et al., 11980; 980; C. D. Thomas 1 996; above) events events in E. editha editha ((Ehrlich Thomas et al. al.,, 1996; above) and and Aphantopus hyperantus Pollard and Yates, 993; Sutcliffe 996b), and and Aphantopus hyperantus ((Pollard Yates, 11993; Sutcliffe et al. al.,, 11996b), Hanski, evidence evidence that that extinction extinction probability probability varies varies between years in M. M. cinxia cinxia ((Hanski, this volume). As As yet, these these events events have have rarely been been shown shown to cause cause metapopula­ metapopulalarge metapopulations metapopulations or tionwide extinction, extinction, but but the above above examples concern concern large or ones habitat ones which contain contain either either some some very large patches patches or or some some relatively safe safe habitat number of type. The The example in which which B. aquilonaris aquilonaris became became extinct from from a number of forest patches, after forest fragments fragments that that contained contained fewer fewer than than 20 habitat habitat patches, after a wet and and cloudy summer summer (T. Ebenhard, Ebenhard, personal personal communication, communication, above), above), shows shows that that ex­ extreme environmental environmental events events can cause cause entire entire metapopulations metapopulations to become become extinct.

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Over Over longer longer time time periods, periods, landscape landscape and and habitat habitat changes changes that that cause cause deterministic deterministic Hesperia extinctions have have often often been been relatively relatively synchronized synchronized over over large large areas. areas. Hesperia extinctions comma comma and and Lysandra Lysandra bellargus bel/argus both both showed showed aa period period of of rapid rapid decline decline in in England England when when myxomatosis myxomatosis killed killed rabbits, rabbits, and and their their short-grass short-grass habitats habitats disappeared disappeared throughout the the UK UK (J. A. Thomas, 1983a; 1 983a; C. D. Thomas Thomas and and Jones, Jones, 1993). 1 993). SimSim­ throughout economic pressures pressures and and technological technological innovations innovations that that cause cause changes changes in ilarly, economic farming or or forestry forestry practices practices usually usually do do so so over over very large areas areas in a relatively farming very large short short space space of of time, causing causing widespread widespread changes changes in the the fortunes fortunes of of associated associated The information information that that exists at the the moment moment suggests suggests that that extinctions extinctions are are species. The least partially synchronized. synchronized. normally at least the absence absence of of better better data data on on the the spatial spatial synchrony of extinctions and and In the synchrony of colonizations (see also also Hanski, Hanski, this this volume), volume), we we must must rely on on analyses analyses of of extant extant colonizations populations, and and presume presume that that different different levels of of synchrony synchrony in in their their dynamics dynamics populations, provide some some insight insight into into the the extent extent to to which which local local extinctions extinctions might might be be synsyn­ provide chronized Analyses of of butterfly, butterfly, moth, and aphid population popUlation chronized over over wide areas. Analyses and aphid over wide wide areas (Britain) suggest popUlations fluctuate in synsyn­ dynamics over areas (Britain) suggest that populations chrony over areas of at least 105 1 05 km km22 ((Pollard Pollard and Yates, Yates, 1993; 1 993; Hanski Hanski and and chrony over areas of ' s metapopuWoiwod, 1993), 1 993), which which is orders orders of of magnitude magnitude greater than anyone metapopu­ Woiwod, greater than anyone's For many of of these species, correlated correlated population popUlation fluctuations fluctuations lation study areas. For occur over over areas much larger distances occur areas that that are are so much larger than than their potential migration distances of year-to-year that climate must be a major major determinant determinant of year-to-year population population variability ((Pollard Pollard and Yates, 1993, 1 993, and references therein). These These conclusions conclusions are based and references are based on counts counts of others) from from on of insects (transect (transect counts counts for for butterflies, traps traps for for the others) scattered locations locations across However, each point widely scattered across the landscape. However, each sampling point together insects from more population, may to some extent extent lump together more than than one one local population, and local popUlations populations may be fluctuating partly out of of synchrony. In the case of of the = ~ 11-- to 3-km butterfly transect transect walks, several habitats habitats are are sampled. For aphids aphids and moths, traps traps may attract insects from more more than than one habitat. If If each sampling location counts insects from more than than one habitat habitat patch, local population population vari­ variability may have been been averaged out, leaving only residual large-scale large-scale variability caused caused by by the climate climate to be be detected. detected. Studies of of population population fluctuations at a smaller scale provide provide a rather different different picture. Small-scale analyses are possible for but­ for butterflies because the British butterfly transects are usually divided into about 10 1 0 sections, and separate counts are made for each section. Populations Populations in individual sections often fluctuate in parallel with regional fluctuations, apparently because of of weather weather effects, but changes in local habitat management can cause deviations from regional trends (Pollard and Yates, 11993). 993). When local habitats improve in quality, the change in local popu­ population size is upward upward relative to the overall regional trend, and downward downward when local habitats deteriorate ((Pollard Pollard and Yates, 11993). 993). Such deviations are partic­ particularly clear in species which are associated with successional vegetation. Plebejus argus fluctuates out of synchrony in areas which are only 500 m apart on suc­ successional habitats 99 1 ), and M. athalia, which habitats in heathland (c. (C. D. Thomas, 11991), inhabits inhabits freshly made made clearings in a forest in southeast southeast England, increases in new

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others as they become overgrown ((Warclearings, but simultaneously declines in others War­ ren, 11987b, 987b, 11991). 99 1 ). There are good and bad years for these species over wide areas, associated associated with climatic fluctuations, fluctuations, but local populations behave idiosyn­ idiosynareas, response to local habitat conditions. cratically in response Even in nonsuccessional species, analyses of local population fluctuations show that there is a great deal of heterogeneity in local dynamics, often but not (mialways associated with different responses of local populations in different (mi­ cro)habitats to annual variation in the climate (Ehrlich et 975; Sutcliffe et et al., al., 11975; et al.,, 11996a,b; al. 996a,b; see Fig. 2 in Hanski, this volume). Such heterogeneity in local E. editha editha above, dynamics may be crucial to long-term persistence. As found with E. habitat in most years may be crucial to per­ perwhat appears to be relatively poor habitat sistence after some extreme environmental event The scale over which systemwide extinction is likely to take place as a result of extreme events is one of the most crucial, but least well understood, aspects of metapopulation biology and needs to be addressed by long-term and large­ largescale field studies.

Conclusions VI. Conclusions Many of the predicted predicted patterns patterns and processes are widely observed in studies of butterfly metapopulations. to realize that a key of metapopulations. However, we have also come cometo identify the critical habitat habitat requirements requirements of of different different empirical challenge is to identify species and the factors causing changes in the distribution of habitats. Species individualistic habitat habitat and host plant plant requirements, requirements, hence the habitat patch have individualistic networks available to each species are specific and not generally congruous with human of general for each human definitions of general vegetation type. The specific habitat habitat mosaic for modem, human-dominated human-dominated landland­ species is likely to be dynamic, especially in modern, scapes, distributions of scapes, and changes in the distributions of species species are often often driven by spatial changes in the distribution of suitable distribution of suitable habitats. habitats. Populations in large habitat habitat patches patches have have low rates of of extinction for Populations for several reasons, including size, high habitat heterogeneity, and including large initial population population size, low risk of extinction from habitat dynamics. There is also some evidence of extinction evidence to show that isolated isolated local populations populations are relatively prone to extinction. extinction. ColoniColoni­ zation probability probability is determined determined by isolation, zation isolation, by the the sizes of of source populations, populations, and of the the patch be colonized colonized (large patches more likely and also by the size of patch to be patches are are more become colonized). colonized). The The dynamic dynamic processes processes of of extinction and colonization colonization can can to become extinction and thus generate generate the the widely widely observed observed pattern pattern in which which large large patches patches that that are located located close to each each other other are are likely to to be populated populated but but in which which small small and and isolated isolated patches are are usually usually empty. It appears appears that that the the flow of of migrants in and and out out of of patches habitat habitat patches patches is also an important important determinant determinant of of patch patch occupancy occupancy and and local population sizes, sizes, and and this needs to to be be addressed addressed more more specifically specifically in the the next population generation of of models. models. Considerable spatially realistic Considerable progress progress has has been been made made with with spatially realistic simulation simulation modmod­ els, which which have have successfully successfully predicted predicted the the observed patterns patterns of of patch patch occupancy

384 384

Chris Thomas and and Ilkka IIkka Hanski Chris D. D. Thomas Hanski

based on the dynamic processes of of extinction and colonization. colonization. These models models based have predicted persistence persistence where where metapopulations metapopulations do survive and extinction extinction where they do not, and qualitatively, the models have predicted which empty networks networks of habitat habitat patches patches butterflies will invade invade and which networks they will between model predictions and fi field not invade. Quantitative differences between eld data have been useful in revealing where further biological information is required, required, for for example example on migration and the effects of habitat habitat quality. Use of the models in specific systems has also helped us to identify some general problems, including the need for for butterfly population population dynamics to be superimposed superimposed on shifting habitat mosaics. We return now to two serious issues which which have not yet been settled and pop­ where much more information is required. required. The first relates to migration and population structure, and the second to the importance of of specifi specificc habitats to persist­ persistence. Metapopulation Metapopulation research research has stimulated a considerable considerable reappraisal of of the migration capacities of butterflies. Results that are presently available suggest that the notion of of a local population, which is at the core of metapopulation ideas, may be under under threat. Exchange rates of individuals among among adjacent but distinct habitat habitat patches may be so high (sometimes > > 20%) that that local populations have only limited demographic independence. independence. If the population population in an individual patch includes many immigrants, stochastic breeding failure will not result in extinction of i ) the habitat im­ of that "local" "local" population population unless ((i) habitat changes in such a way that immigrants no longer enter or remain in the patch or (ii) regional stochasticity produces patches, thus interrupting produces simultaneous breeding breeding failures in a group of of patches, immigration. Observed local extinction rates may therefore be much lower than 995b; Hanski, this volume). Yet, even when the underlying rate (Han ski et (Hanski et at., al., 11995b; dea metapopulation consists of an assemblage of such populations with a high de­ gree of popUlations at of connectance, connectance, direct interactions interactions (migration) between between local populations the opposite ends of the same network im­ network may never occur. In such systems, immigration and emigration are important important determinants determinants of of local dynamics, but the whole network network is certainly not one panmictic population. Migration is vital to local dynamics as well as to metapopulation-wide metapopulation-wide processes. Most current meta­ metapopUlation population thinking thinking (if not modeling) limits the role of migration to one of of seeding empty habitat patches (with little effect of of immigration on abundance abundance after col­ colonization), and to a lesser extent extent as propping up small local populations which are under threat threat from stochastic extinction (rescue effect). The role of migration in local dynamics needs to be explored more fully in structured structured metapopulation metapopulation 1 992; Hanski and Gyllenberg, 1993; 1 993; Gyllenberg models (Gyllenberg and Hanski, 1992; et popUlation variavaria­ et at. al.,, this volume) and through analyses of of spatial patterns in population bility, Hanski, this volume). bility, colonization, colonization, and and extinction extinction ((Hanski, volume). Because Because emigration and and immigration rates vary with patch size and and isolation, real metapopulations metapopu­ metapopulations do not fall easily into the various categories categories of of metapopulation types ((Harrison, Harrison, 11991; 99 1 ; Harrison and Taylor, this volume). In some parts of of a patch patch network, persistence may largely depend on the existence of one or a few large blocks of habitat habitat (mainlands), but other parts of the same system may persist

1155

Butterfly ButterflyMetopopulotions Metapopulations

38S 385

because there there is a high density of small patches. patches. Parts Parts of of metapopulations with a high density of of small patches, each with a high emigration and immigration rate, resemble scaled-down versions of of the population structure of of highly mobile spe­ species ((patchy patchy populations, sensu 99 1 ). In mobile species, such as the sensu Harrison, 11991). nettle-feeding nymphalids in Europe, practically no single patch could support a nettle-feeding local population for for more than a few generations in isolation, and each individual will enter and leave several such patches. The distinction between between relatively sed­ sedentary species which are regarded as existing as metapopulations metapopulations and the more mobile species with "patchy populations" is becoming increasingly vague. In some cases, it is just a matter of of scaling. The relative contribution of of local versus "populations" within metapopuregional population processes in different local "populations" metapopu­ lations and in different metapopulations is a much more important issue than trying to pigeonhole each system and give it an approved name. General models and fi eld studies exploring these notions would be very useful. field The contribution im­ contribution of of specifi specificc habitats habitats to persistence persistence is also becoming an important question. Most metapopulation models and fi eld studies examine proba­ field probabilities of extinction in relation to patch area, local population popUlation size and isolation, but pay limited attention attention to variation variation in habitat habitat quality. Nonetheless, there there is already enough evidence to suggest that the impor­ the type of habitat can be just just as important. If populations respond differently to environmental stochasticity in different habitats or microhabitats, popUlations against microhabitats, habitat habitat heterogeneity can buffer buffer populations large fluctuations fluctuations and extinction. We should ask whether whether large populations in large habitat patches survive best because they are large, or because large patches patches usually contain several microhabitats; microhabitats; and whether large metapopulations persist because they have much habitat habitat or because they have more kinds of of habitats habitats and microhabitats than 0 patches, than small patch networks? networks? A small metapopulation metapopulation in 110 patches, each of of a slightly different different habitat type, might possibly persist for longer than a metapopulation patches. meta population in 50 identical patches. Allied to this is the question of of whether whether some habitats habitats always hold the key L. bel/argus to persistence. J. A. Thomas ((1983a) l 983a) suggested that L. bellargus may spread spread in good years, but is confi ned to population refuges in bad years. The same argument confined has been put forward for for several mobile species which may breed over large areas at favorable times of year, but retract to specifi specificc habitats at other times (Shapiro, 11979; 979; Jordano et al., 11991). 99 1 ). If this phenomenon et al., phenomenon is widespread, widespread, the existence existence of of specifi specificc habitats habitats wit:lin within patches patches or patch patch networks may be more more important to persistence persistence than patch size or number. An entire program of of empirical empirical research is required to evaluate to what extent populations in different different habitats vary in their responses to environmental stochasticity, whether whether habitat heterogeneity buf­ buffers populations populations against extinction, extinction, and whether specific habitats habitats hold the key to persistence. Finally, a metapopulation metapopulation approach approach is becoming important important in several other areas of of butterfly population biology. A metapopulation approach approach provides the potential to bridge the gap between between studies of of local population dynamics and species distributions. Densities, Densities, sizes, and average suitabilities suitabilities of of habitat habitat patches

386 386

Chris Chris D. D. Thomas Thomas and and IIkka Ilkka Hanski HanskJ

may vary geographically, but this aspect of metapopulation biology has barely been considered for butterflies. Comparing central and marginal parts of species ranges is almost bound to reveal interesting results (J. A. Thomas, 1993; 1 993; J. A. Thomas et 994). Another area of interest is the extent to which metapopu­ et ai., al., 11994). which metapopulation structure and migration affect local adaptations to different habitats (C. (c. D. Thomas and Singer, 11987; 987; Thompson, 11993, 993, 11994; 994; Singer and Thomas, 1996; 1 996; Barton and Whitlock, this volume) and levels of of genetic variation (Descimon and Napolitano, 11993a,b; 993a,b; Hedrick and Gilpin, this volume), and whether habitat gege­ 1 976; Dempster, Dempster et ometry itself affects the evolution of of migration ((Dempster et ai., al., 1976; 11991; 99 1 ; Olivieri and Gouyon, this volume). In an ever changing landscape, evo­ evolutionary changes may play an increasingly important role in popUlation persist­ population persistence and extinction.

16

TTritrophic ritrophic Metapopulation Metapopulation Dynamics Dynamics A A Case CaseStudy Study of Ragwort, Ragwort, the Cinnabar Cinnabar Moth, Moth, and the Parasitoid ParasitoidCotesia Cotesiapopularis popularis Ed van der Meiiden Meijden

Catharina Catharina A. M. van der Veen­ Veenvan Wiik Wijk

I. INTRODUGION INTRODUCTION Many short-lived monocarpic plant species have a markedly patchy distri­ distribution. These species reproduce only once in their lifetime, they typically exploit disturbed habitats, and they often have a high rate of local extinction (Harper, 11977; 977; Gross and Werner, 11978; 978; van Baalen, 11982; 982; Reinartz, 1984; 1 984; Grubb, 11977; 977; de long 988; see van der Meijden et 1 992, for a review). Jong and Klinkhamer, 11988; et al. al.,, 1992, We II-known examples are biennial plant Well-known plant species colonizing colonizing windfalls in wood­ woodlocalIy grazed or otherwise disturbed vegetation on sand lands, species exploiting exploiting locally dunes and chalk grasslands, and species of "old fields." To survive over long periods of of time, such extinction-prone extinction-prone biennials depend on regional regeneration through interacting local populations populations (metapopulations). (metapopulations). Critical processes for long-term persistence persistence include seed dispersal and dormancy in variable and patch­ patchdistributed environments (Kuno, 11981; et al. al.,, 11987). of po98 1 ; Klinkhamer et 987). Also of po­ ily distributed tential importance are biotic interactions with species at higher trophic trophic levels. levels. Often the herbivores and their parasites are monophagous, monophagous, and their populations populations too may function as metapopulations. metapopulations. Dynamics of of species at the higher trophic levels are necessarily affected by dynamics of the host plant, but the herbivores and predators can also play a more active role by modifying the extinction prob­ probabilities abilities at other trophic levels (Nee et et al. al.,, this volume; Holt, this volume). Metapoplliariofl Metapopulation B/(J/ogy Biology 1997 by Academic Prc�s. Copyright � 9 1997 Press. Inc. All rights of of reproduction reproduction in any form reserved.

387

388

Ed Ed van van der der Meijden Meijden and and Catharina Catharina A. A. M. M. van van der der Veen·van Veen-van Wijk Wijk

In this chapter, we analyze whether whether and to what extent interactions interactions within a tritrophic system are affected by the spatial distribution of patches. In of habitat patches. doing so, we follow the suggestion of 1 995): "many of Harrison Harrison et al. ((1995): "many populations appear patchy to the human eye, but [that] critical examination examination is required to deduce the dynamic consequences of cally, we review of this patchiness." patchiness." Specifi Specifically, our long-term data data (two decades) on the relationships between between the plant plant ragwort (Senecio jacobaea), jacobaea), its most important herbivore, the monophagous cinnabar cinnabar moth (Tyria jacobaeae), jacobaeae), and the specialist parasitoid of of the herbivore, Cotesia Cotesia

popularis. popularis. Ragwort has been the subject of of intensive studies in several countries over a long time, starting in 11935 935 when Cameron summarized his early biocontrol proj­ proj"Natural Control of of Ragwort." Ragwort seeds had been ect in a study entitled "Natural accidentally introduced introduced into New Zealand, and the plant had grown into a major pest 874 and 1900. 1 900. pest by colonizing the entire country in a few decades, between between 11874 The wort is undoubtedly based The powerful colonizing and and weedy behavior of rag ragwort on its capacity to efficiently exploit scattered scattered disturbed habitat habitat patches. patches. Ragwort is a "pest" thanks to the alkaloids that it produces, which are toxic to cattle but not to the cinnabar 1 957), which was used as a bioconbiocon­ cinnabar moth (Harper and Wood, 1957), trol agent. Subsequently, two additional additional long-term population studies of this plant - moth system have been carried plant-moth carried out in the United United Kingdom Kingdom (Dempster, (Dempster, 11982; 982; Crawley 989). Dempster concluded moth's Crawley and and Gillman, 11989). concluded that that "the moth's population is buffered against extinction by the heterogeneity within the habitat," habitat," indicating that that some sort of of spatial effects are important. In the early 1970s, 1 970s, we commenced our studies of ragwort in The Netherlands on three small local dune populations. Within 3 years, two of the three populations had become extinct. As ragwort density in the dune area area as a whole did not continue to decrease, decrease, we became convinced that population dynamics of this species should be studied on a much broader broader scale and that that spatial aspects are crucial for understanding the mechanism of of persistence, persistence, which is the focus of the present chapter. chapter. Apart from the patchy distribution of the plant and, consequently, of the herbivore herbivore and its parasitoid, typical features of of this tritrophic tritrophic system on sand dunes include frequent complete defoliation of plants by the cinnabar cinnabar moth (not only leaves are consumed, but also buds and fl owers, thus reducing ragwort flowers, seed production to zero). The lifetimes of of local plant populations are restricted, restricted, with the cinnabar cinnabar moth and its parasitoid continuously tracking these ephemeral ephemeral populations (van der Meijden, 11979a). 979a). chapter, we will first give an outline outline of of the the population dynamics of of In this chapter, 1 99 1 , 11992) 992) and describe how they the three three organisms (van der Meijden Meijden et al. al.,, 1991, interact interact with each other. Next, we will calculate parameters describing the degree of of synchrony between between local populations and the metapopulation to reveal to what what extent the dynamics of local populations differ differ from each other. We also pay attention to spatial correlations within and between between the three species. Finally, we will discuss the mechanisms that appear of the appear to play a role role in the persistence of the

116 6

TTritrophic ritrophic Metapopulation Metapopuiation Dynamics Dynamics

389 389

plant, the herbivore, and the parasitoid in their patchy patchy environment. This infor­ infor(Han ski and Gilpin, 11991, 99 1 , mation will be used to infer the type of of metapopulation (Hanski this volume) that best describes the organisms of this tritrophic interaction. tritrophic interaction.

II. MATERIALS AND METHODS MATERIALSAND METHODS A. A. Study StudySystem System Ragwort is a facultative facultative biennial plant. It is native to Europe and has invaded overgrazed 957; Dempster, 11982; 982; overgrazed areas throughout the world (Harper (Harper and Wood, 11957; van 979b). Its weedy character van der der Meijden, 11979b). character is largely due due to its extremely powerful powerful reproductive reproductive potentials. Individual plants may produce up to 20,000 seeds with pappus 1 940) refer to pappus that that enable wind dispersal. dispersal. Poole and and Cairns ((1940) per plant plant ranging from 50,000 to 1150,000 seed numbers per 50,000 in New Zealand. Mowing, plowing, and other other such conditions conditions that that reduce reduce the opportunities opportunities of of generative reproduction stimulate vegetative reproduction (Poole and Cairns, 11940; 940; Harris et al. , 1 978). Even in vivo, small root fragments may develop into mature plants al., 1978). (van der Meijden, 11979b). 979b). Seeds may remain viable for more than 8 years (Poole and Cairns, 11940). 940). Ragwort is a common weed weed of sand sand dunes, dunes, roadsides, and and waste lands. Local disturbances disturbances create suitable circumstances for for establishment, establishment, whereas vegetation of suitable growing sites. Often, however, populations succession may lead to loss of disappear disappear without any changes in the vegetation. Such sites may become recol­ recolonized at a later later time. The cinnabar cinnabar moth is a univoltine insect and monophagous on ragwort (else­ (elsewhere vulgaris; where it has been reported reported to occasionally use the closely related related Senecio Senecio vulgaris; Aplin and Rothschild, 11972). 972). It lays its eggs in small batches of of ca. 30 eggs. (Dempster, 11982). First or second second instar Fecundity varies from l100 Oa to 400 eggs (Dempster, 982). First po­ larvae may become parasitized by the specialist braconid parasitoid Cotesia Cotesia popularis. S parasitoid larvae may develop per host larva. These larvae leave pularis. Up to I15 their host shortly before before it would otherwise have pupated, and the host then dies. From the the third instar onward larvae show a tendency to disperse. This This is especially and fifth instar instar when many larvae larvae leave their original original food plant so in the fourth and before it is fully defoliated. defoliated. This dispersal tendency tendency is related to the numbers of of larvae on the plant and the plant size (van der Meijden, 1 976; Sjerps and Haccou, larvae plant Meijden, 1976; Sjerps 11996). 996). Population data on the the three species were collected collected in a coastal sand dune area near The Hague in The Netherlands. Netherlands. The patchy distribution distribution pattern pattern of of local ragwort populations is brought about by the geomorphology of of the dunes in com­ combination bination with grazing activity of of rabbits. rabbits. The landscape is a mosaic of of north­ northof trees, shrubs, and grasses, poorly vegfacing slopes with a closed vegetation of veg­ etated etated south-facing slopes, and and valleys with a vegetation depending depending on the

390 390

Ed Edvan van der der Meijden Meijdenand and Catharina CatharinaA.A. M. M. van van der der Veen-van Veen-vanWijk Wijk

groundwater groundwater level. level. Ragwort Ragwort can can potentially grow grow anywhere anywhere in in this this landscape landscape pro­ provided that that neither the the grass grass layer layer nor nor the the shrub/tree shrub/tree canopy canopy is is closed. closed. vided An An area area of of about about 66 km2 km 2 of of aa much much larger larger system system was was searched searched for for local local ragwort 973. Plants ragwort populations populations in in 11973. Plants are are considered considered to to belong belong to to one one local local popu­ population lation if they they are are not spatially separated from each each other other by ragwort-free distances distances of 50 local populations, 1102 02 were selected, based on differ­ of more than than 5 m. Of 1150 differences ences in population density and habitat characteristics, such such as the amount amount of of Formica polyc­ shade by woody perennials and the presence of the predatory ant shade and the of the ant polyctena. Local populations covered areas 974 (mean areas ranging from from 8 to 3000 m2 m 2 in 11974 2 ). Plant numbers (from small vegetative rosettes to large fl owering plants) m2). flowering 900 m per population varied from 11.5 .5 to 62 per m2• m 2. The distance between populations ranged 0 to over 200 m, with at least 5 m without any ranged from 110 any ragworts. Local populations were usually separated from each other by barriers like scrubs, for­ forested areas, dune lakes, or blowouts.

B. Census CensusData B_ Census 974 to 11994. 994. Relevant Census data data were collected from 11974 Relevant data data for this paper are:

wort biomass per 11.. The amount of of rag ragwort per local population sample. During During the the per­ period of of cinnabar cinnabar moth egg laying, in May-June M a y - J u n e in each year, the same same permanent manent squares (4 m m 22)) in each local population were visited three to four four times, and ragwort cover cover in dm22 was estimated estimated (a measure measure of of biomass; biomass; van der der Meijden, Meijden, and prior to oviposition by the the cinnabar cinnabar moth. The The highest value per per local 11979a) 979a) just prior population in in each was used used as the fi nal estimate. Herbivory by by cinnabar cinnabar population each year year was as the final estimate. Herbivory moth larvae larvae may may reduce reduce biomass (and seed production) from from June June onward. DeDe­ moth foliated plants plants often produce regrowth foliage shortly after after herbivory. A A popupopu­ foliated often produce regrowth foliage lation of ragwort was supposed to have gone gone extinct if (either of ragwort if no no living plants plants (either plant, or or mature mature plant) plant) were seedling, rosette plant, were present present on any any census census dates dates 11 year year the cinnabar moth. Based Based on on observations observations made made after cinnabar moth. after the the last last herbivory herbivory episode episode by by the in the the vicinity vicinity of of the the sampling sampling squares, squares, we we concluded concluded that that disappearance disappearance from from aa in sample typically typically meant meant extinction of of that that particular particular local population. popUlation. sample number of of cinnabar moth egg egg batches per local population. Egg Egg 2. The The number cinnabar moth batches per local population. batches were were counted counted in in the the above-mentioned above-mentioned 44 m m22 squares squares at four four to to six visits visits batches with with weekly weekly intervals intervals to to each each local local population population in in May-July May-July each each year. year. A A popupopu­ lation of of the the cinnabar cinnabar moth moth was was supposed supposed to to have have gone gone extinct extinct if if no no eggs eggs were were lation found on on the the plants plants following following aa year year in in which which eggs eggs were were present present in in that that population. population. found 3 . Percentage Percentage parasitism parasitism by by Cotesia. Cotesia. Percentage Percentage parasitism parasitism by by Cotesia Cotesia was was 3. determined (from (from 1988 1 988 onward) onward) by by collecting collecting fourth fourth or or fifth fifth instar instar cinnabar cinnabar moth moth determined larvae in in five five census census populations popUlations at at three three moments moments during during the the larval larval season season (from (from larvae the end end of of May May until until the the beginning beginning of of August) August) because because of of aa seasonal seasonal trend trend in in the percentage parasitism parasitism (Soldaat, (Soldaat, 1991). 1 99 1 ). At At every every date, date, 50 50 larvae larvae were were collected collected percentage per site, site, yielding yielding 750 750 larvae larvae which which were were reared reared to to pupation pupation every every year. year. per

1166

Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics

391 391

III. III. POPULATION POPULATIONDYNAMICS DYNAMICSOF OF RAGWORT RAGWORT Local recolonization of Local extinction extinction of of ragwort ragwort and recolonization of empty patches patches are are frequent frequent populations over events events on sand sand dunes. dunes. Figure Figure 11 (A) shows shows the number number of of extant extant populations over time. In two 1 975 - 1 976 and 1 98 1 - 1 982), 40 two extreme extreme seasons seasons ((1975-1976 and 1981-1982), 40 populations populations dis­ disappeared appeared in a single single year year (Fig. 11,, middle). middle). The The cumulative cumulative extinction extinction curve curve over over time demonstrates demonstrates that that not all local populations populations are are equally vulnerable. vulnerable. Eighteen Eighteen 02 populations populations never never became of of 1102 became extinct during during the period period of of 20 20 years, years, whereas whereas 56 populations recolonized) once even populations disappeared disappeared (and were were recolonized) once or twice and 26 even three to five times. Apparently Apparently ragwort ragwort has refuge refuge populations populations with extincthree with a low extinc­ 992) revealed revealed that tion probability. probability. A A habitat habitat analysis analysis (van (van der der Meijden Meijden et et ai. al.,, 11992) this group of populations is located in areas with a mixed vegetation of populations located areas mixed vegetation of of trees and and shrubs and a not fully closed ground vegetation with grasses and/or mosses and and not closed ground vegetation grasses and/or mosses lichens. lichens. The The populations populations with the highest highest extinction extinction risk are situated situated in open sandy sandy areas. areas. To test whether whether the probability probability of of popUlation population extinction was was related related to their their To 974 until 1976 1 976 (Table size, we analyzed analyzed data data for for the period period from from 11974 (Table I). Contrary Contrary to the general expectation had a higher expectation (Hanski, (Hanski, this volume), volume), small popUlations populations had higher probability of of survival survival than than large large populations. populations. The The reason for this is that that popu­ popuprobability reason for lations in areas areas with trees and/or and/or shrubs, shrubs, with a low extinction extinction risk, tend tend to be small (Table I). Apparently Apparently vegetation structure structure detennines determines the size of of suitable suitable habitat habitat patches. patches. In the the open open dune dune areas areas suitable suitable habitat habitat patches patches are considerably considerably larger larger than in areas areas with trees and/or and/or shrubs. shrubs. Even shows huge uctuations Even on a regional regional (metapopulation) (metapopulation) scale, ragwort ragwort shows huge fl fluctuations ground cover The difference difference in ground ground cover cover between between 1980 1 980 in total ground cover (Fig. 11,, A). The

Extinction, and Habitat Extinction,Survival, Survival, and Habitat Type Type of of Ragwort Ragwort Populations Populations between between in Relation to Patch Patch Size Size in in 11974 11974 974 and 11976 976 in Relation to 974

TABLE I

No. No. of of populations populations Fate Fate

Patch Patch size size m') 2) ((m

< < 110 0 110-100 0- 1 00 1101-1000 0 1 - 1000 11001 00 1 -2000 - 2000 > > 2000 2000 X2 X'

Significance Significance level level

Habitat Habitat

Extant Extant

Extinct Extinct (( % % ))

Trees Trees or or shrubs shrubs

9 9 113 3 117 7 113 3 33

2( 18) 2(18) 4(24) 4(24) 117(50) 7(50) 1 0(43) 10(43) 112(80) 2(80)

10 110 0 23 9 88

111.03 l .03 0.026 0.026

Open Open sandy sandy habitat habitat

7 7 I11I 114 4 7 7

1

15.68 0.003 0.003

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16 16 Tritrophic TritrophicMetapopuiation MetapopulationDynamics Dynamics 1 00

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(Continued) (Continued)

and 11987 987 was almost l100-fold. Oa-fold. There There is a fair fair correlation between total ground ground populations, which suggests cover of of the metapopulation and the number number of of extant populations, that it is unlikely that that there there is much asynchrony in fluctuations fluctuations among among local popthat pop­ ulations. As a measure of of synchrony synchrony we used the correlation coefficient coefficient calculated 2) of between the time series of of ragwort ground ccover o v e r ((dm d m Z2//m m 2) rag wort ground of every local popupopu­ No significant significant negative correlations metapopulation (Fig. 2, top). No lation and and the metapopulation correlations were found. significantly found. Fifty-eight local populations populations were were signifi cantly positively correlated correlated with the overall fluctuation, ground cover cover in 44 44 local populations populations was was not not fluctuation, but ground significantly correlated correlated with metapopulation metapopulation fluctuations. fluctuations. Sixteen populations of significantly populations of this are located this latter latter group group are located in sites sites that that were were completely overgrown overgrown by by dense dense vegetation (the (the percentage percentage cover cover of of trees, trees, shrubs, shrubs, and and grasses grasses increased increased from 1 973 vegetation from 1973 1 994 from from 71 to to 100%). 1 00%). Mechanical Mechanical reduction reduction of of immigrating immigrating seeds, seeds, reduction reduction to to 1994 of "safe "safe sites" for germination, and and reduction reduction of of penetrating penetrating light light by by the the closed closed of for germination, vegetation, may may have have rendered rendered these these sites unsuited unsuited for ragwort germination, germination, vegetation, for ragwort growth, growth, or or survival. survival. To test test whether whether sites sites where where ragwort ragwort had had disappeared disappeared had had indeed indeed become become To unsuitable for plant growth, growth, seeds seeds were were added added experimentally experimentally (Table (Table II). II). Seeds Seeds unsuitable for plant germinated successfully successfully in in all all sites sites and and aa number number of of the the rosettes rosettes survived survived in in most most germinated sites, indicating indicating that that ragwort ragwort is is seed seed limited. limited. However, However, twin twin plots plots that that were were cleared cleared sites, of of the the grass-herb grass-herb vegetation vegetation showed showed considerably considerably higher higher germination germination and and sursur­ vival, demonstrating demonstrating that that the the suitability suitability of of growing growing sites sites was was affected affected by by vegevege­ vival, tation tation development. development. Two main main factors factors are are responsible responsible for for the the fluctuations fluctuations in in biomass biomass and and the the Two shortage of of seeds seeds contributing contributing to to local local extinction: extinction: herbivory herbivory by by cinnabar cinnabar moth moth shortage

Ed van van der der MeJjden Meijden and and Catharina Catharina A.A. M. M. van van der der Veen-van Veen-von Wijk Wijk Ed

394 394

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FIGURE22 (Top) (Top) Pearson's Pearson's correlation correlation coefficient coefficient (r) (1")between between the the amount amount of of ground ground cover cover of of FIGURE ragwort in each each local local population population and and the the metapopulation metapopulation (the (the sum sum of of all all local local populations), populations), reflecting reflecting rag wort in the level level of of synchrony. synchrony. (Bottom) (Bottom) Pearson's Pearson's correlation correlation coefficient coefficient (r) (r) between between the the numbers numbers of ofcinnabar cinnabar the moth eggs eggs per per dm2 dm 2of of ragwort ragwort in in local local populations populations and and in in the the metapopulation metapopulation (the (the sum sum of of all all local local moth populations). populations).

16 1 6 Tritrophic Tritrophic Metapopulation Met(]popul(]tion Dynamics Dyn(]mics

395 395

Experimental Seed Seed Addition Addition to to 25 25 Sites Sites in in TABLE IIII Experimental TABLE Different Habitat Habitat Types Types without without Ragwort Ragwortaa Different Rosette Rosette plants plants

Seedlings Seedlings Habitat Habitat

Open woodland woodland Open Grass-herbs Grass-herbs Moss-herbs Moss-herbs

A A

B B

A A

B B

1 93 193 1 23 123 1 75 175

624 624 531 531 348 348

12 12 44 12 12

21 21 37 37 19 19

a

Three thousand thousand seeds seeds were were sown sown per per aa plot plot o 30 by by "Three off 30 30 cm. em. The The table table gives gives the the mean mean numbers numbers of of seedlings seedlings 30 and rosette rosette plants plants per per plot plot after after 11 year. year. Plots Plots were were either either and undisturbed (A) (A) or or cleared cleared of of the the grass-herb grass- herb vegetation vegetation undisturbed (B). (B).

larvae quantitative effect effect of of herbivory herbivory will be below) and drought larvae (the quantitative be described described below) and drought der Meijden Meijden et et al., ai. , 1985). 1 985). Defoliation Defoliation may may not not only mean mean complete complete loss of of (van der above-ground biomass, but but also also lack of of seed seed production over periods periods of of 2 or or even even above-ground biomass, production over more more years. The effects effects of of drought drought on plant performance performance were were also mentioned mentioned by 1 982) and 1 989). Dempster Dempster ((1982) and Crawley Crawley and and Gillman Gillman ((1989). In Figure 11 (C) colonization wort of patches is plotted plotted over colonization by rag ragwort of empty patches over time. Years Years in which which total total defoliation defoliation in the entire metapopulation metapopulation was observed observed are 11975 975 and 1976, 1 976, 11981 9 8 1 and 11982, 982 , and 986 and 1 987. During and 11986 and 1987. During such such periods periods seed production production is reduced reduced to zero. This figure immediately reveals that colonization colonization of empty patches may take place in years following following total defoliation defoliation and destruc­ destruction of of the seed crop. Consequently, Consequently, colonization colonization probably occurs through through ger­ germination mination from from the seed bank. However, relatively high rates of of colonization were found also in the second and third year following following defoliation, defoliation, when plants in the extant populations produced produced again seeds, suggesting that seed dispersal is also important important to to colonization. colonization.

IV. IV. MECHANISMS MECHANISMSOF OF PERSISTENCE PERSISTENCEIN IN RAGWORT RAGWORT Ragwort has three mechanisms that enhance the probability of of survival on local scale: local scale:

11.. Regrowth Regrowth capacity. capacity. Ragwort has has spectacular powers powers of of regeneration, defoliated plants may which make it such aa powerful weed. Even completely defoliated 988). Plants that regenerate regenerate (van der der Meijden et et ai. al.,, 11988). that were were artificially artificially damaged damaged by eld experiment by removing removing the the whole shoot shoot in in aa fifield experiment suffered suffered only 5% more more mor­ mortality tality than than control control plants. Biomass Biomass was was reduced reduced by by 35%. 35%. However, However, the the negative effect effect of of herbivory herbivory may may be be much much greater greater in in combination with with adverse adverse weather weather

396

Ed and Catharina Ed van van der der Meijden Meijden and Catharino A. A. M. M. van van der der Veen-van Veen-van Wijk Wijk

conditions (drought) during or after defoliation (van der 01., 1985; 1 985; der Meijden Meijden et et al., Prins 990). Repeated Repeated defoliation reduces regrowth as Prins and Nell, 11990). defoliation reduces as well (McEvoy, 11985). 985). Despite Despite the the fact that that ragwort is a biennial biennial plant, and and normally dies dies after after a Anal­ single production of of seeds, the life cycle may be prolonged by herbivory. Analogous to to aa seed seed bank bank we can can talk about about aa rosette bank bank in this this species. species. 2. Dormancy. Dormancy. Ragwort has only a small small seed bank, bank, but it seems seems to be effec­ effective, judging judging from germination in many many open open sites after at least least 2 years years in which which seed production was reduced to zero zero by cinnabar cinnabar moth moth herbivory (van der der Meijden Meijden et 985). Fifty 983, after et 01., al., 11985). Fifty populations were sampled in 11983, after two successive successive seasons seasons of total defoliation, by taking eight circular (diameter 1 0 cm) soil cores per pop­ of taking circular (diameter 10 cores per population. Samples of 3 1 populations did not contain any viable seeds. In 9 popu­ of 31 populations one seed was found, than found, in 4 two seeds, seeds, in I1 three three seeds, seeds, and and in 5 more than five seeds (which means . 1 4 m22 soil samples). These numbers five means 55 viable seeds in 33.14 numbers 20,000 seeds owering are small compared compared with a production of of 20,000 seeds per per individual fl flowering plant. Many Many local populations do not seem to have a seed bank bank at all, though one one cannot cannot be sure based based on small small samples samples of of soil. Experiments on the the longevity of of buried buried seeds showed that that viability was positively related related to the depth depth at which they were buried. Viability of 0 cm was hardly of seeds buried buried at a depth of of 110 hardly reduced reduced after 5 years (E. van der Meijden, Meijden, unpublished unpublished results). 1 940) showed 3. Seed Seed dispersal. dispersal. Experiments by Poole and and Cairns Cairns ((1940) showed that that the the majority of of ragwort seeds landed landed within a few meters meters from the the parent parent plant. plant. Yet aa considerable considerable number number of of seeds seeds was was found found to to have have been dispersed dispersed up up to 20 m, and especially in the direction direction of of prevailing winds the dispersal curve seemed seemed to der Meijden Meijden et et 01., al., 1985) demonstrated that that both have a long tail. We (van der 1 985) have demonstrated of dormant seeds and colonization through seed dispersal were imgermination of im­ portant mechanisms mechanisms in the reestablishment reestablishment of of local populations. populations. Because Because no seeds seeds were were formed in the 2 earlier earlier years, years, the occurrence occurrence of of seedlings seedlings in empty patches patches after a period of indicates germination from a seed bank. However, However, of defoliation indicates the level of of germination in empty patches patches in subsequent subsequent years years was much too high high explained by germination of of dormant seeds alone, and and much of of this gerto be explained ger­ mination must have been the result of of seed dispersal. dispersal. Unfortunately, our data data do not allow us to judge refuge populations played an judge whether whether the small number number of of refuge essential essential role role as as sources sources of of dispersing dispersing seeds_ seeds.

V. OF THE THE CINNABAR AND THE THE PARASITOID V. POPULATION POPULATIONDYNAMICS DYNAMICSOF CINNABARMOTH MOTHAND PARASITOID COTfSIA COTESIAPOPULARIS POPULARIS The The cinnabar cinnabar moth moth is conspicuous conspicuous not only only because because of its appearance appearance (a bright bright red and black colored moth and and even brighter yellow and black black colored larva), but also because of of its behavior. Periodically, larvae completely defoliate defoliate their food plants over large areas (in the Dutch coastal dunes dunes defoliation seems to be highly synchronized). During During such years, thousands thousands of of larvae can be observed to disperse fth instar disperse in in search search of of food. food. Fourth Fourth and and especially especially fi fifth instar larvae larvae can can cover cover

1]66

Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics

397 397

hundreds motor­ hundreds of of meters and and even leave leave their their habitat by crossing crossing beaches beaches and and motorways. Eventually they may defoliate all local populations populations of of ragwort. During such episodes Dempster ((1982) 1 982) and Crawley episodes thousands thousands of of larvae die from starvation. Dempster and Gillman ( 1 989) also found food shortage to be the key factor and (1989) found food shortage factor in population population dynamics in the United Kingdom, determining the magnitude of of population population fluc­ fluctuations first sight, the frequent tuations of of the cinnabar cinnabar moth. At first frequent total total defoliation defoliation and and the enormous uctuations in plant and insect populations populations might give the impression enormous fl fluctuations that this insect-plant insect-plant relationship relationship is one of of the best examples for for refuting coad­ coadaptation aptation and regulation. Nevertheless, both ragwort and the cinnabar cinnabar moth are still common species in sand dunes and some other other habitats of western Europe. The pest characteristics of wort, especially its regenerative powers, might well of rag ragwort, be the result result of of selection selection by the the insect. The The insect, on the other other hand, hand, sequesters the alkaloids of of its food food plant, probably as defense defense against potential natural natural en­ enlen and van der 99 1 ; Ehmke et 990). emies (van Zoe Zoelen der Meijden, 11991; et al. al.,, 11990). Figure 3 shows fl uctuations in the metapopulation size cinnabar moth fluctuations size of of the cinnabar expressed uctuated more expressed as numbers of of eggs per plant biomass. Egg numbers fl fluctuated than 200-fold. For comparison biomass fl uctuations of fluctuations of ragwort are plotted in the same graph. Above a level of 0 eggs per dm22 of of 110 of ragwort, total defoliation will take 99 1 ). The numbers of of take place (van der Meijden Meijden et et al. al.,, 11991). The moth moth lays the largest largest numbers eggs per per unit of of plant plant biomass biomass in open open areas, without without ants ants (F. polyctena; polyctena; Table III). These are also the areas that receive most eggs in absolute terms, because there there are many more plant populations in open open than fully shaded sites. Experi­ Experiments on oviposition oviposition demonstrated demonstrated an almost absolute preference preference for for open areas

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, 3 90 84 88 8~4 86 8'6 90 9'2 94 92 year year FIGURE number of [:[~gRl: 3 ~ Metapopulation dynamics of of the number of cinnabar moth eggs (mean number of eggs per population sample of of 4 m2; open squares) and ragwort ragwort ground cover (black squares). 76 7"6

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398 398

Ed Meijden and and Cathmina Ed van van der der Meijden Catharina A. A. M. M. van van der der Veen-van Veen-van Wijk Wiik Mean dm22 Mean Yearly Yearly Egg Egg Load Load of of the the Cinnabar Cinnabar Moth Moth per per dm of of Ragwort Ragwort in in Local Local Populations Populationsof of Ragwort Ragwort in in Different Different Habitat Habitat Types 974 to 994aa Types from from 11974 to 11994 TABLE TABLE III

Habitat Habitat type

No. of of populations populations

Egg numbers/ numbers/ dm2 dm 2 ragwort ragwort

Standard Standard error error

- SS- F- F S S -- F F - S S F F S S F F

65 65 14 14 1166 7 7

9.08 9.08 3.88 3.88 4.79 4.79 0.99 0.99

0.83 0.83 11.03 .03 11.57 .57 0.41 0.41

a

shaded, Formica-free; a_ S S -- F, F, unshaded, unshaded, Formica-free; S S -- F, F, shaded, habitat. Effects Effects of - SF, SF, unshaded, unshaded, Formica; SF, SF, shaded, shaded, Formica Formica habitat. of shade shade and and Formica are are both both significant significant (P< ( P < 0O.OOI . 0 0 1 in in two-way two-way ANaYA); not sig­ ANOVA); the the interaction interaction between between shade shade and and Formica is is not significant. nificant.

over fully shaded areas al., 11991). 99 1 ). A similar result was found areas (van der Meijden et et al., with respect respect to daily light intensity: the moth does not oviposit in cloudy days. The amount amount of ragwort biomass in open open habitat is probably the the most impor­ important parameter parameter determining the population change of the herbivore. The correla­ correlation coefficient between the amount of of ragwort in the open populations and the number cant (Table IV). number of of cinnabar cinnabar moth eggs in the next year is highly signifi significant This correlation was not signifi cant in the shaded plant populations. significant Cinnabar Cinnabar moth population population growth is followed by an inevitable reduction of of plant plant biomass as explained explained above. The The correlation coefficient coefficient between the the number + 11 is negative and significant of eggs in year t and rag ragwort wort biomass in year t + of for the open populations (Table IV), indicating the effect of of herbivory on plant production. The correlation is negative but not signifi cant in the shaded plant significant populations. The interaction between between the plant plant and and the moth produces the striking cycles

Correlations Expressed as Correlationsbetween between the the Amount Amount of of Food Food ((Expressed as Ragwort Ragwort Ground Ground Cover! Cover)inin Year Year tt and and the the Number Number of of Cinnabar Cinnabar Moth Moth and between Amount of Herbivory Eggs Eggs in in Year Year tt + + I1 (A) (A) and between the the Amount of Herbivory ((Expressed Expressed as Eggs) in Year Year tt and as the the Number Number of of Cinnabar Cinnabar Moth Moth Eggs)in and Ragworl Ragwort Ground Ground Cover Cover in in Year Year tt + + I1

TABLE IV IV

B B

A A Ragwort Ragwort population population

Open Open populations populations Shaded Shaded populations populations

r r

p P

r r

p P

0.74 0.74 0.34 0.34

0.00 0.00 NS NS

- 0.74 0.74 - 0.46 0.46

0.02 0.02 NS NS

1166

Tritrophic Metapopulation Dynamics TritrophicMetapopulation Dynamics

399 399

in their their populations shown shown in Fig. 3, with plant plant biomass biomass followed by insect (egg) 2 ragwort in the numbers. 1 0 eggs/dm2 numbers. After overshooting the egg/biomass ratio ratio of of 10 the open populations, populations, a complete complete defoliation follows. Next larvae larvae disperse disperse to the still undefoliated undefoliated plants plants in other other populations. Finally this leads leads to regional defoliation and total loss of seed production. A second season of of total defoliation usually follows, resulting in a strong reduction reduction of of egg numbers in the following year. Reduced plant biomass is now driving insect numbers. The final increase increase in insect numbers in Fig. 3 was unexpectedly ended in 1991, 1 99 1 , without the moth overshoot­ overshooting the carrying capacity and without further reduction in plant biomass. It was only in 11995 995 that that the ragwort population became again completely defoliated. There is no effective regulatory mechanism for for the insect on the scale of of a local plant population. With the numerical advantage for an individual to lay for more eggs than its competitors, this makes food shortages among larvae inevi­ than larvae inevitable. Food shortages in tum reduc­ turn result in mass migration migration and starvation. The reduction in plant biomass leads to local disappearance of the herbivore. Local ragwort disappearance populations with relatively more food are selected by the adult moth for ovipo­ oviposition. Figure Figure 4 (top) illustrates the dynamics of the numbers numbers in local cinnabar cinnabar moth populations over time and (bottom) local moth extinctions. Within 6 years from 980, all local insect populations populations had become from the beginning of this study, in 11980, extinct at least once. Contrary to its food food plant, the cinnabar moth has no refuge populations that never became extinct. Figure 3 shows that that after the crash crash in herbivore herbivore (egg) numbers numbers due to food shortage and starvation it took another another year before egg numbers started to increase again ((1978, 1 978, 11984 984 and 11989). 989). This is unexpected because plant biomass did not appear 978 and 1989. 1 989. This lag appear to be limiting in these years, especially not so in 11978 period implies that during these periods insect numbers were not driven by avail­ available plant plant biomass. It also implies (Fig. 3) that that as herbivory is relaxed for 11 year after a period of heavy herbivory, local ragwort populations can recover recover and biomass can increase. The absence of of moth population growth in the year following the crash crash in egg numbers is probably due to two factors. In the fi rst place adult females are first much smaller than in a normal year (Dempster, 982), because (Dempster, 11982), because they suffered food shortage as a larva in the previous year. Mean pupal size is reduced reduced from 0.52 to 0.45 cm, which corresponds to a reduction in fecundity of of more than 60%. Sec­ Secpopularis is inversely related ond, the effect effect of of the parasitoid C. popularis related to the density of of the cinnabar 982). The highest percentage cinnabar moth (Dempster, 11982). percentage parasitism is found after 1 935) found after a population population crash of of the cinnabar cinnabar moth. Similarly Cameron Cameron ((1935) the highest percentage percentage parasitism after a population crash of of the cinnabar cinnabar moth in the United Kingdom in 11932. 932. We have hypothesized (van der Meijden et al., 11991) 99 1 ) that the the mechanism mechanism underlying this this inverse relationship is that Cotesia Cotesia has has a lower lower fecundity and and a lower lower dispersal capacity than its host. Especially in the years of crash in cinnabar 1 977, 1983, 1 983, and 11988), 988), few new patches cinnabar moth numbers ((1977, are colonized by the moth (Fig. 4, bottom). This allows a buildup of of parasite parasite numbers and consequently an increase in parasitism. parasitism. In the following years the host colonizes other local ragwort populations populations and temporally escapes from from its

Ed van van der der Meijden Meijden and end Cotharino CatharinaA.A. M. AA.von van der der Veen-van Veen-vanWijk Wijk Ed

400 400

parasitoid. If If the the rate rate of of increase increase of of the the host host is is higher higher than than that that of of the the parasitoid, parasitoid, parasitoid. percentage percentage parasitism parasitism should should decrease decrease with with the the rapidly rapidly increasing increasing host host numbers. numbers. Dempster's 1 982) data Dempster's ((1982) data suggest suggest such such aa lower lower rate rate of of increase increase in in the the parasitoid. parasitoid. Table Table V V gives further further information information on on parasitism, parasitism, colonization colonization of of new new sites sites by by the the cinnabar moth, and population change of the moth. The 1 988 crash resulted in aa cinnabar moth, and population change the 1988 crash

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FIGURE 4 (Top) (Top) Time Time course course of ofthe the number number of ofextant extant populations populations of ofthe the cinnabar cinnabar moth moth (= (= ragwortragwort­ FIGURE growing sites sites with with cinnabar cinnabar moth moth eggs eggs in in the the current current year). year). (Middle) (Middle) Time Time course course of of the the number number of of growing extinct extinct populations populations of of the the cinnabar cinnabar moth moth (populations (populations that that went went extinct extinct in in the the current current year; year; open open squares) and and the the number number of of ragwort-growing ragwort-growing sites sites colonized colonized by by the the cinnabar cinnabar moth moth in in the the current current year year squares) (black (black squares). squares).

116 6 Tritrophic Metapopulotion Dynamics TritrophicMetapopulation Dynamics TABLE TABLEV

Parasitism Cinnabar Moth Parasitismof of the the Cinnabar Mothby by Cotesia Cotesia popu/arisa popukil'Js a

Year

Percentage Percentage parasitism parasitism

Moth population population turnover rate

Change in moth population

11988 988 11989 989 11990 990 11991 99 1 11992 992 11993 993 1994

37.9 3.2 0.0 0.4 11.0 .0 .1 55.1 110.5 0.5

25 75 92 53 43 39 37 37

0.08 0.33 117.12 7.12 2.55 11.05 .05 0.52 11.10 . 10

a a

401

Population 1 00 X Population turnover turnover in the cinnabar cinnabar moth moth was calculated calculated as 100 • number number newly newly colonized colonized sites/(number sites/(number of persistent persistent popUlations populations + + newly newly colo­ colonized sites). Change size was calculated nized sites). Change in host host population population size calculated as the number number of cinnabar cinnabar moth moth eggs eggs in year year t - I/number 1/numberof eggs eggs in year year t. -

high percentage percentage of due to the small number of new sites high of parasitism, parasitism, apparently apparently due to the small number of new sites colonized 1 989 and 1 990 colonized by by the the moth moth and and the the reduction reduction in in its its population population size. size. 1989 and 1990 show increase in show a a fast fast decrease decrease in in percentage percentage of of parasitism parasitism together together with with an an increase in population population turnover turnover and and general general increase increase of of the the moth. moth. The The increase increase in in parasitism parasitism in 993 and 994 occurs host population population turnover in 11993 and 11994 occurs concurrently concurrently with with a a reduction reduction in in host turnover rate host. rate and and a a reduction reduction in in population population growth growth of of the the host. The level of The level of synchrony synchrony between between egg egg numbers numbers of of the the cinnabar cinnabar moth moth per per plant plant biomass biomass in in local local plant plant populations populations and and the the pooled pooled egg egg numbers numbers in in the the metapop­ metapopulation (Fig. ulation was was calculated calculated as as the the correlation correlation coefficient coefficient between between these these variables variables (Fig. 2, will only 2, bottom). bottom). As As eggs eggs will only be be found found on on ragwort ragwort plants plants we we skipped skipped those those sites sites that I %. Half Half of that had had no no ragwort ragwort or or had had aa ragwort ragwort cover cover of of < < 1%. of the the correlation correlation coefficients cant, indicating indicating considerable coefficients in in Fig. Fig. 2 2 are are positive positive and and signifi significant, considerable syn­ synchrony local chrony in in the the local local fluctuations fluctuations of of the the numbers numbers of of cinnabar cinnabar moth moth eggs. eggs. The The local populations invariably populations with with a a low low correlation correlation with with metapopulation metapopulation fluctuations fluctuations invariably had values due low ragwort less suitable had many many zero zero values due to to low ragwort density density or or less suitable habitat habitat (shaded (shaded ragwort populations or ragwort populations or populations populations with with the the predatory predatory ant ant F. polyctena). These These results is able to track results confirm confirm the the notion notion that that the the cinnabar cinnabar moth moth is able to track efficiently efficiently suit­ suitable plant. able local local populations populations of of its its host host plant.

VI. MECHANISMS OF PERSISTENCE PERSISTENCE IN IN THE THE CINNABAR VI. MECHANISMSOF CINNABARMOTH MOTH There five mechanisms There are are at at least least five mechanisms that that enhance enhance the the probability probability of of persistence persistence cinnabar moth: moth: in the cinnabar 11.. Dispersal of of adult adult moths and and the capacity to locate isolated isolated spots of of rag­ ragwort. The be strong yet the The female female moths moths do do not not seem seem to to be strong flyers, flyers, yet the distribution distribution of of eggs plant biomass across local ragwort shows good eggs in in relation relation to to plant biomass across local populations populations of of ragwort shows good synchrony, only be be explained casynchrony, which which can can only explained by by an an effective effective dispersal/searching dispersal/searching ca-

402 402

Ed Edvan van der der Meijden Meijdenand and Catharina CatharinaA.A. M. M. van van der der Veen-van Veen-vanWijk Wijk

pacity pacity of of the the female female moths. moths. The The infonnation information in in Fig. Fig. 44 on on newly newly colonized colonized local local sites is is aa clear clear illustration illustration of of the the dispersal dispersal capacity. capacity. Harrison Harrison et et al. al. ((1995) came sites 1 995) came to aa similar similar conclusion. conclusion. to 2. 2. Dispersal Dispersal of of larvae. larvae. During During periods periods of of food food shortage, shortage, larvae larvae are are observed observed to nd food. fth ins tar larvae to move move long long distances distances to to fifind food. Especially Especially fourth fourth and and fififth instar larvae can can cover hundreds hundreds of of meters meters and and reach reach yet yet undefoliated undefoliated sites. sites. A A remarkable remarkable feature feature of of larval behavior behavior is is that that some some of of them them leave their their food food plants before before defoliation defoliation has caused caused any any food food shortage. shortage. Using Using aa gametheoretical gametheoretical model model Sjerps Sjerps and and Haccou Haccou has ((1994) 1 994) demonstrated demonstrated that that such such behavior, behavior, especially when sibs sibs are are involved, may have an evolutionary advantage. advantage. 3. 3. The temporal temporal distribution of of oviposition. oviposition. Oviposition is extended from early May until early JUly. July. There are marked differences in the the beginning of the oviposition season between years which undoubtedly relate to effects of temper­ temperature on the development of pupae. The wide distribution of oviposition guar­ guarature antees that some egg batches have a considerable head start over others. During years of of food shortage shortage larvae from the first egg batches will have the highest of reaching the threshold weight for pupation. chance of 4. Flexible size for for pupation. During periods of of food food shortage pupal size can be considerably reduced 982). Successful pupation is possible only reduced (Dempster, 11982). Successful pupation reached a threshold threshold weight of of 140-150 Meijden, when the larva has reached 1 40- 1 50 mg (van der Meijden, reach in a year of of abundant abundant 11976), 976), which lies far below the weight that a larva may reach food supply (300-500 food (300-500 mg). Distribution of of eggs over expressed as 5. Distribution over different habitats. The The egg load expressed the number of of plant biomass varies varies significantly significantly and the number of eggs per per unit unit of plant biomass and consistently between different habitat types (Table (Table III). Open without the between different Open habitat without the predatory predatory ant is preferred, preferred, shaded shaded habitat habitat with with the the ant is avoided. avoided. This This suggest suggest that that the the less less ant favored habitats habitats may may act act as short-term short-tenn refuges refuges in periods periods of of food food shortage. shortage. As As favored mentioned moth has mentioned earlier, earlier, the the cinnabar cinnabar moth has no no refuge refuge populations populations over over long long periods periods of of time. time.

VII. DISCUSSION DISCUSSION A. Population Population Dynamics Dynamics and Persistence Persistence Populations of ofragwort ragwort show show tremendous tremendous fluctuations fluctuations in in biomass biomass at at both both local local Populations and regional regional scales, scales, with with local local fluctuations fluctuations frequently frequently resulting resulting in in extinction. extinction. HerHer­ and bivory by by the the cinnabar cinnabar moth, moth, especially especially in in open open habitats, habitats, contributes contributes to to the the local local bivory extinction extinction risk. risk. The The metapopulation metapopulation of of ragwort, ragwort, however, however, does does not not seem seem to to be be in in great danger danger of of extinction. extinction. Figure Figure 11 shows shows that that we we never never observed observed more more than than aa great half of of the the local local populations populations to to have have gone gone extinct extinct in in 11 year. year. The The lowest lowest percentage percentage half ground cover cover was was observed observed in in 1987, 1 987, when when the the plant plant was was still still found found in in 62 62 of of 102 1 02 ground patches. On On the the other other hand, hand, the the extinction extinction risk risk of of ragwort ragwort is is not not independent independent of of patches.

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location: the the plant plant has has refuge refuge populations populations that that never never went went extinct extinct during during 20 20 years, years, location: and itit has has local local populations populations with with aa very very high high risk risk of of extinction. extinction. and On aa local local scale scale the the cinnabar cinnabar moth moth has has an an ephemeral ephemeral existence existence of of one one or or On only aa few few years. years. Its Its survival survival is is closely closely linked linked with with the the presence presence of of ragwort. ragwort. By By only numerically overshooting overshooting the the local local carrying carrying capacity, capacity, larvae larvae are are subject subject to to numerically scramble competition competition for for food, food, which which frequently frequently leads leads to to mass mass starvation and scramble starvation and local extinction. extinction. Even Even on on the the regional regional scale, scale, the the cinnabar cinnabar metapopulation metapopulation seems seems local to be in great danger of extinction during such events. In 1 984 and 1 989, we to be in great danger of extinction during such events. In 1984 and 1989, we found only only six six and and five five egg egg batches, batches, respectively, respectively, which which could could have have been been propro­ found duced by a single female, female, in all samples (408 (408 m2). m2). Dempster Dempster (1982) ( 1 982) who who studied studied duced similar system in the United United Kingdom Kingdom found found only one one egg egg batch batch in 1969 1 969 in in a similar 1 50 m m22 samples. samples. He He observed observed immigration immigration of of adult adult moths moths from from outside his study 150 outside his more isolated isolated population, Dempster Dempster (1971) ( 1 97 1 ) observed observed an an system. In a smaller, more extinction of of the the moth moth in 1968, 1 968, and and it was was only 10 1 0 years years later later that that the site site was was extinction reoccupied. Apart of reoccupied. Apart from the risk of of extinction extinction one would expect extreme extreme loss of genetic population bottlenecks genetic diversity to result from such severe population bottlenecks (Hedrick (Hedrick and and Gilpin, this this volume). volume). Gilpin,

Extended Dynamics Probability of Survival? B. Do Spatially Extended Dynamics Affect Affect the Probability Survival? Do these these organisms survive because assemblages assemblages of of local populations populations persist in a balance balance between local extinctions and colonizations, as in the classical meta­ metapopulation scenario (Hanski, this volume)? In some biennial plants with a fairly constant appearance appearance and decay of of growing sites that might indeed be the case (van der Meijden ai., 11992). 992). In species like ragwort, with refuge populations Meijden et et al., and rather synchronous fl uctuations in biomass, the situation appears fluctuations appears to be dif­ different. Although some localities became unsuitable for ragwort, the majority re­ remained suitable and could have been recolonized from the seed bank or through seed dispersal from extant local populations. Survival over long periods of time is enhanced enhanced by these two buffer mechanisms, by the low vulnerability to extinc­ extinction in the refuges and by the capacity capacity of individual plants to recover from ex­ extreme treme damage. damage. The metapopulation type (Hanski and Gilpin, this volume) that fi ts ragwort best would be the sourcesink metapopulation consisting of patches fits source-sink with mostly negative population growth rate in the absence of seed dispersal and patches with a positive growth rate. Even so, seeds from open populations may occasionally disperse disperse to the refuge populations and and contribute contribute to their stability. stability. Regional survival rst of survival of of the cinnabar moth is brought brought about, about, fi first of all, by the the capacity capacity of of its its food plant to recover recover soon after complete complete defoliation. defoliation. Heterogeneity of of the ragwort ragwort habitat, habitat, which leads to to differences differences in egg load per unit of of plant biomass, biomass, and and the the temporally temporally extended extended oviposition oviposition period period also increase increase the the chance chance of of at at least least aa few few larvae larvae to to reach reach the the threshold threshold weight for for pupation pupation ev�n eve.n w!len when food food becomes 979a, becomes completely completely exhausted exhausted on on the the regional regional scale scale (van (van der der Meijden, Meijden, 11979a, 99 1 ; Dempster, 982). The van van der der Meijden, Meijden, et et ai., al., 11991; Dempster, 11982). The positive, positive, and and often often high, high,

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Ed ond Cothorino Ed von van der der Meijden Meijden and Catharina A. A. M. M. von van der der Veen-von Veen-van Wijk Wijk

correlations between between temporal fluctuations in local populations populations and the metapop­ metapopulation indicates that the searching capacity of the female moth is high and ap­ apparently not much hampered by interpatch distances. In this respect, the cinnabar cinnabar moth may be considered to live in only one, but patchy, population and not in a classical metapopulation (Harrison, 11991; 99 1 ; Harrison and Taylor, this volume). An important question in metapopulation theory is whether an interacting persist­ system of of local populations is more stable or has a higher probability of of persistence than the separate 1 99 1 ). This case study has separate local populations (Hanski, 1991). demonstrated that many sites where ragwort went extinct are recolonized recolonized through seed dispersal from existing local populations. populations. Such dynamics must lead to higher regional densities densities and biomass production and, consequently, to a higher proba­ probability of persistence. persistence. This study has also demonstrated that habitat heterogeneity in local ragwort ragwort populations popUlations adds to the probability of of cinnabar cinnabar moth survival. Finally, it seems very plausible that that the parasitoid C. popuiaris, popularis, with its limited limited powers of of dispersal between between local popUlations, populations, causes a delay in the recovery of of the cinnabar cinnabar moth and consequently enables ragwort populations to grow for for one season without herbivory. In conclusion, spatial effects related related to habitat habitat hetero­ heterogeneity and parasitoid dispersal probably contribute contribute greatly to cinnabar moth survival. We would expect that that even if ragwort were were not patchily distributed, distributed, but all patches patches were were combined combined to one large population, population, it would still survive given its specific characteristics characteristics and the present level of spatial heterogeneity. Because dispersal of of cinnabar cinnabar moth moth larvae larvae would not be affected by distances in such a uniform environment, food would become exhausted sooner. This would lead to greater fluctuations in insect numbers and probably in plant plant biomass as well ai., 11991). 99 1 ). As its survival in a network (Sabelis et et al., network of connected connected patches patches seems to be risky, the survival of the cinnabar cinnabar moth would be even more more questionable. questionable. However, in such a system without any interpatch distances distances the parasitoid parasitoid C. popuiaris popularis might might become much much more more effective, effective, because dispersal, its weak point, would be less critical. We thus conclucde conclucde that that every possible outcome, from cinnabar cinnabar moth extinction to regulation by the parasitoid, remains a possibility in a large uniform habitat. Harrison et al. ((1995) 1 995) studied ragwort and a guild of of its herbivorous insects during 3 years in the United Kingdom to test whether whether coexistence coexistence of of these com­ competing species could be explained explained by any spatial effects. They experimentally demonstrated that insect dispersal across local patches was not seriously limited by interpatch interpatch distances. From From these experiments experiments they concluded that that it was un­ unlikely that patches patches were acting acting as separate dynamic entities entities with respect to com­ competition. petition. In other words, they refuted the idea that spatial effects were essential for for coexistence. Our study demonstrates demonstrates that that spatial effects are not limited to those caused by distances distances between local populations: differences differences in habitat quality among local patches may be critical. Although we prefer simple models to test ( 1 99 1 ) that "there ecological principles, we agree agree with Hanski (1991) "there is an urgent urgent need to develop metapopulation models that include variation in habitat quality." This

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study metapopulation-level study has has also also demonstrated d e m o n s t r a t e d that that many many m e t a p o p u l a t i o n - l e v e l processes p r o c e s s e s cannot c a n n o t be be easily easily detected d e t e c t e d in in short-term s h o r t - t e r m studies. studies.

ACKNOWLEDGMENTS ACKNOWLEDGMENTS We enjoyed discussing discussing these ideas with Tom de Jong and Peter Klinkhamer, Rinny Kooi is acknowledged for technical technical assistance, assistance, and we especially thank Ilkka Hanski for his extensive com­ comments on the first version of this chapter.

17

Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulaton Andrew T.T. Smith

Michael Gilpin

I. INTRODUGION INTRODUCTION Because interest in modeling fragmented landscapes landscapes and associated meta­ metapopulation dynamics is relatively recent ((Hanski Hanski and Simberloff, this volume; Wiens, this volume), few long-term empirical empirical studies studies of of metapopulations metapopulations are available to guide theoretical analysis and exploration exploration (but see Thomas and Han­ Hanski, this volume). In this chapter chapter we present the results of of a long-term study of of a metapopulation metapopulation that appears ideal with regard regard to the measurement of of parameters of of a metapopulation. In this study the patches are of roughly the same size, and interpatch spacing is fairly regular. The metapopulation is large relative to lifetime movements of the animals. Not all patches patches have been occupied occupied in any census period. Both numerous numerous extinctions extinctions and recolonizations have been recorded over the more than 20 years of observation of of the metapopulation. metapopulation. The study organism 1 32 g) alpine lagomorph; the is the American American pika (Ochotona (Ochotonaprinceps), princeps), a small ((132 study site is the abandoned gold-mining area of of Bodie, Mono Mono County, California. One of 1 969 (Smith, 11974a,b, 974a,b, of us (A.T.S.) has been studying pikas at Bodie since 1969 11978, 978, 11980), 980), and here here we present present data data from four complete complete population population censuses made at varying intervals intervals since that that time. Results from the first two censuses ((1972 1 972 and 11977) 977) were were interpreted in terms of of area-dependent extinction rates and distance-dependent distance-dependent colonization rates bor-

Metapopularion Metapopulation Biology Biology 1997 by Academic Press. reproduction in any form reserved. Copyright © 9 1997 Press, Inc. All rights of of reproduction

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rowed directly directly from from island island biogeographic biogeographic theory theory (MacArthur (MacArthur and and Wilson, Wilson, 1967; 1 967; rowed Smith, Smith, 1974a, 1 974a, 1980). 1 980). The The populations populations on on patches patches of of habitat habitat at at Bodie Bodie apparently apparently represented aa dynamic dynamic equilibrium equilibrium between between extinction extinction (which (which was was inversely inversely rere­ represented lated lated to to patch patch size) size) and and recolonization recolonization (which (which was was inversely inversely related related to to interpatch interpatch distance; Smith, Smith, 1974a, 1 974a, 1980). 1 980). These These results suggested suggested that that a two-dimensional two-dimensional distance; "stepping-stone" metapopulation model, i.e. one one for which interactions interactions occur occur bebe­ "stepping-stone" metapopulation model, for which tween neighboring patches, could be parameterized to explain these patterns. We tween neighboring patches, could be parameterized to explain these patterns. We introduce this chapter such a spatially spatially explicit explicit model model that that successfully successfully integrates integrates introduce in this chapter such patch-specific population population growth growth with with these these size and and distance distance effects effects for for the the Bodie Bodie patch-specific pika metapopulation. metapopulation. pika addition, qualitative qualitative inspection inspection of of the the map-based map-based pattern pattern of of patch extinc­ In addition, patch extinctions and and recolonizations that occurred between the the first first and and the the second second censuses censuses tions recolonizations that occurred between suggested these events events did did not not occur occur randomly randomly across across the the Bodie Bodie landscape. landscape. suggested that that these Instead, there there appeared appeared to to be be a clustering clustering of of extinction extinction and and recolonization recolonization events; events; Instead, neighborhoods patch patch occupancy markedly, while while in other in some some neighborhoods occupancy declined declined markedly, other neigh­ neighborhoods patch increased (Smith, 1 980). Our Our stepping-stone stepping-stone model model borhoods patch occupancy occupancy increased (Smith, 1980). inherently incorporates observations by accounting accounting for certain correlations correlations inherently incorporates such observations for certain population growth growth on patches patches within within neighborhoods, neighborhoods, particularly particularly as a result result of of in population recolonization from from neighboring neighboring patches. patches. Thus, Thus, the consideration recolonization the model model takes takes into into consideration that that immigration immigration onto onto vacant vacant patches patches and and recurrent recurrent colonization colonization of of occupied occupied patches effect;" Brown Brown and Kodric-Brown, 1977; 1 977; Smith, Smith, 1980) 1 980) in patches (the "rescue "rescue effect;" and Kodric-Brown, regions of high average average patch occupancy can can lower lower the the probability probability of of extinction extinction regions of high patch occupancy neighboring patches patches and result in the increased persistence of of clusters clusters of of on neighboring and result increased persistence patches. patches. We empirical investigation conducting a third third and and fourth We renewed renewed the empirical investigation by conducting fourth census 1 989 and 11991) 99 1 ) with the intention intention of census ((1989 of examining examining closer closer the spatial structure structure of of local population dynamics in the system. These These recent censuses censuses paint a picture of rather is different of the Bodie Bodie pika metapopulation metapopulation that rather different from from that presented presented earlier by Smith ((1974a, 1 974a, 11980), 980), with the scale of turnover events dramatically of correlated correlated turnover populations in one subregion half widened. Almost all populations subregion at Bodie (covering almost almost half of the area of of the metapopulation) metapopulation) had had gone extinct, while the remaining remaining subregion subregion of of poppop­ hardly suffered suffered any change change in distribution distribution or abundance. abundance. The "collapse" of ulations in the one subregion subregion represents represents a departure departure from from the supposed dynamic equilibrium and clustering effects effects seen in the earlier censuses. censuses. Thus, the full data data set (four censuses censuses and and three census census intervals) challenges challenges Thus, our ability to construct construct a metapopulation metapopulation model built exclusively around the ideas of island biogeography: area-dependent area-dependent extinctions and distance-dependent distance-dependent re­ recolonizations. The inability of the stepping-stone stepping-stone model to explain completely the regional effects observed in our map-based map-based observations observations of pika populations populations at Bodie led us to consider more fully elements of "regional stochasticity" of"regional stochasticity" (Hanski and Gilpin, 11991) 99 1 ) that have been explored theoretically by Gilpin ((1988, 1 988, 11990) 990) and Hanski 1 989). We explored these Hanski ((1989). these properties of the Bodie metapopulation metapopulation quantitatively with an examination examination of the autocorrelational autocorrelational properties of our spa-

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tially referenced data. The results demonstrate the need to understand the land­ landmetapopulations in nature to understand fully their behavior. scape properties of metapopulations

RELEVANTPIKA PIKA NATURAL NATURALHISTORY HISTORY II. RELEVANT Characteristics of American American pikas allow measurement measurement of the most important important Characteristics variables 974a,b, 11978, 978, 11980, 980, 11987; 987; variables needed needed for metapopulation metapopulation analysis (Smith, 11974a,b, and Ivins, Ivins, 11984; Smith and 984; summary in Smith and Weston, 11990). 990). Pikas are diurnally hibernate, are active all year, and spend active, thus easy to observe. Pikas do not hibernate, gathering vegetation that is stored in haypiles to serve as much of their summer gathering food overwinter. Haypiles form form the figurative center of activity for each individ­ individround feces that that can be distinguished distinguished ual. Pikas, being lagomorphs, have small round readily from those of of all other mammals. mammals. They also deposit deposit soft soft feces (black and readily animal.. Unlike Unlike most lagomorphs, lagomorphs, pikas are elongate) unlike those of any other animal and utter utter both short and long calls calls (males (males only) that can can be be heard highly vocal and distances. Vocalizations, haypiles, feces, and urine urine stains stains can can each each over long distances. Vocalizations, fresh haypiles, feces, and used to assess current current occupancy occupancy of of pika pika habitat. habitat. Because alpine alpine climates climates are be used haypiles and the round round fecal pellets pellets do not decompose decompose readily; at Bodie Bodie dry, both haypiles they feces may they may may persist persist for many many years. years. Old haypiles haypiles and and old feces may be used used to determine sites that have have been been occupied occupied previously by pikas, even if if they they no no determine pikas, even longer occur occur there. there. longer American pikas pikas are are habitat-specifi habitat-specificc to talus or piles of of broken rock adjoining adjoining American suitable vegetation vegetation for for grazing grazing and and gathering gathering forage forage for for their their haypiles. haypiles. As talus talus suitable characteristic habitat habitat type which which is easily distinguished distinguished from from surrounding surrounding is a characteristic vegetative define vegetative habitat, habitat, it is possible to defi ne precisely the the habitat habitat area area available available for for pika occupancy. occupancy. Additionally, distributed patchily, Additionally, talus talus is usually distributed patchily, and and at the Bodie Bodie site is highly highly fragmented fragmented (see (see below). below). American are individually individually territorial. and females sepAmerican pikas pikas are territorial. Males Males and females maintain maintain sep­ arate and there significant difference arate territories, territories, and there is no no significant difference in territory size by gender. gender. Population and averages averages about Population density is low and about four four to eight eight animals animals per per hectare hectare on suitable on suitable habitat habitat throughout the geographic range range of of the the American American pika. WithinWithin­ patch (Smith and and Weston, patch nearest-neighbor nearest-neighbor distances distances average average approximately approximately 20 20 m m (Smith Weston, 1990). 1 990). As As a result result of of these these factors factors it is possible to obtain obtain a good good estimate estimate of of carrying capacity, capacity, or of each each pika or percentage percentage saturation saturation by pikas, pikas, of pika habitat habitat patch. patch. It is likely likely that that carrying carrying capacity capacity can can be be determined determined more more accurately accurately for for pikas pikas than any than any other other animal. animal. Spacing Spacing among among territories territories does does not not vary vary among among years, years, and and most are traditional and built after year most haypile haypile localities localities are traditional and built on on the the same same site site year year after year (Smith (Smith and and Weston, Weston, 1990). 1 990). The sex-ratio pikas is near near unity, unity, and and animals The sex-ratio of of adult adult pikas animals tend tend to to reside reside on on territories an animal and Ivins, territories adjoining adjoining an animal of of the the opposite opposite gender gender (Smith (Smith and Ivins, 1984). 1 984). Male and body mass Male and female female pikas pikas are are remarkably remarkably similar; similar; body mass is not not significantly significantly different different and and even even their their external external reproductive reproductive morphology morphology is barely barely distinguishdistinguish-

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Andrew T. Smith and Michael Michael Gilpin Andrew

able (Smith (Smith and and Weston, Weston, 1990). 1 990). American American pikas pikas are are relatively relatively long-lived long-lived for for small small able mammals. Survivorship Survivorship normally normally exceeds exceeds 50% 50% per per year, year, and and almost almost 10% 1 0% of of the the mammals. Bodie population population was was 5 or or 6 years years old old (Smith, (Smith, 1978). 1 978). Bodie Throughout the the range range of of the the American American pika, pika, all all females, females, including including all all yearyear­ Throughout lings, initiate initiate two two litters litters per per summer summer breeding breeding season and and successfully successfully wean wean only lings, one of of these these litters litters (Smith, (Smith, 1978; 1 978; Smith Smith and and Ivins, Ivins, 1983a). 1 983a). Litter Litter size size (determined (determined one from embryo embry'o counts counts of of pregnant pregnant females) females) is relatively small for lagomorph and and from for a lagomorph averages 3 throughout throughout the range range of of the the American American pika pika (Smith (Smith and and Weston, Weston, 1990). 1 990). averages The mean mean litter the highest yet studied The litter size size of of 3.7 3 .7 at at Bodie Bodie is the highest of of all populations populations yet studied (Smith, 1978). 1 978). Litter Litter size does does not not vary vary significantly with with age age of of mother, mother, habitat habitat (Smith, productivity, or or between between first first and and second second litters. Litter Litter size at weaning weaning is normally productivity, than litter litter size size determined determined by embryo embryo counts counts (Smith and and Weston, Weston, 1990). 1 990). less than Weaned young grow grow rapidly rapidly and and reach reach adult adult size their summer summer of of birth. birth. Weaned size in their Juveniles must must successfully colonize colonize a vacant vacant territory to survive to become become Juveniles adults. As adults adults tend tend to be be long-lived, long-lived, vacancies vacancies are are rare rare and and occur occur only sporadsporad­ ically, and and one one would expect pikas to exhibit high vagility in their search search for for available territories. However, there there are two severe constraints constraints on the movements movements available territories. of juvenile juvenile pikas. First, it is difficult for juveniles to move freely in saturated saturated pika pika of for juveniles habitat; apprehend and unfamiliar juveniles juveniles from their territories habitat; adults adults apprehend and chase chase unfamiliar their territories 1 983b, 1984). 1 984). Second, Second, pikas are cold-adapted and cannot cannot toltol­ (Smith and and Ivins, 1983b, are cold-adapted erate warm warm temperatures. temperatures. At At Bodie, which which is near near the lower distributional distributional boundbound­ erate ary of temperatures severely restrict restrict the of pikas in the Sierra Nevada, high daytime temperatures movements of of pikas (Smith, 1974b). 1 974b). movements Dispersal has been been observed in two investigaDispersal by marked animals animals at Bodie Bodie has investiga­ tions. In one study 58 animals were individually marked animals were marked (J. (1. D. Nagy, personal personal communication). Of Of 34 adults, 25 were resighted the following year. Only one 1 8 m to a neighboring neighboring patch. Of Of 24 adult dispersed, and this movement was 18 juveniles, juveniles, only 5 were were resighted the following year. Two Two of of these 5 juveniles juveniles dispersed dispersed from their their natal patch, patch, in each each case to the next closest patch (60 m and 150 of 105 1 50 m, respectively). In a second study, only one of 1 05 marked adults dispersed; it moved to a neighboring patch patch (Peacock, 1995). addition, Peacock Peacock ((1995) 1 995). In addition, 1 995) observed dispersal by 15 marked juveniles juveniles that occurred during the summer summer months. Three been Three juveniles dispersed dispersed from their their natal natal patches patches after they had been tagged; the other 112 2 were trapped postdispersal, and their natal patches were were identified by genetic identified genetic paternity exclusion analysis. Four of of these animals dispersed within a large patch (moving an average of 45 m from their natal home range), the other 1111 dispersed between patches. Nine of of the 1111 originated from saturated patches, the other 2 came from unsaturated patches. Nine colonized the the next closest unsaturated patch, while 2 "passed up" available neighboring patches to settle on patches farther farther away. The average distance moved by these 1111 juveniles that dispersed between patches was 1132.5 32.5 m (range 70396 m; Peacock, 11995). 995). 70-396 These results indicate that juvenile dispersal among patches occurs at Bodie, but that these movements are usually limited to nearby and/or neighboring neighboring patches.

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This restricted restricted ability of of pikas to disperse between between patches has profound impli­ implications for the metapopulation dynamics at this site. the

III. III. THE THE BODIE BODIE SITE SITE American American pikas live in a patchy distribution throughout their their geographic range in the mountains of of western North America. America. Most expanses expanses of of talus are disjunct and of of of a size to harbor harbor pika populations in the neighborhood of of tens of 00 animals. It is rare rare for for continuous populations of of pikas to contain contain more than 1100 animals. animals. The The habitat habitat at Bodie presents presents an ideal situation situation for for assessing the metapopu­ metapopulation dynamics of ). Bodie is an old mining area, and most of of pikas (Fig. 11). of the habitat occupied occupied by pikas there are the tailings and scree left by prior mining mining activity. Average Average size of of rock rock in each tailing patch patch is similar, hence potential aspects of habitat habitat selection by substrate these tailing substrate are minimal. In general, these patches habitat -for patches are smaller and contain fewer fewer pikas than than natural natural habitat m for example example in the Sierra Nevada 1 859 and Nevada mountains 35 km away. Mining at Bodie began began in 1859 has continued, with bursts of of activity, until the present day. Roughly 1100 00 mine mine tailing tailing patches dot the landscape landscape at Bodie, and they vary in size and and distance distance from one another. another. These patches are separated separated by a sea of of sagebrush sagebrush and other other typical Great Great Basin sage community vegetation. Additionally, pikas have been found in several of of the higher natural natural bluffs surrounding surrounding Bodie, Bodie, habitat that was Pleistocene when it was cooler in the area most likely colonized during the Pleistocene area (Smith, 11974a). 974a). We have no indication of of any recent recent dispersal dispersal from these these natural habitats to the mine tailing patches at Bodie; in fact, the nearest nearest natural natural population on Sugarloaf 0) went extinct between Sugarloaf Hill (estimated (estimated K = 110) between the second and third censuses. In summary, the Bodie pika metapopulation system fits the patchpatch ­ matrix paradigm paradigm discussed by Wiens (this volume). The intervening sagebrush habitat habitat is quite homogeneous, and animals dispersing through it would not face a series of of choices between between subhabitats. The pikas at Bodie were "discovered" "discovered" by the biologist Joye Harrold Severaid in the mid1 940s. We know from Severaid' 1 955) 4 years of mid-1940s. Severaid'ss ((1955) of intensive obser­ observations, historical accounts, and more recent 974a, 11980) 980) that recent censuses (Smith, 11974a, of available pikas at Bodie have at one time occupied occupied every patch of available habitat at Bodie. Severaid ((1955) 1 955) reported reported that that three long-time residents residents of of Bodie knew knew about the pikas, going back to their boyhood around the tum turn of of the century. We also know from inspection of of maps, photographs, and descriptions of of the mine tailing patches (Severaid, 11955; 955; Bodie California State Historical Park records), as well as direct observation over the past 25 years, that these patches are permanent permanent and have not changed quantitatively or qualitatively since the mid1 940s. mid-1940s. Severaid 1 955) was the fi rst to observe Severaid ((1955) first observe that that the pika population at Bodie "was never equal to the carrying capacity of of the habitat." habitat." He trapped out some =

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FIGURE FIGURE 1! The configuration of mine tailing patches occupied by pikas at Bodie, California. The filled rectangles rectangles represent represent habitable patches, with the size of the rectangle rectangle proportional to the carrying Major dirt roads roads are portrayed with the stippled lines. Principal natural topo­ topocapacity of the patch. Major graphic features and mine tailings are named.

of the tailing patches and observed that they were colonized slowly over the course of his studies. Thus, there were early indications that pika populations populations on patches patches at at Bodie Bodie were were linked linked by by occasional occasional dispersal. dispersal.

IV. IV. METHODS METHODS Censuses 972 (Smith, 974a), Censuses of the pika pika population at at Bodie were made in 11972 (Smith, 11974a), (Smith, 11980), and 11991. census was conducted conducted in in late late summer, summer, 11977 977 (Smith, 980), 11989 989 and 99 1 . Each census for for two two reasons. reasons. First, First, at at this time time those those juveniles with with aa chance chance of of surviving surviving to to

1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulation Metopopulation 17

413 41 3

become adults adults have have reached reached adult adult size size and and become become established established on on available available terter­ become ritories. ritories. Thus, Thus, at at this this time time the the resident resident population population is is at at its its highest highest level. level. Second, Second, in in late late summer summer there there is is more more available available sign sign (haypiles, (haypiles, feces, feces, etc.) etc.) to to facilitate facilitate an an accurate population census. accurate population census. The The four four censuses censuses included included almost almost all of of the the available available pika pika habitat habitat in the the Bodie region. region. More More than than 70 70 isolated isolated small small patches patches (effectively (effectively islands) islands) of of talus talus Bodie were included included in each each census. census. In In the the area of High High Peak Peak (Fig. 1), 1 ), where where mining mining were area of was extreme, extreme, we we sampled sampled three large expanses expanses of of tailings, tailings, but but did did not not activity was three large take take a complete complete census. census. We We estimate estimate that that this this large large area area harbors harbors aa population population of of approximately 50 pikas pikas (see (see also Peacock, Peacock, 1995). 1 995). Regrettably, Regrettably, one one area area at at Bodie Bodie approximately was not not censused: censused: the the expanse expanse near near the the top top of of Bodie Bodie Bluff Bluff (several (several small small patches) patches) was and the downslope downslope from from the the bluff in the the direction of the the Syndicate Syndicate Mine Mine (a (a small small and the bluff in direction of number of of medium-sized medium-sized patches; patches; Fig. Fig. 1). l ). Access Access was was denied denied to to this this area for the the number area for first two two censuses. censuses. first Patch size (measured by perimeter perimeter in meters), meters), degree degree of of isolation (measured (measured Patch size (measured by the nearest or more more pikas pikas in meters), meters), and and by distance distance to the nearest patch patch inhabited inhabited by three three or number of of resident resident pikas pikas were determined for each patch patch (following Smith, Smith, the number determined for 1 974a, 1980). 1 980). Patch Patch perimeter perimeter provided provided the the most most meaningful meaningful assessment assessment of of patch patch 1974a, because territories territories on the mine mine tailings are spaced linearly and and adjacent adjacent to size, because are spaced rock- vegetation interface. interface. None None of of the the patches are are so tall as to contain contain a the rock-vegetation of pikas. Interpatch Interpatch distances distances were were measured measured between between patches patches "second story" of inhabited by three three or or more more pikas pikas because those those with with fewer fewer animals were were unlikely inhabited of colonizing colonizing individuals individuals (Smith, 1974a, 1 974a, 1980). 1 980). to be sources sources of Percent saturation of pikas pikas on patches patches was was defined defined as Ni/Ki, NJK; , where where N; the Percent saturation of Ni is the number of of pikas pikas found found on the ith patch, patch, and and K; is the carrying carrying capacity capacity of of each each number based on patch size and the number number of of potential territories that would fi fitt patch, based into this this area 974a). into area (following (following Smith, Smith, 11974a). model that A spatially explicit structured metapopulation model that incorporates incorporates our data on patch patch locations, population population change change on patches patches between between census census intervals, data observations of of extinction of of populations and recolonization of of and observations populations on patches patches and vacant patches independent sto­ vacant patches was created. created. Conceptually the model assumes assumes independent stochastic growth, A, A, on each each of of the habitat habitat patches patches and the possibility of of rescue rescue chastic nearby patches. (sensu Brown and Kodric-Brown, 11977) 977) or recolonization from nearby patches. patch has has a population ceiling, K, based based on its size (perimeter in meters), Each patch patch populations populations that that rise above K in a single time step are are truncated truncated back back and patch to to K K at at the the end end of of the the time time step. step. For For the the one one area area of of High High Peak Peak that that was was sampled, sampled, rather than completely censused, we extrapolated our population samples to in­ incorporate corporate the the entire entire patch patch area area (new (new K K = = 50). 50). We We calibrated calibrated our our stochastic stochastic growth growth on the turnover seen between the 11972 972 and the 11977 977 censuses censuses (when the meta­ metapopulation was assumed to be in equilibrium, given that the occupancy remained near 60%), and accordingly our time step was set at 5 years. Immigration from nearby is based the number nearby patches patches is based on on the number of of individuals individuals living living within within an an "effects "effects of the target patch. We utilized a rectangular rectangular dispersal function such that radius" of dispersal dispersal probabilities probabilities drop drop to to zero zero beyond beyond the the maximal maximal dispersal dispersal (effects) (effects) radius. radius.

4 14 414

Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin

The steps ((100 l 00 years) this was was The model model was was run run for for 20 20 time time steps years) into into the the future, future, and and this radius. A A sensitivity investigation repeated repeated 20 20 times times for for each each A A and and effects effects radius. investigation was biologically perfonned performed by varying both A A and the effects effects radius radius over over a range range of of biologically plausible values, based pikas at Bodie. plausible based on the ecology ecology of of pikas Bodie. While While there are are a number number of directions directions that that such a model model may may take, we limited ourselves ourselves to an investigation investigation of of within 1100 00 years, pop­ of the the following following question: question: what what is the probability probability that, within years, all populations in a subregion will go extinct? In other words, our model is designed ulations subregion go extinct? other words, our model designed to allow us to detennine, for a variety of combinations of model the determine, for variety of combinations of model parameters, parameters, the extent extent to which the landscape landscape properties properties of of the patches patches at Bodie may may contribute contribute to regional regional persistence persistence or or collapse collapse of of pika pika populations. populations. Copies of of the compiled compiled model model are are available available by contacting contacting M.E.G. M.E.G. at [email protected]. [email protected]. We popu­ We have have used used two two approaches approaches to quantify quantify spatial spatial autocorrelation autocorrelation of of population-level lation-level events events on patches patches and, most most important, important, integrated integrated population population changes changes in neighborhoods neighborhoods or or subregions subregions centered centered on each each patch. First, First, we analyzed analyzed net population population growth growth or or decline in neighborhoods neighborhoods surrounding surrounding each each focal focal patch, excluding excluding the change change on the focal patch patch itself. Positive spatial autocorrelations autocorrelations were population size increased increased on both focal patch were said to have have occurred occurred when when population both the focal patch and neighborhood, or if both and in the surrounding surrounding neighborhood, or if both the focal patch patch and and surrounding surrounding patches patches decreased decreased in population population size. Negative Negative spatial autocorrelations autocorrelations occurred occurred when population growth when population growth or decline decline experienced experienced on a focal patch patch was was the the opposite opposite of We ran this analysis of the net population population trend in the surrounding surrounding neighborhood. neighborhood. We for .0, and for three three "effects "effects radii" radii" (0.5, 11.0, and 2.0 km) surrounding surrounding each focal patch. patch. Results Results of of these analyses analyses include include only those those focal focal patches patches that that exhibited exhibited a change change in population population size during during the respective interval. Second, we we quantified quantified spatial autocorrelation autocorrelation of of discrete discrete growth growth rates among among neighboring 990). neighboring patches patches more more precisely using Moran's Moran's I statistic (Haining, (Haining, 11990). autocorrelation occurs occurs when when occupancy occupancy patterns patterns of of pikas pikas on on patches within Spatial autocorrelation patches within specific distances distances are are signifi significantly associated. For we examined examined the specific cantly associated. For this analysis, we the spatial autocorrelation autocorrelation of of the discrete discrete growth growth rate rate per per census interval. This This growth growth A, is defined as (n/ patches. The rate, A, ( n , ++ I1 - n/)/n/ n , ) / n , for for each each of of the i patches. The general general approach approach is to look fonn look at a kind kind of of a spatially weighted weighted cross cross product product tenn term of of the form -

1I = 2: WijLl;j', = 2: EEw;/x, Ii

j J

where Llij is a measure where A~j measure of of the proximity of of the the variate, variate, the discrete discrete growth growth rate, between the ith and jth jth spatial positions, positions, and where where wij w;j is an arbitrary spatial weighting function. function. On On aa regular regular grid, grid, wij w 0 is is sometimes sometimes taken as as 1I for for nearest nearest neighbors and structure was irregular, irregular, we neighbors and 0 otherwise. otherwise. Because our our patch patch structure we ap­ approached the problem differently. differently. We We set wij equal to 1I if if the Euclidean Euclidean distance distance between points i and and jj was less than d, an "effects "effects radius," radius," and 0 otherwise. otherwise. For For the Moran (A) is the mean mean value value Moran statistic, Llij Aij is taken taken as (A;-(A») (A;-(A)) (\-(A»), (Aj-(A)), where (A) of Moran of the discrete discrete growth growth rate for for all patches patches during during the the time period. period. The The Moran coefficient tenn divided divided by the coefficient is then calculated calculated as this weighted weighted covariance covariance term variance of of the growth growth rates.

17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics inin a Pika Metapopulation Metapopulation

415 41 5

We computed computed the the Moran Moran statistic statistic for for each each of of the the census census period period intervals intervals and and We for aa range range of of different different effects effects radii, radii, from from 0.1 0. 1 to to 1.5 1 .5 km, km, in in 0.1-km O. l -km increments. increments. for The 95% 95% confidence confidence limit limit against against which which our our data data were were contrasted contrasted was was produced produced The by creating creating 200 200 independent independent realizations realizations of of uncorrelated uncorrelated discrete discrete growth growth over over our our by grid of of habitat habitat patches. patches. grid

RESULTS V. RESULTS Patch Occupancy Occupancy A. Patch As was was initially initially observed observed by by Severaid Severaid (1955), ( 1 955), not not all all habitable habitable mine mine tailing tailing As patches at at Bodie Bodie contained contained pikas pikas during during any any of of our our four four censuses. censuses. Approximately Approximately patches 60% of of the the patches patches were were occupied occupied during during the the first first two two censuses censuses (Smith, (Smith, 1974a, 1 974a, 60% 1 980), but but this this level level of of occupancy fell to to about 45 % for for the the latter latter two two censuses censuses 1980), occupancy fell about 45% (Table Correspondingly, more pikas were located on on the study area (Table I). Correspondingly, more pikas were located the study area during during the the two earlier censuses than latter two two (Table I). two earlier censuses than the the latter (Table I). Typical of a metapopulation system, there was frequent Typical of metapopulation system, there was frequent turnover turnover (extinctions (extinctions of populations on patches and of vacant vacant patches patches from from of populations on patches and subsequent subsequent recolonization recolonization of existing local populations) populations) between between censuses (Table II). In In addition, addition, roughly roughly 50% 50% existing local censuses (Table of of patches patches varied varied in in popUlation population size size during during each each of of the the three three census census intervals intervals (Table (Table II). II). The patches at Bodie The most most striking difference difference in the the pattern pattern of of occupancy occupancy of of patches Bodie among among the the censuses censuses was was the decline decline and and near near collapse collapse of of pika pika populations populations on patches Bodie patches in the southern southern half half of of the study area. area. The The distribution distribution of of patches patches at Bodie is is roughly roughly in in the the shape shape of of an an hourglass; hourglass; aa saddle saddle in in the the area area of of aa dirt dirt road road extending extending up from the Bodie township guratively divides the northern township fi figuratively northern and southern southern half half of the study area (Fig. 11). there was a mixture of of occupied occupied of ). In the first two censuses, there and and unoccupied unoccupied patches patches in in both both the the northern northern and and the the southern southern parts parts (Fig. (Fig. 2). 2). By By 11989 989 we noted a significant drop drop in the percentage percentage of of occupied occupied patches patches in the southern half (Fig. 2). This decline included the extinction of the pika population population on the Red Cloud tailing, a site that harbored 972 and 11977. 977. Only harbored nine pikas in 11972 22 years years later later pikas pikas were were absent absent from from nearly nearly all all patches patches in in the the southern southern half. half. In In the the extreme southern southern part of the study study area, only only one animal was found (on the relatively relatively large large Noonday Noonday tailing; tailing; Fig. Fig. 2). 2).

B. B. Area Area Effects Effects Size Size of of habitat habitat patch patch appeared appeared to to be be the the most most important important factor factor governing governing the the occurrence occurrence of of pikas on the habitat patches at at Bodie (Table I), an effect effect apparently due due to to the the relatively relatively low low probability probability of of extinction extinction of of popUlations populations on on large large patches patches (Smith, 980). In (Smith, 11980). In all all censuses, censuses, average average size size of of occupied occupied patches was greater greater than than the cant differences the average average size of of vacant patches. patches. There There were, however, however, signifi significant differences in rst two in the the apparent apparent effect effect of of patch patch size size among among the the four four censuses. censuses. In In the the fifirst two

4 16 416

Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin

TABLE I Descriptive DescriptiveMeasurements Measurements and and Correlational Correlational Statistics Statistics of of Pikas Pikas on on Mine Tailing Tailing Patches Patches at at Bodie Bodie Mine 1972" 1972 a

A verage size Average size of of patches' patches" (perimeter m) (perimeter in in m) Occupied patches Occupied patches Vacant Vacant patches patches A verage inter-patch Average inter-patch distances' (m) distances" (m) Occupied Occupied patches patches Vacant Vacant patches patches Correlation Correlation of of percentage percentage saturation with patch saturation with patch size size All All patches patches Occupied Occupied patches patches Correlation Correlation of of percentage percentage saturation saturation with with interpatch interpatch distance distance All All patches patches Occupied Occupied patches patches Number N u m b e r of of patches patches censused censused Percentage Percentage occupied occupied of of mine tailing tailing patches mine patches Total number of Total number of pikas pikas censused censused

1977b 1977 ~

1989 1989

1991 1991

96.0 96.0 29. 29.11

90.5 90.5 4 1 .3 41.3

96.9 96.9 48.2 48.2

85.9 85.9 55.2 55.2

1101.5 O l .5 1184.8 84.8

1119.1 19. 1 238.8 238.8

1102.6 02.6 267.7 267.7

193.4 193.4 11065.2 065.2

r, r~"a = =

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0.53*** 0.53*** 0.43** 0.43**

r~ = = r,

0.37*** 0.37*** - 00.07 .07

r, = = r,

r, rs = =

- 0.47*** 0.47*** 0.03 0.03

r, r~ = =

- 0.57*** 0.57*** - 00.48** .48**

r, r, = =

r, rs = =

r, rs = =

0.65*** 0.65*** 0.47** 0.47**

- 0.30** 0.30** 0.02 0.02

r, rs = =

r, rs = =

r, r~ = =

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=

r, r~ = =

r, rs = =

78 78

78 78

77 77

78 78

60.3 60.3

57.7 57.7

44.2 44.2

43.6 43.6

1164 64

140 140

1118 18

1129 29

- 00.69*** .69*** - 00.01 .01

a

Data Smith ((1974a). l 974a). "D a t a from from Smith from Smith from Smith ((1980). 1 980). '' Excluding Excluding three three High High Peak Peak samples. samples. d d Spearman Spearman rank rank correlation correlation coefficient coefficient corrected corrected for for tied tied observations. observations. **p < 1. **P < 0.0 0.01. ***P ***P < < 0.001 0.001.. h Data Data h

censuses 1 972 and 977) there cant correlations censuses ((1972 and 11977) there were were signifi significant correlations between between patch patch size and percentage percentage saturation saturation for for both both all patches patches and just those those that that were were occupied. occupied. This cant) for This correlation correlation was weaker weaker (although (although still highly highly signifi significant) for all patches patches in 11989, 989, but there was no 1 989. In 1991, 1 99 1 , no correlation correlation for for occupied occut, ied patches patches only in 1989. neither 1 989, neither all patches patches nor nor occupied occupied patches patches silOwed si~owed a correlation correlation to area. In 1989, when metapopulation was was when the "collapse" "collapse" of of the southern southern half half of of the Bodie Bodie pika pika metapopulation underway, underway, a number number of of large large patches patches harbored harbored very few few pikas. The The resulting resulting low percentage resulted in the lack of percentage saturation saturation values on large large patches patches apparently apparently resulted of a correlation 99 1 , there correlation between between these these variables variables in that year. In 11991, there was no no correla­ correlation between between size and and percentage percentage saturation saturation for for all patches patches because because nearly all of southern half unoc­ of the patches patches in the southern half of of the Bodie metapopulation were were unoccupied, independent of remained no correlation cupied, independent of their their size. Interestingly, Interestingly, there remained correlation

17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Metapopulotion Metapopulation

417 417

Between-Census Extinctions Extinctions and ond Recolonizations Recolonizations and and Changes Changes in in Percentage Percentage Saturation Saturation on on TABLE III I Between-Census Mine Tailing Patches at Bodie Mine Tailing Patches at Bodie 1972- 1977 1972-1977

Patch extinctions extinctions Patch Patch recolonizations recolonizations Patch Percentage saturation saturation Percentage patches increased increased No. patches patches decreased decreased No. patches N o change change (including ( including unoccupied unoccupied No patches) patches) Percentage of patches that that experienced experienced a Percentage of patches change change

1977-1989 1977- 1989

1989-1991 1989- 1991

10 10 88

14 14 44

8 7

17 17 30 30

20 20 21 21

15 15 21 21

30 30

35

41

6 1 .0 61.0

53.9 53.9

46.8 46.8

between percentage saturation saturation for the occupied occupied patches the northern northern between size and and percentage for the patches in the half of of the the study area. While While the the southern southern half half was was nearly nearly void of of pikas, pikas, the the half study area. absolute number number of of pikas pikas increased increased in 1991 1 99 1 over over 1989 1 989 (Table (Table I). Thus, Thus, while while there there absolute were fewer fewer occupied 1 99 1 (Table (Table I), those those that that were were occupied occupied were occupied sites at Bodie in 1991 had relatively high percentage percentage saturation saturation values values independent of size of patch. patch. all had relatively high independent of size of

C Isolation Isolation Effects Effects C. There was correlation for There was a strong strong negative negative correlation for all patches patches of of percentage percentage satusatu­ ration ration with interpatch interpatch distance distance across across all years (Table (Table I), apparently apparently because the percentage percentage of of patches patches that that were were unoccupied unoccupied increased increased with with interpatch interpatch distance distance (see Smith, 11980). This correlation each subsequent subsequent 9 80). This correlation was more more pronounced pronounced in each census year year (Table (Table I). The The strongest strongest correlation, correlation, in 11991, resulted from from the long census 99 1 , resulted long inter-patch unoccupied patch inter-patch distances distances from from each each unoccupied patch in the southern southern half half of of the study area to the closest patch patch with three three pikas pikas in the northern northern half half (Table (Table I). area At the same time, in 3 of of 4 years there was was no no relationship relationship between between interpatch interpatch At distance distance and percentage percentage saturation saturation for for occupied occupied patches patches (Table I). Apparently, Apparently, once occupied, occupied, factors factors other other than than isolation play a major major role in the determination determination once of 989 when of percentage percentage saturation saturation on tailing patches. patches. The The one exception was was in 11989 when there was a signifi cantly negative correlation between these two significantly two variables (Table I). At this time the collapse of of the southern southern half of of the Bodie pika metapopulation metapopulation percentage saturation among among occupied patches patches apparently was had started, and percentage not being determined determined by patch size as in other other years (see above). Instead, the effect effect of of isolation isolation was was beginning beginning to to show show among among occupied occupied patches; patches; those those that that occupied yet declining in percentage percentage saturation were not receiving new were occupied propagules propagules (i.e. no "rescue effect"). Thus Thus the the data data on correlation of of percentage percentage saturation with patch size and interpatch distance portrayed in Table I show a "wave" effect from 972 and from 11972

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Spatially SpatiallyCorrelated CorrelatedDynamics Dynamicsinin aa Pika PikaMetapopulation Metapopulation

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The Spatially Spatially Explicit ExplicitStructured StructuredMetapopulation Metapopulation Model Model DD.. The There are are many ways of constructing and parameterizing a metapopulation metapopulation There model (Hanski and Gilpin, 11991). 99 1 ). We developed a spatially explicit structured metapopulation model because we were interested in exploring a specific ques­ quessouthern region region of the Bodie metapopulation metapopulation more more extinction extinction prone prone tion: is the southern northern region? To approach approach this we have examined A, A, which governs than the northern "rescue" and recolonization, recolonization, rate of extinction, and the effects radii, which govern "rescue" reasonable, ranges of of values (Fig. 3). Our response response over wide, but biologically reasonable, of pikas in the northern region to the number number of pikas in the variable is the ratio of southern region at the end of 20 time steps, or 1100 00 years (Fig. 3). The outcome

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FIGURE 3 The The difference FIGURE difference between between projected projected population population behavior behavior of of pikas pikas in the north north and and south south subregions two parameters parameters of subregions at Bodie versus versus the two of the the stepping-stone stepping-stone model. One One independent independent axis axis shows shows aa set set of of values values for for the the "effects "effects radius" radius" (see (see text text for for explanation), explanation). while while the the second second independent independent axis from nearby axis shows shows values values for for the the expected expected discrete discrete growth growth rate rate (which (which is absent absent any any immigration immigration from nearby patches). Over Over the the full set set of of parameter parameter values, values. the the full full metapopulation metapopulation behavior behavior goes goes from full patches). from full occupancy occupancy and and full full saturation saturation to to complete complete regional regional extirpation. extirpation. The The dependent dependent axis, axis. plotted plotted vertically vertically as as histogram histogram cells, cells. shows shows the the ratio ratio of of the the number number of of animals animals in the the northmost north most half half of of the the metapopumetapopu­ lation to to the the number number of of animals animals in the the southmost southmost half half of of the the metapopulation metapopulation after 1 00 years years (20 (20 time time lation after 100 steps) based based on on 50 50 replications. replications. At At full full saturation (high effects effects radius radius and and high high expected expected growth growth rate), rate). steps) saturation (high there there are are about about 20% 20% more more animals animals in the the northern northern subregion. subregion. As As the the parameters parameters are are each each reduced reduced in in value, so that the value. so does does this this north-to-south north-to-south ratio, ratio, which which indicates indicates that the northern northern half half of of the the metapopulation metapopulation is predicted predicted to to be be the the more more vulnerable vulnerable subregion subregion to to subregional subregional extirpation. extirpation.

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Andrew Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin

shows shows that the the north-to-south north-to-south ratio declines declines with declining declining parameter parameter values. The model appears appears most sensitive to variation in the effects effects radii. The north and south subregions paramet.er values values are subregions at Bodie Bodie appear appear to behave similarly only when when parameter are jointly high (leading (leading to (leading to full occupancy occupancy of of patches) or jointly low (leading regional regional extinction). extinction). At intermediate intermediate values values the north-to-south north-to-south ratio ratio falls below unity, indicating indicating that that the northern northern half half of of the metapopulation metapopulation is the more vulner­ vulnerable subregion subregion to subregional subregional extirpation extirpation (Fig. 3). This finding is the exact exact op­ op1 99 1 . posite of of our our empirical empirical result in 1991. One One potential potential reason reason for the discrepancy discrepancy between between the model and empirical empirical results is that average nearest-neighbor distance between each pair of patches that average nearest-neighbor distance between pair of patches is shorter 1 69 km) than Perhaps shorter in the southern southern region (0. (0.169 than in the north north (0.204 km). Perhaps more more important important is Fig. 4, a demonstration of of the degree degree of of connectivity connectivity (number (number of of potential potential patches patches within within sweeps sweeps of of each each of of four four effects effects radii) for for each each patch. patch. Clearly, au­ Clearly, there there are are significantly fewer fewer patches patches available available to influence influence spatially autocorrelated longer tocorrelated changes changes at the shortest shortest distance distance (0.25 km) than than at each each of of the longer .0, and 2.0 km, respectively (Fig. 4). At distances, 0.5, 11.0, At an effects effects radius of of 0.25 0.25 km, kin, very few of of the the patches patches show connectivity, connectivity, and extinction extinction of of popUlations populations on patches regionally or throughout the Bodie Bodie metapopulation metapopulation is not a surprising surprising result. At 0.5 km km many of of the patches patches in the southern region region are are connected, connected, and and there there are two distinct distinct clusters clusters with high connectivity. At this distance distance the the north is still very loosely connected. connected. The 2.0-km 2.0-km effects radius figuratively "joins" "joins" each each patch patch with a very large large subset of of patches patches within the Bodie metapopulation metapopulation system (Fig. 4).

E. Regional Effects Effects (Spatially Autocorrelated Patterns) The The spatial pattern pattern of of patch patch occupancy, occupancy, and and area area and isolation effects effects (Fig. 2; Tables 980) indicate Tables I and and II; see also Smith, Smith, 11980) indicate that the dynamics dynamics of of extinction extinction and recolonization recolonization of of patches patches at Bodie has not occurred occurred uniformly across the the landscape. landscape. In addition, addition, our model based based on stepping-stone stepping-stone metapopulation metapopulation dy­ dynamics namics failed failed to explain explain our observation observation that that the southern half half of of the Bodie Bodie meta­ metapopulation 99 1 census, population collapsed collapsed in the 11991 census, while the northern northern half half remained remained close to fully saturated. saturated. Thus, we examined examined this putative putative spatially nonrandom nonrandom pattern pattern by determining determining the extent to which there was spatial autocorrelation autocorrelation of of popula­ population-level tion-level events among patches. patches. Between-census Between-census correlated correlated changes changes in percentage percentage saturation were determined determined between focal patches patches and neighboring neighboring patches patches at three three effects effects radii for for each each of of the 972 - 1 977 and 1 989 - 1 99 1 interinter­ the three three census census intervals intervals (Table III). For For the the 11972-1977 and 1989-1991 vals, there patches that there were no significant differences differences between the number number of of patches exhibited exhibited positive positive spatial autocorrelation than negative spatial autocorrelation autocorrelation at the 2.0-km effects effects radii (Table III). Both intervals intervals showed showed significant significant differences differences (P . 1 ) at the 0.5-km radius; and for 1 989 - 1 99 1 interval, interval, there (P < < 00.1) for the 1989-1991 there was a .0-km radius. The highly significant significant difference difference at the 11.0-km The opposite result obtained obtained for 977 - 1 989 census interval: for the 11977-1989 interval: there there were significant significant differences differences at effects effects

17 1 7 Spatially Spatially Correlated Correlated Dynamics Dynamics in a Pika Pika Metapopulation Metapopula�on

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422 422

Andrew T.T. Smith Smith and and Michael Michael Gilpin Gilpin Andrew

TABLE III III Between-Census Between-Census Correlated Correlated Changes Changes inin Percentage Percentage Saturation Saturation between between Focal Focal Mine Mine Tailing Tailing TABLE Patches and and Neighboring Neighboring Patches Patches at at Various Various Fixed Fixed Distances Distances ("Effects ("Effects Radii") Radii")a~ Patches Effects Effects radius (km) (km) radius

Census Census interval interval

Positive spatial spatial Positive autocorrelations autocorrelations

Negative spatial spatial Negative autocorrelations autocorrelations

G-test G-test value value

2.00 2.00

1 972- 1 977 1972-1977 1 977- 1 989 1977-1989 1989- 1 99 1 1989-1991 1972- 1 977 1972-1977 1 977 - 1 989 1977-1989 1 989- 1 99 1 1989-1991 1972- 1 977 1972-1977 1977- 1 989 1977-1989 1 989- 1 99 1 1989-1991

18 18 27 27 18 18 25 25 26 26 26 26 26 26 23 23 25 25

21 21 13 13 16 16 14 14 14 14 88 13 13 17 17 99

0.101 0.101 4.302** 4.302** 0.028 0.028 2.59 1 2.591 3.062* 3.062* 8.895*** 8.895*** 3.75 1 * 3.751" 0.627 0.627 6.849*** 6.849***

1 .00 1.00

0.50 0.50

a

Positive spatial spatial autocorrelations autocorrelations occurred occurred when when the the focal focal patches patches increased increased or or decreased decreased in in perper­ a Positive centage saturation saturation and and the the average average change change in in percentage percentage saturation saturation in in neighboring neighboring patches patches (within (within centage respective radius) correspondingly increased increased or or decreased. decreased. Negative Negative spatial spatial autocorrelations autocorrelations ococ­ aa respective radius) correspondingly curred when when percentage percentage saturation saturation between focal patch patch and and its its neighboring neighboring patches patches were were of of difdif­ curred between aa focal ferent sign. sign. Only Only those those patches patches that that exhibited exhibited aa change change in in percentage percentage saturation saturation during during the the respective respective ferent census interval interval were were analyzed. analyzed. census p< 0. 1 0. ** P < 0.10. p< < 0.05. 0.05. ** P ** *** p < 0.0 1. *** P < 0.01.

radii ooff 2.0 and 11.0 .0 km, whereas difference between number of whereas there there was no difference between the number of positive positive and and negative negative spatially autocorrelated autocorrelated patches patches at the 0.5-km radius radius (Table III). III). The sensitivity of spatial autocorrelation autocorrelation at varying radii was was examined also using Moran's 972 - 1 977 census interval, Moran's I statistic (Fig. 5). During the 11972-1977 interval, there was positive spatial autocorrelation only for effects radii up to 0.3 km (Fig. 5), which is a distance that includes on on average very few nearest neighbors (Figs. 11,, 4). During this time overall saturation was high, there was an interspersion of occupied occupied and and unoccupied patches patches (see Fig. 2), and and population population extinctions and subsequent subsequent recolonizations on on patches appeared to to be in aa dynamic dynamic equilibrium (Smith, 11974a, 974a, 11980). 980). This result is consistent with the reported low vagility of of pikas at Bodie (Smith, 974b)- population sizes on focal patches were (Smith, 11974b)mpopulation were likely to be be influenced influenced only only by by nearby nearby patches patches (either (either growing growing in in part part due due to to immigration immigration of of surplus surplus animals from from nearby nearby patches, patches, or or declining declining and and concomitantly concomitantly not not re­ receiving ceiving immigrants immigrants from from nearby nearby patches patches that that were were also declining). declining). The 977 - 1 989 census The 11977-1989 census interval interval shows shows aa more more interesting interesting pattern. pattern. During During this 1 40 to 1 8; Table this interval interval there there was was aa net net loss loss of of animals animals ((140 to 1118; Table I), I), and and this this loss loss was was confined confined mostly mostly to to the the southern southern end end of of the the study study area area (Fig. (Fig. 2). 2). This This regional regional decline decline accounts accounts for for much much of of the the positive positive spatial spatial autocorrelation autocorrelation (largely (largely negative negative patch patch growth growth associated associated with with negative negative regional regional growth growth in in the the south) south) over over distances distances from .5 km from 0.3 0.3 to to 11.5 km (Fig. (Fig. 5). 5). ItIt is is also also possible possible that that the the longer longer time time interval interval between between censuses uenced the censuses infl influenced the autocorrelation autocorrelation statistic statistic by by allowing allowing more more "averaging" "averaging"

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over pikas at Bodie. Bodie. In over regions regions of of the cumulative cumulative small dispersal dispersal movements of of pikas this increased in population population size the census this way, many many patches patches in in the the north north increased size during during the census interval interval (Fig. (Fig. 4). 4). A A lack lack of of significant positive positive autocorrelation autocorrelation occurred occurred over over the the shortest shortest time interval 1 989- 1 99 1 ; Fig. 5). The remaining animals interval ((1989-1991; The few remaining animals in the southern southern portion portion high overall died out, and and the the northern northern region showed stability stability at at aa relatively high overall saturation. saturation.

VI. DISCUSSION DISCUSSION Our Our long-term long-term multigenerational multigenerational study study of of the the Bodie Bodie pika pika metapopulation metapopulation has has provided provided insight insight into into the the dynamics dynamics of of aa real real world world metapopulation. metapopulation. We We have have gained mechanisms of gained understanding understanding of of the the underlying underlying mechanisms of the the metapopulation metapopulation dy­ dyhave learned namics, and and we have learned something something about about the different different spatial spatial scales that that are are relevant these mechanisms relevant to to these mechanisms and and to to their their associated associated dynamics. dynamics. The which metapopulation dynamics are likely to has The regional regional scale scale at at which metapopulation dynamics are likely to occur occur has been that "at place been defined defined as as that "at which individuals individuals infrequently infrequently move from from one one place

424

Andrew and Michael Andrew T.T. Smith Smith and Michael Gilpin Gilpin

(population) (population) to another, typically across across habitat types which are not suitable for their feeding and breeding activities, activities, and often with substantial risk of failing to locate 99 1 ). locate another another suitable habitat habitat patch patch in which to settle" (Hanski and Gilpin, 11991). The American pika population at Bodie fits this definition closely. Pikas cannot and do not occupy the Great Basin sagebrush habitat that separates separates their obliga­ obligatory talus habitat -the mine tailing patches habitat~the patches at Bodie. Physiologically, pikas are at risk due to high temperatures temperatures and inability to find ameliorating microclimates (such as the cool interstices deep 974b). Pikas deep in talus) while dispersing (Smith, 11974b). are more vulnerable 974a; Ivins and Smith, vulnerable to predation predation when off off of of talus (Smith, I1974a; 1 983), and predators of 1983), and a high diversity of of potential predators of pikas pikas occupy occupy the the study area (Smith, 11979). 979). Juveniles, the primary dispersers in this system, normally settle Juveniles, dispersers on the closest unsaturated patch (1. A. Nagy, personal communication; Peacock, unsaturated (J. 1 995). 1995). Metapopulations are characterized characterized also by their between extinction Metapopulations their balance between of habitat of local popUlations populations and and establishment establishment of of new populations populations in vacant habitat patches (Hanski and Gilpin, 1991). 1 99 I ). Such extinction-colonization extinction-colonization dynamics, as well as a pattern of incomplete patch occupancy (in the range of 60-40%), 60-40%), have been a characteristic characteristic of of the Bodie pika population for several decades decades (Tables I, II; Severaid, 1955; 1 955; Smith, 11974a, 974a, b, 1980). 1 980). During the 1970s, 1 970s, extinction of pop­ of populations ulations on patches patches was a function of patch patch area, area, largely due to demographic stochasticity, and immigration, which was a function of nearest ' of the distance distance to the nearest' possible source patch (Table I; Smith, 11974a, 974a, 1980). 1 980). Indeed, the average small patch size at Bodie, thus the small carrying carrying capacity of pikas on these patches, appears appears to be a central central feature of this metapopulation. metapopulation. The strong effect of isolation of patches on occupancy rates of pikas in the of patches Bodie system indicates that that a basic assumption of of the spatially implicit Levins ((1970) 1 970) meta population model--that model- that dispersal occurs metapopulation occurs with equal equal probability be­ between any pair of patches, patches, independent independent of of their location~is tween location -is incorrect. incorrect. Smith ( 1 974a, 1980) 1 980) formulated his early analysis on a spatially (1974a, spatially explicit framework based 967). In the based on island biogeographic theory (MacArthur (MacArthur and Wilson, 11967). present analysis, we have extended parameterization of present extended this with the parameterization of a spatially explicit "stepping-stone" "stepping-stone" metapopulation model. There There are are many many forms, varying considerably in complexity, that that such a step­ steppingstone pingstone metapopulation model can take. We chose to work work with the simplest form of of model that that could include the the most important features of of the the Bodie pika metapopulation system. The of all patches. metapopulation The model includes the size and location location of We have have provided for distance-dependent distance-dependent rescue rescue and and recolonization through an "effects radius" that limits single time-step time-step dispersal to a maximal distance, which, "effects radius" distance, which, in our sensitivity analyses, we have kept to below I1 km. We modeled occupancy and al., this volume, for and extinction in a structured structured manner manner (see Gyllenberg Gyllenberg et et al., for a review) in which we distinguish the integer population size on a patch, most of of which fall in the 0 to 10 IO range. range. Transitions between states states on a patch patch are are governed ' by demographic demographic stochasticity, which is is" independent independent between between individuals. Based on data 1 970s, our analysis shows the metapopulation system data from the 1970s, to be stable, with continuing turnover and with patch patch occupancy occupancy in the range range of of

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Spatially Dynamics in SpatiallyCorrelated Correlated Dynamics in aa Pika Pika Metapopulation Metapopulation

425 425

60%. If prediction, then If these these results were were to be taken as a model model prediction, then the the actual system -the near sub­ behavior of of the systemwthe near extirpation of of populations in the southern southern subregionwould be quite surprising. surprising. Of region--would Of interest, then, is whether whether the model was misparameterized or incompletely structured. structured. have explored the possibility of of misparamterization. We have performed performed We have a wide sensitivity analysis by varying the effects radius and the expected expected growth rate per patch. We varied these parameters over a range that takes the metapop­ metapopulation from full saturation and occupancy to extirpation. What we found found is that it is highly improbable improbable (although not impossible) that the southern subregion should go extinct, while the northern northern subregion subregion should remain highly saturated. Thus, Thus, we conclude conclude that the the structure of of the the model is incomplete. Because of pre­ of the apparent inadequacy of of our stepping-stone stepping-stone model model for predicting the future of spa­ of the Bodie metapopulation, metapopulation, we analyzed the system for for spatially correlated patterns among neighborhoods neighborhoods (or clusters) of of patches. We did this in two ways: with analyses analyses of of correlated changes in population population growth within within neighborhoods, neighborhoods, and with the development development of of Moran' Moran'ss I statistic for our map­ mapbased data in which multiple effects radii were examined. The autocorrelation/neighborhood autocorrelation/neighborhood analysis indicated indicated that most patches patches were strongly influenced by the average level of patches­ of occupancy in surrounding patchesw thus thus that entire neighborhoods neighborhoods rather rather than distance to a single single potential source patch was important in determining the probability of of patch occupancy (Table III). These neighborhoods ned for neighborhoods were well-defi well-defined for the shortest effects radius, 0.5 km, for 1 972 - 1 977, 11989-1991). 989 - 1 99 1 ). The for the the two two censuses taken close together ((1972-1977, .0 km yielded the highest overall difference between positive and radius of of 11.0 negative autocorrelations autocorrelations of of patches for all censuses. These results highlight the influence of of neighboring patches at greater distances from target patches than is apparent from the spatially explicit metapopulation model model based based upon stepping­ steppingstone (or nearest-neighbor) nearest-neighbor) dispersal distances. The longest census interval was between 11977 977 and 11989, 989, and the strongest strongest effect (difference (difference between number number of of positive and negative autocorrelated patches) was found found at 2.0 km. Apparently, given more more time and the likelihood of of more more cumulative nearest-neighbor nearest-neighbor dispersal movements among patches, the radius affected by such movements increases. The results of of Moran' Moran' s I statistic also confirm the spatially correlated correlated pattern of occupancy of of pikas at Bodie and indicate that there may be distinct differences differences in the nature nature of of autocorrelation autocorrelation of of patch occupancy occupancy among census intervals (Fig. 5). Unfortunately, Unfortunately, because because of of the different intervals between censuses, censuses, we cannot cannot discriminate between between whether whether these different different responses responses were caused by time of of census interval or the actual spacing spacing of of occupied occupied and unoccupied unoccupied patches census patches during the census interval. Parsimoniously, the 1972-1978 1972 - 1 978 interval showed high levels of of autocorrelation autocorrelation at relatively relatively short effects radii radii distances, distances, indicative indicative of of the the ex­ extinction-colonization tinction-colonization dynamics that that characterized characterized this time frame. At the the other other 1 989 - 1 99 1 interval, and this extreme, no autocorrelation autocorrelation was evident for for the 1989-1991 result could be due to either (or both) the short time between censuses censuses or or the total half of popUlation. The 1 977 - 1 989 interval collapse of of the southern southern half of the study population. The 1977-1989 cant autocorrelations range of yielded signifi significant autocorrelations across across the full range of effects radii, again

426

Andrew Andrew T.T. Smith and Michael Michael Gilpin Gilpin

indicating neighborhoods due to indicating that that there may have been been more more "averaging" of of neighborhoods the cumulative effects effects of of stepping-stone stepping-stone dispersal dispersal over this period. We We do do not not know know the the actual actual mechanism(s) mechanism(s) that that may may have have produced produced the the spatial spatial nonrandom ness of nonrandomness of extinction as demonstrated in this investigation. In his book book on 1 990) shows on spatial spatial statistics, statistics, Haining Haining ((1990) shows that that very very different different dynamics, dynamics, and and dif­ different ferent underlying mechanisms, mechanisms, can cause similar patterns of of spatial autocorrela­ autocorrelaThe tion. We We pose two general hypotheses about what occurred occurred in this system. The first is that the results could have been due to "position" effects. The second is results The of the southern half of the study that extinction events leading to the collapse of half of population were driven by spatial autocorrelation effects. effects. The of the The position position effects effects hypothesis hypothesis holds that the southern half half of the study study area is qualitatively different different from the northern half half or that external forces forces (such as climate, predation, competition) could affect affect pikas pikas in the southern and northern northern halves differentially. differentially. Although we have no direct measurements, available data data that we outline below indicate that there is no demonstrable impor­ demonstrable difference difference in imporparameters that may affect affect pika pika viability in the southern and northern halves tant parameters halves of of the study area. area. The The Great Basin sage plant community community is roughly roughly similar throughout the study area, area, and and each of of the habitat patches patches on which pikas pikas live was "thrown up" in the homogeneous habitat. There the middle middle of of this homogeneous There is very very little difference patch, and of the patches patches are difference in size of of rock rock in each patch, and the structure of are remarkably similar throughout the study area. Similarly, the area area is too small to have have been been affected affected differentially differentially by by climatic climatic patterns. patterns. All available observations on potential potential predators and competitors (Severaid, indicate that that they they occur throughout the study study area. area. The The most most 11955; 955; Smith, 11979) 979) indicate likely predators weasels (Mustela predators of of pikas pikas on their their preferred preferred talus habitat are are weasels -both of 979), as Jrenata frenata and M. erminea ermineamboth of which are common at Bodie; Smith, 11979), they can gain pikas (Ivins and Smith, 1983). 1 983). Weasels at gain entrance entrance to the dens of of pikas Bodie have the tendency to hunt repeatedly on the same patch (A. T. Smith, unpublished predation is the unpublished data), and and it is likely that that weasel predation the most common common cause cause of of extinction extinction of of a population of of pikas pikas on a patch. patch. If If a family of of weasels then moved to the next closest occupied patch (as they do when hunting Microtus; occupied patch 977), a cluster of How­ Fitzgerald, 11977), of patches patches could show "correlated extinctions." extinctions." However, as mentioned mentioned above, there there is no reason that such predator-caused predator-caused extinctions would occur occur only in one region of of the study area. Instead, such extinction extinction events could be said to operate as a form of of environmental stochasticity stochasticity that that would impact impact neighborhoods neighborhoods of of patches patches (see (see below). below). final position effect is that the the largest single single patch (High Peak; Peak; A fi nal potential position Fig. 11)) was found on the northern northern half half of of the study area area (although there there were were large patches-the l ] -found patches--the Noonday Noonday and the Red Red Cloud mine tailings [Fig. [Fig.1] - - found in the south). It could be that that the northern northern area area operates operates more more like a mainland mainland-­ island population, in which persistence depends depends on the existence of of one or more more extinction-resistent 99 1 ). Two extinction-resistent populations populations (Harrison, 11991). Two lines of of evidence ar­ argue against against this effect. First, the large patch patch (K = = 50) was included our spa­ spaincluded in our tially explicit model, and in spite of its size the northern northern population was more more

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Spatially Correlated Spatiolly CorrelatedDynamics Dynamicsinin aQ Pika Pika Metapopulation Metapopulafion

427 427

extinction-prone than the 988 to 11991 99 1 all the southern population. Second, from 11988 pikas marked (Peacock, pikas in a portion of of the the High High Peak patch were individually marked (Peacock, 11995). 995). Although several patches patches are found in the neighborhood of of this site, none of them were colonized by any of 1 995). Thus, of the marked marked animals animals (Peacock, (Peacock, 1995). within within the time scale of of our censuses, the continued continued occupancy occupancy of a large large number number of patches patches in the northern half does not appear appear to be explained explained by mainland­ mainlandisland colonization colonization dynamics. The alternative alternative to the position effects effects hypothesis is that extinction extinction events leading half of leading to the collapse of of the southern southern half of the study population were were driven driven by spatial autocorrelation effects. In this case the southern and northern portions of of the study area are considered equivalent, equivalent, and a stochastic cause of of extinction may have impacted patch populations in the south rather than in the north. impacted rather than north. As fewer fewer and fewer fewer southern southern patches patches were occupied, occupied, then then there were fewer fewer source patches to produce number of produce propagules for for the increasing number of unoccupied patches. Ultimately, we believe a threshold may have been reached that the system itself was incapable -colonization equilibrium, incapable of of remaining remaining in extinction extinction-colonization equilibrium, and and the result was its total 989 census, which showed a greatly reduced number total collapse. The The 11989 of 99 1 . of occupied islands in the south, was a harbinger harbinger of the collapse observed in 11991. The 11989 989 census is important in that it shows that the southern half half did not collapse over a short period (such as might be expected expected should there have been an epidemic), epidemic), but rather rather that there there was a gradual gradual decline decline in the number number of of occupied islands. Effectively, what we observed in the southern half half of of the study area during the 11991 99 1 census was a regional extension of of the spatial autocorrelation among 1 99 1 census (Table patches seen throughout the Bodie landscape landscape leading up to the 1991 III; Fig. 5). We We have no direct evidence of of the stochastic event(s) that may have initiated half of of patch occupancy in the southern half of the Bodie study this downward spiral of area. Demographically, pikas are long-lived. It is possible that the age structure structure on some patches could have been skewed to older tum prohib­ older animals, animals, which in turn prohibited local settlement of 1 983a). A few of their offspring (see Smith and Ivins, 1983a). few key patches could have been structured in this way, followed by the death death of of all adults adults from "old age." Also possible is that that populations on individual patches patches became became genetically inbred, inbred, leading to inbreeding depression and local patch patch extinctions (Gilpin, 11991). 99 1 ). However, a detailed detailed study of of the genetics of of pikas in a subset of the Bodie popUlation patches population indicated indicated that enough movement took place among patches (with juveniles primarily colonizing the closest available territory territory as well as oc­ occasionally dispersing greater greater distances) that that habitat habitat fragmentation resulted in only limited genetic subdivision within this population (Peacock, 11995). 995). Another Another pos­ possibility, as mentioned above, is that that weasels cleared cleared out a few few key patches, patches, be­ beginning the decline. Naturally, there could also have been multiple mUltiple causes of of extinction of of patches patches in the south. The results of of this investigation show clearly the need for for metapopulation models to incorporate incorporate explicit spatial dimensions and to examine examine a range of of spatial and temporal scales in their analyses. Accordingly, efforts to understand understand and to

428

T. Smith and Michael Gilpin Andrew T.

model further further the the Bodie Bodie pika pika metapopulation metapopulation continue. continue. First, First, we we desire desire to to underunder­ model stand dispersal dispersal as as aa function function of of source source patch patch density, density, degree degree of of patch patch isolation, isolation, stand intervening habitat habitat structure structure (Wiens, (Wiens, this this volume), volume), and and size size and and occupancy occupancy of of intervening target patches. patches. Second, Second, we we want want to to see see to to what what degree degree demographic demographic stochasticity stochasticity target plus rescue rescue m may explain local local patch patch fluctuations fluctuations in in population population size. size. Third, Third, we we plus a y explain want to explore alternative hypotheses for spatial autocorrelation of extinction. want to explore alternative hypotheses for spatial autocorrelation of extinction. For us, us, the the metapopulation metapopulation of of A American pikas at at the the ghost ghost town town of of Bodie Bodie remains remains For m e r i c a n pikas living laboratory. laboratory. aa living

ACKNOWLEDGMENTS ACKNOWLEDGMENTS Bodie California State Historical Historical Park for for their support and assistance We thank the staff staff of the Bodie California State Parks over the years. Permission for for access to our field site was kindly granted granted by the California of Land Management, and various mining companies holding patent rights rights at Association, Bureau of Peacock (1989 ( 1 989 and 1991), 1 99 1 ), and David Bodie. We appreciate the contributions of Chris Ray and Mary Peacock ( 1 991), who helped us conduct the censuses. Lyle Nichols assisted assisted in the statistical statistical analanal­ Quammen (1991), grateful. We thank Jan Bengtsson, Steve Dobson, Ilkka Hanski, John Nagy, yses, for for which we are grateful. Harriet Smith, and Chris Thomas for for their conscientious reviews of the manuscript.

18

A Case Case Study Study of Genetic Genetic Structure Structure in a

Plant Metapopulation Plant Barbara Barbara E.E. Giles Giles

Jerome J&~me Goudet Goudet

I. INTRODUGION INTRODUCTION Habitats that are suitable for for the establishment and maintenance of most across landscapes. Environ­ Environspecies of plants and animals are distributed patchily across mental patchiness populations patchiness forces species to be structured into systems of of local populations within which cs are more likely to interact which conspecifi conspecifics interact with each other other than than with conspecifi cs from other 995). Isolation, however, conspecifics other populations populations (McCauley, 11995). however, is not complete and since most real organisms have some power power of of dispersal, members of in­ of a local population have a low but positive probability of of interacting with individuals from other localities. Depending Depending on the rate rate of of migration, demographic demographic and genetic dynamics will be influenced by this migration as well as by local birth rates. birth and and death death rates. Local populations populations are seldom immortal. Demographic, environmental, and genetic stochasticities (Shaffer, 98 1 ; Lande, 11988b), 988b), and deterministic processes processes (Shaffer, 11981; such as succession (Olivieri et 1 990; 11995; 995; Harrison, 99 1 ) may cause the et al. al.,, 1990; Harrison, 11991) extinction of of local populations, populations, although some of of these habitats may be colonized colonized again again by dispersing dispersing propagules. As a consequence, consequence, local populations populations come and go on temporal scales that do not allow demographic demographic and/or and/or genetic equilibria to be attained, and the age structure structure of of the local populations populations reflects the time elapsed since 990). since they they were were formed formed (Whitlock (Whitlock and and McCauley, 11990).

Metapopulation Metapopulation Biology Biology Copyright © 997 by Academic Press. Inc. All rights of of reproduction 9 11997 Academic Press, reproduction in any form reserved.

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Barbara E.E. Giles Giles and and Jerome J6rSmeGaudet Goudet Barbaro

Patches Patches may also differ in quality, size, and and spatial spatial arrangement (Hanski, this volume). Since (re)colonization and migration depend on the distances between habitat patches relative to the dispersal range of a species, patch dispersion is for the dynamics in individual patches and for the maintenance maintenance of the critical for metapopulation as a whole (Harrison, 11991; metapopulation 99 1 ; Fahrig, 11992; 992; Slatkin, 11993; 993; Hastings Harrison, 11994; Where local popu­ popuand Harrison, 994; McCauley, 11995; 995; Hanski, this volume). Where connected by migration migration that is not extensive enough to entirely oblit­ oblitlations are connected erate 992), erate the local generation to generation dynamics (Gyllenberg and and Hanski, 11992), the system as if it were one one large homogeneous homogeneous population population at equilibrium equilibrium treating the is not appropriate 990). Instead, understanding appropriate (Olivieri et al. al.,, 11990). understanding population population and evolutionary processes in temporally and spatially structured populations populations requires requires evolutionary metapopulation level (Antonovics (Antonovics et al. al.,, 11994; work at a regional or metapopulation 994; Hanski, 11996a). 996a). metapopulations is now reasonably well developed developed While the theory of metapopulations (Hanski and and Simberloff, Simberloff, this volume) and many many ecological ecological and genetic genetic predic­ predic(Hanski tions have been been made, made, empirical empirical tests of theoretical predictions predictions lag behind. The tions of theoretical behind. The primary purpose purpose of of this chapter chapter is to present present a case study of of a plant metapopuprimary plant metapopu­ lation. The plant, plant, Si/ene Silene dioica, dioica, is a dioecious dioecious perennial perennial and a component component of of early of primary succession in northern northern Scandinavia. This study was carried out stages of islands in an area area of of the Baltic Sea subject subject to land uplift uplift so that that new on islands new islands islands continuously, though though slowly, being being formed. rate of of land land uplift allows the the are continuously, formed. The rate of the island island populations populations to be estimated; estimated; the successional successional processes processes together together ages of continual creation creation of of new new islands islands imply that that population population turnover turnover must with the continual occur. We We have have proceeded proceeded by constructing constructing groups groups of of islands differing differing in their spatial, temporal temporal or demographic demographic characteristics, characteristics, and compared compared the observed observed changes in genetic differentiation among groups with those changes genetic differentiation among these these groups those predicted predicted by metapopulation have been able to test test and, and, in many many cases, cases, metapopulation models. models. In this way, we have been able confirm effects of spatial and and temporal confirm the predicted predicted effects of spatial temporal heterogeneity on genetic genetic structuring resulting metapopulation dynamics. We short restructuring resulting from from metapopulation We begin begin with a short re­ view of of the genetic genetic theory theory of of metapopulations metapopulations to contrast contrast the differences the differences in the assumptions and and genetics genetics models. assumptions and questions questions of of the ecological ecological and models. After After presenting presenting the results other genetic genetic studies of metapopularesults of of our our study, we briefly briefly review other studies of metapopula­ tions. tions.

II. THEORY THEORY The consequences The consequences of of genetic genetic differentiation differentiation and and gene gene flow flow among among local local poppop­ ulations for the ulations for the rate rate and and pattern pattern of of evolutionary evolutionary change change has has been been studied studied for for a long long time time in in population population genetics. genetics. In In large large randomly randomly mating mating populations, populations, the the major major factor affecting affecting allele and and genotypic frequencies frequencies is selection. selection. In assemblages assemblages of of factor small small populations, populations, however, however, other other factors factors come come into into play, play, the the main main ones ones being being genetic drift (the genetic drift (the random random fluctuations fluctuations of of allele allele frequencies) frequencies) and and the the degree degree to to

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which which drift is counterbalanced counterbalanced by migration. migration. The evolutionary evolutionary role of of genetic drift long-standing debate in population drift is at the center center of of a long-standing population biology, biology, and and many theories 1 990). For For example, theories rely upon upon this force (Barton and Clark, 1990). example, in the shifting shifting balance evolution (Wright, 1977; 1 977; Barton Barton and Whitlock, this volume), volume), balance theory of of evolution the populations can landscape the general idea is that many many small small populations can explore explore the the fitness fitness landscape (where fitnesses, respectively) more (where valleys and peaks peaks represent represent low and and high fitnesses, more extensively than larger pressures need larger populations. populations. This This is because because the selective selective pressures to be much random genetic much stronger stronger in small populations populations to be effective effective against against random drift. consequences of drift. While the consequences of small population population sizes are much much debated debated in the literature in terms of ( 1 977, Chapter 1 3) showed showed of inbreeding inbreeding depression, depression, Wright Wright (1977, Chapter 13) that new combinations could arise. In this process, process, individuals individuals new favorable favorable gene gene combinations with gene combinations from large combinations that that selection selection would would eliminate eliminate from large populations populations may survive populations increases, ne survive to reproduce. reproduce. As the size of of these small populations increases, fi fine tuning population to reach tuning of of these gene gene combinations combinations may allow the population reach a higher higher adaptive adaptive peak. Migration Migration could could then spread spread these combinations combinations to other other localities (Barton Whitlock, this volume). however, poppop­ (Barton and Whitlock, volume). For shifting shifting balance balance to work, however, ulations need isolated from one another. would ulations need to be fairly isolated another. Too Too much much gene flow would prevent genetic prevent the spread genetic differentiation; differentiation; too little would would prevent spread of of favorable favorable combinations. combinations. Much effort effort has therefore therefore been been put into measuring the extent to which which populations populations are differentiated, differentiated, and and these measurements measurements have have been been used to infer populations. infer the level of of gene flow between between local populations. The The modeling modeling of of spatially structured structured populations populations was was pioneered pioneered by Sewall Wright 1 93 1 , 11943). 943). His first first model, model, the island island model 93 1 ), assumed Wright ((1931, model (Wright, (Wright, 11931), assumed a set of finite and equal sizes with equal probabilities migrant of populations populations of of finite and equal probabilities of of migrant exchange. exchange. At equilibrium equilibrium between the forces forces of of migration and genetic drift, drift, Wright ' s model showed that the degree of Wright's of genetic genetic differentiation differentiation among among local populations, variance of frequencies standardized populations, measured measured as FST (the variance of allele frequencies standardized by the maximum possible possible variance), variance), was a simple function numbers function of of the the effective effective numbers of populations (Wright, 11940, 940, 1943, 1 943, 1951): 1 95 1 ): of migrants migrants among among populations FST = FST ~

11/4Nm /4Nm + + 11..

((1) 1)

Further Further mathematical mathematical development development of of the model model allowed the effects effects of of spatial variance variance in migration migration rate rate to be incorporated incorporated and led Kimura Kimura to propose propose his stepping-stone 955; Kimura 964; Weiss stepping-stone model model (Kimura, (Kimura, 11955; Kimura and Weiss, 11964; and Kimura, 964), where adjacent populations migrants. Both Kimura, 11964), where only adjacent populations exchange exchange migrants. Both the island and the stepping-stone members stepping-stone models models assume assume that individuals individuals are members of populations within which For of discrete populations which mating mating occurs occurs at random random (panmixia). (panmixia). For many species, however, however, it is possible possible that truly random random mating units do not exist at all. To another set To account account for for this type of of spatial spatial popUlation population structure, structure, another of isolation by distance of models models without without panmictic panmictic units was developed. developed. In the isolation distance or neighborhood 1 943), the genes of individual disdis­ neighborhood model model of of Wright ((1943), of each individual perse as a decreasing function of of distance, neighborhood being ned decreasing function distance, with a neighborhood being defi defined for each each individual. individual. The size of of the neighborhood neighborhood corresponds for corresponds to the area from

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which the parents parents of of the central central individual could have been drawn at random (Wright, 11943). 943). This area is defined defined as a circle of 20' around the central of radius radius 2o" o" is the standard standard deviation of of a Normal distribution of of parent­ parentindividual, where 0' offspring migration distances. Both isolation by distance and stepping stone modmod­ els of of migration, in which there there is spatial variance in migration rate, have been shown to increase the degree of of population population differentiation differentiation relative to the island 964; Crow and Aoki, 1984; 1 984; Whitlock, 11992a; 992a; GouGou­ model (Kimura and Weiss, 11964; det, 11993). 993). The island, stepping-stone, and isolation by distance models, however, however, still assume that that popUlations populations have have constant sizes and and are immortal. Although Although Wright ((1940) 1 940) was the fi rst to suggest that extinction could increase population popUlation differ­ first differentiation relative to an island model, the effects of of extinction and (re)colonization (re)colonization (population turnover) on among-population (population turnover) among-population variance variance in gene frequencies frequencies were not not studied before 1977, 1 977, when meta­ when Slatkin introduced introduced the ecological concept of of the metapopulation popUlation genetics. Population turnover implies that local pop­ population into population populations, ulations, or demes, vary in their their degree degree of of demographic maturity. Newly estab­ established populations are not demographically mature, and (re)colonization (re)colonization represents founder effects at the time represents an additional source of of genetic drift due to founder of introduction of of colonization. colonization. With the introduction of population population turnover turnover into models of of spa­ spatial population population structure, structure, spatial variance in migration rate rate is ignored ignored (i.e., migra­ migration patterns patterns are those assumed in the island model) and the focus lies entirely on the temporal temporal relationships among habitat habitat patches. Slatkin's 1 977) models are variations of Wright's island model and consist Slatkin's ((1977) of Wright's consist of of a collection collection of of local populations populations of of diploid monoecious monoecious individuals individuals of of identical m. Gen­ and fixed size N, which exchange exchange migrants migrants (gametes) at a common rate rate m. Generations represent the carrying capacity, or erations do not overlap overlap and and N is assumed to represent the number number of of organisms which the resources resources of of a habitat can support. Each generation, a proportion populations goes extinct, where the prob­ proportion e of of the local populations probability of recolo­ of extinction is equal for for all age classes. The extinct sites are then recolonized immediately by a fixed number number of of (diploid) colonizers, k, which in turn reproduce reproduce and give rise to N offspring offspring within one generation. This model differs from that of Levins ((1970) 1 970) in that: ((1) I ) patch size is a variable in the model (Levins was only concerned vol­ concerned with the proportion proportion of of occupied occupied patches) patches) (Hanski, this volume), (2) extant populations populations exchange migrants (they do not matter in the the Levins Levins model), and and (3) no site is empty at any time. Slatkin cast his model model in terms of of the variance in gene gene frequencies frequencies among populations populations and and considered two two forms of of colonization: the migrant migrant pool model, in which the k colonizers are drawn drawn as a random random sample from from the metapopulation; metapopulation; and the propagule pool model, where the k colonizers population chosen at random. His results colonizers are drawn drawn from a single population indicated that that under under the propagule propagule pool model, genetic genetic drift resulting from sampling effects during population differ­ population establishment would would increase the differentiation among local populations, populations, while under under the migrant pool model, mixing prior ow reducing reducing the prior to colonization resulted resulted in additional gene fl flow the degree degree of of dif-

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ferentiation. 1 985, 1987) 1 987) later concluded concluded that if if the ferentiation. Slatkin ((1985, the average average time (in generations) to extinction of populations was less than the ef­ generations) of local populations than or or equal to the effective size of of populations, populations, then, even even in the absence absence of of migration, migration, extinction extinction and recolonization populations by recolonization would would prohibit prohibit the genetic genetic differentiation differentiation of of local populations genetic genetic drift. drift. Wade and McCauley ((1988), 1 988), who This latter view was was challenged challenged by Wade and McCauley who recast FST> and the model in terms of standardized variance of frequencies, FST, of the standardized of allele frequencies, and asked: asked: "under what conditions conditions do do extinction, extinction, colonisation, colonisation, and and dispersion dispersion bind an array of of subdivided subdivided populations populations into a single evolutionary evolutionary unit, and and when do do they permit trajectories?" permit local populations populations to assume more more independent independent evolutionary evolutionary trajectories?" For ST would would be For the migrant pool pool model, model, these these authors authors found found that that F FST be increased increased compared compared to an island island model model if if the number number of of colonists was was less than than or equal equal to ST was was always increased propagule twice the number number of of migrants, migrants, whereas whereas F FST increased in the the propagule model. 1 990) generalized pro­ model. Whitlock Whitlock and and McCauley McCauley ((1990) generalized Slatkin's Slatkin's migrant migrant and and propagule pagule pool model by including including a new parameter, parameter, > 0 (Giles (Giles and Goudet, For Fis and and F~, permuted within within and individual may and among among populations, populations, respectively. Since the two two alleles in an individual may not ST were not be independent, independent, the permutation permutation units for for F FsT were the genotypes genotypes permuted permuted among populations. among populations.

among Islands B. Genetic Genetic Structure Structure among Islands While While it is essential essential that the the local populations populations composing composing a metapopulation metapopulation are connected If connected by migration, migration, migration migration must must also be shown shown to be restricted. restricted. If no assumed that that no evidence evidence of of population population structure structure can be be obtained, obtained, then then it must must be be assumed the local populations populations belong to the same same panmictic evolutionary evolutionary unit. Using Using the data rst asked data from from all study sites within within the archipelago, we fi first asked whether whether there there was population structure ow arising from was any evidence evidence of of population structure and and restricted restricted gene gene fl flow arising from habitat habitat subdivision subdivision within and among among islands. islands. As Table Table Ia shows, shows, the values values Fit loci, are Fit = = 0. 0 . 11 1100, , FST FST = = 0 0 ..00338 8 , and a n d Fis Fis = = 00.075, . 0 7 5 , calculated c a l c u l a t e d over o v e r all all loci, are signifi signifi-­ cantly 0.0002), providing restricted c a n t l y greater greater than than zero zero (P (P < < 0.0002), providing strong strong evidence evidence of of restricted gene ow among gene fl flow among (FST) (FsT) and and within within (Fis (Fis)) islands in in the archipelago archipelago (Fit (Fit).) . Note Note that that no no spatial or or temporal temporal dynamics have have been been taken taken into into account; account; this analysis re­ reveals only whether populations are structured structured where structured. whether the populations where the habitat is structured. ,

C. C. Genetic Genetic Differentiation Differentiation among among Cohorts Cohorts of Plant Plant Populations Populations Varying in Age The The geological geological and and successional successional processes processes occurring occurring in the Skeppsvik Skeppsvik Ar­ Archipelago chipelago clearly indicate indicate that that our our populations populations are are characterized characterized by colonization/ colonization/ F-Statistics F-StatisticsAveraged Averaged over over All All loci Loci and Populations Populations (a) (a) and over over All All loci Loci for for the the Three Three Age Age Classes Classesof of Populations Populations ((b)b)

TABLE I

(a) (b)

a

Class Class

N N

All Young Intermediate Old

52 13 13 30 9

Fit FIt 0. 1 10 0.110 0. 1 05 0.105 0. 1 07 0.107 0. 1 58 0.158

(.039) (.06 1) (.061) (.033) (.068)

Fsv FST

Fis Fis

0.038 (.0 1 0) (.010) 0.057 (.028) 0.030 0.030 (.006) 0.066 0.066 (.009)

0.075 (.035) 0.052 (.046) (.046) 0.080 0.080 (.032) 0.098 0.098 (.068)

a Standard Standard errors in parentheses. All F F values values are are significantly greater than than zero (P < < 0.0002). 0.0002).

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extinction dynamics dynamics and and differ differ in in terms terms of of their their demographic demographic maturity. maturity. The The quesques­ extinction tion then then is is whether whether these these particular particular dynamics dynamics increase increase genetic genetic differentiation differentiation tion among populations populations relative relative to to an an island island model model that that does does not not take take turnover turnover into into among account. account. The The predicted predicted increase increase in in differentiation differentiation in in the the model model arises arises solely solely from from founder effects effects during during recolonization recolonization since since the the effect effect of of extinction extinction is is seen seen as as founder recolonization (Section (Section III.D). III.D). This This also also partly partly explains explains why why the the FST FST values values of of recolonization only two two age age groups groups have have been been used used to to show show whether whether population population turnover turnover inin­ only creases differentiation differentiation (see (see Whitlock, Whitlock, 1992b; 1 992b; Dybdal, Dybdal, 1994; 1 994; McCauley McCauley et et al., ai., creases 1 995). In In these these comparisons, comparisons, the the younger younger group group contains contains the the newly newly founded founded poppop­ 1995). ulations, while all other populations are combined into a single older group since ulations, while all other populations are combined into a single older group since these populations populations are are assumed assumed to to be be approaching approaching equilibrium equilibrium between between migration migration these and genetic genetic drift. drift. Where Where extinction in in natural natural systems systems is stochastic, stochastic, dividing poppop­ and ulations into into two two groups groups may may be be appropriate. In contrast, contrast, our our populations popUlations decline decline ulations appropriate. In become extinct extinct as as succession succession proceeds proceeds (Section (Section III.C). III.C). The The slowly and and eventually eventually become decrease in size size is likely to be due due to a reduction reduction in germination success in in the the decrease germination success presence of of later later successional successional species. This reduction reduction in in germination germination appears to presence species. This appears to have two two effects on these these populations; populations; the the age age distribution distribution shifts shifts toward toward adult adult have effects on individuals (Carlsson, (Carlsson, 1995), 1 995), and be recruited individuals and since since migrants migrants can be recruited only by the cannot compensate the reduction reduction in N N which which growth of of seeds, seeds, migration (m) cannot compensate for for the is expected under model which which assumes assumes that is constant constant at at equilibrium. equilibrium. It It is expected under aa model that Nm N m is thus appears appears that that not only are the oldest populations moving away from demo­ demographic and genetic numbers and mi­ genetic equilibria in our system, the reductions reductions in numbers migration populations. Since gration are are likely likely to to increase increase the the F Fsv among these these populations. Since our our declining declining ST among populations do not fulfill the the criterion criterion for membership membership in a single group group called "older" populations 990; Whitlock, 11992b; 992b; Dybdal, populations (Whitlock (Whitlock and and McCauley, 11990; 11994; 994; McCauley et 995), we divided our populations into three groups, called et ai., al., 11995), young, intermediate, and old, where where our young and intermediate populations populations cor­ correspond to the younger and older populations in the papers cited above. To test whether colonization increases increases differentiation, we compared the F Fsv of ST values of intermediate populations; to test whether the population decline as­ asyoung and intermediate sociated with succession also infl ates levels of differentiation at the metapopu­ inflates metapopulation lation level, level, we we looked looked to to see see whether whether F FST values increased increased in in the the old old popUlations. populations. ST values We We proceeded proceeded as as follows. follows. The The young young group group contains recently founded populations. These These populations still still occupy occupy the the centers centers of of the the islands islands and and are are less less than than 30 30 years years old. old. The The inter­ intermediate group contains the populations which were well into the expansion expansion phase. These These had had the thick thick ring form and and were were between between 30 and and 250 years years old. The old populations populations were were decreasing in size, had had the the broken ring ring form, and and were were older older than than 250 years. The average average census sizes of of popUlations populations in the young, interme­ intermediate, and 3 000, and 300 individuals, respectively. The and old groups were 670, 670, 113000, and 11300 classifi cation of classification of each each population population into into an an age age group group is is given as as "stage class" class" in in the the Appendix. We We then then calculated the the F F statistics for for each each group group separately separately and and tested tested whether FST of whether the the Fsv of young young populations populations was was larger larger than than that that of of intermediate intermediate pop­ populations, ulations, and and whether whether the the F Fsv of intermediate intermediate populations populations was was smaller smaller than than that that ST of

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of method, in of old populations. populations. We We tested tested our our hypotheses hypotheses using using a Monte Monte Carlo Carlo method, which partitions partitions of of the total total sample sample into into three three classes were generated at random. random. which were generated Each partition was made up 1 2, and Each three-class three-class partition was made up of of 30, 12, and 9 islands, islands, sampled sampled without without groups were were calculated repli­ replacement. replacement. Differences Differences in FST between between groups calculated for for 5000 5000 repliI ) FST,y FST,y = cates. cates. The The distributions distributions of of these these differences differences are are our our null null hypotheses; hypotheses; ((1) = FST,i (2) FST,i which we we tested FsT, i,, and and (2) FsT.i = = FST,o FsT,o,, which tested against against the the alternative alternative hypotheses; hypotheses; ((1) 1 ) FST FST,y > FST,i FST,i,, and and (2) (2) FST,i < < FST,o (y, (y, i,i, 0 o = = young, young, intermediate, intermediate, and and old, old, ,y > respectively). respectively). All F intermediate and F statistics estimated for for the the young, young, intermediate and old old age age classes classes are are among young significantly greater than than zero zero (Table (Table Ib). FST is high among young islands islands (0.057), decreases decreases to 0.030 0.030 in the intermediate intermediate age age class, and and reaches reaches its highest highest (0.057), value of of 0.066 0.066 among among the the old old islands. islands. The The FST of of the the young young populations populations was was (P < which significantly larger larger than than that that of of the intermediate intermediate populations populations (P < 0.05), 0.05), which suggests that that colonization colonization increases increases differentiation differentiation (Whitlock (Whitlock and McCauley, McCauley, 11990). 990). Continued migration among populations also appears to contribute Continued migration among populations also appears contribute to the the ST observed intermediate class. This This inference inference is supported decrease decrease in F FST observed in the intermediate supported by noting noting that that the means means and and standard standard deviations of of the numbers numbers of of alleles alleles ob­ observed served at all loci (maximum 20) in the the young, intermediate, intermediate, and and old old groups groups were were 114.33 4.33 ± 8.56 ± 5 .25 ± ___ 3.39, 3.39, 118.56 __ 0.73, 0.73, and and 115.25 ___ 2.05, 2.05, respectively. The The FST FsT for for the intermediate cantly greater intermediate class class remains remains signifi significantly greater than zero, suggesting that that gene fl ow is not popUlations into a single evolutionary evolutionary flow not high high enough enough to link these these populations unit. In the second model, the second test, which which is not not part of of the the model, the FST for for the old class class was significantly significantly higher higher than than that that for the the intermediate intermediate class (P (P < < 0.04). We was 0.04). We strongly suspect that suspect that that this increase increase is associated associated with extinction extinction processes, processes, and and that in our both founding our metapopulation metapopulation system, both founding events events and population population decay may island model at equilibrium. increase levels levels of of differentiation differentiation relative relative to an island equilibrium. We We know been reported, remains know of of no no other other study in which which such such an increase increase has has been reported, and and it remains to be seen whether metapopulation systems whether these results will be found found for for other other metapopulation with similar similar extinction extinction dynamics. However, However, in studies studies where where the the aim is to see whether which whether colonization colonization increases increases differentiation differentiation using the FST relationships relationships which follow - McCauley model, it may be wise to check follow from from the the Whitlock Whitlock-McCauley check whether whether FST values for for old or decaying decaying popUlations populations are are exceptionally exceptionally high. high. If If they are, group of assumed to these populations populations should should be removed from from the group of populations populations assumed be approaching their inclusion inclusion could could obscure approaching equilibrium equilibrium since since their obscure the the expected expected de­ decrease between between younger younger and and older older populations. populations.

D. Genetic Genetic Structure within Islands While While we have have demonstrated demonstrated that that extinction/colonization extinction/colonization processes processes increase increase the the genetic genetic variance variance among among island island populations, populations, this this is is not not the the entire entire story. story. The The within-island obtained in this study are large and within-island and and overall values values of of Fis Fis obtained and sig­ significantly greater implications of greater than than zero zero (Appendix (Appendix and and Table Table I). One One of of the implications of these these significant significant Fis values values is that that the the island populations populations themselves themselves consist consist of of more more than than one one random random mating mating unit. To To test this, a pilot study was was carried carried out out on

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island 23 ((Fig. Fig. 11,, Appendix), in which all S. dioica dioica individuals within six 50 cm22 patches, separated by 11.5 .5 to 35 m, were surveyed electrophoretically. electrophoretically. Fis values FST values cal­ for all patches patches were were not statistically different from zero, zero, but all Fsacalculated culated for for different combinations of patches were were significantly greater greater than than zero. Converting the distance distance between between the two nearest patches patches in which FST was sig­ significant nificant into area revealed that that random mating mating units could be as small as 0.2 0.2-E. Lundqvist, J. Goudet, and B. E. Giles, unpublished). Some of of this within­ within6 m22 ((E. island differentiation differentiation probably results from restrictions to gene flow. Seed dis­ dispersal by gravity may lead to the establishment establishment of groups of of offspring near near the mother mother plant plant which are likely to be at least half-sibs. half-sibs. Nearest-neighbor Nearest-neighbor pollination has also been observed (Carlsson, 11995), 995), and the compounding of of these two processes over a number number of of generations could lead to an apparently continuous population consisting of of a patchwork patchwork of of differentiated differentiated family groups (see Turner Turner et al., 11982). 982). Spatial subdivision, however, does does not not appear to be the the only dynamic af­ affecting within-island within-island population structure. Because Because an island is rising all the time, the middle is always older than than the outer outer parts. parts. With With the exception of of shoreline species, each successional successional stage invades the middle first and moves outward. Close observation of of any single patch of of ground on an island reveals reveals that the plant species colonizing it go through a series of of successional successional and demographic demographic changes similar similar to those described at the island level (Section III.C). Ill.C). A patch is first colonized by S. dioica first dioica when conditions are right. fight. Initially, we observe that species may still share a few flowering adults, few if any seedlings, and shoreline species the habitat. When When seedlings begin to be produced within the patch, the numbers and densities of of the S. dioica dioica "population" "population" increase, plants plants at all life-history stages dioica is the most common plant. As 1. J. communis, communis, V. myrtillus, myrtillus, are present, and S. dioica B. pendula, pendula, and P. abies abies invade, fewer fewer and fewer S. dioica dioica seedlings are are observed, the number of of adults decreases, decreases, and S. dioica dioica eventually disappears disappears from the patch patch (Carlsson, 11995; changes associated with patch patch 995; B. E. Giles, unpublished). These changes colonization and extinction suggest that each island population is itself an age­ agestructured metapopulation. metapopulation. In a recent study of three islands (23, 35, and 39; Fig. dioica from patches of of different different ages, the FST values 11)) in which we collected S. dioica from young, intermediate, intermediate, and old patches patches within an island followed followed the same between young, intermediate, intermediate, and old islands (Giles and and Goudet, trend as those between 11996b). 996b). Clearly a new model will be needed to investigate how the dynamics occurring occurring at these two nested spatial and temporal temporal scales could affect affect each other other and the system as a whole. Nonetheless, fi nding that the degree of finding of demographic maturity of of mating units at two spatial and temporal scales has a consistent effect on the degree phe­ degree of genetic differentiation differentiation among those units suggests that that this phenomenon may be general.

E. E. The Effects Effects of Distance Distance on Genetic Genetic Differentiation among among Islands Islands If an organism's organism's potential for migration is limited limited relative to the the typical isolation of the habitat habitat patches patches it occupies, occupies, the arrangement arrangement and distances distances among

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Barbaro E.E. Giles and J6rSme Jerome Goudet Barbara Giles and Goudet

patches can affect population population differentiation differentiation even even in the absence absence of of turnover. turnover. We patches We next looked looked for evidence evidence of of isolation isolation by distance distance which has been shown to inin­ Whitlock, 11992a; 992a; crease the the degree of of differentiation differentiation among among populations populations (e.g., Whitlock, Goudet, 993; Slatkin, 993). Since Goudet, 11993; Slatkin, 11993). Since population population turnover turnover and and spatial spatial variance variance in migration migration have have not not been been jointly jointly investigated investigated in in aa single single model, model, it is difficult difficult to to know how how to disentangle disentangle their their effects. To To reduce reduce the confounding confounding effects of of age age know and possible, and concentrate concentrate on on the the effects effects of of spatial spatial variance variance in migration migration as as much much as as possible, we islands which, the we restricted restricted our our analyses analyses to to specific specific groups groups of of neighboring neighboring islands which, on on the basis of flow, we we believed believed could could not) basis of information information about about vectors vectors of of gene gene flow, could (or (or could not) exchange genes. exchange genes. Age Age was was ignored ignored in in these these analyses. analyses. From From the the time time of of seed seed release and and onward onward into into the the autumn, autumn, the the prevailing prevailing winds blow from the southwest, storms blowing from the southeast are winds blow from the southwest, storms blowing from the southeast are common, common, and cycle in and water water moves moves into into the the archipelago archipelago in in the the course course of of the the annual annual water water cycle in the 98 1 ). The forces will move seeds the Baltic Baltic (Ericson, (Ericson, 11981). The joint joint action action of of these these forces will be be to to move seeds from the south south to the north in the archipelago. archipelago. To To investigate the spatial patterns patterns of ow via seeds, we constructed of gene gene fl flow constructed three three "chains" "chains" of of islands, called called the outer, outer, middle, and -north direction middle, and inner inner chains, chains, which which are are oriented oriented in in aa south south-north direction (A,B,C, (A,B,C, ). Isolation three chains. respectively, respectively, Fig. Fig. 11). Isolation by by distance distance was was expected expected in in all all three chains. We We also also looked looked for for isolation isolation by by distance distance where where wind wind and and water water are are not likely agents east-west transects. Since bumblebee agents of of seed flow by setting setting up two two east-west bumblebee movement frequent in the inner inner movement occurs occurs among among islands islands and appears appears to be more more frequent (unshaded (unshaded part part in in Fig. Fig. 11)) than than in the outer outer archipelago archipelago (L. (L. Ericson, personal personal communication), isolation by unexpected in communication), the the occurrence occurrence of of isolation by distance distance is is not not unexpected in the the north D, Fig. 1). 1 ) . The second E, Fig. 1), 1 ), was set north transect transect ((D, second transect, transect, south south transect transect ((E, up in the most exposed exposed part part of of the archipelago archipelago as a contrast contrast to the other other four four linked by groups. Since Since we we suspect suspect that that south south transect transect islands islands are are not not directly directly linked by wind and expect to wind and water, water, we we do do not not expect to detect detect isolation isolation by by distance. distance. We We looked looked for for evidence evidence of of isolation by by distance distance within within these these chains chains and and transects using Mantel Mantel, 11967) 967) of transects using Mantel tests tests ((Mantel, of the the correlations correlations between between the the ma­ matrices physical distances. cant trices of of pairwise pairwise genetic genetic (based (based on on FST) FST) and and physical distances. A A signifi significant correlation correlation is taken taken as evidence of of isolation isolation by distance. distance. Since there there is little in­ information ST and is appropriate, formation as as to to what what type type of of transformation transformation of of F FST and distance distance is appropriate, we used three: 1 ) untransformed vs untransformed physical distances, we used three: ((1) untransformed FST FST vs untransformed physical distances, (2) (2) Slatkin' 1 993) In[ /4( l /FsT )] (the in an island Slatkin' ss ((1993) In[ l1/4(1/FsT -- 11)] (the log log of of the the number number of of migrants migrants in an island model) vs vs log log physical physical distances, distances, and vs ranked physical distances. model) and (3) (3) ranked ranked FST FST vs ranked physical distances. Table II shows cients (R) between two matrices matrices Table II shows that that the the correlation correlation coeffi coefficients between the the two were north transect were high, high, 0.40 0.40 or or greater, greater, for for the the outer outer and and inner inner chains chains and and the the north transect for cients for for for the the untransformed untransformed and and ranked ranked data. data. In In contrast, contrast, the the correlation correlation coeffi coefficients the middle the south were very our predictions the middle chain chain and and the south transect transect were very low. low. Thus, Thus, our predictions are are supported supported by these these tests for the outer outer chain, chain, the southern southern transect, transect, and marginally for for the the inner inner chain chain and and the the northern northern transect, transect, but but not not for for the the middle middle chain. chain. These These results results suggest suggest that that genetic genetic differentiation differentiation among among islands islands in in the the Skeppsvik Skeppsvik Ar­ Archipelago is inflated due to chipelago is inflated relative relative to to an an island island model model due to the the dispersion dispersion of of the the islands, islands, although although it it is as as yet yet impossible impossible to to estimate estimate the the relative relative contributions contributions of of

Genetic Plant Metapopulation GeneticStructure Structure inin aa Plant Metapopulation

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445 44S

Mantel in Groups Groups of Mantel Tests Tests of of Isolation Isolation by by Distance Distance in of Islands Islands Formed Formed on on the the Basis Basis of of Hypotheses Hypotheses about about Vectors Vectors of of Gene Gene Flowa Flow~

TABLE II II

Group Group

N N

Inside chain Middle chain Outer chain North transect South transect (b) Inside chain Middle chain Outer chain North transect South transect (c) Inside chain Middle chain Outer chain North transect South transect

7 9 I11I 6 5 7 9 1111 6 5 7 9 I11I 6 5

(a)

R

0.4669 0.0767 0.4296 0.5629 - 0.5034 0.3947 -0.3947 - 0.2548 0.3289 -0.3289 - 0.2275 0.2922 0.3961 0 . 1 68 1 0.1681 0.3980 0.5673 - 0.3697

P P

sdr

Threshold Threshold

0.01 0.0199 0.327 0.008 0.054 0.932 0.052 0. 1 28 0.128 0.021 0. 1 96 0.196 0.759 0.057 0.268 0.0 11 0.011 0.024 0.868

0.26 0.27 0 .16 0.16 0.29 0.36 0.24 0.21 0 .14 0.14 0.27 0.34 0.24 0.25 0 .15 0.15 0.29 0.33

0.01 25 0.0125 0.01

0.01 0.01

0.01

a

were used: ST against ~,Three transformations of pairwise genetic and physical distances were used: (a) F Fsx against distance, 1/4 FST (b) In ((1/4 Fsv - 1/4) 1/4) against against In distances, distances, (c) ranked FST Fsx against ranked distances. Threshold specifies sequential Bonferroni level of significance significance required for rejection of the null null hypothesis. specifies the sequential N, R, P, and sdr are sample size, size, correlation coefficient, coefficient, probability, and standard deviation, respectively. -

temporal levels of temporal and and spatial spatial heterogeneity heterogeneity to to the the total total levels of differentiation differentiation in in this this metapopulation. metapopulation. The The occurrence occurrence of of isolation isolation by by distance distance in in this this archipelago, archipelago, how­ however, to ever, also also allows allows us us to to examine examine another another factor factor which which is is critical critical for for the the degree degree to which which colonization colonization and and extinction extinction dynamics dynamics will will increase increase differentiation, differentiation, namely, namely, the Whitlock the parameter parameter � ~b and and the the degree degree of of propagule propagule mixing mixing at at colonization colonization (( W hitlock and McCauley, 990; McCauley, 99 1 ). and M c C a u l e y , 11990; McCauley, 11991).

F. Numbers Sources of Colonists Numbers and Sources Colonists The depend not not only typical The genetic genetic consequences consequences of of founding founding events events depend only on on the the typical numbers individuals involved involved in numbers of of individuals in a a colonization colonization (k), (k), but but also also on on the the nnumber u m b e r of of sources sources (�) (~b) from from which which the the individuals individuals entering entering empty empty patches patches are are drawn drawn (see (see Eq. 977; Wade 1 988; W Whitlock McCauley, Eq. (2); (2); Slatkin, Slatkin, 11977; W a d e and and McCauley, M c C a u l e y , 1988; h i t l o c k and and McCauley, 11990; 990; McCauley 995). The M c C a u l e y et et al., al., 11995). The observation observation that that groups groups of of seedlings seedlings are are found found on 10 individuals representing both sexes have have arrived on an an island island when when 5 5 to to 10 individuals representing both sexes arrived and and flowered first attempt attempt to to find find out out the the flowered simultaneously simultaneously suggests suggests that that k k is is small. small. As As aa first probability origin of the colonists, have looked looked for isolation by by probability of of common c o m m o n origin of the colonists, we we have for isolation distance groups. Isolation young popupopu­ distance within within age age groups. Isolation by by distance distance detected detected among a m o n g young lations, or populations, will be taken lations, or among a m o n g both both young young and and intermediate intermediate populations, will be taken as as evidence that propagules come come from small number of nearby islands evidence that propagules from one one or or a a small n u m b e r of nearby islands

Barbara E. Giles Giles and J6rOme Jerome Goudet Gaudet Barbara

446 446

(high ( high ~b) 4» and and aa propagule propagule pool pool model model will will best best describe describe the the archipelago. archipelago. If, If, howhow­ ever, isolation isolation by by distance distance is is detected detected among among intermediate intermediate but but not not among among young young ever, populations, it is is more more likely likely that that colonizers colonizers come come from from aa wide wide range range of of source source populations, populations rather rather than than neighboring neighboring ones, ones, and and aa migrant migrant pool pool model model will will be be inin­ populations ferred. We We carried out Mantel Mantel tests tests of of the the correlations correlations between between the the genetic genetic and and ferred. carried out distances of of the the young young and and intermediate intermediate island island populations. populations. As As above, above, aa physical distances significant correlation correlation between between the the two two matrices matrices was was taken taken as as evidence evidence of of isolation isolation significant by distance. distance. No No effect of distance distance was was detected any of of the the tests tests for the young young by effect of detected in any for the popUlations, but but all three Mantel tests tests were were highly highly significant significant for islands in the the populations, three Mantel for islands intermediate age class (Table (Table III). These These results suggest that that colonizers colonizers are are aa intermediate mixture from from several several locations in the the archipelago archipelago and and that that the the migrant migrant pool pool model model mixture an appropriate appropriate description description of of founding founding events. is an Finding isolation isolation by distance distance among islands in the the intermediate intermediate age age class class Finding among islands raises some some interesting interesting problems problems and questions. According According to existing existing genetic genetic raises and questions. metapopulation models, postcolonization migration among populations populations is supsup­ metapopulation posed to follow Finding isolation isolation by distance distance among interme­ posed follow an island model. Finding among intermemeans that their degree degree of diate-age islands in Skeppsvik Archipelago means of genetic differentiation these distance means that this this differentiation will be inflated by these distance effects. It also means metapopulation system pool colonization subsequent dis­ metapopulation system combines combines migrant pool colonization with subsequent distance-dependent differentiation is lower migrant tance-dependent migration. Since the the degree of of differentiation lower in migrant than in propagule propagule pool colonization, colonization, and and distance-dependent migration inin­ pool than distance-dependent migration differentiation relative to an island island model, the first problem problem is that that this flates differentiation of colonization and and migration modes modes is a "worst-case scenario" scenario" for for combination of detecting overall increases increases in differentiation differentiation arising arising from colonization. colonization. Too few few detecting studies of of metapopulation metapopulation systems have been carried carried out to know whether whether this combination FST of combination is a common common one. In our case, the FsT of intermediate populations populations was still significantly lower lower than than that that of of the young populations populations (Table Ib) as predicted by the model (which also strengthens strengthens our belief that the conditions conditions specified in Eq. (2) are are met in this metapopulation). metapopulation). It is, however, possible that these expected ST with age may not be detectable in other systems expected changes changes in F FsT showing showing a similar combination of of colonization colonization and migration migration patterns. Further Further

TABLE III III Mantel Mantel Tests Tests for for Isolation Isolation by by Distance Distance among among Islands Islands in in Young Young and and Intermediate Intermediate Age Age (Iassesa Classes"

(a) (b) (c)

a

Group

N

R

P P

sdr

Threshold

Young Intermediate Young Intermediate Intermediate Young Intermediate Intermediate

113 3 30 113 3 30 113 3 30

0.0561 -0.0561 0.3662 0.00 1 1 -0.0011 - 0.323 0.32311 0.0032 -0.0032 0.3298 0.3298

0.405 0.405 0.002 0.002 0.43 0.00 0.00 0.461 0.461 0.002 0.002

0.207 0.207 0. 1 18 0.118 0. 1 59 0.159 0.093 0.093 0. 161 0.161 0. 1 06 0.106

0.05 0.05 0.025 0.025 0.05 0.05 0.025 0.025 0.05 0.05 0.025 0.025

a See Table Table 2 legend for description of transformations (al, (a), (b), (c) and column column entries.

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development of of isolation by of the theoretical theoretical framework to include the effect of distance, distance, in addition to the effect of turnover turnover dynamics is clearly required. A second question is whether whether colonization and migration are really different different phenomena. In plants at least, the answer answer is probably yes, since migrants come in two forms, as diploid seeds and haploid pollen. Colonization can only occur through seed migration, migration, while migration into and and out of of established established populations may occur as seeds and pollen. While we do not know which form of of migration is more responsible for the gene flow which reduces the genetic differentiation differentiation among maturing popUlations, populations, we strongly suspect that gene flow via pollen be­ becomes more prevalent as populations age. As the islands age and rise, seed dep­ deposition even on the shores becomes more diffi cult. In addition to the large difficult. large pro­ proportion of flowers from uninfected M. violaceum uninfected populations which bore M. violaceum spore "pollen-markers" "pollen-markers" (see Section III.D), other evidence suggests that the potential potential Agren, 1996; 1 996; Handel, 11983; 983; for pollen carryover increases with population size ((,~gren, Fritz and Nilsson, 11994). 994). At At this stage of of the study, we can only speculate. speculate. The conclusive evidence will come from a study using markers markers which enable enable gene flow via pollen to be distinguished from that via seed. This can be done by comparing the patterns of gene flow inferred from maternally inherited organelle organelle markers markers (e.g., mtDNA mtDNA or cpDNA) with the patterns inferred from biparentally inherited nuclear nuclear markers will nuclear markers (e.g., allozymes), since only the nuclear be carried from one population to another Ennos, 1994; 1 994; McCauley, another as pollen pollen ((Ennos, 11994). 994). FST bladder campion, Fsv values of of newly founded populations populations of the white bladder Silene Silene alba, alba, have been found to be higher than than those of of older populations be­ belonging to the same metapopulation ((McCauley McCauley et 995), and a comparison et al. al.,, 11995), of estimates from allozymes and cpDNA suggested that the rate of of gene flow in nuclear nuclear genes was greater than the effective rate of organellar gene flow in this McCauley, 11994). 994). species ((McCauley,

G. Environmental Environmental Heterogeneity Heterogeneity Our studies have also revealed revealed that that the degree of of genetic differentiation is influenced by environmental factors. The values of Fjs , which describe of Fis, describe the degree degree of of structuring structuring within single islands (Appendix), (Appendix), were not only significantly greater greater than zero, they were also highly heterogeneous and ranged from from - 00. . 1188 to 0.38. It is therefore of of interest to know whether whether particular particular factors associated with high levels of within-island ed. Successional changes occur within-island structuring can be identifi identified. more more quickly in the protected protected than than in the exposed parts parts of of the archipelago. archipelago. Ex­ Exposure is likely to increase increase both the difficulty with which migrants (of any plant species) are deposited on islands and the degree of local patch destruction by salty waves and ice-blocks blown onto the islands during storms. We thus sussus­ pected that exposure could increase Fj" Fi~, but we could not ignore population age because of of the differences differences in rates of succession. We therefore therefore used age (stage class, Appendix) Appendix) and exposure exposure as factors in a two-way analysis of of variance on -

448 448

Barbaro and Jerome Barbara E.E. Giles Giles ond J6rOmeGoudet Goudet TABLE IV IV Two-Way Two-Way Analysis Analysis of of Variance Variance of of Ranked Ranked Within-Island Within-Island f;, F~ Values Values for for Exposure Exposure and and Stage Stage Classes Classes(Appendix) (Appendix) Source Source

DF DF

Exposure Exposure Age Age Interaction Interaction Residual Residual

2 2 46

1

MS MS

F F

P P

11819 819 284.3 284.3 2 [7.9 217.9 1175.5 75.5

110.36 0.36 11.62 .62 11.24 .24

0.002 0.002 0.209 0.299

the ranked ranked Fis Fis values values for the the islands. Islands Islands were grouped grouped into two two exposure classes, classes, protected protected and exposed, according to their position in the archipelago archipelago relative to the shaded and unshaded Appendix). unshaded sections of of Fig. 11 (see exposure, Appendix). The islands The shading shading marks marks the the boundary boundary between between those those islands islands in in the the lee lee of of other other islands (un shaded) and those that (unshaded) that are are directly exposed exposed to wind and and wave actions actions by autumn autumn storms (shaded) (shaded) and the area which is frozen solid during the winter (unshaded) (shaded) (Ericson, (unshaded) and and the the area area subject subject to to pack pack ice ice driven driven by by winds winds (shaded) (Ericson, 11981). 98 1 ). Table cant (P < Table IV indicates that exposure is signifi significant < 0.05), but age and and the interaction between age Fis values values interaction between age and and exposure exposure are are not. not. With With few few exceptions, exceptions, Fis for We have con­ for islands islands in the exposed exposed area area are are greater greater than than 0.09 (Appendix). (Appendix). We confi rmed this firmed this effect effect in in our our recent recent study study of of genetic genetic structure structure within within three three of of the the islands islands 996b); when (Giles and and Goudet, Goudet, 11996b); when islands islands were were ranked ranked according to to their their degree degree of exposure, within-island within-island FST Fsv values increased increased with exposure. We strongly strongly sus­ susof pect that local local patch destruction by wave and and ice action followed by recoloniza­ recolonization increases increases the genetic differentiation differentiation over and and above that induced by the suc­ successional dynamics within individual exposed islands, but this is currently currently a working hypothesis.

H. Conclusion Conclusion The populations of The results of of our study of of isozyme isozyme markers markers in 52 island populations of S. dioica popula­ dioica in Skeppsvik Archipelago Archipelago show show that migration cannot bind bind these these populations populations are tions into into aa single single evolutionary evolutionary unit and and that that each each of of the the island island populations are independent evolutionary likely to to have have independent evolutionary trajectories. trajectories. Furthermore, overall overall levels levels of inflated relative of genetic genetic differentiation differentiation in in this metapopulation will will be be inflated relative to an an island island model model at at equilibrium. equilibrium. In our our case, case, the the founder founder events events dealt dealt with with in in the the genetic 1 977; Wade 1 988; Whit­ genetic metapopulation metapopulation models models (Slatkin, (Slatkin, 1977; Wade and and McCauley, McCauley, 1988; Whitlock 1 990) do inflation lock and and McCauley, 1990) do not not appear appear to be the the only only cause cause of of this this inflation since we detected detected additional additional affects affects arising arising from other other temporal, spatial spatial and and en­ environmental rmed model predictions vironmental factors. factors. Our Our study study has has both confi confirmed predictions and and shed shed light on other areas areas which need further further theoretical theoretical and empirical development. We populations were more We found that that newly colonized populations more differentiated differentiated than than those nearer nearer their their demographic demographic and and genetic genetic equilibria, implying implying that that founder events events

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could increase differentiation at the metapopulation level as predicted predicted by the WhitlockMcCauley model. However, we also observed in­ Whitlock-McCauley observed that differentiation differentiation increased among the oldest populations undergoing demographic populations undergoing demographic and genetic changes as they approached approached extinction by succession. While these "old" popu­ populations will contribute contribute to the total increase increase in levels of of differentiation measured measured at a metapopulation metapopulation level, it is not yet known known what kind of of long-term influences these old populations populations could exert on the system (e.g., on the maintenance maintenance of of genetic genetic diversity) since the relative contributions contributions of of old, intermediate, and and young populations to migrant and colonizer colonizer pools are unknown. unknown. It will be interesting to see whether whether these results will be found found for for other other metapopulation metapopulation systems with similar successionally driven extinction dynamics. A spatial factor factor known to in­ increase differentiation, namely isolation by distance, a form of of spatial variance variance in migration rate 992a; Goudet, 1 993; Slatkin, 11993), 993), was also detected rate (Whitlock, 11992a; Goudet, 1993; in Skeppsvik Archipelago neighboring islands and Archipelago among several small groups groups of of neighboring and of the intermediate age group. For field among the islands of For the past 30 years, fi eld population consequences of subdi­ population geneticists have been focusing focusing on the consequences of spatial subdivision on popUlation population differentiation. differentiation. Our Our study indicates that spatial and and temporal temporal factors interact interact in a manner manner which may be complex, and the few few other other studies 992a,b; which account account for for temporal temporal as well as spatial effects (e.g., Whitlock, 11992a,b; 1 995) imply that McCauley McCauley et et al. al.,, 1995) that both are are crucial crucial for understanding understanding population population structure. structure. Clearly, the interplay of of these two factors needs to be more more thoroughly investigated in theoretical and empirical studies. Our results showing that the degree to which habitats are exposed to storm driven wind, water, and ice infl u­ influences the degree degree of of genetic differentiation differentiation also suggests that more attention should should be paid to environmental factors likely to affect the dynamics of ow (see of gene fl flow Whitlock, 11992a). 992a).

V. V. OTHER OTHERGENETIC GENETICMETAPOPULATION METAPOPULATIONSTUDIES STUDIES Studies of of single species metapopulation dynamics involving plants are rare rare (Silvertown, 1 99 1 ; Husband 995), as are empirical studies designed (Silvertown, 1991; Husband and Barrett, 11995), designed to test the predictions predictions of of genetic metapopulation models models regardless of of organism. To our knowledge, knowledge, there are are three such genetic studies studies in addition to our our own own (Giles and 1 996a). and Goudet, Goudet, 1996a). Bolitotherus cornutus, Whitlock l 992b) studied the Whitlock ((1992b) the forked-fungus forked-fungus beetle, beetle, Bolitotherus cornutus, which lives on patchily distributed fruiting fruiting bodies of of several species of of polypore bracket fungi growing on dead and dying trees. Whitlock was able to estimate migration and extinction rates, numbers numbers of of colonizers, and their probabilities probabilities of of common -recap­ common origin using a combination combination of of methods methods based based mainly on mark mark-recapture. He also created new habitat patches to estimate colonization rate and numnum­ bers and resurveyed habitat patches known to be occupied 10 1 0 years prior patches known prior to his specified by study. These These experiments enabled enabled him to show that the conditions specified Eq. (2) were met. To confirm confirm that these conditions conditions led to increased genetic dif-

450 450

Barbara Barbara E.E. Giles Giles and and Jerome J6rSrne Gaudet Goudet

ferentiation, Whitlock divided the populations populations of of forked forked fungus beetles into young and older groups using the size of of the fungal resource resource as an indicator indicator of of demo­ demographic maturity. An electrophoretic study confirmed that the young populations were more more differentiated than the older populations. populations. Dybdal ((1994) 1 994) studied populations populations of of the bottom-dwelling bottom-dwelling marine marine copepod, copepod, Tigriopus Tigriopus californicus, californicus, which inhabits inhabits intertidal splash-zone splash-zone tidepools on rocky shores shores of of the Pacifi Pacificc coast of of North North America. In contrast to the studies of of Whitlock ((1992b), 1 992b), Giles and 1 996a), and 1 995, see below), he and Goudet Goudet ((1996a), and McCauley et al. ((1995, found populations of found that that the older older populations of T. californicus californicus were were genetically more more differ­ differentiated than the younger populations populations and concluded concluded that extinction/recoloniza­ extinction/recolonization dynamics decreased decreased genetic differentiation in this metapopulation system. Dybdal ((1994), 1 994), suggested that this was due to violations of of two of of the assumptions assumptions of the metapopulation model, namely, not all extant populations populations were equally likely to be sources populations had prob­ sources of of colonists, and and not all extant extant populations had equal probof extinction. In the T. californicus californicus system, the extinction probabilities probabilities abilities of were were related related to the size and and location of of the the tidepools; tidepools; there was a tendency for the older older tidepools to be larger and more isolated than the younger younger pools, which reduce, according to Dybdal, the likelihood likelihood of of older older populations populations acting acting as sources sources of of colonists. In the light of of our our earlier earlier discussions discussions about about the importance of of migrant exchange among local populations, Iso­ populations, we suggest an alternative interpretation. Isolated populations populations are less likely to exchange migrants after colonization, colonization, and therefore some of populations may be more influenced of the older, more isolated populations influenced founder events that the genetic metapopulation metapopulation model suggests are are by the very founder responsible higher FST popula­ responsible for for the higher FsT values normally observed observed among among young young populations. tions. McCauley et al. ((1995) 1 995) have alba, a shorter have worked on S. alba, shorter lived but close Mountain relative to S. dioica, dioica, which which grows grows along along roadsides roadsides in the the vicinity of of Mountain Lake Biological Station in Giles County, Virginia. The dynamics of meta­ of this metapopulation system differ differ in several ways from from the S. dioica dioica system described unin­ above. The S. alba alba populations populations chosen chosen for for study are not separated separated by an uninhabitable matrix. As a consequence, the metapopulation ned as an area metapopulation was defi defined area (25 X • 25 km) and the local populations populations as those individuals inhabiting each 40-m segment of 50 km of of 1150 of roadside roadside contained contained in this area (Antonovics (Antonovics et et al. al.,, 11994). 994). These populations, followed since 11988, 988, are often much populations, which have been been followed 1 994), and the system smaller than ours (see Appendix and Antonovics et al. al.,, 1994), can can be be thought thought of of as as aa large large series series of of small small patches, patches, not not necessarily necessarily very very far far apart, apart, 994; D. E. McCauley, which show high rates of of turnover (Antonovics (Antonovics et al. al.,, 11994; personal personal communication). Extinction Extinction rates rates have been shown shown to be higher higher in small than in large large populations populations and and the extinction-colonization extinction-colonization dynamics dynamics of of the S. alba alba system, which are infl uenced by human activities (e.g., mowing, road mainte­ influenced maintenance), 994; D. E. McCauley, personal nance), are highly stochastic stochastic (Antonovics (Antonovics et al. al.,, 11994; personal 1 995) showed that communication). McCauley et al. ((1995) that the FST Fsv values calculated from both both isozyme and and cpDNA cpDNA data were higher higher for newly colonized populations populations than than those for for established populations. populations. Thus, Thus, even though though there are differences differences in

Genetic Structure Structure in i n aa Plant Plant Metapopulafion Metapopulation 18 1 8 Genetic

4S 1 451

the nature nature and and rates rates of of the the turnover turnover dynamics dynamics in in the the S. S. dioica dioica and and S. S. alba alba metameta­ the populations, the the two two systems show show qualitatively qualitatively similar similar results. results. populations, The degree degree of of genetic genetic differentiation differentiation between between 11 1 1 established established and and 77 newly newly The founded populations populations of of the the perennial perennial cowslip, cowslip, Primula Primula veris, veris, was was studied studied by by AnAn­ founded trobus trobus and and Lack Lack (1994). ( 1 994). While While this this study is concerned concerned with with the the effects effects of of subsub­ division per per se se and and whether whether founder founder effects effects associated associated with with colonization colonization may may lead lead division of genetic genetic variation, variation, the the metapopulation metapopulation concept concept is never never mentioned. mentioned. to a loss of These authors concluded that that there there were were no no differences differences in the the genetic genetic structure structure These authors concluded of new new and and established established populations popUlations because because the the genetic genetic distances distances and and degree degree of of of = differentiation among all the populations (F 0.039) were small and because T differentiation among all the populations (FsT S = 0.039) were small and because new populations populations did did not not possess possess fewer fewer rare or uncommon uncommon alleles alleles than than established established new rare or populations. However, However, Antrobus Antrobus and and Lack Lack (1994) ( 1 994) report FST values values of of 0.046 0.046 for for populations. report FST 0.033 for for established established populations. populations. While While the FST FST values values have have not been been and 0.033 new and tested for for differences differences from from zero zero or or from each each other, other, the the relative relative magnitudes magnitudes of of tested values are with the predictions predictions of of the metapopulation metapopulation model. model. these values are consistent consistent with Metapopulation dynamics dynamics have been been shown to account for changes Metapopulation account for changes in the the relative proportions of hermaphroditic plants plants in patchily distribdistrib­ relative of male sterile and and hermaphroditic of the gynodioecious, perennial perennial shrub, Thymus Thymus vulgaris uted populations of vulgaris (Couvet al. , 1986; 1 986; Belhassen et 989; Olivieri et al., aI., 1990; 1 990; see also Frank, Frank, this et al., et al., 11989; Male sterility is determined determined by cytoplasmic, maternally inherited inherited genes, volume). Male Thymus vulgaris vulgaris lives in but specific nuclear nuclear genes may restore male male fertility. Thymus but areas which are While fire leads leads to the extinction of of areas which are regularly destroyed by fire. While populations, the burned burned sites are recolonization. Newly colocolo­ local populations, are available available for for recolonization. nized and young populations populations contain of male male sterile individuals, individuals, nized contain high proportions of but as the popUlations hermaphrodites increase. but populations age, the the proportions of of hermaphrodites increase. New sites are colonized by seeds from different sources sources and and there is a strong probability that that the founding propagules will contain contain male steriles steriles and hermaphrodites which which contain the matching matching nuclear nuclear genes required to restore the fertility of of the do not contain male steriles in just that patch. Since male sterile plants produce more more seeds (Gouyon and Couvet, 11987), 987), they increase hermaphrodites, increase more more quickly than than hermaphrodites, leading to young populations populations with high proportions of females. With time and migration among populations populations in the form of of pollen, the restorer genes specific to the various male sterile forms eventually appear in the population. Since most pollination occurs within local populations, the matching restorer gene(s) spread(s) through the population and and the proportion of of hermaphrodites hermaphrodites increases as the local populations approach equilibrium. In a similar manner, recurrent extinction and recolonization dynamics have also been shown to affect sex ratio in local populations popUlations of the African butterfly, Acraea Heuch, 11978). 978). Two kinds of females occur in these populations, Acraea encedon encedon ((Heuch, normal females females which produce produce equal numbers numbers of sons and daughters daughters and abnor­ abnor-linked gene causing meiotic drive of the Y mal females (thought to contain a Y Y-linked chromosome) which only produce daughters. Since females are the heterogametic sex, daughters always have the same phenotype as their mothers. Population and eventually species extinction is expected since the numbers of females only pro-

452 452

Barbara E. E. Giles Giles and and J6rSme Jerome Goudet Gaudet Barbara

ducing daughters daughters increase increase in in aa population population with with time. time. Species Species extinction extinction is is not not ducing observed in in nature, nature, even even though though population population extinction extinction has has been been recorded. recorded. Heuch Heuch observed ( 1 978) showed showed that that turnover turnover within within the the metapopulation metapopulation could could explain explain the the perper­ (1978) sistence of of these these butterfly butterfly populations populations by by allowing allowing recolonization recolonization by by normal normal fefe­ sistence males. males. Selection associated associated with with high high rates rates of of population population turnover turnover may may also also account account Selection Carduus pycnocephalus pycnocephalus and and C. for the the maintenance maintenance of of aa seed seed polymorphism polymorphism in in Carduus for tenuifiorus (Olivieri (Olivieri et et al., al., 1983, 1 983, 1990, 1 990, 1995; 1 995; Olivieri Olivieri and and Gouyon, Gouyon, this this volume). volume). tenuiflorus These thistles thistles produce produce two two kinds kinds of of seeds: seeds: those those produced produced in in the the center center of of the the These capitulum, which which are are not not dormant dormant and and have have a pappus pappus allowing allowing wide dispersal, dispersal, capitulum, and the the outer outer seeds, seeds, which which have have no no dispersal dispersal mechanism mechanism but but show show dormancy. dormancy. and The proportion proportion of of seeds producing pappuses pappuses is under under partial partial genetic genetic control control (Oli(Oli­ The seeds producing 1 985). Older populations have been observed to produce higher vieri and Gouyon, vieri and Gouyon, 1985). Older populations have been observed to produce higher proportions of of nondispersed seed, and and it has has been been proposed that there there is selection selection proportions nondispersed seed, proposed that within populations populations against against interpopulation dispersal since since most most dispersing dispersing seeds seeds within interpopulation dispersal 1 985). However, However, their source source populations popUlations (Olivieri (Olivieri and Gouyon, Gouyon, 1985). will be lost to their popUlation extinction and occur frequently frequently and because all new new poppop­ population extinction and colonization colonization occur and because ulations must must be founded founded by migrants, aiding dispersal cannot ulations migrants, traits aiding cannot be lost at at a metapopulation level if the species is to persist. persist. This suggests that these these traits are metapopulation subject different ages. Olivieri subject to different selection pressures pressures in populations populations of of different et al. ((1995) 1 995) and Olivieri and Gouyon Gouyon (this volume) showed that between-deme selection for for high migration migration and and within-deme selection for for low migration could maintain maintain this this seed seed polymorphism. polymorphism. As evidenced by the metapop­ the papers papers written and cited in this volume, the the metapopulation concept is more than just a "rage." This concept concept recognizes and studies the the consequences consequences of of movement movement among among the the spatially spatially separated separated and and smaller-than­ smaller-thaninfi nite populations that typify nature. It also allows us to start to let go of our infinite comfortable view of of nature nature as an immortal, homogeneous, and constant constant entity and to examine the consequences of extinction, fl uctuation, and randomness fluctuation, randomness where it exists. At some level, metapopulation dynamics probably probably influence most species ((Husband Husband and Barrett, 11995). 995). In our case study and and several of the others that we have described, the focus has been on neutral genetic variation. These are important studies. That their results support the predictions of existing models suggest that the concept is not simply an empty fashion, but one that describes nature. For any particular particular species, studies involving neutral neutral characters are are an important fi rst step first step in establishing establishing the spatial and temporal scales at at which turn­ turnover over occurs occurs and and for for coupling the intimate details of of the biology of each species species to to the the magnitude of of the metapopulation effect. It is, however, however, too much to believe that ected in that the the effects of of metapopulation dynamics dynamics we have seen seen so so well refl reflected neutral neutral characters characters will not effect effect the rates rates and and patterns patterns of of evolution (see (see Barton and and Whitlock, Whitlock, this this volume). volume). The The difficult difficult but but necessary necessary theoretical theoretical and and empirical empirical next next step step demands demands that that we we learn learn to to understand understand and and measure measure the the consequences consequences of of this this dynamic dynamic population population structure structure for for adaptation and and speciation. speciation.

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Index

Age, Age, metapopulation effect effect on genetic variability, 440-442 estimation, 434 evolution of age structure at equilibrium, 3313 13

Basic reproduction number calculation for metapopulation, 1100-103, 00-1 03, 1121 21 mathematical derivation, 1113-114, 1 3- 1 14, 118-119 1 1 8- 1 19 steady state assumption, 100, 100, 1113 13 Butterfly metapopulation applications of metapopulation models, 385386 386 colonization, 367-368 conservation management, 359-360, 372-373 genetic structure, 1 -452 structure, 45 451-452 habitat change, population response change, meta metapopulation climatic variation, 379-380 heterogeneity of habitat, 379-38 379-381I loss of habitat, 377, 379

minimum habitat fragments for survival, 88-89, 3371-372 7 1 -372 vegetation alteration, 377 habitat identification, 361-362 incidence function modeling, 369-370 local extinction factors affecting extinction, 365-367, 385 rate, 365 time to extinction, 370, 379 migration area-dependent emigration, 375-376 capacity, 384 data collection, 373-374 estimation, 374-375 patch occupancy mapping, 362-364 population size in isolated patches, calculation, 375 rampant population turnover and and classical dynamics, 70-72 rescue effect, 280-282, 368 spatially explicit modeling, 368-369 spatial spatial synchrony of population fluctuations. fluctuations, 38 I -383 381-383

505 sos

506 506

Index Index

Butterfly metapopulation metapopulation (continued) (continued) Butterfly structure, 360-361 360-361 structure, usefulness in in metapopulation metapopulation studies, studies, 354354usefulness 355, 359 359 355,

Catastrophe Catastrophe examples in in nature, nature, 236 examples extinction scenarios, scenarios, 235 235 extinction genetic variability variability effects, effects, 236-237 236-237 genetic rate modeling, modeling, 235 235 rate Cinnabar moth Cinnabar 389 life history, 389 401-402 persistence mechanisms, 401 -402 ragwort-cinnabar moth-Cotesia moth-Cotesia popularis popularis ragwort-cinnabar metapopulation census data, 390 389-390 data collection, 389-390 extinction rates, 402-403 extinction population dynamics, 396-399 spatially extended dynamics and survival probability, 403-405 Climate, see Weather see Cytoplasmic male sterility CMS, see Colonization metapopulations, 367-368 butterfly metapopulations, capture- recapture estimation of immigration capture-recapture 260-261 interpopulation immigration, 260-261 259-260 single populations, 259-260 248-251 defined, 248-251 local extinction extinction and recolonization, recolonization, 33-34, 33-34, 57, 240 240 modeling bifurcation bifurcation parameter, 105-106 1 05-1 06 combining with local dynamics, 99, 99, 110 1 10 mathematical mathematical derivation, 108-109 1 08- 1 09 redistribution redistribution kernels, 97-98, 97-98, 108-109, 1 08- 1 09, 119 1 19 probability, probability, 81 81 rate rate estimation estimation difficulty, difficulty, 251 25 1 equation, equation, 74, 74, 105 1 05 experimental manipulation manipulation of of populations, populations, 263-264 263-264 gene gene flow estimation, estimation, 253-254 253-254 indirect indirect global global approach, approach, 251-253 25 1 -253 parameters, parameters, 252 252 patch-specific patch-specific estimation, estimation, 264-265 264-265 Community, Community, see see Metacommunity Metacommunity Conservation, Conservation, metapopulation metapopulation modeling modeling butterfly butterfly management, management, 359-360, 359-360, 372-373 372-373 ecosystem ecosystem management, management, 26 26 genetic genetic modeling, modeling, 25-26 25-26 island island biogeography biogeography theory, theory, 17-19 1 7- 1 9

landscape landscape ecology ecology impact, impact, 58-60 58-60 local 17 local extinction extinction parameters, parameters, 2217 minimum minimum viable viable metapopulation metapopulation size, size, 22, 22, 24, 24, 65 65 nonequilibrium nonequilibrium metapopulations metapopulations and and time time to to extinction, extinction, 85-86, 85-86, 88 88 parasitic parasitic impact, impact, 24-25 24-25 refuge refuge design, design, 22-23 rescue effect effect relevance, relevance, 27 2711 rise 9-2 1 rise in in popularity popularity of of models, models, 119-21 species 1 -22 species conforming conforming to Levins Levins model, 221-22 structured versus unstructured models, 1107071108 08 Cotesia Cotesia popularis popularis life history, 389 ragwort-cinnabar moth-Cotesia moth- Cotesia popularis popularis metapopulation census data, data, 390 data collection, 389-390 extinction extinction rates, 402-403 population dynamics, 399-401 spatially extended dynamics and survival probability, 403-405 Cowslip metapopulation, genetic structure anal­ analysis, 45 4511 Cytoplasmic male sterility colonization-extinction colonization-extinction dynamics of alleles, 343-347 cytotypes, cytotypes, 340 defined, 340 varia­ genotype dimensionality and and spatial variation, 341-343, 35 1 tion, 341-343, 351 343-344 natural history of of alleles, 343-344 nature, 340 prevalence in nature,

Demographic stochasticity, stochasticity, modeling modeling of of extincextinc­ Demographic tion tion demographic and and environmental stochasticity demographic stochasticity modeling, 228-231 228-23 1 modeling, diffusion approximation, approximation, 231 23 1 diffusion extinction, 232 time to extinction, Dispersal, see see Migration Migration Dispersal, Dormancy Dormancy evolution, 311-312 3 1 1-312 evolution, ragwort, 396 396 ragwort,

Effective population population size size Effective defined, 166 1 66 defined, estimation, 170, 1 70, 172, 1 72, 179 1 79 estimation, gene flow flow rate rate between between populations, populations, effect effect gene on, 178-180, 1 78-180, 188 188 on,

Index Index modeling of of genetic effects approaches, 167-169 167-169 colonization rate, 169, 172, 1 76, 1186 86 172, 174174-176, extinction rate, 169, 172, 172, 174-176, 86 rate, 169, 174-176, 1186 general pattern pattern of of results, 173 heterozygosity estimation, 170-172 170-172 patch carrying capacity, capacity, 169, 174 number of 80 of founders, effects on, 176, 176, 1180 simulation simulation of of effects on heterozygosity, 166167 subdivision, 87-1 88 subdivision, effects on, 1187-188 subpopulation subpopulation number, effects on, 177 variance, 1187-188 87- 188 Emigration butterfly metapopulations metapopulations area-dependent area-dependent emigration, 375-376 375-376 data data collection, 373-374 373-374 estimation, 374-375 374-375 defined, defined, 248-249 248-249 rate estimation capture-recapture capture-recapture estimation between populations, 260-261 260-261 experimental manipulation manipulation of populations, 262 patch-specific patch-specific estimation, 264-265 Environmental stochasticity defined, 2 1 8, 355 218, modeling of extinction autocorrelation, 232-233 232-233 ceiling adjustment, adjustment, 234-235 234-235 ceiling model, model, 220-223, 220-223, 225-227, 225-227, 242 demographic and environmental stochasticity modeling, 228-23 228-2311 diffusion approximation, approximation, 220, 228 extinction parameter estimation estimation from time series data, 223, 225 logistic density dependence, 233-234 233-234 time to extinction, 221-222, 221-222, 234 modes, 19 modes, 2219 regional stochasticity and correlated environ­ environmental fl uctuations, 238-240 fluctuations, species-area species-area curve dependence, 226-227 226-227 Evolution, metapopulations metapopulations biological traits, see see Metapopulation effect deleterious mutations fixation probability, 9 1 - 193 probability, 1191-193 manifestations, 191 manifestations, 191 spatial subdivision effects, 192-193 favorable allele establishment extinction effects, 190 fixation probability, 1189-190 89-190 mechanisms, 84 mechanisms, 1184 rate of 19 I of spread, spread, 191

S07 $01

stages, 1189 89 gene mutation 191 mutation rate, rate, 190190-191 genetic variation maintenance, maintenance, 208-209 208-209 limitations gene flow, 197-1 98, 200 197-198, population 99 population density, 198-1 198-199 local adaptation adaptation dispersal range of genes, 195 extinction 96 extinction effects, 195-I 195-196 gene flow rate, 193, 195 migration effects, 196 neutral variation effective population size variation, 1187871188 88 extinction/recolonization 187 extinction/recolonization models, 185185-187 migration effects, 186-187 186-187 stable local population 85 population models, 1185 shifting shifting balance balance model adaptive peaks, peaks, 201-202 201-202 extinction effects, 206-209 206-209 hybrid hybrid zone, 207-208 207-208 migration migration effects, 202-204, 202-204, 206-209 206-209 peak shift modeling, 202-204, 202-204, 206 phases, 201 speciation, 201 Extinction, see see Local extinction

Forked-fungus beetle metapopulation, metapopulation, genetic structure analysis, 449-450 449-450 F ST, see FST, see Genetic variability Genetic Host-parasite geGenetic variability, see see also also Host-parasite netics adequacy 10 adequacy in population modeling, 2210 calculation, 1171, 7 1 , 1185, 85, 4 3I 431 effects on F ST Fsx catastrophe, 236-237 colonization rate, 1169, 69, 172, 172, 174-176, 86 174-176, 1186 effective population population size, 166-167 166-167 extinction rate, 169, 172, 172, 1174-176, 74-176, 1186 86 rate, 169, gene flow rate, 178 habitat fragmentation, 65-66, 1181 81 number ooff founders, 176, 176, 180 patch 169, 174 patch carrying capacity, 169, subpopulation subpopulation number, 177 local extinction dependence, 169, 1172, 72, 1741176, 76, 1186 86 maintenance, maintenance, 208-209 208-209 measurement 165, 170measurement of heterozygosity, 165, 172 migration rate estimation, 28 1 , 287-288, 433 281,287-288,

508

Index Index

variability (continued) (continued) Genetic variability plant metapopulations effects on genetic differentiation, distance, effects 443-445 of environmental environmental heterogeneity, heterogeneity, effect of 447-448 effect of population age, 440-442 founder effects, 445-449 interisland genetic structure, 440 structure, 442-443 intraisland genetic structure, statistical analysis, 439-440

see also also Patch Habitat, see classification, 115 5 destruction 144, 146 effects on migration, 144, 143-145 multispecies models, 1126, 26, 1431 45 fragmentation effects genetic variability of population, 65-66 metapopulation dynamics, 296-297 minimum fragments for butterfly survival, 88-89 Host--parasite parasite genetics Host bacteria-phage bacteria -phage interactions, 348-349 cytoplasmic male sterility as host-parasite model colonization-extinction dynamics of alleles, 343-347 cytotypes, 340 defined, 340 defined, genotype dimensionality and spatial variavaria­ 3511 tion, 341-343, 341 -343, 35 natural history of of alleles, 343-344 prevalence in nature, 340 factors influencing spatial spatial variation, 350-351 350-35 1 genotype dimensionality and colonizationextinction dynamics complexity of natural dynamics, 330-332 four host-parasite pairs, pairs, 329-330 migration effects, 330 single-patch model, 327-330 two two host-parasite host-parasite pairs, pairs, 328-329 328-329 genotype variability, 325-327 insect insect herbivores, 349-350 349-350 major major histocompatibility complex, 349 349 plant-pathogen plant -pathogen interactions gene-for-gene systems, systems, 333-334 dimensionality of of host resistance, 334334336 flax and and flax rust, rust, 334-336, 338-339 338-339 crops, 335

colonization-extinction colonization-extinction dynamics dynamics of of al­ alleles, 336-339 legume legume and and fungus, 336-338

IF, see see Incidence function model Immigration, see see Colonization Incidence function model applications, applications, 83, 85 butterfly metapopulations, 369-370 characteristics, 14 colonization probability, 80-8 80-811 extinction probability, 80 incidence of species species in a patch, calculation, incidence 80-811 80-8 regression analysis for for parameter determina­ determination, 881-82 1 -82 testing, 82-83 Island biogeography theory comparison to metapopulation models, 116 6 conservation biology, 117-19 7-19 fall ooff popularity, 117-19 7-19 refuge design, 24 Isolation by distance model, 43 1 -432 431-432

Landscape defined, 45 heterogeneity, effects on local community composition colonization rate, 152-154 1 52-1 54 1 52 extinction rate, 152 growth rate of of rare species, species, 153-154 1 53- 1 54 of species, 1 52- 1 55, habitat specialization of species, 152-155, 1 64 164 1 5 1 - 1 52 metapopulation modeling, 151-152 species distribution, 150-151 1 50- 1 5 1 spillover between habitats, 154 1 54 high coverage landscapes and metapopulametapopula­ tion applicability, 4-5 low coverage landscapes and metapopulation applicability, 4-5 Landscape Landscape ecology 1 5, comparison with metapopulation ecology, 15, 48, 50 48, defined, 45-48 defined, development, 45-47 fusion with metapopulation dynamics conservation impact, 58-60 58-60 conservation impact, 61 importance, 58, 61 local extinction and recolonization, 57 effects, 51-57 5 1 -57 migration effects,

Index Index patch characteristics, 50-5 50-511 success, 43 patch theory, 44-45, 47, 58 Levins metapopulation assumptions of model, 9, 113, 3, 73-74, 94-95 basic reproduction number, calculation, 1101, 01, 1114-115 14-1 15 colonization, rate equation, 74 1 -72 conditions for application, 771-72 conservation biology, 221-22 1 -22 definition and synonyms, I11I fraction of occupied patches, 93 homology with susceptible- infected-suscepinfected- susceptible model, 94 migration and local dynamics, 95-96 patch area and extinction risk, 74-75 species candidates, 75-76 stochastic version, calculation of time to extinction, 76-78 Levitt-Boorman metapopulation, see see Main­ Mainland land--island island metapopulation Life history, evolution of traits age structure at equilibrium, 3313 13 annuals versus perennials, 3313-317 1 3-3 1 7 migration effects, 320-321 Local extinction butterfly metapopulations factors affecting extinction, 365-367, 385 rate, 365 time to extinction, 370, 379 catastrophe and genetic impoverishment, 235-237 causes, 30-3 1 , 429 30-31,429 defined, 17 defined, 30, 2217 effects favorable allele establishment, 1190 90 genetic variability, 1169, 69, 1172, 72, 1174-176, 74-1 76, 1186 86 local adaptation, 1195-196 95- 1 96 mUltispecies 57-1 58, multispecies metapopulation, 36, 1157-158, 1161-163 6 1 - 1 63 nonequilibrium metapopulations, 85-86, 88 shifting balance model, 206-209 size of population, 2 1 6, 237-238 216, landscape mixtures mixtures and metapopulation ex­ extinction, 308-3 11 308-311 migration and persistence, 32-33, 40, 57, 272, 275, 290-291 09- 1 1 0, 2 1 5-21 6 modeling, 98-99, 1109-110, 215-216 conservation parameters, 2 17 217 features of a good model, 2 16-21 7 216-217 metacommunities, 240-241 population growth, 2217, 1 7, 2 19 219

509 509

stochastic modeling, see see Demographic sto­ stochasticity; Environmental stochastic­ stochasticity time to extinction, 2218, 221-222, 1 8, 221 -222, 229, 234 recolonization, 33-34, 57, 240 Local population definition and synonyms, 1111 migration and metapopulation approaches, 9110 0 modeling of growth, 98-99, 1115-118, 15 - 1 1 8, 2 1 7, 217, 2 19 219

Mainland-island metapopulation, definition and synonyms, 1111 Metacommunity, see see also also Multispecies metapopulation apparent competition modeling, 1161-164 6 1 - 1 64 50 defined, 1150 extinction modeling, 240-24 240-2411 food web complexity, 1161 61 landscape heterogeneity, effects oonn local community composition colonization rate, 1152-154 52- 1 54 extinction rate, 1152 52 growth rate of rare species, 1153-154 53 - 1 54 habitat specialization of species, 1152-155, 52-1 55, 1164 64 metapopulation modeling, 1151-152 5 1 - 1 52 50- 1 5 1 species distribution, 1150-151 spillover between habitats, 1154 54 Metapopulation, see see also also specific specific models m o d e l s and and populations populations conservation biology ecosystem management, 26 genetic modeling, 25-26 minimum viable metapopulation determi­ determination, 22, 24, 65 non equilibrium metapopulations and time nonequilibrium to extinction, 85-86, 88 parasitic impact, impact, 24-25 refuge design, 22-23 rise in popularity of models, 119-21 9-21 species conforming to Levins model, 221-22 1 -22 structured versus unstructured models, 107-1 08 107-108 definition and synonyms, 9-1 1 , 48, 50 9-11, density-dependent regulation, 297 growth in literature, 5-6, 2 211 history of studies, 9 studies, 5, 7-8, 119 homology with epidemiology, 94, 1124-126 24-126

5510 10

Index Index

Metapopulation Metapopulation (continued) (continued) mathematical derivation, 1111-113 1 1- 1 1 3 rampant rampant population population turnover, butterfly exam­ example of of classical dynamics, 70-72 steps in modeling, 96-100 96-100 Metapopulation Metapopulation effect defined, 294 donnancy, 1 1-312 dormancy, 3311-312 interacting genomes and reproductive sys­ systerns, 1 7-3 1 9 tems, 3317-319 interacting species, 3317 17 life history age structure at equilibrium, 3313 13 annuals versus perennials, 3313-317 1 3-3 1 7 migration effects, 320-321 migration rate adaptation adaptation to landscape heterogeneity, 308-3 11 308-311 conditional conditional migration, 303-304 evolutionarily stable migration migration rate balancing with optimal migration rate, 305-308 balancing with polymorphism, 300-301 calculation, 302 landscape influence, 301 survival rate effects, 302-303 302-303 metapopulation metapopulation effect, 295 polymorphism, 298-301 294-295 rationale for selection, 294-295 1 -322 selection levels, 32 321-322 thyme, 3317-319 1 7-3 1 9 Metapopulation structure, definition and syno­ synoI , 267 nyms, I11,267 Migration, see see also also Rescue effect bifurcation 04- 1 05 bifurcation parameter, 1104-105 butterfly metapopulations metapopulations area-dependent area-dependent emigration, 375-376 capacity, capacity, 384 data data collection, 373-374 estimation, estimation, 374-375 374-375 computer computer simulation, simulation, 257 connectivity of habitats, habitats, 55 defined, 248-250 248-250 effects genetic variability, 28 1 , 287-288 281,287-288 habitat 46 habitat destruction, 144, 144, 1146 life history trait evolution, 320-321 local adaptation, adaptation, 196, 196, 268 local extinction, 32-33, 32-33, 40, 57, 272, 275, 290-291 , 304 290-291,304 neutral variation models, 1186-187 86- 1 87 shifting balance model, 202-204, 206-209 206-209 38- 1 4 1 spatially explicit models, 1138-141

evolution adaptation to landscape heterogeneity, adaptation 308-3 11 308-311 conditional migration, 303-304 evolutionarily evolutionarily stable migration migration rate balancing with optimal migration rate, 305-308 balancing with polymorphism, 300-301 calculation, 302 landscape influence, influence, 301 survival rate effects, 302-303 metapopulation effect, 295 polymorphism, 298-301 rationale for selection, 294-295 selection levels, 321-322 frequency, 268, 28 2811 gene selection, 2213 13 ground insects, methods of study, 257-258 257-258 insect model, 53-54 landscape ecology effects, 54-55, 6 611 mating probability, 290 modeling in metapopulations, 91 0, 551-57, 1 -57, 9-10, 1104-105, 04-105, 272, 274 mortality during migration, 258-259 258-259 optimization of of rate, 305-308 percolation theory, 55-56, 61 plant studies, 255 rate estimation experimental manipulation manipulation of of populations, 262-263 genetic variability, 433 patch-specific patch-specific estimation, 264-265 264-265 timing, 268-269, 275 Minimum Minimum viable metapopulation, detennina­ determination, 22, 24, 65, 76-78 Moran Moran statistic, statistic, computation for pika metapop­ metapopulation, 4 1 4-41 5 , 425 414-415,425 Multispecies metapopulation, metapopulation, see see also also Host­ Hostparasite genetics; Metacommunity; Spa­ Spatially explicit metapopulation model classical dynamics, 38-39 competition modeling conditions conditions for for inferior species existence, 1128 28 habitat destruction effects, 1128-130 28- 1 30 Levins model, 1127-130 27-1 30 rate equations for population growth, 1271 271128 28 mutualism modeling eradication threshold, 1135 35 habitat destruction effects, 135-136 1 35- 1 36 rate 1 34rate equations for population growth, 1341135 35

Index Index persistent refuges, refuges, 37 37 persistent predation modeling modeling predation apparent competition competition modeling, modeling, 161-164 1 6 1 - 1 64 apparent biogeographic donor donor control, control, 157 157 biogeographic colonization-extinction dynamics, dynamics, 157, 1 57, colonization-extinction 1 6 1 - 1 63 161-163 habitat destruction destruction effects, effects, 131-133 1 3 1 - 1 33 habitat laissez-faire predators, predators, 133 1 33 laissez-faire prey colonization colonization scenarios, scenarios, 158-159 1 58- 1 59 prey prey extinction extinction scenarios, scenarios, 157-158, 1 57-1 58, 163 1 63 prey rate equations equations for for population population growth, growth, 131 131 rate two-prey modeling, modeling, 163 1 63 two-prey ragwort-cinnabar moth -Cotesia Cotesia popularis popularis ragwortcinnabar mothsystem system census data, data, 390 390 census data collection, collection, 389-390 389-390 data extinction rates, rates, 402-403 402-403 extinction persistence mechanisms mechanisms persistence cinnabar moth, moth, 401-402 40 1 -402 cinnabar ragwort, 395-396 395-396 ragwort, population dynamics dynamics population cinnabar moth, moth, 396-399 396-399 cinnabar Cotesia popularis, popuiaris, 399-401 399-40 1 Cotesia ragwort, 39 1 , 393, 395 ragwort, 391,393, properties of of individual individual species, species, 389 properties spatially dynamics and and survival survival spatially extended extended dynamics probability, probability, 403-405 stability stability of of populations populations predator, predator, 37 37 prey, prey, 35-37 35-37 subdivision subdivision of of populations, populations, 38 38 theory, theory, 29 29 three-level three-level food food chain chain colonization55colonization- extinction extinction dynamics, dynamics, 11551156, 56, 1160 60 occupancy 60 occupancy of of intermediate intermediate competitor, competitor, 1160 trophic 1 , 1155, 55 , 1161 61 trophic complexity, complexity, 441,

Patch, Patch, see see also also Habitat; Habitat; Landscape Landscape approaches approaches to to patchiness, patchiness, 44-45 44-45 definition definition and and synonyms, synonyms, 1111 metapopulation metapopulation effects effects configuration configuration of of edges, edges, 50-5 50-511 linking linking elements. elements, 5511 number 25 number of of patches, patches, 1125 size. size, 50 50 spacing, 1 , 89-90, spacing, 50-5 50-51, 89-90, 430 430 weather weather effects, effects, 90-91 90-91 Persistence I Persistence time, time, definition definition and and synonyms, synonyms, I11 Pika Pika metapopulation metapopulation Bodie 1 1 -4 1 2 Bodie site site properties, properties, 407, 407, 4411-412 census 1 2-4 1 3 census data data collection, collection, 408, 408, 4412-413

51 Sl !1

competitors, competitors, 426 426 natural natural history history of of pikas, pikas, 409-411 409-4 1 1 patch patch area effects, effects, 415-417 4 1 5-4 1 7 area isolation isolation effects, effects, 417-418, 4 1 7-4 1 8, 424 424 occupancy, occupancy, 415, 4 1 5, 424 424 size size assessment, assessment, 413 413 predators, predators, 424, 424, 426 426 rescue rescue effect, effect, 283,408 283, 408 spatial spatial autocorrelation autocorrelation of of population population events, events, 414-415,420, 414-4 1 5, 420, 422-423,425-428 422-423, 425-428 spatially spatially explicit explicit modeling, modeling, 413-414, 4 1 3-414, 419-420 4 19-420 Plant Plant metapopulation, metapopulation, see see also also Ragwort; Ragwort; Red Red bladder bladder campion campion metapopulation Carduus, 452 452 Carduus, metapopulation metapopulation effect effect in in thyme thyme populations, populations, 3 1 7-3 1 9 317-319 Primula veris, veris, 451 Primula Thymus Thymus vulgaris, vulgaris, 451 45 1

Ragwort life history, 388-389 persistence mechanisms, 395-396 ragwort-cinnabar mothmoth- Cotesia Cotesia popularis popularis metapopulation census data, 390 data collection, 389-390 extinction rates, 402-403 population dynamics, 39 391,393, 1 , 393, 395 spatially extended dynamics and survival probability, 403-405 Red bladder campion metapopulation age estimation, 434 colonization, 436-439. 436-439, 445-447 445-447 colonization, genetic genetic variability variability distance, effects effects on genetic differentiation, distance, 443 -445 443-445 effect of of environmental environmental heterogeneity, heterogeneity, effect 447-448 447-448 effect of of population population age, age, 440-442 440-442 effect founder effects, effects, 445-449 founder interisland genetic genetic structure, structure, 440 440 interisland intraisland genetic genetic structure. structure, 442-443 442-443 intraisland statistical analysis, analysis, 439-440 439-440 statistical life history history of of plant, plant, 436 436 life migration, 438-439, 438-439, 442 442 migration, patch carrying carrying capacity. capacity, 438 438 patch population population growth, growth, 437 437 Silene alba alba metapopulation, metapopulation, genetic genetic structure structure Silene analysis, 450-451 450-451 analysis, Skeppsvik Archipelago Archipelago characteristics, characteristics, 434, 434, Skeppsvik 438 438

5512 12

Index Index

see also also Conservation Refuge, see 22-24 design, 22-24 and management, 59-60, landscape ecology and 1123-24 23-24 effect Rescue effect 271I conservation relevance, 27 defined, 2213 13 effect ooff population growth rate, 278 evidence within metapopulations butterflies, 280-282 chipmunks, 285 chipmunks, mice, 284-285 natterjack toads, 287-288 pikas, 283 pool frogs, 288-289 prairie dogs, 284, 286 proof requirements, requirements, 279 proof red-spotted newts, 287 285-286 squirrels, 285-286 283-284 voles, 283-284

Shifting balance model 201-202, 4311 adaptive peaks, 201 -202, 43 206-209 extinction effects, 206-209 hybrid zone, 207-208 4311 migration effects, 202-204, 206-209, 43 peak shift modeling, 202-204, 206 phases, 201 speciation, 20 201I Single species metapopulation metapopulation local extinction causes, 30-31 30-3 I defined, 30 migration migration and prevention, 32-33, 40 recolonization, recolonization, 33-34 low turnover, 35 mixed metapopulation metapopulation structures, 34-35 34-35 persistence, 331-32 persistence, I -32 theory, 28-29 theory, Source-sink metapopulation Source-sink metapopulation definition and synonyms, 10-11 1 0- I I genetic genetic drift, drift, 207

migration, 269-270 pseudosinks, 380-38 380-3811 explicit metapopulation model Spatially explicit advantages and disadvantages, 14 14 advantages butterfly metapopulations, 368-369 definition and synonyms, 112-13 2- 1 3 migration effects, 1138-141 38-141 muitispecies multispecies modeling complex spatial dynamics, 140-143, 146147 habitat destruction and and spatial dynamics, 143-145 host-parasitoid host-parasitoid modeling, 140-143 local stability, 1137-140 37-140 three-species systems, 145I 46 145-146 pika metapopulation, 4 1 3-414, 4 19-420 413-414, 419-420 population size, calculation, 1138, 38, 140 Spatially implicit metapopulation model advantages, 113 3 definition and synonyms, 112 2 see Spatially realistic metapopulation model, see also Incidence function model; State trantran­ also sition model applications, 78-79 2, 114 4 definition and synonyms, 112, State transition model applications, 79 characteristics, 79 Stochastic modeling, see see Demographic stochas­ stochasStochastic ticity; Environmental stochasticity

Tigriopus Tigriopus calif californicus amicus metapopulation, metapopulation, genetic

structure analysis, 450 Transfer, see see Colonization; Emigration; MigraMigra­ tion Turnover, II Turnover, definition and synonyms, 11

Weather, variation within habitats, habitats, 90-91 90-91,379, 379380