Interplay between Local Dynamics and Dispersal in Discrete-time Metapopulation Models

doi:10.1006/jtbi.2002.3075

Journal of Theoretical Biology

Volume 218, Issue 3, 7 October 2002, Pages 273-288

https://doi.org/10.1006/jtbi.2002.3075 Get rights and content

Abstract

The effects of synchronous dispersal on discrete-time metapopulation dynamics with local (patch) dynamics of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are explored. Single-species metapopulation models behave as single-species single-patch models, whenever all local patches are governed by compensatory dynamics. Dispersal gives rise to multiple attractors with complex basin structures, whenever some local patches are under overcompensatory dynamics. In mixed systems, dispersal is capable of altering the local dynamics from compensatory to overcompensatory dynamics and vice versa. Examples are provided of metapopulation models supporting multiple attractors with intermingled basins of attraction.

References (56)

C. CASTILLO-CHAVEZ et al.
Dispersal, disease and life-history evolution
Math. Biosc.
(2001)
J.P. CRUTCHFIELD et al.
Fluctuations and simple chaotic dynamics
Phys. Rep.
(1982)
M. DOEBELI
Dispersal and dynamics
Theor. Popul. Biol.
(1995)
J.E. FRANKE et al.
Geometry of exclusion principles in discrete systems
J. Math. Anal. Appl.
(1992)
J.E. FRANKE et al.
Extinction and persistence of species in discrete competitive systems with a safe refuge
J. Math. Anal. Appl.
(1996)
J.L. GONZALEZ-ANDÚJAR et al.
Chaos, metapopulations and dispersal
Ecol. Model.
(1993)
M. GYLLENBERG et al.
Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model
Math. Biosci.
(1993)
J.G. MILTON et al.
Chaos, noise, and extinction in models of population growth
Theor. Popul. Biol.
(1990)
H.L. SMITH
Cooperative systems of differential equations with concave nonlinearities
Nonlinear Anal. Theor. Methods Appl
(1986)
J.C. ALEXANDER et al.
Riddled basins
Int. J. Bifurc.
(1992)

L.J. ALLEN

Persistence, extinction, and critical patch number for island populations

J. Math. Biol.

(1987)

M. BEGON et al.

Ecology: Individuals, Populations and Communities. Populations and Communities

(1996)

BEST, J. PASOUR, V. TISCH, N. CASTILLO-CHAVEZ, C. 2001, Delayed density dependence and the dynamic consequences of...

F. BRAUER et al.

Mathematical models in population biology and epidemiology

C. CASTILLO-CHAVEZ et al.

Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments

COHEN, D. LEVIN, S. A. 1987, The interaction between dispersal and dormancy strategies in varying and heterogeneous...

M. DOEBELI et al.

Evolution of dispersal rates in metapopulation model: branching and cyclic dynamics in phenotype space

Evolution

(1997)

D.J. EARN et al.

Coherence and conservation

Science

(2000)

P.L. ERRINGTON

Some contributions of a fifteen year local study of the northern bobwhite to a knowledge of population phenomena

Ecol. Monogr.

(1945)

J.E. FRANKE et al.

Mutual exclusion versus coexistence for discrete competitive systems

J. Math. Biol.

(1991)

M. GADGIL

Dispersal: Population consequences and evolution

Ecology

(1971)

C. GREBOGI et al.

Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics

Science

(1987)

HAIRSTON, N. G. Diapause dynamics of 2 Diaptomid copepod species in a large lake, Hydrobiologia, 293, 209,...

I. HANSKI et al.

Multiple equilibria in metapopulation dynamics

Nature

(1995)

I. HANSKI

Single-species metapopulation dynamics-concepts, models and observations

Biol. J. Linn. Soc.

(1997)

I.A. HANSKI et al.

Metapopulation Biology: Ecology, Genetics, and Evolution

(1997)

M.P. HASSELL

The dynamics of competition and predation

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There is a copious amount of literature on patch models (e.g., Maynard Smith and Slatkin, 1973; Allen, 1975; Holt, 1985; Reeve, 1988; Taylor, 1988; Adler, 1993; Gyllenberg et al., 1993; Hastings, 1993; Doebeli, 1997; Yakubu, 2000; Amarasekare, 2000; Yakubu and Castillo-Chavez, 2002; Holt et al. 2004; also see a review paper by Briggs and Hoopes, 2004; and references therein; Selgrade and Roberds, 2005; Kang and Lanchier, in press). Only a few are discrete two-species-interacting models in a patchy environment (e.g., Maynard Smith and Slatkin, 1973; Allen, 1975; Reeve, 1988; Taylor, 1988; Adler, 1993; Doebeli, 1997; Yakubu, 2000; Yakubu and Castillo-Chavez, 2002; Selgrade and Roberds, 2005). Maynard Smith and Slatkin (1973), as well as Allen (1975) and Reeve (1988) show that the unstable equilibrium of a single-patch predator–prey model cannot be stabilized by diffusive coupling with identical patches, since the coupled system behaves identically to the single-patch system if the patches are synchronized.
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¹: Corresponding author. Mathematical and Theoretical Biology Institute (MTBI), Department of Biometrics for Undergraduate Research, Cornell University College of Agriculture and Life Sciences, 432 Warren Hall, Ithaca NY 14853-7801, U.S.A.. Tel.: +1-607-255-81-03; fax: +1-607-255-4698. E-mail addresss: [email protected] [email protected] (A.-A. Yakubu).

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Regular ArticleInterplay between Local Dynamics and Dispersal in Discrete-time Metapopulation Models

Abstract

Math. Biosc.

Phys. Rep.

Theor. Popul. Biol.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Ecol. Model.

Math. Biosci.

Theor. Popul. Biol.

Nonlinear Anal. Theor. Methods Appl

Riddled basins

Int. J. Bifurc.

Persistence, extinction, and critical patch number for island populations

J. Math. Biol.

Ecology: Individuals, Populations and Communities. Populations and Communities

Mathematical models in population biology and epidemiology

Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments

Evolution of dispersal rates in metapopulation model: branching and cyclic dynamics in phenotype space

Evolution

Coherence and conservation

Science

Some contributions of a fifteen year local study of the northern bobwhite to a knowledge of population phenomena

Ecol. Monogr.

Mutual exclusion versus coexistence for discrete competitive systems

J. Math. Biol.

Dispersal: Population consequences and evolution

Ecology

Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics

Science

Multiple equilibria in metapopulation dynamics

Nature

Single-species metapopulation dynamics-concepts, models and observations

Biol. J. Linn. Soc.

Metapopulation Biology: Ecology, Genetics, and Evolution

The dynamics of competition and predation

Regular Article
Interplay between Local Dynamics and Dispersal in Discrete-time Metapopulation Models