# Convergence of a structured metapopulation model to Levins's model

Barbour, A D; Pugliese, A (2004). Convergence of a structured metapopulation model to Levins's model. Journal of Mathematical Biology, 49(5):468-500.

## Abstract

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting régime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Stein's method.

## Abstract

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting régime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Stein's method.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Modeling and Simulation Life Sciences > Agricultural and Biological Sciences (miscellaneous) Physical Sciences > Applied Mathematics Metapopulations - Levinsrsquos model - Convergence of equilibrium distribution - Poisson approximation English 2004 07 Apr 2010 09:19 03 May 2024 01:38 Springer 0303-6812 The original publication is available at www.springerlink.com Closed Related URL. An embargo period may apply. https://doi.org/10.1007/s00285-004-0272-8 http://www.math.uzh.ch/fileadmin/math/preprints/22_03.pdf 15549310

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