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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.

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.

Citations

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Metapopulations - Levinsrsquos model - Convergence of equilibrium distribution - Poisson approximation
Language:English
Date:2004
Deposited On:07 Apr 2010 09:19
Last Modified:05 Apr 2016 13:24
Publisher:Springer
ISSN:0303-6812
Additional Information:The original publication is available at www.springerlink.com
Free access at:Related URL. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00285-004-0272-8
Related URLs:http://www.math.uzh.ch/fileadmin/math/preprints/22_03.pdf
PubMed ID:15549310

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