Reconciling theoretical approaches to stochastic path-occupancy metapopulation models
type: Item
instance of: scholarly article
title: Reconciling theoretical approaches to stochastic path-occupancy metapopulation models
zbMATH Open document ID: 0968.92021
author: Frédéric Gosselin
published in: Bulletin of Mathematical Biology
publication date: 2001-09-16T00:00:00Z
review text: The paper attempts to reconcile asymptotic results on stochastic patch-occupancy matapopulation (Markov chain) models obtained by \textit{J.N. Darroch} and \textit{E. Seneta} [J. Appl. Probab. 88-100 (1965; Zbl 0134.34704)] (DS) and by \textit{M. Gyllenberg} and \textit{D.S. Silvestrov} [J. Math. Biol. 33, No. 1, 35-70 (1974; Zbl 0816.92016)] (GS) in the case when there is a ''quasi-mainland'' (a patch so large that only rarely goes extinct). In these models there are n patches, which can be either extinct (state 0) or not (state 1), the state space being the n-tuples of zeros and ones. (0,0,...0) is an absorbing state. The DS approach uses spectral properties of stochastic and substochastic matrices and, conditioned on non-extinction, proves (under appropriate conditions) convergence of the state space distribution to a stationary conditional distribution (SCD) as time goes to infinity. The GS approach considers the ''quasi-mainland'' model as a perturbation (given by a small \(\varepsilon\)) of the mainland case (\(\varepsilon=0\)) and uses perturbation techniques from renewal theory and the theory of perturbed Markov chains to show (under appropriate conditions and conditioned on non-extinction) convergence, as \(\varepsilon\) converges to zero, to a quasistationary distribution (QSD) of the state space distribution for large \(t\) (with \(t\) going to infinity in an appropriate way as \(\varepsilon\) converges to zero). The author proves (under appropriate conditions), based on the spectral theory of linear operators, that, in the ''quasi-mainland'' case, the SCD distribution converges, as \(\varepsilon\) converges to zero, to the QSD, thus linking the two approaches. Some examples are given.
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 47N60
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 1281791
zbMATH Keywords: matapopulation models
zbMATH Keywords: stochastic patch-occupancy models
zbMATH Keywords: quasistationary distributions
reviewed by: Carlos A. dos Santos Braumann
MaRDI profile type: Publication
full work available at URL: bulm.1998.0052
OpenAlex ID: W1991069016
DOI: 10.1006/BULM.1998.0052
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description: scientific article; zbMATH DE number 1281791