Reconciling theoretical approaches to stochastic path-occupancy metapopulation models
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.
scientific article; zbMATH DE number 1281791