Limiting conditional distributions for stochastic metapopulation models

Phil Pollett

Abstract: We consider a Markovian model proposed by Gyllenberg and Silvestrov [J. Math. Biol. 33, 35-70, 1994] for studying the long-term behaviour of a metapopulation. There is considerable empirical evidence reported in the work of Hanski and Gilpin which suggests that a balance between migration and extinction is obtained which enables these populations to persist for long periods. For this reason, there has been considerable interest in developing methods which account for the persistence of these populations and which provide an effective means of studying the long-term behaviour before extinction occurs. We propose a method, based on work of Jacka and Roberts [J. Appl. Probab. 32, 902-916, 1995] on weak convergence of conditioned Markov processes. We compare and contrast this with the methods of Gyllenberg and Silvestrov (quasi-stationary and pseudo-stationary distributions) as well as those of Day and Possingham [Theoret. Pop. Biol. 48, 333-360, 1995], which are based on the classical notion of a quasi-stationary distribution.

Acknowledgement: This work is supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems

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Last modified: 15th September 2004