Limiting-conditional distributions for stochastic metapopulation models

Phil Pollett

Abstract: We consider a Markovian model proposed by Gyllenberg and Silvestrov for studying the long-term behaviour of a metapopulation: a population that occupies several geographically separated habitat patches. Although the individual patches may become empty through extinction of local populations, they can be recolonized through migration from other patches. There is considerable empirical evidence reported in the work of Hanski and Galpin which suggests that a balance between migration and extinction is obtained which enables these populations to persist for long periods. The Markovian model predicts that unless (exceptionally) there is a population explosion, the metapopulation becomes extinct in finite-mean time. 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 new method, based on recent work of Jacka and Roberts 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, which are based on the classical notion of a quasistationary distribution.

Acknowledgement: This worked was funded by the Australian Research Council.

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Last modified: 31st October 1997