(Joint work with Ross McVinish, University of Queensland)
Abstract: We consider a model for the presence/absence of a population in a collection of habitat patches, which assumes that colonisation and extinction of patches occur as distinct phases. Since the local extinction probabilities are allowed to vary between patches, our model permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number n of habitat patches becomes large. When the initial number of occupied patches increases at the same rate as n a law of large numbers ensues. However, here we focus attention on the case where the initial number occupied is fixed, and thus our aim is to determine conditions under which a metapopulation that is close to extinction may recover. By treating the patch survival probabilities of occupied patches at time t as a point process S^{n}(t) on [0,1), we are able to exploit the probability generating functional techniques described in Section 9.4 of Daley and Vere-Jones Vol. II to show that S^{n}(t) converges weakly to a point process S(t). Since extinction of the metapopulation by time t corresponds to the event that S(t) is the empty set, this probability can be calculated from the probability generating functional of S(t) in much the same way that the probability of extinction of a branching process can be calculated from the probability generating function of its offspring 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: 26th November 2011