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Title: Evaluating persistence times in populations that are subject to local catastrophes.

Authors: B.J. Cairns and P.K. Pollett.

Reference: In (Ed. David A. Post) Proceedings of the International Congress on Modelling and Simulation, Vol 2, Modelling and Simulation Society of Australia and New Zealand, Townsville, Australia, pp. 747-752.

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Abstract: The birth-death process is a familiar tool in modelling populations which are subject to demographic stochasticity. However, many populations are also subject to one or more forms of local `catastrophe' (a term usually taken to mean any population decrease of size greater than one). Natural disasters, such as epidemics, and migration to other populations, are all examples of local catastrophes. The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model, in which the transition rates are allowed to depend on the current population size in a completely arbitrary matter. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction (persistence time) and the distribution of the population size conditional on non-extinction (the quasi-stationary distribution) have been evaluated explicitly. However, whilst all of these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models, and indeed it is difficult to imagine instances where catastrophe events would occur at a rate proportional to the population size. Our model addresses this difficulty by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for persistence times.

Keywords. Hitting times; Extinction times; Population processes.

Acknowledgement. The authors would like to thank Hugh Possingham for his suggestions on applications of the model, and the referees for their helpful comments on an earlier version of the manuscript. This work was funded by the Australian Research Council.


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Cancer Epidemiology Unit, University of Oxford, Richard Doll Building, Roosevelt Drive, Oxford OX3 7LF, U.K.