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Title: Extinction times for a general birth, death and catastrophe process.

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

Reference: Journal of Applied Probability 41(2): 1211-1218.

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Abstract: 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 an arbitrary manner. 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, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst 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. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Keywords. Catastrophe process, persistence time, hitting time.

Acknowledgement. The support of the Australian Research Council (grant A00104575) is gratefully acknowledged. The work of the first author was supported by a PhD scholarship from the Australian Resarch Council Centre of Excellence for Mathematics and Statistics of Complex Systems.

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