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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: This worked was funded by the Australian Research Council.
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Last modified: 2nd March 2003