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Extinction times for a general birth, death and catastrophe process

Part of: Mathematical biology in general Markov processes

Published online by Cambridge University Press:  14 July 2016

Ben Cairns  and
P. K. Pollett
Ben Cairns*
Affiliation:
University of Queensland
P. K. Pollett*
Affiliation:
University of Queensland
*
∗ Postal address: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia
∗ Postal address: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia
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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 processpersistence timehitting time

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92B05: General biology and biomathematics 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1211 - 1218
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.Google Scholar
Brockwell, P. J. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 42–52.Google Scholar
Brockwell, P. J., Gani, J., and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.Google Scholar
Chen, A., Pollett, P. K., Zhang, H., and Cairns, B. (2003). Uniqueness criteria for continuous-time Markov chains with general transition structure. Submitted. Available at http://www.maths.uq.edu.au/∼pkp/.Google Scholar
Ēzhov, Ī. Ī., and Reshetnyak, V. N. (1983). A modification of the branching process. Ukranian Math. J. 35, 28–33.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Mangel, M., and Tier, C. (1993). Dynamics of metapopulations with demographic stochasticity and environmental catastrophes. Theoret. Pop. Biol. 44, 1–31.Google Scholar
Pakes, A. G. (1987). Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biol. 25, 307–325.CrossRefGoogle ScholarPubMed
Pakes, A. G. (1989). Asymptotic results for the extinction time of Markov branching processes allowing emigration. I. Random walk decrements. Adv. Appl. Prob. 21, 243–269.CrossRefGoogle Scholar
Pollett, P. K. (2001). Quasi-stationarity in populations that are subject to large-scale mortality or emigration. Environ. Internat. 27, 231–236.Google Scholar
Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on l . Acta Math. 97, 1–46.Google Scholar
Shafer, C. L. (2001). Inter-reserve distance. Biol. Conservation 100, 215–227.Google Scholar
Shaffer, M. L. (1981). Minimum population sizes for species conservation. BioScience 31, 131–134.CrossRefGoogle Scholar
Walker, D. M. (1998). The expected time till absorption when absorption is not certain. J. Appl. Prob. 35, 812–823.Google Scholar
Yang, Y. S. (1973). Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447–450.Google Scholar