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A Note on Extinction Times for the General Birth, Death and Catastrophe Process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Phil Pollett ,
Hanjun Zhang  and
Benjamin J. Cairns
Phil Pollett*
Affiliation:
The University of Queensland
Hanjun Zhang*
Affiliation:
The University of Queensland
Benjamin J. Cairns*
Affiliation:
University of Bristol
*
∗ Postal address: Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia.
∗∗ Email address: [email protected]
∗∗∗ Email address: [email protected]
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Abstract

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We consider a birth, death and catastrophe process where the transition rates are allowed to depend on the population size. We obtain an explicit expression for the expected time to extinction, which is valid in all cases where extinction occurs with probability 1.

Keywords

Population processhitting timecatastrophezeta distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 566 - 569
Copyright
Copyright © Applied Probability Trust 2007 

References

Brockwell, P. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 42–52.CrossRefGoogle Scholar
Cairns, B. and Pollett, P. (2004). Extinction times for a general birth, death and catastrophe process. J. Appl. Prob. 41, 1211–1218.CrossRefGoogle Scholar
Harris, T. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Malamud, B. D., Morein, G. and Turcotte, D. L. (1998). Forest fires: an example of self-organized critical behaviour. Science 281, 1840–1842.CrossRefGoogle 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. and Pollett, P. K. (1989). The supercritical birth, death and catastrophe process: limit theorems on the set of extinction. Stoch. Process. Appl. 32, 161–170.CrossRefGoogle Scholar
Pollett, P. (2001). Quasi-stationarity in populations that are subject to large-scale mortality or emigration. Environ. Internat. 27, 231–236.CrossRefGoogle ScholarPubMed
Yang, Y. (1973). Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447–450.CrossRefGoogle Scholar