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

  • Phil Pollett (a1), Hanjun Zhang (a1) and Benjamin J. Cairns (a2)

Abstract

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.

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Copyright

Corresponding author

Postal address: Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia.
∗∗ Email address: pkp@maths.uq.edu.au
∗∗∗ Email address: hjz@maths.uq.edu.au
∗∗∗∗ Postal address: School of Biological Sciences, University of Bristol, Bristol BS8 1UG, UK. Email address: ben.cairns@bristol.ac.uk

References

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[1] Brockwell, P. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 4252.
[2] Cairns, B. and Pollett, P. (2004). Extinction times for a general birth, death and catastrophe process. J. Appl. Prob. 41, 12111218.
[3] Harris, T. (1963). The Theory of Branching Processes. Springer, Berlin.
[4] Malamud, B. D., Morein, G. and Turcotte, D. L. (1998). Forest fires: an example of self-organized critical behaviour. Science 281, 18401842.
[5] Pakes, A. G. (1987). Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biol. 25, 307325.
[6] 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, 161170.
[7] Pollett, P. (2001). Quasi-stationarity in populations that are subject to large-scale mortality or emigration. Environ. Internat. 27, 231236.
[8] Yang, Y. (1973). Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447450.

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

  • Phil Pollett (a1), Hanjun Zhang (a1) and Benjamin J. Cairns (a2)

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