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The extinction time of a birth, death and catastrophe process and of a related diffusion model

Published online by Cambridge University Press:  01 July 2016

P. J. Brockwell
P. J. Brockwell*
Affiliation:
Kuwait University
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.
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Abstract

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

Keywords

MARKOVIAN POPULATION PROCESSTIME TO EXTINCTIONDIFFUSION PROCESS WITH DOWNWARD JUMPS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 42 - 52
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research carried out while on leave from Colorado State University and partially supported by NSF Grant No. MCS 82 02335.

References

Bak, J. and Newman, D. J. (1982) Complex Analysis. Springer-Verlag, New York.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
Brockwell, P. J., Gani, J. and Resnick, S. I. (1982) Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.CrossRefGoogle Scholar
Brockwell, P. J., Gani, J. and Resnick, S. I. (1983) Catastrophe processes with continuous state-space. Austral. J. Statist. 25.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II. Wiley, New York.Google Scholar
Hanson, F. B. and Tuckwell, H. C. (1978) Persistence times of populations with large random fluctuations. Theoret. Popn Biol. 14, 46–61.Google Scholar
Hanson, F. B. and Tuckwell, H. C. (1981) Logistic growth with random density independent disasters. Theoret. Popn Biol. 19, 1–18.Google Scholar
Kaplan, N., Sudbury, A. and Nilsen, T. (1975) A branching process with disasters. J. Appl. Prob. 12, 47–59.Google Scholar
Murthy, D. N. P. (1981) A model for population extinction. Appl. Math. Modelling 5, 227–230.Google Scholar
Pakes, A. G., Trajstman, A. C. and Brockwell, P. J. (1979) A stochastic model for a replicating population subjected to mass emigration due to population pressure. Math. Biosci. 45, 137–157.Google Scholar
Trajstman, A. C. (1981) A bounded growth population subjected to emigrations due to population pressure. J. Appl. Prob. 18, 571–582.CrossRefGoogle Scholar