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Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.

The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Keywords

LINEAR BIRTH AND DEATH PROCESSCATASTROPHES AND EMIGRATIONLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 2 , June 1989 , pp. 243 - 269
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

This research was partially supported by N.S.F. Grant DMS-8501763 during a visit to Colorado State University.

References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington.Google Scholar
Barbour, A. D. and Pakes, A. G. (1979) Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean. Adv. Appl. Prob. 11, 63–72.CrossRefGoogle 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.CrossRefGoogle Scholar
Brockwell, P. J., Gani, J. and Resnick, S. I. (1982) Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.CrossRefGoogle Scholar
Ewens, W. J., Brockwell, P. J., Gani, J. M., and Resnick, S. I. (1987) Minimum viable population size in the presence of catastrophes. In Viable Populations, ed. Soulé, M., Cambridge University Press.Google Scholar
Ezhov, I. I. and Reshetnyak, V. N. (1983) A modification of the branching process. Ukranian Math. J. 35, 28–33.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. 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.CrossRefGoogle ScholarPubMed
Hanson, F. B. and Tuckwell, H. C. (1981) Logistic growth with random density independent disasters. Theoret. Popn. Biol. 19, 1–18.CrossRefGoogle Scholar
Kaplan, N., Sudbury, A. and Nilsen, T. (1975) A branching process with disasters. J. Appl. Prob. 12, 47–59.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Pakes, A. G. (1974) Some limit theorems for Markov chains with applications to branching processes. In Studies in Probability and Statistics, ed. Williams, E. J., Jerusalem Academic Press and North-Holland, Amsterdam, pp. 21–39.Google Scholar
Pakes, A. G. (1979a) Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean. Adv. Appl. Prob. 11, 31–62.CrossRefGoogle Scholar
Pakes, A. G. (1979b) The age of a Markov process. Stoch. Proc. Appl. 8, 277–303.CrossRefGoogle Scholar
Pakes, A. G. (1986) The Markov branching-catastrophe process. Stoch. Proc. Appl. 23, 1–33.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. (1988a) The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction. J. Math. Biol. 26, 405–420.CrossRefGoogle Scholar
Pakes, A. G. (1988b) The Markov branching process with density independent catastrophes, I. Behaviour of extinction probabilities. Math. Proc. Camb. Phil. Soc. 103, 351–366.CrossRefGoogle Scholar
Pakes, A. G. (1988c) The Markov branching process with density independent catastrophes, II, III The population size. Math. Proc. Camb. Phil. Soc. To appear.CrossRefGoogle Scholar
Pakes, A. G. and Pollett, P. K. (1987) The supercritical birth, death and catastrophe process: Limit theorems on the set of extinction. Stoch. Proc. Appl. To appear.CrossRefGoogle Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Yang, Y. S. (1973) Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447–450.CrossRefGoogle Scholar
Zolotarev, V. M. (1957) More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 245–253.CrossRefGoogle Scholar