Abstract
The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.
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References
Asmussen, S., Herring, H.: Branching processes. Boston: Birkhauser 1983
Athreya, K. B., Ney, P. E.: Branching processes. Berlin Heidelberg New York: Springer 1972
Bailey, N. T. J.: The elements of stochastic processes with applications to the natural sciences. New York: Wiley 1964
Brockwell, P. J.: The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 42–52 (1985)
Brockwell, P. J., Gani, J., Resnick, S. I.: Birth, immigration and catastrophe processes. Adv. Appl. Probal. 14, 709–731 (1982)
Cohn, H.: Almost sure convergence of branching processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 73–81 (1977)
Ewens, W. J., Brockwell, P. J., Gani, J. M., Resnick, S. I.: Minimum viable population size in the presence of catastrophes. In: Soulé, M. (ed) Viable populations. Cambridge: C.U.P. 1986
Ezhov, I. I., Reshetnyak, V. N.: A modification of the branching process. Ukranian Math. J. 35, 28–33 (1983)
Feller, W.: An introduction to probability theory and its applications, vol. II, 2nd edn. New York: Wiley 1971
Hanson, F. B., Tuckwell, H. C.: Persistence times of populations with large random fluctuations. Theor. Popul. Biol. 14, 46–61 (1987)
Harris, T. E.: Some mathematical models for branching processes. Proc. 2nd Berkeley Symposium, pp. 305–328 (1951)
Harris, T. E.: The theory of branching processes. Berlin Heidelberg New York: Springer 1963
Pakes, A. G.: Some limit theorems for the total progeny of branching processes. Adv. Appl. Prob. 3, 176–192 (1971)
Pakes, A. G.: The limit behaviour of a Markov chain related to the simple branching process allowing immigration. J. Math. Phys. Sci. 15, 159–171 (1981)
Pakes, A. G.: The Markov branching-catastrophe process. Stochastic Processes Appl. 23, 1–33 (1986)
Seneta, E., Vere-Jones, D.: On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–436 (1966)
Tweedie, R. L.: Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford (2) 25, 485–495 (1974)
Yang, Y. S.: Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447–450 (1973)
Zolotarev, V. M.: More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 245–253 (1957)
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Research done while on leave at Colorado State University from the University of Western Australia and partially supported by N.S.F. grant DMS-8501763
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Pakes, A.G. Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biology 25, 307–325 (1987). https://doi.org/10.1007/BF00276439
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DOI: https://doi.org/10.1007/BF00276439