Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T23:53:12.316Z Has data issue: false hasContentIssue false

Birth, immigration and catastrophe processes

Published online by Cambridge University Press:  01 July 2016

P. J. Brockwell*
Affiliation:
Colorado State University
J. Gani*
Affiliation:
University of Kentucky
S. I. Resnick*
Affiliation:
Colorado State University
*
Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.
∗∗Postal address: Department of Statistics, University of Kentucky, Lexington, KY 40506, U.S.A.
Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.

Abstract

We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by NSF Grant No. MCS 78 00915-01.

Portions of this work were done during a visit to the Department of Statistics, Colorado State University, to which grateful acknowledgement is made for hospitality and support.

References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Ma.Google Scholar
Hanson, F. B. and Tuckwell, H. C. (1978) Persistence times of populations with large random fluctuations. Theoret. Popn Biol. 14, 4661.CrossRefGoogle ScholarPubMed
Hanson, F. B. and Tuckwell, H. C. (1981). Logistic growth with random density independent disasters. Theoret. Popn. Biol. 19, 118.CrossRefGoogle Scholar
Kaplan, N., Sudbury, A. and Nilsen, T. (1975) A branching process with disasters. J. Appl. Prob. 12, 4759.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Murthy, D. N. P. (1981) A model for population extinction. Appl. Math. Modelling 5, 227230.CrossRefGoogle Scholar
Pares, 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, 137157.Google Scholar
Trajstman, A. C. (1981) A bounded growth population subjected to emigrations due to population pressure. J. Appl. Prob. 18, 571582.Google Scholar