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Random population dynamics under catastrophic events

Part of: Stability theory Difference and functional equations, recurrence relations Partial differential equations Markov processes

Published online by Cambridge University Press:  15 August 2022

Patrick Cattiaux ,
Jens Fischer ,
Sylvie Rœlly  and
Samuel Sindayigaya
Patrick Cattiaux*
Affiliation:
Université de Toulouse
Jens Fischer*
Affiliation:
Université de Toulouse and Universität Potsdam
Sylvie Rœlly*
Affiliation:
Universität Potsdam
Samuel Sindayigaya*
Affiliation:
Institut d’Enseignement Supérieur de Ruhengeri
*
*Postal address: Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse CEDEX 9, France. Email address: [email protected]
**Postal address: Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse CEDEX 9, France; Institut für Mathematik der Universität Potsdam, Karl-Liebknecht-Str. 24–25, 14476 Potsdam OT Golm, Germany. Email address: [email protected]
***Postal address: Institut für Mathematik der Universität Potsdam, Karl-Liebknecht-Str. 24-25, 14476 Potsdam OT Golm, Germany. Email address: [email protected]
****Postal address: Institut d’Enseignement Supérieur de Ruhengeri, Musanze Street NM 155, PO Box 155, Ruhengeri, Rwanda. Email address: [email protected]
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Abstract

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties. In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t. This is completed by some asymptotic results.

Keywords

Birth-and-death processpopulation dynamicsextinction timebirthdeathand catastrophe process

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 65Q30: Recurrence relations 34D45: Attractors 35B40: Asymptotic behavior of solutions
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 962 - 982
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Anderson, W. J. (1991). Continuous-Time Markov chains. Springer.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. (1986). The extinction time of a general birth and death process with catastrophes. J. Appl. Prob. 23, 851–858.CrossRefGoogle Scholar
Brockwell, P. J., Gani, J. M. and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.CrossRefGoogle Scholar
Di Crescenzo, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (2008). A note on birth–death processes with catastrophes. Statist. Prob. Lett. 78, 2248–2257.CrossRefGoogle Scholar
Feller, W. (1939). Die Grundlagen der volterraschen Theorie des Kampfes ums dasein in wahrscheinlichkeitstheoretischer Behandlung. Acta Bioth. Ser. A. 5, 11–40.Google Scholar
Hall, H. S. and Knight, S. R. (1891). Higher Algebra: A Sequel to Elementary Algebra for Schools, 1st edn. Macmillan, London; St Martin’s Press, New York. Available at https://archive.org/details/higheralgebraseq00hall.Google Scholar
Kapodistria, S., Tuan, P.-D. and Resing, J. (2016). Linear birth/immigration-death process with binomial catastrophes. Prob. Eng. Inf. Sci. 30, 79–111.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1958). Linear growth, birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Kendall, D. G. (1948). On the generalized birth-and-death process. Ann. Math. Statist. 19, 1–15.Google Scholar
Meyn, S. and Tweedie, R. L. A survey of Foster–Lyapunov techniques for general state space Markov processes. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.2517.Google Scholar
Norris, J. R. (1997). Markov Chains. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Sindayigaya, S. (2016). The population mean and its variance in the presence of genocide for a simple birth–death–immigration–emigration process using the probability generating function. Internat. J. Statist. Anal. 6, 1–8.Google Scholar
Swift, R. J. (2001). Transient probabilities for a simple birth–death–immigration process under the influence of total catastrophes. Internat. J. Math. Math. Sci. 25, 689–692.CrossRefGoogle Scholar
van Doorn, E. A. and Zeifman, A. I. (2005). Birth–death processes with killing. Statist. Prob. Lett. 72, 33–42.CrossRefGoogle Scholar
van Doorn, E. A. and Zeifman, A. I. (2005). Extinction probability in a birth–death process with killing. J. Appl. Prob. 42, 185–198.Google Scholar