Journal of Applied Probability

Extinction times for a general birth, death and catastrophe process

Ben Cairns and P. K. Pollett

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Abstract

The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Article information

Source
J. Appl. Probab. Volume 41, Number 4 (2004), 1211-1218.

Dates
First available: 30 November 2004

Permanent link to this document
http://projecteuclid.org/euclid.jap/1101840567

Digital Object Identifier
doi:10.1239/jap/1101840567

Mathematical Reviews number (MathSciNet)
MR2122816

Zentralblatt MATH identifier
1063.60106

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92B05: General biology and biomathematics 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Catastrophe process persistence time hitting time

Citation

Cairns, Ben; Pollett, P. K. Extinction times for a general birth, death and catastrophe process. Journal of Applied Probability 41 (2004), no. 4, 1211--1218. doi:10.1239/jap/1101840567. http://projecteuclid.org/euclid.jap/1101840567.


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