## Journal of Applied Probability

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

### Extinction times for a general birth, death and catastrophe process

#### 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 in Project Euclid: 30 November 2004

**Permanent link to this document**

https://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. J. Appl. Probab. 41 (2004), no. 4, 1211--1218. doi:10.1239/jap/1101840567. https://projecteuclid.org/euclid.jap/1101840567.