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Zbl 1063.60106
Cairns, Ben; Pollett, P.K.
Extinction times for a general birth, death and catastrophe process.
(English)
[J] J. Appl. Probab. 41, No. 4, 1211-1218 (2004). ISSN 0021-9002

The birth, death and catastrophe process $X(t), t\geq 0$, is a continuous-time Markov chain taking nonnegative integer values. The process $X(t)$ is interpreted as the number of individuals in the population at time $t$ and evolves in time according transition rates $q_{ij}=f_i\sum_{k\geq i}d_k,$ when $j=0, i\geq 1$; $=f_i d_{i-j},$ when $j=1,2, \ldots, i-1, i\geq 2$; $=-f_i,$ when $j=i, i\geq 1$; $=f_i a,$ when $j=i+1, i\geq 1,$ and $=0,$ otherwise, where $f_i>0$ is the rate at which the population size changes when there are $i$ individuals present; when a change occurs, it is a birth with probability $a>0$ or a catastrophe of size $k$ with probability $d_k, k\geq 1$. Theorem 1 proves that $X(t)$ is nonexplosive (i.e. it cannot reach infinity in a finite time) iff $\sum_{i=1}^{\infty}1/f_i=\infty$ or $\sum_{i=1}^{\infty}i d_i\geq a$. Theorem 2 considers the case when $X(t)$ is subcritical and provides (1) necessary and sufficient conditions which ensure that the expected extinction time is finite; (2) the explicit expression for the expected extinction time. Theorem 2 is illustrated by several examples. In particular, the authors point out explicit expressions for the expected extinction times when (a) $f_i=\rho i, \rho>0$ (the linear case); (b) the catastrophe size has a geometric distribution.
[Aleksander Iksanov (Kiev)]
MSC 2000:
*60J27 Markov chains with continuous parameter
92B05 General biology and biomathematics
60J80 Branching processes

Keywords: catastrophe process; persistence time; hitting time

Cited in: Zbl 1154.60064

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