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Extinction times for a general birth, death and catastrophe process. (English) Zbl 1063.60106
The birth, death and catastrophe process X(t),t0, 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 ki d k , when j=0,i1; =f i d i-j , when j=1,2,...,i-1,i2; =-f i , when j=i,i1; =f i a, when j=i+1,i1, 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 ,k1. Theorem 1 proves that X(t) is nonexplosive (i.e. it cannot reach infinity in a finite time) iff i=1 1/f i = or i=1 id i 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 =ρi,ρ>0 (the linear case); (b) the catastrophe size has a geometric distribution.

60J27Continuous-time Markov processes on discrete state spaces
92B05General biology and biomathematics
60J80Branching processes