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Extinction times for a general birth, death and catastrophe process. (English) Zbl 1063.60106
The birth, death and catastrophe process $X\left(t\right),t\ge 0$, is a continuous-time Markov chain taking nonnegative integer values. The process $X\left(t\right)$ is interpreted as the number of individuals in the population at time $t$ and evolves in time according transition rates ${q}_{ij}={f}_{i}{\sum }_{k\ge i}{d}_{k},$ when $j=0,i\ge 1$; $={f}_{i}{d}_{i-j},$ when $j=1,2,...,i-1,i\ge 2$; $=-{f}_{i},$ when $j=i,i\ge 1$; $={f}_{i}a,$ when $j=i+1,i\ge 1,$ 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 ${d}_{k},k\ge 1$. Theorem 1 proves that $X\left(t\right)$ is nonexplosive (i.e. it cannot reach infinity in a finite time) iff ${\sum }_{i=1}^{\infty }1/{f}_{i}=\infty$ or ${\sum }_{i=1}^{\infty }i{d}_{i}\ge a$. Theorem 2 considers the case when $X\left(t\right)$ 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.

##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 92B05 General biology and biomathematics 60J80 Branching processes
##### Keywords:
catastrophe process; persistence time; hitting time