*(English)*Zbl 0551.92013

The author considers a Markov process with state space ${\mathbb{N}}_{+}$ and generator $\left[{q}_{ij}\right]$ where ${q}_{ij}=i\lambda {I}_{\{i+1\}}\left(j\right)+i\mu {I}_{\{i-1\}}\left(j\right)+i\kappa {d}_{i-j}{I}_{(0,i)}\left(j\right)$ if $j\ge 1$ and $j\ne i$, ${q}_{i0}=i\kappa {\sum}_{k=i}^{\infty}{d}_{k}+\mu {I}_{\{i-1\}}\left(0\right)$ and ${q}_{ii}=-i(\lambda +\mu +\kappa )+i\kappa {d}_{0}$. This defines a linear birth-death process modified to allow ’catastrophic’ decrements in the population size at a rate proportional to population size. This model, with specific catastrophe-size distributions $\left\{{d}_{i}\right\}$, has previously been examined by the author, *J. Gani* and *S. I. Resnick,* ibid. 14, 709-731 (1982; Zbl 0496.92007).

Here the author is interested in the time T to extinction. He derives an expression for its probability generating function, criteria ensuring that ${P}_{i}(T<\infty )=1$ and an asymptotic expression for ${P}_{i}(T<\infty )$, when this is not unity, as $i\to \infty $. In addition he derives a generating function for ${E}_{i}T$ and obtains an asymptotic form of ${E}_{i}T$ for large i when the process drifts to the origin. Finally, he considers the analogous problems for the Feller continuous- state branching process modified to allow downward jumps at a rate proportional to the level of the process.

##### MSC:

92D25 | Population dynamics (general) |

60J80 | Branching processes |

60J85 | Applications of branching processes |