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Zbl 0551.92013
Brockwell, P.J.
The extinction time of a birth, death and catastrophe process and of a related diffusion model. (English)
[J] Adv. Appl. Probab. 17, 42-52 (1985). ISSN 0001-8678

The author considers a Markov process with state space ${\bbfN}\sb+$ and generator $[q\sb{ij}]$ where $q\sb{ij}=i\lambda I\sb{\{i+1\}}(j)+i\mu I\sb{\{i-1\}}(j)+i\kappa d\sb{i-j}I\sb{(0,i)}(j)$ if $j\ge 1$ and $j\ne i$, $q\sb{i0}=i\kappa \sum\sp{\infty}\sb{k=i}d\sb k+\mu I\sb{\{i-1\}}(0)$ and $q\sb{ii}=-i(\lambda +\mu +\kappa)+i\kappa d\sb 0$. 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 $\{d\sb i\}$, has previously been examined by the author, {\it J. Gani} and {\it S. I. Resnick,} ibid. 14, 709-731 (1982; Zbl 0496.92007). \par Here the author is interested in the time T to extinction. He derives an expression for its probability generating function, criteria ensuring that $P\sb i(T<\infty)=1$ and an asymptotic expression for $P\sb i(T<\infty)$, when this is not unity, as $i\to \infty$. In addition he derives a generating function for $E\sb iT$ and obtains an asymptotic form of $E\sb iT$ 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.
[A.Pakes]

MSC 2000:: *92D25 Population dynamics
60J80 Branching processes
60J85 Appl. of branching processes

Keywords: asymptotic behaviour; time to extinction; linear birth-death process; population size; catastrophe-size distributions; generating function; Feller continuous-state branching process; downward jumps

Citations: Zbl 0496.92007

Cited in: Zbl 0633.92014

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