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The extinction time of a birth, death and catastrophe process and of a related diffusion model. (English) Zbl 0551.92013

The author considers a Markov process with state space + and generator [q ij ] where q ij =iλI {i+1} (j)+iμI {i-1} (j)+iκd i-j I (0,i) (j) if j1 and ji, q i0 =iκ k=i d k +μI {i-1} (0) and q ii =-i(λ+μ+κ)+iκd 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 i }, 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<)=1 and an asymptotic expression for P i (T<), when this is not unity, as i. 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.

Reviewer: A.Pakes

MSC:
92D25Population dynamics (general)
60J80Branching processes
60J85Applications of branching processes