The Markov branching-castastrophe process

Abstract

This paper considers a Markov branching process modified to allow decrements which occur randomly at a rate proportional to the population size. The problems considered are (i) Conditions for eventual extinction, (ii) Asymptotic behaviour of the probability of extinction and the mean time to extinction; and (iii) Behaviour of the population size. Special consideration is given to the case where the decrement distribution is geometric. In this case limit theorems for the population size are obtained which are analogues of the basic limit theorems for the Markov branching process.