The compound Poisson immigration process subject to binomial catastrophes

Abstract

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.