Null recurrence and transience for a binomial catastrophe model in random environment

Abstract

We consider a discrete time population model for which each individual alive at time n survives independently of everybody else at time n + 1 with probability β n . The sequence is i.i.d. and constitutes our random environment. Moreover, at every time n we add Z n individuals to the population. The sequence is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts 1994 J. Appl. Probab. 31 48–58). We apply our results to a particular distribution and deterministic β. This particular case shows a rather unusual phase transition in β in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.