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

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 ${\beta }_n$. The sequence $({\beta }_n)$ is i.i.d. and constitutes our random environment. Moreover, at every time $n$ we add $Z_n$ individuals to the population. The sequence $(Z_n)$ is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts). We apply our results to a particular $(Z_n)$ distribution and deterministic $\beta$. This particular case shows a rather unusual phase transition in $\beta$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.