Summary: ``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 $\Bbb Z_n$ 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 1994 {\it J. Appl. Probab}. {\bf 31} 48--58) [MR1260570]. 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.''