  We classify the possible behaviors of a class of one-dimensional stochastic
recurrent growth models. In our main result, we obtain nearly optimal bounds
for the tail of hitting times of some compact sets. If the process is an
aperiodic irreducible Markov chain, we determine whether it is null recurrent
or positive recurrent and in the latter case, we obtain a subgeometric
convergence of its transition kernel to its invariant measure. We apply our
results in particular to state-dependent Galton-Watson processes and we give
precise estimates of the tail of the extinction time.
