  A discrete time branching process where the offspring distribution is
generation-dependent, and the number of reproductive individuals is controlled
by a random mechanism is considered. This model is a Markov chain but, in
general, the transition probabilities are non-stationary. Under not too
restrictive hypotheses, this model presents the classical duality of branching
processes: either becomes extinct almost surely or grows to infinity.
Sufficient conditions for the almost sure extinction and for a positive
probability of indefinite growth are provided. Finally rates of growth of the
process provided the non-extinction are studied.
