The growth of general population-size-dependent branching processes year by year

We study discrete-time population models where the nearest future of an ind ividual may depend on the individual's life-stage (age and reproduction his tory) and the current population size. A criterion is given for whether the re is a positive probability that the population survives forever. We ident ify the cases when population size grows exponentially and linearly and sho w that in the latter population size scaled by time is asymptotically Gamma -distributed.