Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

A population-size-dependent branching process {Z(n)} is considered where the population's evolution is controlled by a Markovian environment process {xi(n)}. For this model, let m(k,theta) and sigma(k,theta)(2) be the mean and the variance respectively of the offspring distribution when the population size is k and a environment theta is given. Let B = {omega : Z(n)(omega) = 0 for some n} and q = P(B). The asymptotic behaviour of lim(n) Z(n) and lim(n) Z(n)/Pi(i=0)(n-1)m(xi n) is studied in the case where sup(theta) /m(k,theta) - m(theta)/ --> 0 for some real numbers {m(theta)} such that inf(theta) m(theta) > 1. When the environmental sequence {xi(n)} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.