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

Abstract

A population-size-dependent branching process {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ > 1. When the environmental sequence {ξ 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.