Population-size-dependent branching processes in Markovian random environments

Abstract

A branching model ¦Zn¦n≥0 is considered where the offspring distribution of the population’s evolution is not only dependent on the population size, but also controlled by a Markovian environmental process ¦ξ¦ n>-0 For this model, asymptotic behaviour is studied such as\(\mathop {\lim }\limits_{n \to \infty } Z_n \) and\(\mathop {\lim }\limits_{n \to \infty } Z_n /m^n \) in the case that the mean mk,0 of the offspring distribution converges tom > 1 as the population sizek grows to ∞. In the case that ¦ξ¦n≥0 is an irreducible positive recurrent Markov chain, certain extinction (i. c.P(Z n= 0 for somen) = 1) and noncertain extinction (i.e.P(Z n = 0 for somen)< 1) are studied.