Density dependent branching processes as randomly perturbed maps of an interval

Fima Klebaner

Dept of Mathematics and Statistics, University of Melbourne, Australia

Abstract

Branching processes with reproduction dependent on the population density are represented as discrete time dynamics plus noise. Such processes have two parameters: time and the level of noise. Results that describe asymptotic behaviour as time goes to infinity and noise goes to zero are given. It is shown that if the deterministic map has limit cycles one of which is stable, then the quasi-stationary distribution of the process approaches the uniform distribution on the stable cycle.