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.