Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology (Q1872476)

Discrete-time Markov chains on a countable state space are considered which include an absorbing set. A Lyapunov-type sufficient condition is given for the Markov chain to almost surely reach the absorbing set and to asymptotically stabilize conditional on nonabsorption, hence generalizing results about the existence of a Yaglom limit. The results are applied to population-size-dependent Bienaymé-Galton-Watson branching processes in which the offspring distribution depends on the current population size. The results are based on linear operator theory, and especially on the notion of quasi-compact operator. This imposes a rather stringent assumption on the Markov chain but leads to the formulation of sufficient conditions that appear to be biologically realistic and verifiable.