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

We find a Lyapunov-ty pe sufficient condition for discrete-time Markov chains on a countable state space including an absorbing set to almost surely reach this absorbing set and to asymptotically stabilize conditional on nonabsorption. This result is applied to Bienayme-Galton-Watson-like branching processes in which the offspring distribution depends on the current population size. This yields a generalization of the Yaglom limit. The techniques used mainly rely on the spectral theory of linear operators on Banach spaces and especially on the notion of quasi-compact linear operator.