Population processes with unbounded extinction rate conditioned to non-extinction

This article studies the quasi-stationary behaviour of population processes with unbounded absorption rate, including one-dimensional birth and death processes with catastrophes and multi-dimensional birth and death processes, modeling biological populations in interaction. To handle this situation, we develop original non-linear Lyapunov criteria. We obtain the exponential convergence in total variation of the conditional distributions to a unique quasi-stationary distribution, uniformly with respect to the initial distribution. Our results cover all one-dimensional birth and death processes which come down from infinity with catastrophe rate satisfying appropriate bounds, and multi-dimensional birth and death models with stronger intra-specific than inter-specific competition.