  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.
