The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain

There are many stochastic systems arising in areas as diverse as wildlife management, chemical kinetics and reliability theory, which eventually ‘die out,’ yet appear to be stationary over any reasonable time scale. The notion of a quasistationary distribution has proved to be a potent tool in modelling this behaviour. In finite-state systems the existence of a quasistationary distribution is guaranteed. However, in the infinite-state case this may not always be so, and the question of whether or not quasistationary distributions exist requires delicate mathematical analysis. The purpose of this paper is to present simple conditions for the existence of quasistationary distributions for continuous-time Markov chains and to demonstrate how these can be applied in practice.