Quasi-stationarity in populations that are subject to large-scale mortality or emigration

We shall examine a model, first studied by Brockwell et al., which can be used to describe the long-term behaviour of populations that are subject to catastrophic mortality or emigration events. Populations can suffer dramatic declines when disease, such as an introduced virus, affects the population, or when food shortages occur, due to overgrazing or fluctuations in rainfall. However, perhaps surprisingly, such populations can survive for long periods and, although they may eventually become extinct, they can exhibit an apparently stationary regime. It is useful to be able to model this behaviour. This is particularly true of the ecological examples that motivated the present study, since, in order to properly manage these populations, it is necessary to be able to predict persistence times and to estimate the conditional probability distribution of population size. We shall see that although our model predicts eventual extinction, the time till extinction can be long and the stationary exhibited by these populations over any reasonable time scale can be explained using a quasi-stationary distribution.