Abstract: We shall examine a model which can be used to describe the long-term behaviour of populations that are subject to large-scale mortality or emigration. Populations can suffer dramatic losses when disease, such as an introduced virus, affects the population, or when food shortages occur, perhaps due to overbrowsing or significant changes in climate. However, perhaps surprisingly, such populations can survive for rather 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 which 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 distribution of population size. Our starting point is a paper by Klein, which studies populations of reindeer and moose. After their introduction to Alaska, these populations suffered substantial reductions in numbers due to overbrowsing and the effects of severe winters; the moose population was additionally subject to Spruce-Budworm infestation and later fire. We shall see that although our model predicts eventual extinction, the time till extinction can be long and the stationarity exhibited by these populations over any reasonable time scale can be explained using a quasistationary distribution.
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Last modified: 6th November 1999