Modelling quasi-stationary behaviour in metapopulations

We consider a Markovian model proposed by Gyllenberg and Silvestrov for studying the behaviour of a metapopulation: a population that occupies several geographically separated habitat patches. Although the individual patches may become empty through extinction of local populations, they can be recolonized through migration from other patches. There is considerable empirical evidence (in the work of Gilpin and Hanski, for example) which suggests that a balance between migration and extinction is reached which enables these populations to persist for long periods. The Markovian model predicts extinction in a finite time. Thus, there has been considerable interest in developing methods which account for the persistence of these populations and which provide an effective means of studying their long-term behaviour before extinction occurs. We shall compare and contrast the methods of Gyllenberg and Silvestrov (pseudo-stationary distributions) and those of Day and Possingham, which are based on the classical notion of a quasi-stationary distribution. We present here a convincing rationale for the latter, using limits of conditional probabilities.