How Levins' dynamics emerges from a Ricker metapopulation model on the brink of extinction

Understanding the dynamics of metapopulations close to extinction is of vital importance for management. Levins-like models, in which local patches are treated as either occupied or empty, have been used extensively for this purpose, but they ignore the important role of local population dynamics. In this paper, we consider a stochastic metapopulation model where local populations follow a Ricker dynamics, and use this framework to investigate the behaviour of the metapopulation at the brink of extinction. As long as dispersal rates are not too large, the system is shown to have a time evolution consistent with Levins' dynamics. We derive analytical expressions for the colonisation and extinction rates (c and e) in Levins-type models in terms of reproduction, survival, and dispersal parameters of the local populations, providing an avenue to parameterising Levins-like models from the type of information on local demography that is available for a number of species. To facilitate applying our results, we provide a numerical implementation for computing c and e.