Fast-mode elimination in stochastic metapopulation models

We investigate the stochastic dynamics of entities which are confined to a set of islands, between which they migrate. They are assumed to be one of two types, and in addition to migration, they also reproduce and die. Birth and death events are later moderated by weak selection. Systems which fall into this class are common in biology and social science, occurring in ecology, population genetics, epidemiology, biochemistry, linguistics, opinion dynamics, and other areas. In all these cases the governing equations are intractable, consisting as they do of multidimensional Fokker-Planck equations or, equivalently, coupled nonlinear stochastic differential equations with multiplicative noise. We develop a methodology which exploits a separation in time scales between fast and slow variables to reduce these equations so that they resemble those for a single island, which are amenable to analysis. The technique is generally applicable, but we choose to discuss it in the context of population genetics, in part because of the extra features that appear due to selection. The idea behind the method is simple, its application is systematic, and the results are in very good agreement with simulations of the full model for a range of parameter values.