Results on asymptotic behaviour for discrete, two-patch metapopulations with density-dependent selection
A 4-dimensional system of nonlinear difference equations tracking allele frequencies and population sizes for a two-patch metapopulation model is studied. This system describes intergenerational changes brought about by density-dependent selection within patches and moderated by the effects of migration between patches. To determine conditions which result in similar behaviour at the level of local populations, we introduce the concept of symmetric equilibrium and relate it to properties of allelic and genotypic fitness. We present examples of metapopulation stability, instability and bistability, as well as an example showing that differentially greater migration into a stable patch results in metapopulation stability. Finally, we illustrate a Naimark-Sacker bifurcation giving a globally asymptotically stable invariant curve for the 4-dimensional model.