Evolution of migration in a periodically changing environment

The ability to migrate can evolve in response to various forces. In particular, when selection is heterogeneous in space but constant in time, local adaptation induces a fitness cost on immigrants and selects against migration. The evolutionary outcome, however, is less clear when selection also varies temporally. Here, we present a two-locus model analyzing the effects of spatial and temporal variability in selection on the evolution of migration. The first locus is under temporally varying selection (various periodic functions are considered, but a general nonparametric framework is used), and the second locus is a modifier controlling migration ability. First, we study the dynamics of local adaptation and derive the migration rate that maximizes local adaptation as a function of the speed and geometry of the fluctuations in the environment. Second, we derive an analytical expression for the evolutionarily stable migration rate. When there is no cost of migration, we show that higher migration rates are favored when selection changes fast. When migration is costly, however, the evolutionarily stable migration rate is maximal for an intermediate speed of the variation of selection. This model may help in understanding the evolution of migration in a broad range of scenarios and, in particular, in host-parasite systems, where selection is thought to vary quickly in both space and time.