The response of a metapopulation to a changing environment

A species that is distributed across heterogeneous environments may adapt to local conditions. Szep et al (Evolution, 2021) modelled this process in the infinite island model, finding the stationary distribution of allele frequencies and deme sizes. We extend this to ask how a metapopulation responds to changes in carrying capacity, selection strength, or migration rate, restricting attention to fixed deme size ("soft selection"). We develop a "fixed-state" approximation (accurate when migration is rare) which assumes that the loci are near fixation. Under this approximation, polymorphism is only possible for a narrow range of habitat proportions when selection is weak compared to drift, but for a much wider range otherwise. When local conditions (Ns or Nm) change in a single deme of the metapopulation, it takes the population a time of order 1/m to reach the new equilibrium. However, even with many loci, there can be substantial fluctuations in net adaptation, due to the bimodal allele frequency distributions at each locus. Thus, in a finite metapopulation, variation may gradually be lost by chance, even if it would persist if there were infinitely many demes. When conditions change across the whole metapopulation, there can be rapid change, which is predicted well by the fixed-state approximation when Nm<<1.