The role of asymmetric migration on local adaptation: an analysis of the classic two-habitat migration model

We explore how migration and selection interact to shape local adaptation, revisiting the notion that increased gene flow typically suppresses genetic differentiation and local adaptation. Instead, we demonstrate that migration can both suppress and facilitate local adaptation, depending on the symmetry or asymmetry of migration rates between populations. This duality is examined in two scenarios within a single-locus theory: one in which local adaptation of a pair of alleles is maintained under arbitrary patterns of antagonistic selection, and another in which local adaptation develops through the recurrent invasion and fixation of alleles with weak selective effects on a continuous phenotype. For both scenarios, we analyze the classic two-habitat migration model, providing a novel examination of its dynamics. When migration is symmetric, an equal increase in migration rates impedes local adaptation by forcing both variant alleles to spend more time in non-local habitats where they are disfavored. In contrast, when migration rates increase asymmetrically, leading to skewed gene flow, both alleles spend more time in the same habitat, disadvantaging the non-local allele but favoring the locally adapted one. Thus, both increasing and decreasing migration rates can favor or hinder local adaptation, depending on the interplay between migration patterns and selection, underscoring the importance of migration heterogeneity in shaping local adaptation dynamics.