Convergent Evolution During Local Adaptation to Patchy Landscapes

Species often encounter, and adapt to, many patches of similar environmental conditions across their range. Such adaptation can occur through convergent evolution if different alleles arise in different patches, or through the spread of shared alleles by migration acting to synchronize adaptation across the species. The tension between the two reflects the constraint imposed on evolution by the underlying genetic architecture versus how effectively selection and geographic isolation act to inhibit the geographic spread of locally adapted alleles. This paper studies the balance between these two routes to adaptation in a model of continuous environments with patchy selection pressures. We address the following questions: How long does it take for a novel allele to appear in a patch where it is locally adapted through mutation? Or, through migration from another, already adapted patch? Which is more likely to occur, as a function of distance between the patches? What population genetic signal is left by the spread of migrant alleles? To answer these questions we examine the family structure underlying migration–selection equilibrium surrounding an already adapted patch, treating those rare families that reach new patches as spatial branching processes. A main result is that patches further apart than a critical distance will likely evolve independent locally adapted alleles; this distance is proportional to the spatial scale of selection (, where σ is the dispersal distance and sm is the selective disadvantage of these alleles between patches), and depends linearly on log(sm/μ), where μ is the mutation rate. This provides a way to understand the role of geographic separation between patches in promoting convergent adaptation and the genomic signals it leaves behind. We illustrate these ideas using the convergent evolution of cryptic coloration in the rock pocket mouse, Chaetodipus intermedius, as an empirical example.