Establishment process of a magic trait allele subject to both divergent selection and assortative mating

Sexual selection and divergent selection are among the major driving forces of reproductive isolation, which could eventually result in speciation. A magic trait is defined such that a single trait is subject to both divergent selection and mate choice through phenotype-based assortative mating. We are here interested in the evolutionary behavior of alleles at a genetic locus responsible for a magic trait in a finite population. We assume that, in a pair of homogeneous subpopulations, a mutant allele arises at the magic trait locus, and theoretically obtain the probability that the new allele establishes in the population, or the establishment probability. We also show an analytical expression for the trajectory of allele frequency along the establishment, from which the time required for the establishment is obtained, or the establishment time. Under this model, divergent selection simply favors the new allele to fix where it is beneficial, whereas assortative mating works against rare alleles. It is theoretically demonstrated that the fate of the new allele is determined by the relative contributions of the two selective forces, divergent selection and assortative mating, when the allele is rare so that the two selective forces counteract. Our theoretical results for the establishment probability and time allow us to understand the relative role of random genetic drift in the establishment process of a magic trait allele in a finite population.