Establishment of locally adapted mutations under divergent selection

We study the establishment probabilities of locally adapted mutations using a multitype branching process framework. We find a surprisingly simple and intuitive analytical approximation for the establishment probabilities in a symmetric two-deme model under the assumption of weak (positive) selection. This is the first analytical closed-form approximation for arbitrary migration rate to appear in the literature. We find that the establishment probability lies between the weak and the strong migration limits if we condition the origin of the mutation to the deme where it is advantageous. This is not the case when we condition the mutation to first occur in a deme where it is disadvantageous. In this case we find that an intermediate migration rate maximizes the probability of establishment. We extend our results to the cases of multiple demes, two demes with asymmetric rates of gene flow, and asymmetric carrying capacities. The latter case allows us to illustrate how density regulation can affect establishment probabilities. Finally we use our results to investigate the role of gene flow on the rate of local adaptation and identify cases in which intermediate amounts of gene flow facilitate the rate of local adaptation as compared to two populations without gene flow.