The fixation probability of a beneficial mutation in a geographically structured population

Abstract

One of the most fundamental concepts of evolutionary dynamics is the ‘fixation’ probability, i.e. the probability that a gene spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among themselves. The topology of the migration patterns is believed to influence the spread of a new mutant, but no general analytical results were known for its fixation probability. We show that, for large populations, the fixation probability of a beneficial mutation can be evaluated for any migration pattern between local communities. Specifically, we demonstrate that for large populations, in the framework of the Voter model of the Moran model, the fixation probability is always smaller than or, at best, equal to the fixation probability of a non-structured population. In the ‘invasion processes’ version of the Moran model, the fixation probability can exceed that of a non-structured population; our method allows migration patterns to be classified according to their amplification effect. The theoretical tool we have developed in order to perform these computations uses the fixed points of the probability-generating function which are obtained by a system of second-order algebraic equations.