Fixation probabilities for multi-allele Moran dynamics with weak selection

Original Abstract

Fixation probabilities are essential for characterizing stochastic evolutionary dynamics, but analytical results remain limited mainly to systems with two competing types. We develop a perturbative framework to compute fixation probabilities in multi-allele Moran processes under weak selection. Exploiting the general structure of the backward Fokker-Planck operator in this regime, we show that fixation probabilities admit a systematic expansion around their neutral solution. We first introduce the framework in a general case with $M$ competing alleles and arbitrary fitness functions, and then apply it to three biologically motivated examples: a simple model of three competing alleles with a constant fitness function, a coordination game in which allele fitness increases with its frequency in the population, and a model of clonal interference between mutualistic alleles. These results extend the analytical understanding of fixation probabilities beyond pairwise interactions, establishing a framework for investigating multi-strategy stochastic evolutionary dynamics.