Ecological interactions can dramatically alter evolutionary outcomes in complex communities. Yet, the classic theoretical results of population genetics (e.g., Kimura's fixation formula) largely ignore ecological effects. Here, we address this shortcoming by using dynamical mean-field theory to integrate ecology into classical population genetics models. We show that ecological interactions between parents and mutants result in frequency-dependent selection and can be characterized by a single emergent parameter that measures the strength of ecological feedbacks. We derive an explicit analytic expression for the fixation probability that generalizes Kimura's famous result and analyze it in various ecological and evolutionary limits. We find that ecological interactions suppress fixation probabilities for moderately beneficial mutants when compared to Kimura's predictions, with the strength of suppression increasing with larger effective population sizes, greater selective differences between parent and mutant, and for ecosystems with a large number of "open niches" (i.e., ecosystems well below the packing bound). Frequency-dependent selection also gives rise to prolonged parent-mutant coexistence in complex communities, a phenomenon absent in classical population genetics. Our study establishes a framework for integrating ecological interactions into population genetics models and helps illuminate how ecology can affect evolutionary outcomes.
Li et al. (Sun,) studied this question.