The fixation time of a strongly beneficial allele in a structured population

Abstract

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately 2 log(α)/α for a large selection coefficient α. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate µ for which the fixation times have different asymptotics as α → ∞. If µ is of order α, the allele fixes (as in the spatially unstructured case) in time ~ 2 log(α)/α. If µ is of order αγ,0 ≤ γ ≤ 1, the fixation time is ~ (2+(1-γ)Δ) log(α)/α, where Δ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If µ = 1/log(α), the fixation time is ~ (2 + S) log(α)/α, where S is a random time in a simple epidemic model. The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone’s ancestral selection graph.