Paper detail

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

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 $α\to \infty$. If $μ$ is of order $α$, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log(α)/α$. If $μ$ is of order $α^γ, 0\leq γ\leq 1$, the fixation time is $\sim (2 + (1-γ)Δ) \log(α)/α$, 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 $\sim (2+S)\log(α)/α$, 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.