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 $2\log (\alpha )\mathrm{/}\alpha$ for a large selection coefficient $\alpha$. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $\mu$ for which the fixation times have different asymptotics as $\alpha \to \mathrm{\infty }$. If $\mu$ is of order $\alpha$, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log (\alpha )\mathrm{/}\alpha$. If $\mu$ is of order ${\alpha }^{\gamma },0\le \gamma \le 1$, the fixation time is $\sim (2 + (1-\gamma)\Delta) \log(\alpha)/\alpha$,where$\Delta$ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $\mu =1\mathrm{/}\log (\alpha )$, the fixation time is $\sim (2+S)\log (\alpha )\mathrm{/}\alpha$, 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.