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 $2log(alpha)/alpha$ 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 $alpha to infty$. If $mu$ is of order $alpha$, the allele fixes (as in the spatially unstructured case) in time $sim 2log(alpha)/alpha$. If $mu$ is of order $alpha^gamma, 0leq gamma leq 1$, 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 $mu = 1/log(alpha)$, the fixation time is $sim (2+S)log(alpha)/alpha$, 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 Krones ancestral selection graph.