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 (\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 2\log (\alpha )/\alpha $. If $\mu $ is of order $\alpha ^\gamma , 0\leq \gamma \leq 1$, 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 Krone’s ancestral selection graph.
Highlights
The goal of this paper is the asymptotic analysis of the time which it takes for a single strongly beneficial mutant to eventually go to fixation in a spatially structured population
The evolution of type frequencies is modelled by a [0, 1]d-valued diffusion process X = (X(t))t≥0, X(t) = (Xi(t))i=1,...,d, where d ∈ {2, 3, . . .} denotes the number of colonies and Xi(t) stands for the frequency of the beneficial allele B in colony i at time t
A synonymous notion is that of a selective sweep, which alludes to the fact that, after fixation of the beneficial allele B, neutral variation has been swept from the population
Summary
The goal of this paper is the asymptotic analysis of the time which it takes for a single strongly beneficial mutant to eventually go to fixation in a spatially structured population. The starting point for the tools developed in this paper is the ancestral selection graph (ASG) of Neuhauser and Krone [18] This process has been introduced in order to study the genealogy under models including selection. A spatial version of the ancestral selection graph is introduced, and its role in the analysis of the fixation probability and the fixation time by the method of duality is clarified This leads to a proof of Theorem 1 in Sec. 3.10, and to the key Proposition 3.1 which relates the asymptotic distribution of the fixation time of the Wright-Fisher system to that of a marked particle system.
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