Electronic Journal of Probability

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

Andreas Greven, Peter Pfaffelhuber, Cornelia Pokalyuk, and Anton Wakolbinger

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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.

Article information

Electron. J. Probab. Volume 21, Number (2016), paper no. 61, 42 pp.

Received: 2 March 2014
Accepted: 19 September 2016
First available in Project Euclid: 4 October 2016

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Digital Object Identifier

Primary: 92D15: Problems related to evolution
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 92D10: Genetics {For genetic algebras, see 17D92}

interacting Wright–Fisher diffusions ancestral selection graph branching process approximation

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Greven, Andreas; Pfaffelhuber, Peter; Pokalyuk, Cornelia; Wakolbinger, Anton. The fixation time of a strongly beneficial allele in a structured population. Electron. J. Probab. 21 (2016), paper no. 61, 42 pp. doi:10.1214/16-EJP3355. https://projecteuclid.org/euclid.ejp/1475586182.

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