Open Access
2016 The fixation time of a strongly beneficial allele in a structured population
Andreas Greven, Peter Pfaffelhuber, Cornelia Pokalyuk, Anton Wakolbinger
Electron. J. Probab. 21: 1-42 (2016). DOI: 10.1214/16-EJP3355


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.


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Andreas Greven. Peter Pfaffelhuber. Cornelia Pokalyuk. Anton Wakolbinger. "The fixation time of a strongly beneficial allele in a structured population." Electron. J. Probab. 21 1 - 42, 2016.


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

zbMATH: 1352.92100
MathSciNet: MR3563889
Digital Object Identifier: 10.1214/16-EJP3355

Primary: 92D15
Secondary: 60J80 , 60J85 , 92D10

Keywords: ancestral selection graph , branching process approximation , interacting Wright–Fisher diffusions

Vol.21 • 2016
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