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**The fixation time of a strongly beneficial allele in a structured population.**
*(English)*
Zbl 1352.92100

Summary: 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 \(\alpha \rightarrow \infty \).

If \(\mu \) is of order \(\alpha \), the allele fixes (as in the spatially unstructured case) in time \(~ 2\log (\alpha )/\alpha \). If \(\mu \) is of order \(\alpha ^\gamma , 0\leq \gamma \leq 1\), the fixation time is \(~ (2 + (1-\gamma )\Delta ) \log (\alpha )/\alpha \), 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 \(~ (2+S)\log (\alpha )/\alpha \), 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.

If \(\mu \) is of order \(\alpha \), the allele fixes (as in the spatially unstructured case) in time \(~ 2\log (\alpha )/\alpha \). If \(\mu \) is of order \(\alpha ^\gamma , 0\leq \gamma \leq 1\), the fixation time is \(~ (2 + (1-\gamma )\Delta ) \log (\alpha )/\alpha \), 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 \(~ (2+S)\log (\alpha )/\alpha \), 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.

### MSC:

92D10 | Genetics and epigenetics |

92D15 | Problems related to evolution |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60J85 | Applications of branching processes |