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2021 Haldane’s formula in Cannings models: the case of moderately weak selection
Florin Boenkost, Adrián González Casanova, Cornelia Pokalyuk, Anton Wakolbinger
Electron. J. Probab. 26: 1-36 (2021). DOI: 10.1214/20-EJP572


We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new Cannings ancestral selection graph in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane’s formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the probability of fixation is asymptotically (as $N\to \infty $) equal to the selective advantage of haploids $s_{N}$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_{N}$ obeying $N^{-1} \ll s_{N} \ll N^{-1/2} $, which is a regime of “moderately weak selection”. It turns out that for $ s_{N} \ll N^{-2/3} $ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.


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Florin Boenkost. Adrián González Casanova. Cornelia Pokalyuk. Anton Wakolbinger. "Haldane’s formula in Cannings models: the case of moderately weak selection." Electron. J. Probab. 26 1 - 36, 2021.


Received: 18 September 2019; Accepted: 14 December 2020; Published: 2021
First available in Project Euclid: 7 January 2021

Digital Object Identifier: 10.1214/20-EJP572

Primary: 60J10
Secondary: 60F05 , 60J80 , 92D15 , 92D25

Keywords: ancestral selection graph , Cannings model , directional selection , probability of fixation , sampling duality


Vol.26 • 2021
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