Haldane’s formula in Cannings models: the case of moderately weak selection

Abstract

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.