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

Abstract

AbstractFor a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.