Probability of fixation under weak selection: A branching process unifying approach

Abstract

We link two-allele population models by Haldane and Fisher with Kimura's diffusion approximations of the Wright–Fisher model, by considering continuous-state branching (CB) processes which are either independent (model I) or conditioned to have constant sum (model II). Recent works by the author allow us to further include logistic density-dependence (model III), which is ubiquitous in ecology. In all models, each allele (mutant or resident) is then characterized by a triple demographic trait: intrinsic growth rate r , reproduction variance σ and competition sensitivity c . Generally, the fixation probability u of the mutant depends on its initial proportion p , the total initial population size z , and the six demographic traits. Under weak selection, we can linearize u in all models thanks to the same master formula u = p + p ( 1 - p ) { g r s r + g σ s σ + g c s c } + o ( s r , s σ , s c ) , where s r = r ′ - r , s σ = σ - σ ′ and s c = c - c ′ are selection coefficients, and g r , g σ , g c are invasibility coefficients ( ′ refers to the mutant traits), which are positive and do not depend on p . In particular, increased reproduction variance is always deleterious. We prove that in all three models g σ = 1 σ , and g r = z / σ for small initial population sizes z . In model II, g r = z / σ for all z , and we display invasion isoclines of the ‘mean vs variance’ type. A slight departure from the isocline is shown to be more beneficial to alleles with low σ than with high r . In model III, g c increases with z like ln ( z ) / c , and g r ( z ) converges to a finite limit L > K / σ , where K = r / c is the carrying capacity. For r > 0 the growth invasibility is above z / σ when z < K , and below z / σ when z > K , showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.