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

We link two-allele population models by Haldane and Fisher with Kimuraʹs 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 formulau=p+p(1-p){grsr+gσsσ+gcsc}+o(sr,sσ,sc),where sr=r′-r, sσ=σ-σ′ and sc=c-c′ are selection coefficients, and gr, gσ, gc 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 and gr=z/σ for small initial population sizes z. In model II, gr=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, gc increases with z like ln(z)/c, and gr(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 zK, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.