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

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 rr, reproduction variance σσ and competition sensitivity cc. Generally, the fixation probability uu of the mutant depends on its initial proportion pp, the total initial population size zz, and the six demographic traits. Under weak selection, we can linearize uu in all models thanks to the same master formulau=p+p(1-p){grsr+gσsσ+gcsc}+o(sr,sσ,sc),u=p+p(1-p){grsr+gσsσ+gcsc}+o(sr,sσ,sc),where sr=r′-rsr=r′-r, sσ=σ-σ′sσ=σ-σ′ and sc=c-c′sc=c-c′ are selection coefficients, and grgr, gσgσ, gcgc are invasibility coefficients (′′ refers to the mutant traits), which are positive and do not depend on pp. In particular, increased reproduction variance is always deleterious. We prove that in all three modelsgσ=1σ,and gr=z/σgr=z/σ for small initial population sizes zz.In model II, gr=z/σgr=z/σ for all zz, 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 σσ than with high rr.In model III, gcgc increases with zz like ln(z)/cln(z)/c, and gr(z)gr(z) converges to a finite limit L>K/σL>K/σ, where K=r/cK=r/c is the carrying capacity. For r>0r>0 the growth invasibility is above z/σz/σ when zKz>K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.