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 sigma 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(sigma)s(sigma) + g(c)s(c)] + o(s(r),s(sigma),s(c), where s(r) = r' - r, s(sigma) = sigma-sigma' and s(c) = c - c' are selection coefficients, and g(r), g(sigma), 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(sigma) = 1/sigma and g(r) = z/sigma for small initial population sizes z. In model II, g(r) = z/sigma 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 sigma 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/sigma, where K = r/c is the carrying capacity. For r > 0 the growth invasibility is above z/sigma when z < K, and below z/sigma when z > K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.