Probability of fixation under weak selection: a branching process unifying approach. (English) Zbl 1121.92051
Summary: We link two-allele population models by J. B. S. Haldane [Proceedings Cambridge 23, 838–844 (1927; JFM 53.0516.05), see also Calcutta Math. Soc. 1958–1959, Part 1, 99–103 (1959; Zbl 0124.12605)] and R. A. Fisher [Proc. Royal Soc. Edinburg, 50, 204–219 (1930; JFM 56.1107.09)] with M. Kimura’s [J. Appl. Probab. 1, 177–232 (1964; Zbl 0134.38103)] 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 \[ u= p+p(1-p) \{g_rs_r+ g_\sigma s_\sigma+ g_cs_c\}+ o(s_r,s_\sigma,s_c), \] 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.
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 \[ u= p+p(1-p) \{g_rs_r+ g_\sigma s_\sigma+ g_cs_c\}+ o(s_r,s_\sigma,s_c), \] 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.
MSC:
| 92D15 | Problems related to evolution |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 60J85 | Applications of branching processes |
Keywords:
Continuous-state branching process; Diffusion theory; Fixation probability; Demographic stochasticity; Mutant allele; Mean-variance trade-off; Weak selectionReferences:
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