Evolutionary and convergence stability for continuous phenotypes in finite populations derived from two-allele models.
The evolution of a quantitative phenotype is often envisioned as a trait substitution sequence where mutant alleles repeatedly replace resident ones. In infinite populations, the invasion fitness of a mutant in this two-allele representation of the evolutionary process is used to characterize features about long-term phenotypic evolution, such as singular points, convergence stability (established from first-order effects of selection), branching points, and evolutionary stability (established from second-order effects of selection). Here, we try to characterize long-term phenotypic evolution in finite populations from this two-allele representation of the evolutionary process. We construct a stochastic model describing evolutionary dynamics at non-rare mutant allele frequency. We then derive stability conditions based on stationary average mutant frequencies in the presence of vanishing mutation rates. We find that the second-order stability condition obtained from second-order effects of selection is identical to convergence stability. Thus, in two-allele systems in finite populations, convergence stability is enough to characterize long-term evolution under the trait substitution sequence assumption. We perform individual-based simulations to confirm our analytic results.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Japan. joe@meiji.ac.jp
Joe
J
Yuichiro Wakano
Laurent
L
Lehmann
D000483:Alleles; D005075:Biological Evolution; D003198:Computer Simulation; D005787:Gene Frequency / Q000235:genetics; D008957:Models, Genetic; D010641:Phenotype; D011157:Population Dynamics; D013269:Stochastic Processes