Evolutionary dynamics in the two-locus two-allele model with weak selection

Abstract

Two-locus two-allele models are among the most studied models in population genetics. The reason is that they are the simplest models to explore the role of epistasis for a variety of important evolutionary problems, including the maintenance of polymorphism and the evolution of genetic incompatibilities. Many specific types of models have been explored. However, due to the mathematical complexity arising from the fact that epistasis generates linkage disequilibrium, few general insights have emerged. Here, we study a simpler problem by assuming that linkage disequilibrium can be ignored. This is a valid approximation if selection is sufficiently weak relative to recombination. The goal of our paper is to characterize all possible equilibrium structures, or more precisely and general, all robust phase portraits or evolutionary flows arising from this weak-selection dynamics. For general fitness matrices, we have not fully accomplished this goal, because some cases remain undecided. However, for many specific classes of fitness schemes, including additive fitnesses, purely additive-by-additive epistasis, haploid selection, multilinear epistasis, marginal overdominance or underdominance, and the symmetric viability model, we obtain complete characterizations of the possible equilibrium structures and, in several cases, even of all possible phase portraits. A central point in our analysis is the inference of the number and stability of fully polymorphic equilibria from the boundary flow, i.e., from the dynamics at the four marginal single-locus subsystems. The key mathematical ingredient for this is index theory. The specific form of epistasis has both a big influence on the possible boundary flows as well as on the internal equilibrium structure admitted by a given boundary flow.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-017-1140-7) contains supplementary material, which is available to authorized users.

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

“…In simpler words, we assume that genotypes at different loci may produce a combined effect on the fitness of the individual is greater than the sum of their single effects. A general treatment of this case in the weak-selection limit (hence neglecting the spatial dependence) has been recently presented in [38].…”

Eco-Evolutionary Feedbacks Between Predator’s Linkage Disequilibrium and Prey Densities Maintain Diversity

“…The latter may have the advantage of increasing the genetic variance, and thus accelerating the response to selection, but it also imposes a load by breaking up haplotypes carrying favorable allele combinations, thus reducing fitness. Additionally, epistasis, i.e., non-additive interactions amongst locus effects on traits or fitness, and differences in locus effects are common (Phillips, 2008) and may sometimes elevate or generate intraspecific variation (e.g., Gimelfarb, 1989;Nagylaki, 1989;Hastings and Hom, 1990;Gavrilets and Hastings, 1993;Bürger, 2000;Hermisson et al, 2003;Turelli and Barton, 2006;Pontz et al, 2018).…”

Bimatrix games that include interaction times alter the evolutionary outcome: The Owner–Intruder game

“…In Section 3, this includes the distribution of pairs and singles 13 The difficulty of doing such an analysis can be appreciated by considering the complete analysis for the two-locus two-allele viability selection model of population genetics Pontz et al (2018). show that this two-dimensional dynamics on the unit square has at least 192 different phase portraits.…”