A criterion for stability--instability at fixation states involving an eigenvalue one with applications in population genetics

Sex-specific meiotic drive and selection at an imprinted locus

We present a one-locus model that breaks two symmetries of Mendelian genetics. Whereas symmetry of transmission is breached by allowing sex-specific segregation distortion, symmetry of expression is breached by allowing genomic imprinting. Simple conditions for the existence of at least one polymorphic stable equilibrium are provided. In general, population mean fitness is not maximized at polymorphic equilibria. However, mean fitness at a polymorphic equilibrium with segregation distortion may be higher than mean fitness at the corresponding equilibrium with Mendelian segregation if one (or both) of the heterozygote classes has higher fitness than both homozygote classes. In this case, mean fitness is maximized by complete, but opposite, drive in the two sexes. We undertook an extensive numerical analysis of the parameter space, finding, for the first time in this class of models, parameter sets yielding two stable polymorphic equilibria. Multiple equilibria exist both with and without genomic imprinting, although they occurred in a greater proportion of parameter sets with genomic imprinting.

Selection and segregation distortion in a sex-differentiated population

We extend the classical model for selection at an autosomal locus in a sex-differentiated population to include segregation distortion. The equations remain the same, but the fitness parameters are interpreted differently and refer to alleles instead of genotypes. We derive conditions for internal and external stability of the equilibria, i.e., stability with respect to perturbations of alleles that are already present at equilibrium and stability with respect to invasion attempts by newly arising alleles. We show that, in a sex-differentiated population, external stability of an equilibrium can be judged on the basis of Shaw--Mohler criteria. Throughout, we compare the situation in populations with and without sex differentiation. Interestingly, internal stability is more difficult to achieve in a population without sex differentiation than in a population in which selection and segregation distortion are restricted to one sex. In a companion paper we show how the general results of the present paper can lead to new insights into specific systems such as the t complex of the house mouse.

Optimal sex ratios in structured populations

In this paper, we develop a general method to determine evolutionary equilibrium sex ratios and to check evolutionary stability, continuous stability and invadability in exact genetic models with or without dominance. This method is then applied to three kinds of models for structured populations: the first one concerns Hamilton's LMC model, except that only a fraction beta of female offspring mate with male offspring born in the same colonies, while a fraction 1-beta mate with male offspring chosen at random within the whole population; in the second model, it is assumed that partial dispersal of inseminated females occurs after mating; in the third model, partial dispersal of male and female offspring occurs before mating. In the first model, the effect of population regulation is studied while, in the other models, two kinds of dispersal are considered: proportional and uniform.

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.

The role of recombination and selection in the modifier theory of sex-ratio distortion

The equilibrium configurations for a two-locus multialle model of sex-linked meiotic drive are studied with regard to the recombination fraction:limit cycles can occur in the case of small recombination while stable equilibrium points associated with linkage equilibrium can exist for an intermediate range of recombination values depending on the equilibrium sex ratio, linkage disequilibrium at nearby equilibrium points taking turn with loser linkage. The evolutionary dynamics in two-locus sex-ratio distortion systems is enlightened: while equilibria with a sex ratio closer to 1/2 are more likely to be stable with respect to perturbations on the frequencies of sex-ratio distorters that are represented at equilibrium, such equilibria are also more vulnerable to the invasion of mutant distorters when there is some degree of linkage with the sex-determining locus. For X-linked multimodifier systems of sex-ratio distortion, differential fertilities and viabilities are incorporated and a maximum principle is suggested.

Social selection in human populations: sufficient conditions for protection of deleterious alleles in a subdivided population

Population dynamics of wild type (A1) and the deleterious genes (A2) under social selection have been studied by considering a subdivided population where the i-th subpopulation consists of Ni individuals with relative size ci (= Ni/sigma i Ni, i = 1,2, ..., n). A social selection model is constructed by assuming that the fitness of an individual is determined by its own as well as the parental phenotypes and that the number of migrants (M) from the ith subpopulation is divided equally into other subpopulations including the ith subpopulation itself. It has been shown that the gene frequency change depends on the loss of fitness of an individual due to the trait (gamma), an affected parent in the ith subpopulation (beta i), the probability that the heterozygote develops the trait (h), and the migration rates mi (= M/Ni). For 0 less than h less than or equal to 1, a sufficient condition for protection of the deleterious allele from extinction also depends on all of these parameters. However, when mi much less than 1 for all i, the condition is beta i less than gamma/(1 - gamma) for some i, whereas when mi much greater than h[gamma + beta i(1 - gamma)] for all i it is given by sigma i ci beta i less than -gamma/(1 - gamma). When h = 0, that condition is given by sigma ici beta i less than - gamma/(1 - gamma). Analyses also show that, when the deleterious alleles in a population are rare, the relative fitnesses of A1A1, A1A2, and A2A2 are given approximately by 1, 1-hS, and 1 - S, respectively, where S is the harmonic mean of Si = gamma + beta i(1 - gamma). Thus, under mutation-selection balance, the equilibrium frequency of deleterious alleles in the entire population is given by alpha/hS for 0 less than h less than or equal to 1 and square root alpha/S for h = 0, where alpha is the irreversible mutation rate from A1 to A2 in each generation. Population dynamics of rare deleterious genes under social selection can readily be studied by considering a finite population size.