Fixation of a homozygote and dynamical characterization of the Price equation

Price equation and genotype dynamics are two methods for studying the fixation of one allele by natural selection in a diploid population. There are two strict monotonicity conditions that imply the fixation of one allele. The genotype dynamics is called Haldane monotone if the relative frequency of one allele strictly increases along all solutions of the genotype dynamics, so this allele is fixed. In this paper, we show that the genotype dynamics is Haldane monotone if and only if the right-hand side of the Price equation is always strictly positive. The other strict monotonicity condition requires that the relative frequency of a homozygote strictly increase according to the genotype dynamics. For example, in a model where the genotype dynamics is governed by interactions between individuals, the cost-accepting homozygote is fixed by natural selection if the other genotypes always receive a smaller average gain from all interactions than the cost-accepting homozygote. Both monotonicity conditions require that the interaction is not well-mixed in the population. These two conditions are not equivalent. In addition, we give a non-monotonicity condition, which also implies the fixation of a homozygote. The fixation of a homozygote depends on the phenotypic payoff of the interaction, the genotype-phenotype mapping, and the interaction scheme. In a sexual population, the interaction scheme of siblings depends on the mating system, and so do the conditions of fixation of the cost-accepting homozygote. We present examples showing that if we only change the monogamous mating system, assuming panmixing or mating assortativity, then the condition for the fixation of the cooperator homozygote is b > 2c and b > c, respectively.