Evolutionarily stable sets in the single-locus frequency-dependent model of natural selection

Recent developments in the static theory of evolutionarily stable sets (ESSets) are applied to the single-locus frequency-dependent model of natural selection. Particular emphasis is paid to the ESSet properties of the preimage of an ESS (or ESSet) under the genotype-phenotype map. When an ESS is realized in genetic equilibrium with redundancy in a diploid sexual population, the basic problem in biological terms is whether the corresponding set of allele frequencies is an evolutionarily stable set. The interesting question of the dynamic stability of this preimage is also discussed and a geometric condition developed which implies its evolutionary and dynamic stability.