Further analysis of negative assortative mating.

HE paper “Ana1ys.k of models with homozygote x heterozygote matings“ was devoted to the complete mathematical analysis of the equilibrium behavior of some models proposed by SCUDO (1964) and FINNEY (1952) and some of their qualitative implications for population genetics in general. In this paper we repeat this process with certain models described by WORKMAN ( 1964), NAYLOR (1962) and others. Originally, WORKMAN used the standard methods of linear infinitesimal approximations (see OWEN 1953) to verify local stability of certain equilibria. The inadequacies of local linear analysis were mentioned in the previous paper. As emphasized in the previous paper, a global analysis, whenever possible, provides more information, not only in the strictly mathematical sense but qualitatively as well. WORKMAN was interested in the fact that models involving (a) exclusively negative assortative mating and (b) mixed self-fertilization and negative assortative mating and (c) mixed random mating and negative assortative mating could explain the maintenance of polymorphic variation in the absence of heterozygote advantage. In this note we will be concerned with those models having exclusively negative assortative mating. Some of the theoretical analysis and implications of the mixed models will be treated elsewhere (BODMER, FELDMAN and KARLIN, manuscript in preparation). We are concerned with a single locus at which the alleles A and a occur. The models are specified by the matings which are forbidden: In model I only unlike genotypes may mate; in model 11, like homozygotes may not mate but the heterozygotes can mate at random; in model I11 matings between like members of one homozygote genotype, as well as those between pairs of heterozygotes are forbidden; in model IV only matings between pairs of heterozygotes are prohibited; in model V matings between like members of one homozygote genotype are forbidden. In the terminology of FINNEY (1952) the models discussed here are of the pollen elimination kind. Zygotes are produced by a union of two gametes, the first one is chosen at random, while the second is chosen at random from among only those individuals with whom mating is permitted. This could be the case in a plant population producing excess pollen, where pollen whose phenotype makes it incompatible fails to germinate. For the biological justification of this type of model the reader is referred to FINNEY (1952), MORAN (1962, Chap. 8)