Limiting Distribution under Assortative Mating*

A multi-locus model for complete positive assortative mating is discussed. For a two-locus model, if the gene frequencies for the two loci are different, as they are likely to be, it is shown that in equilibrium the population is not composed of only two homozygous types, as is usually thought. The limiting distribution will have three homozygous genotypes depending upon the initial gene frequencies. If there are m-loci such that gene frequencies at all loci are different, there will be (m+l) such homozygous genotypes present in the equilibrium population, one in each phenotypic group. SSORTATIVE mating is a form of nonrandom mating where mating is based A on the phenotypic properties of mates. This tendency to mate assortatively is known to occur in certain bird, mammal, and insect populations. In human populations where homogamy prevails, conventional barriers greatly restrict the choice of a mate. Hence human parents tend to resemble one another in heritable characters, e.g., height, intelligence, etc. The simplest model of complete positive assortative mating for a single locus with complete dominance was discussed initially by JENNINGS (1916). These results were later extended by WENTWORTH and REMICK (1916) who gave a more general formula. Positive assortative mating leads to complete homozygosity of a population, though very slowly. In the case of a single locus with alleles A and a and complete dominance, the frequency of Aa individuals after n generations of positive assortative mating is given by (e.g., LI 1955, p. 234) where H , is the initial heterozygosity, and P A is the gene frequency of A which is invariant over time. If there is no dominance the results are similar to those under selfing. In the limit as n -+ m, N, -+ 0, and AA and aa individuals occw with frequency pa and pa, respectively. Consider now a population segregating with respect to m loci, each with two alleles. We assume that there is no dominance at either locus, and that the effects of different loci are equal and additive. Let us define {fEi. . ,}, n = 0,1,2, . . . , to be the probability distribution of genotypes in generation n, where i = 0,1,2, de* Journal Paper No. J-7399 of the Iowa Agriculture and Home Economics Experiment Station, Project 1669. Partial support provided by National Institute of Health Grant No. GM 13827. Genetics 71: 777-732 December, 1973.