Selection and mutation at a diallelic X-linked locus.

Equilibria and convergence of gene frequencies are studied in the case of a diallelic X-linked locus under the influence of selection and mutation. The model used is that of an infinite diploid population with nonoverlapping discrete generations and random mating. It is proved that if the mutation rates and fitnesses are constant and the mutation rates are less than one-third, then global convergence of gene frequencies to equilibria occurs. The phase portraits of the dynamical system describing the change of allelic frequencies from one generation to the next are determined. Convergence of gene frequencies is monotone from a certain generation on if every other generation is skipped. In the case without mutation, our proof of this monotone convergence simplifies G. Palm's original proof.

Selection-mutation at a diallelic autosomal locus in a dioecious population.

The population is assumed to be infinite dioecious with nonoverlapping discrete generations and random mating. It is assumed that the fitnesses and mutation rates are constant, heterozygotes are viable and the mutation rates are less than one-half. It is proved that the allelic frequencies converge to equilibria as the number of generations tends to infinity. The a priori types of phase portraits are determined. The method employed is elementary. The results extend those of [1, 2, 5, 8] to the case of selection-mutation rather than pure selection and those of [7] to the case of an autosomal rather than a sex-linked locus.