The evolution of a sex-linked genetic factor

The genetical theory of evolution is best understood by a knowledge of the distribution of gene frequencies. The sequence of changes in genes, primarily due to effects of mutation, selection, migration, and many other environmental influences, would also gradually change gene frequencies after a long period of time. The distribution of gene frequencies is determined by application of theories of probability and mathematics. In particular, Wright's diffusion theory (14) and early works of Fisher (6,7) and Kolmegorov (9) play a central role. The aim of this thesis is to discover the evolutionary significance of mutation, selection, and random mating in the case of sex-linked factors when the generation structure of the population is overlapping. To facilitate the application of mathematical theory, we assume the population size at any time is large and constant denoted by N. Instead of discussing the individual genotype frequencies, we introduce a properly defined random variable U, approximately the proportion of "a" genes in the population. The first and second moments of the change in U during the birth-death event are obtained. For the diffusion process to work out, we let the time be a function of N, and by moment generating functions the diffusion equation (or Fokker-Planck equation) is justified when N tends to infinity. Following methods of solution given by Barucha-Reid (2), Kimura (8), Li (10), Moran (11,12), Watterson (13), and Wright (14), the density function for the "a" gene frequency is obtained.