Stochastic analysis on population genetics models with multiple factors of stochastic force

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The following stochastic models in population genetics were studied by means of the theory of stochastic processes.1. The concept of asymptotic conditional distribution proposed for some diffusion model in population genetics was generalized to the class of generalized one-dimensional diffusion processes and limit theorems on the asymptotic conditional distribution were proved. These theorems were applied to Bessel processes and birth and death processes. These theorems were also used to obtain asymptotic conditional distributions for population genetics models with stochastic factors of random drift and/or stochastic selection.2. The coalescent model and the Wright-Fisher model with mutation were formulated when the population sizes are stochastic processes. Effective sizes of population were defined to these models and their properties were investigated. The relation between these effective sizes and those for the diffusion model with mutation and the Wright-Fisher model without muta … More tion when population sizes are stochastic processes were studied. Effective sizes of the diffusion model and the Wright-Fisher model with mutation were shown to converge to that of the Wright-Fisher model without mutation when the mutation rate tends to zero.3. Extensive computer simulations of compensatory neutral mutation models in population genetics were performed to study the distribution function, the average and the variance of the time until fixation of double mutant to a diploid population. Three-dimensional diffusion models that approximate the discrete-time compensatory neutral mutation models were identified and partial differential equations for the average time until fixation of double mutant were obtained. Since it is difficult to solve these partial differential equations analytically, further approximations were considered for the three-dimensional diffusion models by one-dimensional diffusion models when, the rates of mutation and recombination are much smaller than the intensity of selection. Less