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

本文运用随机过程理论研究了种群遗传学中的下列随机模型：将群体遗传学中某些扩散模型的渐近条件分布概念推广到广义一维扩散过程，并证明了渐近条件分布的极限定理。这些定理应用于贝塞尔过程和生灭过程。这些定理也被用来获得具有随机漂移和/或随机选择的随机因素的群体遗传模型的渐近条件分布。当种群规模是随机过程时，建立了聚结模型和带突变的Wright-Fisher模型。定义了这些模型的有效种群大小，并对其性质进行了研究。研究了种群规模为随机过程时，这些有效规模与突变扩散模型和无突变Wright-Fisher模型的有效规模之间的关系。当突变率趋于零时，扩散模型和突变的Wright-Fisher模型的有效尺寸收敛于无突变的Wright-Fisher模型。对群体遗传学中的补偿中性突变模型进行了广泛的计算机模拟，研究了双突变体固定到二倍体群体所需时间的分布函数、平均值和方差。建立了近似于离散时间补偿中性突变模型的三维扩散模型，得到了双突变体固定平均时间的偏微分方程。由于这些偏微分方程难以解析求解，当突变和重组的速率远小于选择的强度时，可以考虑用一维扩散模型进一步逼近三维扩散模型。少

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