A study on stochastic models for the interaction between random drift and selection in population genetics

本文研究了种群规模的随机波动对种群有效规模的影响。为此，建立了三个种群规模波动的模型(Wright-Fisher模型、扩散模型、合并模型)，并研究了它们的性质。在文献中，有人说，当人口规模不固定时，有效人口规模等于波动人口规模的调和平均值。然而，加州大学戴维斯分校的J·H·吉莱斯皮发现，当人口规模随机波动时，有效规模不等于调和平均值的一些例子。在种群规模随机波动的情况下，重要的是要知道有效规模与调和平均值和算术平均值的关系，以及有效规模等于调和平均值的条件。结果表明，对于扩散模型和合并模型，有效尺寸大于调和平均，但小于算术平均。对于Wright-Fisher模型，有效尺寸较大(分别为如果种群数量的波动是正的，则小于调和平均值。负)自相关。当且仅当种群规模的波动与世代无关时，有效种群规模等于调和平均值。给出了模型参数中有效尺度渐近等于调和均值或算术均值的条件。

In this study, the effects of the stochastic fluctuation of population size on the effective size of population are investigated. For this end, three models (Wright-Fisher mode, diffusion mode, coalescent model) with the fluctuation of population size are formulated and their properties are studied. In the literature, it is said that the effective size is equal to the harmonic mean of fluctuating population size when population size is not constant. J. H. Gillespie (University of California at Davis) found, however, some examples where the effective size is not equal to the harmonic mean when the population size fluctuates stochastically. In the case where population size fluctuates stochastically, it is important to know how the effective size is related to the harmonic mean and the arithmetic mean and the condition under which the effective size is equal to the harmonic mean. It is shown that the effective size is larger than the harmonic mean but it is smaller than the arithmetic mean for the diffusion model and the coalescent model. For the Wright-Fisher model, the effective size is larger (resp. smaller) than the harmonic mean if the fluctuation of population size is positively (resp. negatively) autocorrelated. The effective size is equal to the harmonic mean if and only if the fluctuation of population size is independent from generation to generation. The conditions on the parameters of the models under which the effective size is asymptotically equal to the harmonic mean or the arithmetic mean are obtained.