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

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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.