When the average number of progeny born to the genotypes at a locus are linear functions of a random variable such that, on the average, the number of progeny produced by each genotype is the same (quasineutrality), then there will be a systematic pressure on gene frequency that tends to favor the genotype with the smallest variance in the number of progeny. In particular, there will be a stable equilibrium attained whenever the variance in the number of progeny produced by the heterozygous genotype is less than that of both homozygous genotypes. We assume that the random variable affecting fitness is some function of an underlying environmental variable that tends to smooth out the day-to-day fluctuations in the environmental variable, such as the mean or the range or the variance. The effect of the environment on fitness can, therefore, be assumed to be approximately constant for periods corresponding to one generation. If the logarithm of the average number of progeny is a linear function of a random variable, then in contrast to the foregoing conclusions, there will be no tendency for any change in gene frequency at the locus (Kimura, M. (1954), Process leading to quasi-fixation of genes in natural populations due to random fluctuations of selection intensities, Genetics 39, 280-295; (1964), Diffusion models in population genetics, J. App. Prob. 1, 177-232.).
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* Work supported by National Science Foundation Grant number GB-18786 awarded to D. L. H. Our thanks to Drs. J. F. Crow, R. Comstock, F. Enfield, R. Levins, and R. Nassar for their helpful comments. We are also grateful to the referees for having pointed out several inaccuracies in an early version of the manuscript.