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Keywords:

  • diffusion process;
  • environmental variance;
  • expected relative fitness;
  • Haldane–Jayakar model;
  • long-run growth rate;
  • stochastic environment

Abstract

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), inline image, where r is the mean Malthusian fitness in the average environment and inline image is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane–Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.