• Diffusion;
  • environmental variance;
  • expected selection gradient;
  • frequency dependence;
  • long-run growth rate;
  • Malthusian fitness

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, inline image. The expected evolution of p or inline image is a product of genetic variability and the gradient of the long-run growth rate of the population, inline image, with respect to p or inline image. This shows that the expected evolution maximizes inline image, the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, inline image as a function of p or inline image represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of inline image. Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.