Fitness patterns and phenotypic plasticity in a spatially heterogeneous environment

We analyse patterns of the means and variances of genotypic fitnesses across different niches in a randomly mating haploid population. The population inhabits a spatially heterogeneous environment where it is subject to mutation and weak multilocus additive selection, with different election coefficients in different niches. Approximate analytical expressions are derived for the stationary mean and variance of genotypic fitnesses among the niches in terms of environmental and genetic parameters. As a special case, we analyse an environment described by a variable t, distributed among the niches with mean t(star) and variance D(star) and quadratic decrease in correlation between environments as a function of the difference in values of t. If the niches have the same qualities, the mean and variance of genotypic fitnesses evolve to be quadratic functions of t that achieve their maximum and minimum, respectively, at t(star). With unequal niche qualities, these are non-polynomial functions that attain their extrema at different, usually intermediate values of t, although the coefficient of variation of the genotypic fitnesses still attains its minimum near t(star). The functions involve the total mutation rate, the combination of the loci to genotypic fitnesses, and the frequency and quality distributions of the niches. Thus, for this relatively simple model the norms of reaction may be calculated in terms of the detailed properties of the environmental heterogeneity, and the genetic system.