Natural Selection for a Quantitative Character over Several Generations

Abstract

THERE are many plausible models to describe how selection might affect the values of a quantitative character. Perhaps the most generally useful is a quadratic model in which fitness decreases in proportion to the square of the deviation of the character from an optimum value. If w is the fitness of an organism when the character has the value x, then w=1−α− K(θ−x)² where θ is the optimum value of the character and K is a constant. Thus at x=θ, w=1−α and fitness is at a maximum. From this model the values of Δx̄ and ΔV_x, the changes in the mean and variance after selection, can be predicted¹. Conversely, if Δx̄ and ΔV_x are known, then the values of the parameters, α, K and θ, and the change in mean fitness can be calculated². The proportionate change in mean fitness, Δw̄/w̄, is a general measure of the effect of selection. While selection is acting only on phenotypes and there has been no mating, Δw̄/w̄=V_w/w̄². But if sexual reproduction takes place after the selection, the formulae for Δx̄, ΔV_x and Δw̄/w̄ may no longer be valid. Changes that affect environmental variations will not be passed on to the next generation: changes in the numbers of genotypes will be subject to genetic segregation and recombination. The variance may increase or decrease depending on the frequencies of the genes. In this communication, formulae are derived for the changes in the mean and variance of a character after selection from one generation to the next. They depend on a simple model of the inheritance of the character.