The Rate of Change of a Character Correlated with Fitness

TLDR: An extended form of Fisher's Fundamental Theorem of Natural Selection gives the rate of change of the mean value of a measured character, which reduces to Kimura's generalization of Fisher’s Fundamental The theorem.

Abstract: An extended form of Fisher's Fundamental Theorem of Natural Selection gives the rate of change of the mean value, [Formula: see text], of a measured character. For a character determined by multiple alleles at two loci, this is [Formula: see text] where the Newtonian superior dot means the time derivative and the circle is the time derivative of the logarithm. Covg (m, γ) is the genic (additive genetic) covariance of the character and fitness. Specifically, it is the covariance of the average excess of an allele for fitness and its average effect on the character. [Formula: see text] is the average rate of change of the value of the character for individual genotypes, weighted by their frequencies. The value could be nonzero because of changing environments or change in the age distribution of the population. The third term on the right is the average over all pairs of alleles at both loci of the product of the dominance deviation and the rate of change of ln θ(n), where θ(n) is a measure of departure from random proportions. The last term is a similar expression for epistatic interactions. If selection is much weaker than recombination, after several generations, the last two terms are much smaller than the first. When the measured character is fitness, our result reduces to Kimura's generalization of Fisher's Fundamental Theorem of Natural Selection.