Mean fitness and the equilibria in multilocus polymorphisms

Abstract

A precise theorem is given for the increase in fitness due to natural selection on diploids subject to random mating, non-overlapping generations and not more than two loci; the method of extension to more loci is given by Kojima & Kelleher, and a precise theorem is given here for any number of loci when there is no recombination. The increase is equal to the haploid (or genic) variance in fitness, multiplied by a factor which is equal to two in the absence of dominance, but which otherwise is a function of gene frequency and dominance. The theorem is compared with that of Kimura, which is more general but harder to apply, and to those of Kojima & Kelleher and Fisher, which are respectively restricted to slow selection and absence of epistasis. The new theorem is used to predict the equilibria in populations polymorphic for two loci, and to deal especially with the quasi-stable equilibrium, for which the critical value of recombination is formulated, and the through point, at which a stable and unstable equilibrium meet and annihilate each other. The effect of this in space is to produce a stepped cline, in which gene frequencies and gametic excess change suddenly over a short distance; in time, the through point brings a new slant to Wright’s multiple peak theory of evolution, as populations can move precipitately from peak to peak without the help of random processes. Mean fitness is related only indirectly to population density. By distinguishing carefully between mean absolute fitness (which is the rate of population growth) and mean relative fitness (which is more useful than the absolute parameter for predicting genetical equilibria) we can show the effects of various types of density control on the genetical composition of the population; density dependent selection may appear to be gene-frequency dependent. The fundamental law of evolution is probably a thermodynamic law of increasing matter energy, which is related only tenuously to the law of increasing genetical fitness.