The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.
W. J. Ewens, following G. R. Price, has stressed that Fisher's fundamental theorem of natural selection about the increase in mean fitness is of general validity without any restrictive assumptions on the mating system, the fitness parameters, or the numbers of loci and alleles involved, but that it concerns only a partial change in mean fitness. This partial change is obtained by replacing the actual genotypic fitnesses by the corresponding additive genetic values and by keeping these values fixed in the change of the mean with respect to changes in genotype frequencies. We propose an alternate interpretation for this partial change which uses partial changes in genotype frequencies directly consequent on changes in gene frequencies, the fitness parameters being kept constant. We argue that this interpretation agrees more closely with Fisher's own explanations. Moreover, this approach leads to a decomposition for the total change in mean fitness which explains, unifies, and extends previous decompositions. We consider a wide range of models, from discrete-time selection models with nonoverlapping generations to continuous-time models with overlapping generations and age effects on viability and fecundity, which is the original framework for Fisher's fundamental theorem.] 1997 Academic Press
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