Error bounds for the fundamental and secondary theorems of natural selection

Abstract

Error bounds are derived for the approximation delta Z approximately C/W, where delta Z, C, and W denote the rate of change of the mean of a character under selection, the genic (or additive genetic) covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character), and the mean fitness, respectively. Generations are discrete and nonoverlapping; the monoecious population mates at random. The character is determined by arbitrarily many multiallelic loci without epistasis; the linkage map is also arbitrary. The genotypic values of the character are constant. Rounds on the absolute error in the above approximation are deduced for an arbitrary character, and these are converted to bounds on the relative error when the character is fitness itself. In that case, C is the genic variance in fitness, and the relative error cannot exceed one half the greatest genotypic selection coefficient.