Effective size of populations under selection.

Equations to approximate the effective size (Ne) of populations under continued selection are obtained that include the possibility of partial full-sib mating and other systems such as assortative mating. The general equation for the case of equal number of sexes and constant number of breeding individuals (N) is Ne = 4N/[2(1 - alpha I) + (Sk2 + 4Q2C2) (1 + alpha I + 2 alpha O)], where Sk2 is the variance of family size due to sampling without selection, C2 is the variance of selective advantages among families (the squared coefficient of variation of the expected number of offspring per family), alpha I is the deviation from Hardy-Weinberg proportions, alpha O is the correlation between genes of male and female parents, and Q2 is the term accounting for the cumulative effect of selection on an inherited trait. This is obtained as Q = 2/[2 - G(1 + r)], where G is the remaining proportion of genetic variance in selected individuals and r is the correlation of the expected selective values of male and female parents. The method is also extended to the general case of different numbers of male and female parents. The predictive value of the formulae is tested under a model of truncation selection with the infinitesimal model of gene effects, where C2 and G are a function of the selection intensity, the heritability and the intraclass correlation of sibs. Under random mating r = alpha I = -1/(N - 1) and alpha O = 0. Under partial full-sib mating with an average proportion beta of full-sib matings per generation, r approximately beta and alpha O approximately alpha I approximately beta/(4 - 3 beta). The prediction equation is compared to other approximations based on the long-term contributions of ancestors to descendants. Finally, based on the approach followed, a system of mating (compensatory mating) is proposed to reduce rates of inbreeding without loss of response in selection programs in which selected individuals from the largest families are mated to those from the smallest families.