INBREEDING AND VARIANCE EFFECTIVE POPULATION NUMBERS

Abstract

In this paper, a correction and extension of earlier work, we derive expressions for the inbreeding effective number, Nel, and the variance effective number, NeV, with various models. Diploidy, random mating, and discrete generations are assumed and formulas for NeI are given for six situations: isogamous monoecious populations with self-fertilization permitted or excluded; monoecious populations, male and female gametes distinguished, with self-fertilization permitted or excluded; and separate sexes with or without male and female progeny distinguished. NeV is given for monoecious and separate-sexed populations. Most higher animals have separate sexes, and male and female progeny are distinguished. Letting the subscript t represent the generation in which the inbreeding effect is manifest, the inbreeding effective number is 1/Nel=.sigma.mm,mf+.mu.mm.mu.mf/4Nm,t-2.mu.mm.mu.mf+.sigma.fm,ff+.mu.fm.mu.ff/4Nf,t-2.mu.fm.mu.ff in which .sigma.sm,sf is the covariance of the number of male (m) and female (f) progeny of a parent of sex s (s = m or f), .mu.sm and .mu.sf are the mean numbers of sons and daughters of a parent of sex s, and Ns,t-2 is the number in the grandparental generation. For the variance effective number 1/NeV.apprxeq.1/4NeVm+1/4NeVf in which 1/NeVs=1/2(2Ns,t-1-1)[1/.mu.sm+1/.mu.sf+.sigma.sm2/.mu.sm2+2.sigma.sm,sf/.mu.sm.mu.sf+.sigma.sf2/.mu.sf2] and .sigma.sm2 and .sigma.sf2 are the variances in the number of sons and daughters of a parent of sex s. Observations are made at the same age in each generation.