Table 2 Definition of parameters

\({BV}_{sire}\)

Breeding value of a sire

\({BV}_{dam}\)

Breeding value of a dam

\({BV}_{animal}\)

Breeding value of an animal

\({\sigma }_{{gamMS}_{sire}}^{2}\)

Gametic Mendelian sampling variance of a sire

\({\sigma }_{{gamMS}_{dam}}^{2}\)

Gametic Mendelian sampling variance of a dam

\({\sigma }_{{gamMS}_{animal}}^{2}\)

Gametic Mendelian sampling variance of an animal

\({\sigma }_{BV}^{2}\)

Variance of breeding values in the population in the current generation

\(\overline{BV }\)

Average breeding value of the population in the current generation

\({i}_{p}\)

Selection intensity belonging to selected proportion \(p\) of the population

\({x}_{p}\)

Normalized truncation selection point belonging to selected proportion \(p\) of the population

\({{\tau }_{p}}_{O}\)

Absolute truncation selection point in the offspring generation based on selected proportion \(p\) of the population; is equal to \(\overline{BV }+\left({i}_{p}+{x}_{p}\right)\sqrt{{\sigma }_{BV}^{2}}\)

\({{\tau }_{p}}_{GO}\)

Absolute truncation selection point in the grand-offspring generation based on selected proportion \(p\) of the population; is equal to \(\overline{BV }+\left({2i}_{p}+{x}_{p}\right)\sqrt{{\sigma }_{BV}^{2}}\)

\({\overline{BV} }_{sel \, off}\)

Average breeding value of selected offspring of a mating. This is identical to ExpBVSelOff

\({\overline{BV} }_{{sel \, ind}_{t}}\)

Average breeding value of selected individuals of the population in the current generation (\(t\)); is equal to \(\overline{BV }+{i}_{p}\sqrt{{\sigma }_{BV}^{2}}\)

\({\overline{BV} }_{{sel \, ind}_{t+1}}\)

Average breeding value of selected individuals of the population in the next generation (\(t+1\)); is equal to \(\overline{BV }+{2i}_{p}\sqrt{{\sigma }_{BV}^{2}}\)

\({\sigma }_{{gam}_{sel \, off}}^{2}\)

Variance of breeding values of gametes produced by the group of selected offspring of a specific mating

\({\overline{\sigma }}_{{gamMS}_{off}}^{2}\)

Average gametic Mendelian sampling variance of offspring of a mating

\({\sigma }_{FS}^{2}\)

Variance of breeding values of full sib offspring of a mating; is equal to \({\sigma }_{{gamMS}_{sire}}^{2}+{\sigma }_{{gamMS}_{dam}}^{2}\)

\({\sigma }_{{gam}_{sel \, ind t}}^{2}\)

Variance of breeding values of gametes produced by the group of selected individuals of the population in the current generation (\(t\))

\({\sigma }_{{gam}_{sel \, ind t+1}}^{2}\)

Variance of breeding values of gametes produced by the group of selected individuals of the population in the next generation (\(t+1\)). Here, we assume \({\sigma }_{{gam}_{sel \, ind t+1}}^{2}={\sigma }_{{gam}_{sel \, ind t}}^{2}\)

\({\sigma }_{GO}^{2}\)

Variance of breeding values of grand-offspring produced by selected offspring of a mating when mated to a random partner of selected animals of the population from the same generation; it is equal to \({\sigma }_{{gam}_{sel \, off}}^{2}+{\sigma }_{{gam}_{sel \, ind}}^{2}\)