Quantitative traits under selection: derivations of distributions, higher moments, and the effects of recombination

The mathematical theory of quantitative traits is over one hundred years old but it is still a fertile area for research and analysis. However, the effects of selection on a quantitative trait, while well understood for the effects on the mean and variance, have traditionally been difficult to attack from the perspective of analyzing the probability density of the breeding values and deriving higher (third and fourth) moments as well as analyzing the impact of recombination. In this paper, the exact formula for the breeding value distribution after selection is derived and, using new integral tables, the first four moments are given exact expressions for the first time. In addition, the effects of recombination on the full distribution of breeding values are demonstrated. Finally, the changes of GXE covariance in the selected parent population caused by factors similar to the Bulmer Effect are also investigated in detail. Mathematics Subject Classification (2020)MSC code 92D10 {middle dot} MSC code 92D15