An alternative to the breeder's and Lande's equations

TL;DR

This paper introduces an exact linear equation relating response to selection in quantitative genetics, valid under Gaussian fitness functions and independent of parental genotype distribution, offering a robust alternative to traditional equations.

Contribution

It derives a new linear relation between phenotypic response and selection differential that does not depend on parental genotype distribution, extending the classical breeder's and Lande's equations.

Findings

The new equation holds regardless of parental genotype distribution when fitness is Gaussian.

Numerical simulations confirm the validity of the derived relation.

The approach generalizes to multivariate cases, extending classical models.

Abstract

The breeder's equation is a cornerstone of quantitative genetics and is widely used in evolutionary modeling. The equation which reads R=h^{2}S relates response to selection R (the mean phenotype of the progeny) to the selection differential S (mean phenotype of selected parents) through a simple proportionality relation. The validity of this relation however relies strongly on the normal (Gaussian) distribution of parent's genotype which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian, an alternative, exact linear equation in the form of R'=j^{2}S' can be derived, regardless of the parental genotype distribution. Here R' and S' stand for the mean phenotypic lag behind the mean of the fitness function in the offspring and selected populations. To demonstrate this relation, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between these quantities. These computations, which are confirmed by individual based numerical simulations, generalize naturally to the multivariate Lande's equation \Delta\mathbf{\bar{z}}=GP^{-1}\mathbf{S} .