The Breeder's Equation

What does it mean and where does it come from
R is the response to selection, defined as the difference in mean phenotype between offspring and the parent generation (before selection).

h² is the (narrow sense) heritability, defined as the ratio between the additive genetic variance of the trait under consideration and the total phenotypic variance of the same trait.

S is the selection differential, defined as the difference in mean phenotype of the parent generation before and after selection.

This classic relationship in quantitative genetic is the simplest selection model, with its roots deep in the early writings of Karl Pearson and also Lush (1937) although the exact origin is somewhat obscure. Another version of the same model is
Δz = β_w,z / w · cov(z_offspring,z_midparent)
where Δz is the change in mean trait value (R above), w is mean fitness, β_w,z is the regression coefficient between fitness and the trait value and the covariance term describes the covariation between (mid)parent trait value and offspring trait value, also called the additive genetic variance of the trait (if the trait is fitness itself, this equation is equivalent to Fisher's Fundamental Theorem). The relationship between the two equations above is given by
h² = cov(z_offspring,z_midparent) / var(z_parent)
and
S = β_w,z var(z_parent) / w.

The multivariate version of the breeders equation was worked out by Lande and others (e.g., Lande 1982, Lande and Arnold 1983). The breeder´s equation is one of the backbones of the quantitative genetics approach in evolutionary theory.

Importance
If traits aren’t heritable (h²=0) or if there is no selection (S=0), then there can be no selective change in trait values, and no adaptive evolution. Quite obvious, perhaps, but nevertheless fundamentally important. The breeder’s equation in its many versions does have some important limitations, though. It can only talk about the change in (mean) trait values from one generation to the next (OK in plant and animal breeding) and says very little about long-term evolution. There are also some serious problems when it comes to observational data, for example, from populations in the wild (e.g., Morrissey et al. 2010). There are also some more fundamental problems. It can be shown (Morrissey et al 2010) that the relationship between fitness and genes must be the same as the relationship between fitness and phenotypes in order for the predictions of the breeder’s equation and another important quantitative genetics equation (“the second theorem of natural selection”), the Robertson–Price identity, to be the same. That is to say that there are some basic issues with the various relationships between traits, fitness, selection, and heritability that are unresolved (in natural population) and that the breeder’s equation has limited value when predicting microevolution in wild populations. This is illustrated when the breeder´s equation is expanded (Heywood 2005, Morris and Lundberg 2011) to
R = h²S + σ_wz',z + E(Δz),
where σ_wz',z is the partial covariance – the covariance controlled by the midparent value – between fitness and the mean offspring trait value (the components of the difference in the mean trait value between generations that is caused by factors influencing differential fitness among parents, but that is not related to the trait value). is the expected change in trait value in the absence of fitness differences among parents, including for example drift. See Morris and Lundberg (2011) for elaborations in the problems with the breeder´s equation.

Per Lundberg

Literature 
Lande, R. 1982. A quantitative genetic theory of life history evolution. Ecology 63: 607-615.

Lande, R. and Arnold, S. J. 1983. The measurement of selection on correlated characters. Evolution 37: 1210-1226.

Lush, J. 1937. Animal breeding plans. Iowa State College Press.

Lynch M. and Walsh, B. 1998. Genetics and analysis of quantitative traits. Sinauer.

Morris, D. W. and Lundberg, P. 2011. Pillars of evolution. Oxford Univ. Press.

Morrissay, M. B., Kruuk, L. E. B. and Wilson, A, J. 2010. The danger of applying the breeder’s equation in observational studies of natural population. J. Evol. Biol. 23: 2277- 2288.