Darwin's theory of natural selection, with a little Neo-Darwinian jargon thrown in.

1. Individuals within a population vary in phenotype. V_P > 0, where V_P is phenotypic variance.
2. Some of this phenotypic variation is heritable, or due to genes. V_A> 0, where V_A is additive genetic variance. Heritability is given by, h² =V_A/V_P.
3. Individuals produce excess offspring, which leads to a struggle for existence. In stationary populations, each female replaces herself on average with one daughter. Birth rates that are higher than this lead to the 'struggle for existence.' This idea comes from economist Thomas Malthus, and is a key component of Darwin's theory.
4. Some individuals, because of their particular phenotypic characteristics have higher rates of survival or reproduction than others. This is selection within the parental generation. In quantitative genetics, this is measured by the selection differential, S, which is the difference in average phenotype before versus after selection, within the parental generation.
5. Those favored phenotypes will increase in frequency in future generations. This phenomenon is realized in the offspring of the parental generation. In quantitative genetics, this is known as the response to selection, R, which is the change in average phenotype between the parental generation before selection and offspring generation after selection.

In the breeder's equation,

R = h² S.

In animal and plant breeding, S is determined by the breeder. In natural selection, S is determined in part by the intensity of the struggle for existence in step 3 above, and by the adaptive landscape.

The breeder's equation can be rewritten as

R = V_Aβ ,

where β  is the selection gradient, i.e. the slope of the regression line of fitness on phenotype. This univariate model can be extended to the multivariate case, where...

ΔZ= Gβ

...where ΔZ is a vector of changes in mean phenotypic traits between the parent and offspring generations, G is a matrix of additive genetic variances on the diagonals, and additive genetic covariances on the off diagonals, and β is a vector of partial regression coefficients of fitness on phenotypic traits.

Note that the G matrix can consider multivariate evolution of phenotypic traits with all sorts of interactions built in.

References

4. Short-term Changes in the Mean: The Breeders' Equation

Measuring selection differentials and gradients

19. Individual Fitness and the Measurement of Univariate Selection

20. Measuring Multivariate Selection