Evolutionary and ecological processes determining properties of the $\bm G$-matrix

The $\bm G$-matrix is the matrix of additive genetic variances and covariances for a vector of phenotypes. Here we apply the classical theory for the balance between selection drift and mutations to find the contributions to $\bm G$ from each locus. The fitness is approximated by a linear function of phenotypes. Fluctuations in the environment generate variation in the coefficients of the fitness function. We show that the $\bm G$-matrix can be decomposed into 4 additive components generated by selection, drift, mutations and environmental fluctuations. Selection is on average counteracted by the other three processes included in Fisher's concept of the deterioration of the environment, in accordance with Frank's approximate conservation law proposing that the response to selection at stasis on average is canceled by effects of drift and mutations. The theory illustrates that Fisher's fundamental theorem cannot be used to accumulate selection through time to describe adaption unless the other effects are corrected for. Another implication of the analyses is that the factor loadings to the eigenvector of the $\bm G$-matrix with the least eigenvalue are likely to indicate which characters contributing the most to the fitness function. This is information notoriously difficult to obtain in natural populations.