Evolutionary dynamics of functionally constrained phenotypic characters.

A nonlinear analysis is performed, employing the theory of Lyapunov functions, to examine the relative importance of genetic and phenotypic covariance matrices for the evolution of functionally coupled quantitative traits in an adaptive topography with several directions of increasing fitness. The analysis is based on Lande's evolution equations for phenotypic characters. It is supposed that evolution of a set of functionally constrained characters far from equilibrium corresponds to evolution along a ridge in the fitness landscape. It is shown that the pattern of variation and covariation restricts the possible directions of evolutionary change in the following sense. Any population starting sufficiently near the ridge will evolve along it, provided that one eigenvector of the genetic covariance matrix and one eigenvector of the phenotypic covariance matrix point into the direction of the ridge. Otherwise, the set of initial positions of a population enabling evolution along the ridge is more or less restricted, depending on the degree of deviation of the eigenvectors from the direction of the ridge. Moreover, too much phenotypic variance of the characters under stabilizing selection may inhibit any evolution along the ridge. Thus, the present analysis establishes population-genetic prerequisites and constraints for the evolution of functionally constrained phenotypic traits.