Multivariate trait evolution: models for the evolution of the quantitative genetic G-matrix on phylogenies

Genetic covariance matrices (G-matrices) are a key focus for research and predictions from quantitative genetic evolutionary models of multiple traits. There is a consensus among quantitative geneticists that the G-matrix can evolve through deep time. Yet, quantitative genetic models for the evolution of the G-matrix are lacking. In contrast, the field of macroevolution has several stochastic models for univariate traits evolving on phylogenies. However, analytical models of how multivariate trait matrices might evolve on phylogenies have not been considered. Here, we show how three analytical models for matrix evolution can be combined to unify quantitative genetics and macroevolutionary theory in a coherent mathematical framework. The models provide a basis for understanding how G-matrices might evolve on phylogenies. We fit models to data via simulation using Approximate Bayesian Computation. Such models can be used to generate and test hypotheses about the evolution of genetic variances and covariances, together with the evolution of the traits themselves, and how these might vary across a phylogeny. This unification of macroevolutionary theory and quantitative genetics is an advance in the study of phenotypes, allowing for the construction of a synthetic quantitative theory of the evolution of species and multivariate traits over deep time.