Universal rules for the interaction of selection and transmission in evolution.

The Price equation shows that evolutionary change can be written in terms of two fundamental variables: the fitness of parents (or ancestors) and the phenotypes of their offspring (descendants). Its power lies in the fact that it requires no simplifying assumptions other than a closed population, but realizing the full potential of Price's result requires that we flesh out the mathematical representation of both fitness and offspring phenotype. Specifically, both need to be treated as stochastic variables that are themselves functions of parental phenotype. Here, I show how new mathematical tools allow us to do this without introducing any simplifying assumptions. Combining this representation of fitness and phenotype with the stochastic Price equation reveals fundamental rules underlying multivariate evolution and the evolution of inheritance. Finally, I show how the change in the entire phenotype distribution of a population, not simply the mean phenotype, can be written as a single compact equation from which the Price equation and related results can be derived as special cases. This article is part of the theme issue 'Fifty years of the Price equation'.