Developments of the Price equation and natural selection under uncertainty.

Many approaches to the study of adaptation, following Darwin, centre on the number of offspring of individuals. Population genetics theory makes clear that predicting gene frequency changes requires more detailed knowledge, for example of linkage and linkage disequilibrium and mating systems. Because gene frequency changes underlie adaptation, this can lead to a suspicion that approaches ignoring these sophistications are approximate or tentative or wrong. Stochastic environments and sexual selection are two topics in which there are widespread views that focusing on number of offspring of individuals is not enough, and that proper treatments require the introduction of further details, namely variability in offspring number and linkage disequilibrium, respectively. However, the bulk of empirical research on adaptation and a great deal of theoretical work continue to employ these approaches. Here, a new theoretical development arising from the Price equation provides a formal justification in very general circumstances for focusing on the arithmetic average of the relative number of offspring of individuals.