The extended Price equation: migration and sex

The Price equation provides a general partition of evolutionary change into two components. The first is usually thought to represent natural selection and the second, transmission bias. Here, we provide a new derivation of the generalised equation, which contains a largely ignored third term. Unlike the original Price equation, this extension can account for migration and mixed asexual and sexual reproduction. The notation used here expresses the generalised equation explicitly in terms of fitness, rendering this otherwise difficult third term more open to biological interpretation and use. This re-derivation also permits fundamental results, derived from the Price equation, to be more easily generalised. We take Hamilton’s rule as a case study, and provide an exact, total expression that allows for population structures like haplodiploidy. Our analysis, more generally, makes clear the previously hidden assumptions in similar fundamental results, highlighting the caution that must be taken when interpreting them.