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 derivation 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.

Inclusive Fitness and Hamilton’s Rule

This chapter evaluates the long and intimate association between the gene’s-eye view and the work of W.D. Hamilton. Hamilton’s key insight was that individual organisms can affect the transmission of their genes through personal reproductive success, as well as through the success of close relatives. Inclusive fitness provides a way to view this process from the perspective of individual organisms, but it can also be seen from a gene’s-eye view. Dawkins and others have repeatedly emphasized the formal equivalence of the two perspectives. Yet, this chapter shows there is an underappreciated tension between the two perspectives. It demonstrates how this tension is expressed in both the current kerfuffle over the value of inclusive fitness theory stemming from Martin Nowak and colleagues and in Alan Grafen’s ongoing Formal Darwinism Project. The chapter ends by discussing two recent attempts to resolve this tension.