Derivation of a Generalised Replicator Equation in the Limit of Weak Selection

Paper in proceeding, 2023

The replicator equation is often applied for describing the change in frequency of competing subpopulations, but has only been formally derived under strong limitations on the dynamics of the total population size. We show that a generalised replicator equation can be derived from a wide class of population dynamical models that allow for a factorisation into density-dependent and frequency-dependent birth and death rates. The method relies on weak selection (i.e. small fitness differences) and the generalised replicator equation is obtained as the zeroth order approximation in a perturbation expansion. We apply our theoretical results to a specific model with linear frequency-dependence and logistic dynamics for the population size, and show that the error introduced by considering a zeroth order composite solution scales as the fitness difference.