Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection

In this paper we consider class-structured populations in discrete time in the limit of weak selection and with the inverse of the intensity of selection as unit of time. The aim is to establish a continuous model that approximates the discrete model. More precisely, we study frequency-dependent growth in an infinite haploid population structured into a finite number of classes such that individuals in each class contribute to a given subset of classes from one time step to the next. These contributions take the form of generalized fecundity parameters with perturbations of order 1 / N that depends on the class frequencies of each type and the type frequencies. Moreover, they satisfy some mild conditions that ensure mixing in the long run. The dynamics in the limit as N → ∞ with N time steps as unit of time is considered first in the case of a single type, and second in the case of multiple types. The main result is that the type frequencies as N → ∞ obey the replicator equation with instantaneous growth rates for the different types that depend only on instantaneous equilibrium class frequencies and reproductive values. An application to evolutionary game theory complemented by simulation results is presented.