Weak Selection and the Separation of Eco-evo Time Scales using Perturbation Analysis

Abstract

AbstractWe show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka–Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for $$k=1$$ k = 1 and 2, that the error scales according to $$\varepsilon ^{k+1}$$ ε k + 1 , where $$\varepsilon $$ ε is the strength of selection and k is the order of the approximation.