Feedback Controlled Bifurcation of Evolutionary Dynamics with Generalized Fitness

Coexistence and interaction of multiple strategies in a large population of individuals can be observed in a variety of natural and engineered settings. In this context, replicator-mutator dynamics provide an efficient tool to model and analyze the evolution of the fractions of the total population committed to different strategies. Although the literature addresses existence and stability of equilibrium points and limit cycles of these dynamics, linearity in fitness functions has typically been assumed. We generalize these dynamics by introducing a nonlinear fitness function, and we show that the replicator-mutator dynamics for two competing strategies exhibit a quintic pitchfork bifurcation. Then, by designing slow-time-scale feedback dynamics to control the bifurcation parameter (mutation rate), we show that the closed-loop dynamics can exhibit oscillations in the evolution of population fractions. Finally, we introduce an ultraslow-time-scale dynamics to control the associated unfolding parameter (asymmetry in the payoff structure), and demonstrate an even richer class of behaviors.