Limit cycles in replicator-mutator network dynamics

The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multi-agent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles for the dynamics in the case of circulant fitness matrices, and illustrate their existence in the noncirculant case. For decision-making networks, these limit cycles correspond to sustained oscillations in decisions across the group.