On Permanence of Conservative Replicator Dynamics with Four Strategies

Abstract

In this paper, we study four-strategy conservative replicator dynamics induced by constant payoff matrices. We establish necessary and sufficient conditions for permanence to occur by associating the payoff matrix with its digraph, revealing exactly five distinct digraph classes governing the global behavior. We further show that, whenever the dynamics is permanent, every non-equilibrium trajectory in the relative interior of the simplex is a Lyapunov-stable periodic orbit. Together with the classification of the boundary phase portraits, these results provide a complete characterization of the global dynamics in the four-strategy case with permanence.