On Permanence of Conservative Replicator Dynamics with Four Strategies

TL;DR

This paper characterizes the global behavior of four-strategy conservative replicator dynamics, identifying conditions for permanence and describing the stability of interior trajectories.

Contribution

It provides necessary and sufficient conditions for permanence and classifies the global dynamics based on payoff matrix digraphs.

Findings

Five distinct digraph classes govern the dynamics.

Interior trajectories are Lyapunov-stable periodic orbits when permanent.

Complete characterization of four-strategy dynamics with permanence.

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.