Higher Order Dynamics in the Replicator Equation Produce a Limit Cycle in Rock-Paper-Scissors

TL;DR

This paper introduces higher-order interactions into replicator dynamics for rock-paper-scissors, revealing the emergence of unstable limit cycles through a subcritical Hopf bifurcation, which are absent in traditional pairwise models.

Contribution

It extends the replicator equation to include triadic interactions via a rank-three tensor, demonstrating new complex dynamics such as limit cycles.

Findings

Higher-order dynamics induce unstable limit cycles.

Subcritical Hopf bifurcation occurs with added interactions.

Traditional pairwise replicator cannot produce these behaviors.

Abstract

Recent work has shown that pairwise interactions may not be sufficient to fully model ecological dynamics in the wild. In this letter, we consider a replicator dynamic that takes both pairwise and triadic interactions into consideration using a rank-three tensor. We study {these} new nonlinear dynamics using a generalized rock-paper-scissors game whose dynamics are well understood in the {standard} replicator sense. We show that the addition of higher-order dynamics leads to the creation of a subcritical Hopf bifurcation and consequently an unstable limit cycle. It is known that this kind of behaviour cannot occur in the pairwise replicator in any three strategy games, showing the effect higher-order interactions can have on the resulting dynamics of the system. We numerically characterize parameter regimes in which limit cycles exist and discuss possible ways to generalize this approach to studying higher-order interactions.