Completely Integrable Replicator Dynamics Associated to Competitive Networks

TL;DR

This paper introduces an infinite family of integrable replicator equations linked to competitive networks, providing explicit conserved quantities and classifying low-dimensional cases, with applications to ecological dynamics.

Contribution

It presents a new class of integrable replicator equations with explicit conserved quantities and Poisson structures, and classifies low-dimensional cases.

Findings

Produced an infinite family of Liouville-Arnold integrable replicator equations.

Classified all tournament replicators up to dimension 6 and most of dimension 7.

Demonstrated quasiperiodic dynamics in ecological network models.

Abstract

The replicator equations are a family of ordinary differential equations that arise in evolutionary game theory, and are closely related to Lotka-Volterra. We produce an infinite family of replicator equations which are Liouville-Arnold integrable. We show this by explicitly providing conserved quantities and a Poisson structure. As a corollary, we classify all tournament replicators up to dimension 6 and most of dimension 7. As an application, we show that Fig. 1 of ``A competitive network theory of species diversity" by Allesina and Levine (Proc. Natl. Acad. Sci., 2011), produces quasiperiodic dynamics.