Completely integrable replicator dynamics associated to competitive networks

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 Allesina and Levine [Proc. Natl. Acad. Sci. USA 108, 5638 (2011)10.1073/pnas.1014428108] produces quasiperiodic dynamics.