Conserved Quantities, Quasi-Periodicity and Liouville-Arnold Integrability of Ecological Networks

Complete odd tournaments are frequently used as abstract models of ecological systems via replicator dynamics. In these models, the payoff matrix is a skew-symmetric +1/-1 matrix and all species interactions result in a non-zero interaction payoff. These matrices correspond to directed graphs with species as vertices and edge direction giving the sign of the corresponding matrix entries. A circulant tournament is defined by a graph in which every vertex has the same in/out degree. It is known that the replicator dynamics derived from these tournaments admit polynomial conserved quantities. In this talk we extend this result to show that these circulant tournaments produce quasi-periodic dynamics and are Liouville-Arnold integrable by showing they commute under the action of a non-linear Poisson bracket (the quadratic bracket). Furthermore, we show that all tournaments constructed by embeddings are Liouville-Arnold integrable. By an embedding we mean a tournament constructed by recursively replacing vertices in an outer circulant tournament with other circulant tournaments and adding appropriate edges. We numerically illustrate that tournaments not constructed in this manner produce chaotic dynamics and classify all dynamics generated by any tournament with up to seven species.