Replicator Dynamics with Higher Order Interactions and Hyper-Graphs

Recent work has shown that pairwise interactions may not be sufficient to fully model ecological dynamics in the wild. In this short talk, we discuss a variant of the replicator dynamic that models both pairwise and triadic interactions among species. We review results by Griffin & Wu that show the addition of these higher-order dynamics can lead to the creation of a subcritical Hopf bifurcation and consequently an unstable limit cycle in the well-known rock-paper-scissors dynamic, implying that the addition of higher-order terms fundamentally alters the underlying phase space of the replicator. We then discuss how these higher-order interactions can be modeled using hyper-graphs, suggesting a generalization of recent work by Visomirski and Griffin relating graph structures to qualitative properties of dynamical systems. Instead, we propose a generalized mapping from hyper-graph structures to dynamics.