Stable Coexistence in Ecologies and Games

Consider a community of $n$ interacting species, modeled by a generalized Lotka-Volterra system. Suppose we only know whether each species grows on its own and the qualitative type of interaction between each pair of species: competition, mutualism, or predation. Can we decide whether some choice of parameters of the system allows all species to coexist at a stable equilibrium? Interaction networks for which this is never possible are called "impossible ecologies". Meng, Horvát, Modes, and Haas classified impossible ecologies with two and three species and, based on numerical evidence, conjectured a list of eleven impossible four-species ecologies. In this talk, we prove that all eleven are indeed impossible, completing the four-species classification, and we show that several of these obstructions extend to infinite families of ecologies with arbitrarily many species. We then show that these obstructions are specific to pairwise interactions: once higher-order interactions are allowed, every interaction network admits a feasible asymptotically stable equilibrium. Through the correspondence with replicator dynamics, feasible equilibria of higher-order Lotka-Volterra systems become totally mixed symmetric Nash equilibria of symmetric multiplayer games: this allows us to prove that every impossible ecology determines a nonempty open class of symmetric two-player games with no totally mixed evolutionarily stable strategy. This is joint work with Türkü Özlüm Çelik, Irem Portakal, and Giulio Zucal.