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Abstract Recent work has shown that pairwise interactions may not be sufficient to fully model ecological dynamics in the wild. In this letter, we consider a replicator dynamic that takes both pairwise and triadic interactions into consideration using a rank-three tensor. We study these new nonlinear dynamics using a generalized rock-paper-scissors game whose dynamics are well understood in the standard replicator sense. We show that the addition of higher-order dynamics leads to the creation of a subcritical Hopf bifurcation and consequently an unstable limit cycle. It is known that this kind of behaviour cannot occur in the pairwise replicator in any three-strategy games, showing the effect higher-order interactions can have on the resulting dynamics of the system. We numerically characterize parameter regimes in which limit cycles exist and discuss possible ways to generalize this approach to studying higher-order interactions. 

We study a dynamical system defined by a repeated game on a 1D lattice, in which the players keep track of their gross payoffs over time in a bank. Strategy updates are governed by a Boltzmann distribution, which depends on the neighborhood bank values associated with each strategy, relative to a temperature scale, which defines the random fluctuations. Players with higher bank values are, thus, less likely to change strategy than players with a lower bank value. For a parameterized rock–paper–scissors game, we derive a condition under which communities of a given strategy form with either fixed or drifting boundaries. We show the effect of a temperature increase on the underlying system and identify surprising properties of this model through numerical simulations.