Studies of topological bands and their associated low-dimensional boundary modes have largely focused on linear systems. This work reports robust dynamical features of three-dimensional (3D) nonlinear systems in connection with intriguing topological bands in 3D. Specifically, for a 3D setting of coupled rock-paper-scissors cycles governed by the antisymmetric Lotka-Volterra equation (ALVE) that is inherently nonlinear, we unveil distinct characteristics and robustness of surface-polarized masses and analyze them in connection with the dynamics and topological bands of the linearized Lotka-Volterra equation. Our results show that insights learned from Weyl semimetal phases with type-I and type-II Weyl singularities based on a linearized version of the ALVE are still remarkably useful, even though the system dynamics is far beyond the linear regime. This work indicates the relevance and importance of the concept of topological boundary modes in analyzing high-dimensional nonlinear systems and hopes to stimulate further topological studies where various topological gapless phases of matter may emerge in higher-dimensional nonlinear systems.

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• Complex systems
• Nonlinear waves
• Topological materials
• Weyl semimetal
• Cycles
• Game theory models