Rich collective behaviors in nonreciprocal multispecies systems: The interplay between nonreciprocity and permutation symmetry among species

Nonreciprocity and permutation symmetry, despite their mutual exclusivity in two-component systems, coexist in multicomponent biological and synthetic systems, ranging from neural circuits to active metamaterials. While these phenomena have been extensively studied independently in the contexts of nonreciprocal phase transitions (e.g., flocking, synchronization, and pattern formation) and cluster synchronization (e.g., oscillator networks), their synergistic effects remain largely unexplored. This study introduces a framework, integrating mean-field theory and group representation theory, to investigate collective behaviors in synchronization and pattern formation systems exhibiting both nonreciprocity and permutation symmetry. Analyzing two three-species scenarios with S_{2} (transposition) and C_{3} (cyclic) permutation groups, we identify distinct symmetry-constrained traveling wave phases in synchronization system. Both phases exhibit parity-time symmetry breaking, but their dynamics and phase transitions diverge due to differing symmetry constraints. Spontaneous permutation symmetry breaking transforms the phase transition points of the traveling wave phase from exceptional points to Hopf bifurcations. Furthermore, we discover a time-dependent phase in scenario with S_{2}, absent in two-species systems. This phase features a dynamic interplay between unidirectional, time-crystal-like oscillations in one species and symmetrical pendulumlike motions in the other two. Our results demonstrate that permutation symmetry significantly expands the repertoire of symmetry-constrained collective behaviors and phase transitions in nonreciprocal systems, exceeding the constraints of two-species systems. The identified mechanism, symmetry-constrained organization of nonreciprocal fluxes, provides a design principle for engineering collective behaviors in neuronal circuits and active metamaterials.