The simplest chaotic Lotka-Volterra system with reflection, rotation, and inversion symmetries

This study presents the simplest known three-species Lotka–Volterra system capable of exhibiting chaotic dynamics. The model is constructed with nonlinear growth and mortality terms defined as products of population densities and quadratic functions of species concentrations, capturing essential ecological nonlinearities in a minimal framework. Unlike many classical three-species Lotka–Volterra models, which typically exhibit only stable or periodic behavior, this system displays rich dynamical behaviors, including chaos, under specific parameter regimes and seven terms. Bifurcation analysis and Lyapunov exponent calculations confirm transitions between periodic oscillations and chaotic attractors. Notably, the chaotic attractor possesses a rare combination of reflection, rotation, and inversion symmetries, despite the system's structural simplicity. These results demonstrate that even the most minimal Lotka–Volterra formulations can generate multiple symmetric chaotic attractors, establishing a new benchmark in the study of simple yet chaotic ecological models.