Complex spatiotemporal dynamics in Lotka–Volterra ring systems

Mathematical models in ecology often need to incorporate spatial dependence to accurately model real-world systems. We consider competitive Lotka–Volterra models modified to include this spatial dependence through organization of the competing species into a one-dimensional ring by an appropriate choice of the interaction matrix. We show that these systems can exhibit complex dynamics, spatiotemporal chaos, and spontaneous symmetry breaking. A high-dimensional, spatially homogeneous, nearest-neighbor example with interaction strengths decreasing with distance is characterized including an analysis of how the dynamics of the system vary with dimension. We also show the existence of Lyapunov functions that arise from this method of including spatial dependence and how they prohibit complex dynamics for certain regions of the parameter space. We utilize these Lyapunov functions to reduce the required calculation time in brute-force searches of parameter space. A short comparison is given to derived line systems including a contrast between the eigenvalues of the two systems..