Reduced order modeling for spatio-temporal pattern approximation in diffusive Lotka-Volterra equations

This paper presents an efficient reduced order modeling (ROM) framework for simulating spatio-temporal pattern formation in three-species diffusive Lotka-Volterra systems. To alleviate the high computational cost associated with long-time simulations of the high-dimensional full order model (FOM), we apply proper orthogonal decomposition (POD) to project the solution onto a low-dimensional subspace. Further efficiency is achieved through tensorial POD (TPOD), which preserves the quadratic nonlinear structure and enables offline-online decomposition. Numerical experiments demonstrate that both POD and TPOD accurately replicate the key features of spatial segregation patterns while substantially reducing computation time, whereas the TPOD is faster. Additionally, we demonstrate accurate long-time pattern prediction using limited training data.