Quantitative Nonlocal-to-Local Limits in Multispecies Interaction Systems via Relative Entropy and Modulated Energy

We derive quantitative convergence rates for nonlocal-to-local limits in a class of multispecies interaction systems with finite-range kernels. The nonlocal model consists of coupled aggregation–diffusion equations in which intra- and interspecies interactions are mediated by short-range convolution operators. As the interaction radius tends to zero, the system converges to a local cross-diffusion model. Going beyond qualitative convergence, we obtain rigorous error estimates by combining a multispecies relative entropy functional with a modulated energy adapted to the flux defect between the nonlocal and local systems. The analytical framework incorporates a weak-solution theory for the nonlocal model, a strong-solution setting for the limiting cross-diffusion system, and quantitative consistency estimates based on kernel moment expansions. Under symmetry and moment assumptions on the interaction kernels, we establish weak–strong stability estimates and derive second-order convergence in L2. When the kernels are not symmetric, we identify the first-order transport correction induced by the kernel asymmetry and show that the convergence rate drops to first order relative to the uncorrected local model, while second-order convergence is recovered for a suitably corrected local equation. Numerical experiments in one and two space dimensions confirm the theoretical rates and demonstrate convergence not only of densities but also of the associated spatial patterns. Additional mesh and time-step refinement studies show that the observed asymptotic behavior is not contaminated by discretization effects.