ASYMPTOTIC BEHAVIOR OF A DELAYED NONLOCAL DISPERSAL LOTKA-VOLTERRA COMPETITIVE SYSTEM

This paper investigates the asymptotic behavior of a nonlocal dispersal Lotka-Volterra competitive system with time delay across the entire $\mathbb{R}^N$. We establish $L^\infty-$decay estimates of solutions of linear systems converging to equilibria utilizing the Fourier transform method applied to the fundamental solution and the Fourier splitting technique. For the nonlinear time-delayed nonlocal dispersal Lotka-Volterra competitive system, we leverage the results from linear systems and obtain the long-time behavior of solutions of the nonlinear system manifesting as the form of time-exponential. More precisely, we further deduce $L^\infty-$decay estimates of solutions of the original nonlinear system through the properties of convolution and Hölder inequality. Additionally, numerical simulations are presented to bolster the principal theoretical results and illustrate that the time delay impedes species growth.