In this article, an initial-/boundary-value problem for a nonlocal diffusion system is treated. In the first part of the article, the unique solvability of the problem is established, and the asymptotic behavior of solutions is discussed by means of some key estimates. For the problem, it is also proved that the corresponding initial-value problem for a system of ordinary differential equations are of crucial importance. The second part of the article is devoted to the analysis of some nonlocal reaction-diffusion systems which are obtained as the special case where the coefficient matrices in the original system are diagonal. Lotka-Volterra prey-predator interaction and competitive interaction for two species are taken as fundamental examples of reaction kinetics; the stability and asymptotic behavior of solutions to these ecological models are inspected.
"Asymptotic behavior of solutions to nonlocal diffusion systems driven by systems of ordinary differential equations." Adv. Differential Equations 12 (8) 841 - 866, 2007.