Spatial pattern formation in activator-inhibitor models with nonlocal dispersal

The stability of a constant steady state in a general reaction-diffusion activator-inhibitor model with nonlocal dispersal of the activator or inhibitor is considered. It is shown that Turing type instability and associated spatial patterns can be induced by fast nonlocal inhibitor dispersal and slow activator diffusion, and slow nonlocal activator dispersal also causes instability but may not produce stable spatial patterns. The existence of nonconstant positive steady states is shown through bifurcation theory. This suggests a new mechanism for spatial pattern formation, which has different instability parameter regime compared to Turing mechanism. The theoretical results are applied to pattern formation problems in nonlocal Klausmeier-Gray-Scott water-plant model and Holling-Tanner predator-prey model.