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.

Automated summary

The study explores the stability of a constant steady state in a general reaction-diffusion model with nonlocal dispersal of activator or inhibitor, and finds that Turing instability and spatial patterns can be induced by fast nonlocal inhibitor dispersal and slow activator diffusion. It also shows that slow nonlocal activator dispersal causes instability without producing stable spatial patterns. The existence of nonconstant positive steady states is demonstrated through bifurcation theory, suggesting a new mechanism for spatial pattern formation different from the traditional Turing mechanism. The theoretical results are applied to pattern formation problems in nonlocal water-plant and predator-prey models.