Dynamics of a reaction–diffusion equation with nonlocal inhibition

Abstract

In this paper we investigate the dynamics of a nonlocal scalar reaction–diffusion equation which corresponds to the limiting equation for a singularly perturbed activator–inhibitor system. The equation is a nonlocal perturbation of the Chafee–Infante problem. It is shown that the nonlocal equation generates a gradient flow whose dynamics can differ from the local problem in several interesting ways. Stable heterogeneous equilibria having any number of zeros on the spatial domain can be obtained by an appropriate choice of parameters. The steady-state equation exhibits an interesting family of secondary bifurcations near double critical points. By analyzing the center manifolds dynamics near the double critical points, it is found that a sequence of secondary bifurcations can result in a transfer of stability between two equilibria by reversing a heteroclinic connection between them. As a consequence the number of zeros of a solution, the lap-number, can increase along trajectories.