Stationary periodic and homoclinic solutions for nonlocal reaction-diffusion equations

We study spatially periodic patterns for 1-D nonlocal reaction-diffusion equations that arise from various biological models. The problem reduces to study periodic and homoclinic solutions of differential equations with perturbations containing convolution terms. We consider the case that the system is time-reversible. Assuming that the unperturbed system has a family of periodic orbits surrounded by a homoclinic orbit, we establish the persistence of these solutions for the perturbed equations. We apply this result to the important Gray-Scott autocatalytic model.