STATIONARY PERIODIC AND HOMOCLINIC SOLUTIONS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS SHANGBING AI DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF ALABAMA IN HUNTSVILLE HUNTSVILLE, AL 35899 Abstract. We study spatially periodic patterns for 1-D nonlocal reaction- diffusion equations that arise from various biological models. The problem re- duces to study periodic and homoclinic solutions of differential equations with perturbations containing convolution terms. We consider the case that the sys- tem is time-reversible. Assuming 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. 1. Introduction In this paper we are concerned with the existence of periodic and homoclinic solutions of the equation u ′′ − g(u)= f (u(x), (J ε ∗ u)(x), [(K ε ∗ u)u] ′ (x),ε), (1.1) where ε> 0 is a sufficiently small parameter, g and f are continuous functions of their arguments, and (J ε ∗ u)(x)=  ∞ −∞ J ε (x − y)u(y) dy and (K ε ∗ u)(x)=  ∞ −∞ K ε (x − y)u(y) dy, and the kernels J ε and K ε are always assumed to satisfy J ε (x)= J ε (−x), K ε (x)= −K ε (−x), J ε ,K ε ∈ L(R). (1.2) 1991 Mathematics Subject Classification. 34C25, 34C37, 35K57, 47G20, 92D25, 92E20. Key words and phrases. nonlocal reaction-diffusion equations; stationary periodic and homo- clinic solutions; persistence. 1