  In this paper, we consider a general reaction-diffusion system with nonlocal
effects and Neumann boundary conditions, where a spatial average kernel is
chosen to be the nonlocal kernel. By virtue of the center manifold reduction
technique and normal form theory, we present a new algorithm for computing
normal forms associated with the codimension-two double Hopf bifurcation of
nonlocal reaction-diffusion equations. The theoretical results are applied to a
predator-prey model, and complex dynamic behaviors such as spatially
nonhomogeneous periodic oscillations and spatially nonhomogeneous
quasi-periodic oscillations could occur.
