Effect of Spatial Average on the Spatiotemporal Pattern Formation of Reaction-Diffusion Systems

Some quantities in reaction-diffusion models from cellular biology or ecology depend on the spatial average of density functions instead local functions. We show that such nonlocal can induce instability constant steady state, which is different classical Turing instability. For a general scalar equation with average, occurrence state bifurcation rigorously proved, and formula to determine direction stability bifurcating given. two-species model, spatially non-homogeneous time-periodic orbits could arise due Hopf equilibrium. Examples cooperative Lotka-Volterra model Rosenzweig-MacArthur predator-prey are used demonstrate patterns.