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