Spatially Nonhomogeneous Hopf and Turing Bifurcations Driven by the Nonlocal Effect With the Spatial Average Kernel

ABSTRACT In this paper, we first derive the algorithm of calculating the normal form of spatially nonhomogeneous Hopf bifurcation and Turing bifurcation for the general reaction–diffusion system with spatial average. Then we investigate the spatiotemporal dynamics of nonlocal Lotka–Volterra competitive model and nonlocal Holling–Tanner predator–prey model. It has been shown that the spatial average is a new mechanism to induce the patterns. By calculating the normal form, the type of bifurcation and stability of spatiotemporal patterns bifurcating from the constant equilibrium are investigated. For the nonlocal Lotka–Volterra competitive model, the coexistence of two spatially nonhomogeneous spatial patterns is found. For the nonlocal Holling–Tanner predator–prey model, we found not only the coexistence of two spatially nonhomogeneous spatial patterns but also the stable spatially nonhomogeneous periodic patterns.