Stability and bifurcation analysis for a general nonlocal predator–prey system with top-hat kernel function

In this paper, we investigate the influence of the nonlocal intraspecific competition by the prey in a general predator–prey system on the stability and spatiotemporal dynamics, where the nonlocal kernel function is characterized by the top-hat function. It has been shown that when the nonlocal competition radius is sufficiently small, the nonlocal item has no impact on the stability of the spatially homogeneous steady state, but when the nonlocal competition radius is larger than some critical value, various types of spatiotemporal patterns occur via the bifurcation. The analytical conditions for the occurrence of Turing bifurcation, Hopf bifurcation and Turing–Hopf bifurcation are clearly determined. The spatially inhomogeneous steady states and inhomogeneous periodic patterns are found for the nonlocal Rosenzweig–MacArthur and Lotka–Volterra models with the top-hat kernel function.