Impacts of fear effect and nonlocal competition on a diffusive prey–predator model with delay

Nonlocal prey competition, which describes that intra-prey competition is not only dependent on a location in space but also related to entire space, is introduced to a prey–predator model involving fear effect and digest delay. In the nonlocal prey competition model, the critical delay threshold increases with the increasing of the fear level or the intra-prey competition coefficient. In addition, in the case that the intra-prey competition coefficient is less than the threshold, local and nonlocal prey competition models admit the same critical delay threshold. However, in the case that the intra-prey competition coefficient is beyond the threshold, the critical delay threshold for nonlocal prey competition is less than local prey competition. Moreover, nonlocal prey competition term can drive Hopf bifurcation for spatially inhomogeneous form, and the spatially inhomogeneous periodic solution emerges. It is worth noting that in the absence of delay, nonlocal prey competition model can undergo spatially inhomogeneous Hopf bifurcation and Turing instability by diffusion, but local prey competition can not occur. Numerical simulations verify the theoretical analysis. Also, under the influence of nonlocal effect, the amplitude of the spatially homogeneous periodic solution becomes larger. Meanwhile, nonlocal effect may increase the risk of extinction for two species to a certain extent.