Pattern formation of a Beddington-DeAngelis predator-prey model with prey-taxis and nonlocal competition

In this paper, a nonlocal Beddington-DeAngelis reaction-taxis-diffusion model is formulated to explore the impacts of prey-taxis and the nonlocal intraspecific competition on spatial pattern formation. For the reaction-diffusion model without the nonlocal competition, the qualitative analyses and numerical simulations show that attractive prey-taxis compresses the spatial patterns, and repulsive prey-taxis contributes to pattern formation. For the nonlocal scenario, sufficient criteria for the nonlocal system undergoing Hopf bifurcation and Turing instability are derived. The main findings demonstrate that nonlocal competition broadens the range of the reaction-diffusion model for spatial pattern formation. In addition, the pattern dynamics of the nonlocal model indicates that the nonlocality conduces to the coexistence of predator and prey by keeping the population at an optimal total population size, which provides a novel insight into the mechanism of the group defenses of some species.