Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $d_2$ or cross-diffusion $d_4$ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $d_2$ and the fractional type cross-diffusion $d_4$ can create spatial patterns.