@article{Xue2023, 
author = {Pan Xue and Cuiping Ren},
title = {Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {8},
pages = {19413-19426},
keywords = {stationary solution, Leray-Schauder degree, predator-prey system, Beddington-DeAngelis functional response, fractional type cross-diffusion},
url = {https://www.sciopen.com/article/10.3934/math.2023990},
doi = {10.3934/math.2023990},
abstract = {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.}
}