On a predator–prey system with cross diffusion representing the tendency of predators in the presence of prey species

In this paper, a predator–prey system with cross-diffusion, representing the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species, under homogeneous Dirichlet boundary condition, is considered. Using the method of upper and lower solutions developed by Pao [C.V. Pao, Strongly coupled elliptic systems and applications to Lotka–Volterra models with cross-diffusion, Nonlinear Anal. 60 (2005) 1197–1217], sufficient conditions for the existence of positive solutions are provided when the induced cross diffusion coefficient is sufficiently small. Furthermore, the investigation of non-existence of positive solutions is also presented.