Nonconstant positive steady states for a ratio-dependent predator–prey system with cross-diffusion

We have investigated a ratio-dependent predator–prey system with diffusion in [X.Z. Zeng, A ratio-dependent predator–prey system with diffusion, Nonlinear Anal. RWA 8 (6) (2007) 1062–1078] and obtained that the system with diffusion can admit nonconstant positive steady-state solutions when a0(b)m1a>m1, the system with diffusion has no nonconstant positive steady-state solution. In the present paper, we continue to investigate a ratio-dependent predator–prey system with cross-diffusion for a>m1a>m1, where the cross-diffusion represents that the predator moves away from a large group of prey. We obtain that there exist positive constants D10 and D30 such that for max{m1−m22,0}D30, the system with cross-diffusion admits nonconstant positive steady-state solutions for some (d1,d2,d3)(d1,d2,d3); whereas, for b≥2m1b≥2m1 or a≥a2(b)a≥a2(b) or d1≥D10 or d3≤D30, the system with cross-diffusion still has no nonconstant positive steady-state solution. Our results show that this kind of cross-diffusion is helpful to create nonconstant positive steady-state solutions for the predator–prey system.