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 a(0)(b) < a < m(1), whereas for a > m(1), 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 > m(1), where the cross-diffusion represents that the predator moves away from a large group of prey. We obtain-that there exist positive constants DO and DO such that for max{m(1)-m(2)/2, 0} < b < 2m(1), m(1) < a < a(2)(b), d(1) < D(1)(0) and d(3) > D(3)(0), the system with cross-diffusion admits nonconstant positive steady-state solutions for some (d(1), d(2), d(3)); whereas, for b >= 2m(1) or a >= a(2)(b) or d(1) >= D(1)(0) or d(3) >= D(3)(0), 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. (C) 2008 Elsevier Ltd. All rights reserved.