Summary: We have investigated a ratio-dependent predator-prey system with diffusion in [X. Zeng [Nonlinear Anal., Real World Appl. 8, No. 4, 1062–1078 (2007; Zbl 1124.35027)] and obtained that the system with diffusion can admit nonconstant positive steady-state solutions when , whereas for , 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 , where the cross-diffusion represents that the predator moves away from a large group of prey. We obtain that there exist positive constants and such that for and , the system with cross-diffusion admits nonconstant positive steady-state solutions for some ; whereas, for or or or , 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.