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

Abstract

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 D 1 0 and D 3 0 such that for max { m 1 − m 2 2 , 0 } < b < 2 m 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 ≥ 2 m 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.