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Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion. (English) Zbl 1230.35050

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 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<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 b2m 1 or aa 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.

MSC:
35K51Second-order parabolic systems, initial bondary value problems
92D25Population dynamics (general)
35J57Second-order elliptic systems, boundary value problems
35K58Semilinear parabolic equations
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