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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}\left(b\right), 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\left\{\frac{{m}_{1}-{m}_{2}}{2},0\right\} and ${d}_{3}>{D}_{3}^{0}$, the system with cross-diffusion admits nonconstant positive steady-state solutions for some $\left({d}_{1},{d}_{2},{d}_{3}\right)$; whereas, for $b\ge 2{m}_{1}$ or $a\ge {a}_{2}\left(b\right)$ or ${d}_{1}\ge {D}_{1}^{0}$ or ${d}_{3}\le {D}_{3}^{0}$, 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:
 35K51 Second-order parabolic systems, initial bondary value problems 92D25 Population dynamics (general) 35J57 Second-order elliptic systems, boundary value problems 35K58 Semilinear parabolic equations