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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 \(\max \{\frac{m_1 -m_2}{2},0\}<b <2m_1, m_{1}<a<a_{2}(b), d_1< D_1^0\) 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 \(b\geq 2m_{1}\) or \(a\geq a_{2}(b)\) or \(d_1\geq D_1^0\) or \(d_3\leq D_3^0\), 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 Initial-boundary value problems for second-order parabolic systems
92D25 Population dynamics (general)
35J57 Boundary value problems for second-order elliptic systems
35K58 Semilinear parabolic equations

Citations:

Zbl 1124.35027
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References:

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