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Zbl 1230.35050
Zeng, Xianzhong; Liu, Zhenhai
Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion. (English)
[J] Nonlinear Anal., Real World Appl. 11, No. 1, 372-390 (2010). ISSN 1468-1218

Summary: We have investigated a ratio-dependent predator-prey system with diffusion in [{\it 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. \par 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 2000:: *35K51
92D25 Population dynamics
35J57
35K58

Keywords: predator-prey system; ratio-dependent; cross-diffusion; nonconstant positive steady states; degree theory

Citations: Zbl 1124.35027

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