期刊
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
卷 11, 期 1, 页码 372-390出版社
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.nonrwa.2008.11.010
关键词
Predator-prey system; Ratio-dependent; Cross-diffusion; Nonconstant positive steady states; Degree theory
资金
- National Natural Science Foundations of China [10671064, 510604008]
- Post-Doctor Foundation of Central South University
- Post-Doctor Foundation of China [20070410994]
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 DO and DO 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 b >= 2m(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. (C) 2008 Elsevier Ltd. All rights reserved.
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