期刊
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
卷 34, 期 12, 页码 5271-5298出版社
AMER INST MATHEMATICAL SCIENCES
DOI: 10.3934/dcds.2014.34.5271
关键词
Existence; stability; steady states; spectral analysis; cross diffusion; shadow system
资金
- NNSF of China [11071172, 11226178]
- Beijing Municipal Education Commission [KZ201310028030]
- Beijing NSF [1132003]
- SRFDP [20101108110001]
- East China Normal University
- NSF [DMS-1210400]
- Beijing Union Univ. [ZK 201206]
This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion { ut = [(d(1) + rho(12v))(u)](xx) + u(a(1) - b(1)u - c(1)v), x is an element of (0, 1), t > 0, v(t) = d(2)v(xx) + v(a(2) - b(2)u - c(2)v), x E (0, 1), t > 0, (1) u(x) = v(x) = 0, x = 0,1,t > 0. First, by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as rho(12) -> infinity and when d(2) is near a(2)/pi(2), which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when rho(12) is large enough and d(2) is near a(2)/pi(2).
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