Positive solutions for Lotka–Volterra competition systems with large cross-diffusion
Positive solutions for Lotka–Volterra competition systems with large cross-diffusion
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DOI：
10.1080/00036811003627534
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发表时间：
2010-06
影响因子：
1.1
通讯作者：
Kousuke Kuto;Yoshio Yamada
中科院分区：
文献类型：
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作者：
Kousuke Kuto;Yoshio Yamada
本文讨论了具有交叉扩散的Lotka-Volterra竞争系统在齐次Dirichlet边界条件下的平稳问题。虽然已知正解存在的一些充分条件，但关于其结构的信息还远不完整。为了更好地理解具有交叉扩散的竞争系统，我们研究了大交叉扩散对正解结构的影响，并通过使其中一个交叉扩散系数为无穷大来研究正解的极限行为.特别地，它将表明，竞争系统的正解收敛到一个适当的限制系统的正解。我们也将得到一些关于这个极限系统正解的满意结果。这些结果给我们提供了重要的信息时，竞争系统的交叉扩散系数是足够大的正解的结构。
This paper discusses the stationary problem for the Lotka–Volterra competition systems with cross-diffusion under homogeneous Dirichlet boundary conditions. Although some sufficient conditions for the existence of positive solutions are known, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we study the effects of large cross-diffusion on the structure of positive solutions and focus on the limiting behaviour of positive solutions by letting one of the cross-diffusion coefficients to infinity. Especially, it will be shown that positive solutions of the competition system converge to a positive solution of a suitable limiting system. We will also derive some satisfactory results on positive solutions for this limiting system. These results give us important information on the structure of positive solutions for the competition system when one of the cross-diffusion coefficients is sufficiently large.