Asymptotic behavior of solutions for a system of reaction-diffusion equations with density-dependent diffusion.

具有密度相关扩散的反应扩散方程组解的渐近行为。

• 批准号: 12640213
• 负责人: KAN-ON Yukio
• 金额: $ 1.6万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640213/
• 关键词: bifurcation theory; Lotka-Volterra model; comarisen pnnciple; 競争モデル

In this research, we consider a system of reaction-diffusion equations with density-dependent diffusion, which describes the dynamics of the population for two competing species community, and we intend to understand the mechanism of the coexistence by studying the existence and stability of stationary solutions for the system.In general, we comparatively easily determine the global attractor for the scalar reaction-diffusion equation by using the comparison principle and the energy. function, so that we can understand the precise asymptotic behavior of solutions for the equation. However, since the comparison principle does not always hold for the system, we have the considerable complexity for discussing the bifurcation structure of stationary solutions for the system.In this research, we consider a system such that the comparison principle holds at some values of a certain parameter, and we study the global bifurcation structure of stationary solutions with respect to such a parameter by employing the mathematical method such as the comparison principle and the bifurcation theory, and the numerical verification method such as the interval arithmetic built into Mathematica. As a result, it is shown that the global bifurcation structure of stationary solutions for the system under a certain condition is similar to that for the scalar reaction-diffusion equation shown by Chafee and Infante (1974/75). Moreover throughout this research, it is recognized again that the numerical verification is effective for analyzing the property of the bifurcation equation.Since the numerical verification does not succeed in some of regions, we have not determined the bifurcation structure of stationary solutions in such regions so far. In the future, it will be necessary to improve the algorithm for the numerical verification and its programming.

在本研究中，我们考虑了一个具有密度依赖扩散的反应扩散方程组，它描述了两个竞争种群群落的种群动力学，我们打算通过研究该系统稳定解的存在性和稳定性来理解共存的机制。利用比较原理和能量，我们比较容易地确定了标量反应扩散方程的整体吸引子。函数，这样我们就可以理解方程解的精确渐近行为。然而，由于比较原理对系统并不总是成立的，因此讨论系统定态解的分支结构具有相当的复杂性.在本研究中，我们考虑了一个系统，使得比较原理在某个参数的某些值处成立，并利用数学方法研究了稳态解关于这一参数的全局分支结构，比较原理和分歧理论，以及Mathematica内置的区间算法等数值验证方法。结果表明，在一定条件下，该系统的定态解的全局分支结构与Chafee和Infante（1974/75）所给出的标量反应扩散方程的全局分支结构相似.此外，通过研究，再次认识到数值验证对于分析分歧方程的性质是有效的，由于数值验证在某些区域没有成功，我们至今还没有确定这些区域内定态解的分歧结构。在今后的工作中，有必要对算法进行改进，以便进行数值验证和编程。

项目成果

Z.-C.Li, T.Yamamoto, Q.Fang: "Superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation. I. Smoothness problems"J. Comput. Appl. Math.. 151(2). 307-333 (2003)

Z.-C.Li，T.Yamamoto，Q.Fang：“泊松方程的 Shortley-Weller 差分近似解导数的超收敛。I. 平滑问题”J.

Qing Fang and Tetsuro Yamamoto: "Superconvergence of finite difference approximations for convection-diffusion problems"Numerical Linear Algebra with Applications. (掲載予定).

Qing Fang 和 Tetsuro Yamamoto：“对流扩散问题的有限差分近似的超收敛”数值线性代数及其应用（即将出版）。

Y.Kan-on: "Global bifurcation structure of positive stationary solutions for a classical Lotka-Volterra competition model with diffusion"Japan J. Indus. Appl. Math.. 20(3). 285-310 (2003)

Y.Kan-on：“具有扩散的经典 Lotka-Volterra 竞争模型的正平稳解的全局分叉结构”日本 J. Indus。

Q.Fang: "A note on the condition number of a matrix"J. Comput. Appl. Math.. 157(1). 231-234 (2003)

Q.Fang：“关于矩阵条件数的注释”J.

Z.-C.Li, H-Y.Hu, Q.Fang, T.Yamamoto: "Superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation. II. Singularity problems"Numer.Funct.Anal.Optim.. 24・3-4. 195-221 (2003)

Z.-C.Li、H-Y.Hu、Q.Fang、T.Yamamoto：“泊松方程 Shortley-Weller 差分近似解导数的超收敛。II. 奇异性问题”Numer.Funct.Anal.Optim.. 24・3-4。195-221（2003）