The search for the structure which can appear in singular limits of reaction-diffusion systems

寻找可以出现在反应扩散系统奇异极限中的结构

• 批准号: 15540194
• 负责人: IIDA Masato
• 金额: $ 2.3万
• 依托单位: Iwate University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540194/
• 关键词: reaction-diffusion system; singular limit; nonlinear diffusion; competition system

What differential equations can be obtained as singular limits of reaction-diffusion systems? In the singular limits of the reaction-diffusion systems some effects by diffusion are generally degenerate in some sense : in fact, some types of systems which exhibit spatial singularities like interfaces are known to be certain singular limits of appropriate reaction-diffusion systems. Can a differential equation which exhibit no singularities become a singular limit of an appropriate reaction-diffusion system? In this research we investigated whether some non-degenerate quasi-linear diffusion equations appearing in mathematical biology can be obtained as such limits. In particular, we focused on the quasi-linear diffusion equations called the Shigesada-Kawasaki-Teramoto model which describes spatial segregation phenomena for the population of two competitive species.We succeeded in constructing a reaction-diffusion system whose appropriate singular limit becomes the Shigesada-Kawasaki-Teramoto model. We proved the convergence of such a reaction-diffusion system to the Shigesada-Kawasaki-Teramoto model by an energy method and clarified the fact that the linear stability of a constant equilibrium in such a reaction-diffusion system coincides with that in the Shigesada-Kawasaki-Teramoto model. Such a reaction-diffusion system we constructed are also regarded as a reasonable model describing the mechanism of biological diffusion.Also we held a workshop on applied mathematics at Morioka three times to bring together information basic in this research. The results of the workshop are summarized in three proceedings which are delivered to more than one hundred researchers who investigate related topics.

什么微分方程可以作为反应扩散方程组的奇异极限？在反应扩散系统的奇异极限中，某些扩散效应在某种意义上是退化的：事实上，某些类型的系统表现出空间奇异性，如界面，已知是适当的反应扩散系统的某些奇异极限。一个没有奇异性的微分方程能成为一个适当的反应扩散方程组的奇异极限吗？本文研究了数学生物学中出现的一些非退化拟线性扩散方程是否可以得到这样的极限。特别地，我们研究了描述两个竞争种群空间分离现象的拟线性扩散方程Shigesada-Kawasaki-Teramoto模型，成功地构造了一个反应扩散系统，其适当的奇异极限为Shigesada-Kawasaki-Teramoto模型.用能量方法证明了这类反应扩散方程组收敛于Shigesada-Kawasaki-Teramoto模型，并阐明了这类反应扩散方程组常数平衡点的线性稳定性与Shigesada-Kawasaki-Teramoto模型一致的事实.我们构建的反应扩散系统也被认为是描述生物扩散机制的合理模型。此外，我们在盛冈举办了三次应用数学研讨会，汇集了本研究的信息基础。研讨会的结果总结在三个会议记录，提供给一百多名研究相关主题的研究人员。

项目成果

西山公直, 三浦康秀: "相乗平均について"日本数学教育学会誌. 85. 472-472 (2003)

Kiminao Nishiyama、Yasuhide Miura：“关于几何平均数”日本数学教育学会杂志 85. 472-472 (2003)。

Decomposition of an order isomorphism between matrix ordered Hilbert spaces.

矩阵有序希尔伯特空间之间的阶同构的分解。

• 发表时间: 2004
• 期刊: Proc.Amer.Math.Soc. 132
• 作者: M.Iida;M.Mimura;H.Ninomiya;Y.Miura;大西良博;Y.Miura;Y.Miura
• 通讯作者: Y.Miura

Wave-Partide complementarity and reciprocity of Gauss sums on Talbot effects

塔尔博特效应上的波粒互补性和高斯和的互易性

• 发表时间: 2003
• 期刊: Foundations of Physics Letters 16巻4号
• 作者: 高橋真映;三浦康秀;三浦 毅;塚田 真;西山公直 他;S.Matsutani et al.
• 通讯作者: S.Matsutani et al.

Diffusion, Cross-diffusion and Competitive Interaction

• DOI: 10.1007/s00285-006-0013-2
• 发表时间: 2006-06
• 期刊: Journal of Mathematical Biology
• 影响因子: 1.9
• 作者: M. Iida;M. Mimura;H. Ninomiya

高橋真映, 三浦康秀, 三浦 毅, 塚田 真: "数例X_<n+1>=X_<n-1>/(1+X_n)の面白さと関連する話題"京都大学数理解析研究所講究録「情報科学としての関数解析とその周辺」. 1340. 104-108 (2003)

Shinei Takahashi、Yasuhide Miura、Takeshi Miura、Makoto Tsukada：《X_<n+1>=X_<n-1>/(1+X_n) 及相关主题的有趣例子》京都大学数学分析研究所研究记录《泛函分析》及其周围环境作为信息科学”。 1340. 104-108 (2003)。