Congestion phenomena of pulses in reaction-diffusion systems with excitability

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17594
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Meiji University
• Principal Investigator: Ikeda Kota 明治大学, 総合数理学部, 専任准教授 (50553369)
• Research Collaborator: Suematsu Nobuhiko J. ; Nagayama Masaharu
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 応用数学一般 / 中心多様体縮約理論

Outline of Final Research Achievements

We obtained two kinds of results in a mathematical model for the collective motion of camphor boats. First, we demonstrated that the numerical results obtained in our models for camphor boats are quite similar to those in a car-following model, the OV model, but there are some different features between our reduced model and a typical OV model. Those results are summarized in a manuscript published in a scientific journal. Secondly, we developed a center manifold theory to reduce the mathematical model of a PDF form for camphor boats into an ordinary differential system.
We also studied a reaction-diffusion system with three components to generate congestion phenomena of pulses. Since the reaction-diffusion system is one of excitable systems, it naturally has a solution with a pulse shape. We conclude that a bifurcation of a standing pulse can generate a traveling pulse and congestion of pulses.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

自己駆動する粒子は生物、非生物系に関わらず広く存在し、直接的、あるいは間接的に影響を及ぼすことで自己組織的に構造を形成する。渋滞現象は粒子系が呈する集団運動の一種と考えられており、そのメカニズムの理解は重要である。近年、車の渋滞を数理科学的に説明する試みが近年数多くなされており、特にOVモデルが良く調べられている。一方、反応拡散方程式系に現れる渋滞現象について、その数理的なメカニズムはほとんど未解明である。本研究で解明した反応拡散方程式系における渋滞現象の数理メカニズムや、開発した解析手法により、渋滞現象の理解がさらに深まったであろう。