Dynamics of reaction-diffusion-ODE system

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03354
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Ibaraki University
• Principal Investigator: Suzuki Kanako 茨城大学, 理工学研究科(理学野), 准教授 (10451519)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 反応拡散系

Outline of Final Research Achievements

We study a general reaction-diffusion-ODE system, which consists of several ordinary differential equations coupled with a reaction-diffusion equation, in a bounded domain and with Neuman boundary condition. There are a lot of mathematical models in the form of a reaction-diffusion-ODE system, for example, models of pattern formation, and models of ecological dynamics.
It has been proved that all near-equilibrium (regular) patterns of a general reaction-diffusion-ODE system are unstable, regardless of the particular structure assumption on nonlinearities. This observation suggests that stable stationary solutions arising in models with non-diffusive components must be far-from-equilibrium exhibiting singularities. We have presented the existence of stationary solutions with jump-discontinuities and have provided sufficient conditions for their stability.

Free Research Field

非線形解析学

Academic Significance and Societal Importance of the Research Achievements

拡散－非拡散系はパターン形成の数理モデルとしても多く用いられているが，個々のモデルに対して数値実験などが行われ，ダイナミクスに関する体系的な研究はほとんどなされていなかった．本研究により，拡散－非拡散系のダイナミクスは，古典的な反応拡散系のそれとはまったく異なることが明らかとなった．特に，拡散－非拡散系の初期値問題の解の挙動は特異的なものに限られるため，本研究結果は数値実験結果を正しく理解するために重要な役割を果たす．これはモデルの再考や現象の理解につながる大変意義のある結果であると考えられる．