Control of patterns by multicomponent reaction-diffusion systems of degenerate type

2017 Fiscal Year Final Research Report

• Project/Area Number: 26610027
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: Takagi Izumi 東北大学, 理学研究科, 名誉教授 (40154744)
• Co-Investigator(Renkei-kenkyūsha): YAMAMOTO Hiroko 明治大学, 研究・知財戦略機構, 研究推進員 (10759153)
• Research Collaborator: LI Ying Harbin Institute of Technology
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 反応拡散方程式 / パターン形成 / 多成分 / 非拡散性成分 / 拡散性成分 / 不連続定常解

Outline of Final Research Achievements

A standard approach to modeling biological pattern formation has been based on the idea that the interaction between a slowly diffusing activator and a rapidly diffusing inhibitor results in spontaneous formation of spatial patterns. However, if the activator is a receptor attached on the cell membrane, it does not diffuse. Lately, reaction-diffusion models with non-diffusive activator and diffusive inhibitor have been proposed. In this project we studied rigorously the existence and stability of steady-state solutions of such model systems. We proved that there exist two types of steady states. Type I: the activator has jump discontinuity (but the inhibitor is continuous), and Type II: the activator is continuous. In addition, there exist continua of discontinuous steady states. Moreover, discontinuous steady states are stable, while continuous steady states are stable.

Free Research Field

非線型解析学