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

Research Project

Project/Area Number 26610027
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionTohoku 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
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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 / 退化型反応拡散系 / 第一種不連続性をもつ定常解 / 安定性 / 拡散成分と非拡散成分 / 反応拡散系 / 不連続定常解の安定性 / 分岐解 / 非拡散性種 / 受容体―結合基反応 / 非拡散性物質
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.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (27 results)

All 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (5 results) Journal Article (7 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 7 results,  Open Access: 2 results,  Acknowledgement Compliant: 4 results) Presentation (13 results) (of which Int'l Joint Research: 9 results,  Invited: 10 results) Remarks (2 results)

  • [Int'l Joint Research] Harbin Institute of Technology(China)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Heidelberg University(Germany)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Heidelberg University(Germany)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Harbin Institute of Technology(China)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Heidelberg University(Germany)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Locator function for concentration points in a spatially heterogeneous semilinear Neumann problem2019

    • Author(s)
      Izumi Takagi and Hiroko Yamamoto
    • Journal Title

      Indiana University Mathematics Journal

      Volume: 掲載決定

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor2019

    • Author(s)
      Ying Li, Anna Marciniak-Czochra, Izumi Takagi and Boying Wu
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 掲載決定

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis2017

    • Author(s)
      Ying Li, Anna Marciniak-Czochra, Izumi Takagi and Boying Wu
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 47 Pages: 217-247

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stable patterns with jump discontinuity in systems with Turing stability and hysteresis2017

    • Author(s)
      S. Haerting, A. Marciniak-Czochra and I. Takagi
    • Journal Title

      Discrete Contin. Dyn. Syst. Series A

      Volume: 37 Issue: 2 Pages: 757-800

    • DOI

      10.3934/dcds.2017032

    • Related Report
      2017 Annual Research Report 2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis2017

    • Author(s)
      Y. Li, A. Marciniak-Czochra, I. Takagi and B.Y. Wu
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 47

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Pattern formation in a diffusion -ODE model with hysteresis2015

    • Author(s)
      Anna Marciniak-Czochra, Madoka Nakayama and Izumi Takagi
    • Journal Title

      Differential Integral Equations

      Volume: 28 Pages: 655-694

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Pattern formation in a diffusion-ODE model with hyteresis2015

    • Author(s)
      Anna Marciniak-Czochra, Madoka Nakayama and Izumi Takagi
    • Journal Title

      Differential and Integral Equations

      Volume: 印刷中

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Steady States of the FitzHugh-Nagumo equations with non-diffusive activator and diffusive inhibitor2018

    • Author(s)
      Izumi Takagi
    • Organizer
      The Third Hale Conference on the Dynamics of Differential Equations
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 分岐点から遠く離れて2017

    • Author(s)
      高木 泉
    • Organizer
      拡散成分と非拡散成分が共存する反応拡散系がつくるパターン
    • Place of Presentation
      東北大学理学研究科
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
  • [Presentation] Pattern formation by FitzHugh-Nagumo equations with non-diffusive activator and diffusive inhibitor2017

    • Author(s)
      Izumi Takagi
    • Organizer
      International Workshop on Nonlinear Analysis and Reaction-Diffusion Equaations
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Control of patterns by way of spatial heterogeneity2017

    • Author(s)
      Izumi Takagi
    • Organizer
      International workshop on nonlinear elliptic and parabolic PDEs
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Continua of stable discontinuous steady-states in receptor-based models of biological pattern formation2017

    • Author(s)
      Izumi Takagi
    • Organizer
      Conference on PDEs and Applications to Physical and Biological Sciences
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Patterns generated by FitzHugh-Nagumo equations with nondiffusive activator and diffusive inhibitor2017

    • Author(s)
      Izumi Takagi
    • Organizer
      Stepping Stone Symposium on Theoretical and Numerical Analysis of PDEs 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Locator function for concentration points of singularly perturbed stationary solutions to a reaction-diffusion equation2016

    • Author(s)
      Izumi Takagi
    • Organizer
      The Second Workshop on Differential Geometry and Differential Equations
    • Place of Presentation
      Renmin University of China, 北京, 中国
    • Year and Date
      2016-11-14
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Patterns generated by a nondiffusive activator and a diffusive inhibitor2016

    • Author(s)
      Izumi Takagi
    • Organizer
      Mathematics of Pattern Formation
    • Place of Presentation
      Mathematical Research and Conference Center in Bedlewo, Poland
    • Year and Date
      2016-09-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical beauty of the activator-inhibitor system2016

    • Author(s)
      Izumi Takagi
    • Organizer
      日本数理生物学会2016年会
    • Place of Presentation
      九州大学椎木ホール,福岡
    • Year and Date
      2016-09-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Pattern formation in homogeneous media vs in heterogeneous media: Cutting and grafting experiments on hydra2016

    • Author(s)
      Izumi Takagi
    • Organizer
      日本数理生物学会2016年会
    • Place of Presentation
      九州大学椎木ホール,福岡
    • Year and Date
      2016-09-08
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Influence of spatial heterogeneity on a reaction-diffusion-ODE system2016

    • Author(s)
      Izumi Takagi
    • Organizer
      JSMB 2016 annual meeting
    • Place of Presentation
      九州大学
    • Year and Date
      2016-09-07
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 受容体・結合子反応モデルについての一考察2015

    • Author(s)
      高木 泉
    • Organizer
      Turing機構に関連するパターンとダイナミクス
    • Place of Presentation
      広島大学
    • Year and Date
      2015-12-19
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 空間的不均一性に基づく点凝集パターンの制御2014

    • Author(s)
      高木 泉
    • Organizer
      岡山駅前セミナー:現象を捉える新しい数学的基盤の構築に向けて
    • Place of Presentation
      岡山国際交流センター, 岡山
    • Year and Date
      2014-12-13
    • Related Report
      2014 Research-status Report
    • Invited
  • [Remarks] 生物の形づくりの数理

    • URL

      http://morphos.jp/

    • Related Report
      2017 Annual Research Report 2015 Research-status Report 2014 Research-status Report
  • [Remarks] 生物の形づくりの数理

    • URL

      http://morpho.sci.tohoku.ac.jp/~morpho/

    • Related Report
      2014 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2019-03-29  

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