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Mathematical analysis and reaction diffusion approximation for pattern formations with nonlocal interactions

Research Project

Project/Area Number 17K14228
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionFuture University-Hakodate (2018-2019)
Hokkaido University (2017)

Principal Investigator

Yoshitaro Tanaka  公立はこだて未来大学, システム情報科学部, 准教授 (80783977)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords反応拡散系 / 非局所相互作用 / 非局所発展方程式 / 反応拡散近似 / 応用数学 / 解析学 / パターン形成
Outline of Final Research Achievements

Recently the existence of the spatially nonlocal interactions which can change the influence on the objects depending on the distance has been reported in fish cells. As this interaction influences spatially globally, it can be modeled by the convolutions with suitable kernel, and various nonlocal evolution equations were proposed. Although the nonlocal evolution equations can reproduce various patterns and many applications are expected, the technique of analyzing a nonlocal evolution equations were not rich. Motivated by these backgrounds, we proposed a method to approximate the nonlocal evolution equations into reaction diffusion system in which the various theories have already been established. We revealed that the nonlocal evolutions equations with any even kernels can be approximated by a reaction diffusion system in one-dimensional spaces.

Academic Significance and Societal Importance of the Research Achievements

動物や魚の表皮等に観察されるパターン形成や,昆虫の脳における神経形成,また細胞接着現象など,積分付きの相互作用をもつ発展方程式はさまざまな現象を記述することができる.広い分野の現象に応用が期待できるため,解析手法を整備することが求められるが,現状発展途上である.そこで我々は,すでに多くの理論が整備されている反応拡散系という方程式に,積分つきの相互作用をもつ発展方程式を近似する方法を提案し,理論的に近似できることを示した.このことから,積分つきの相互作用をもつ発展方程式を解析することが期待でき,さらなる応用例や理論的な知見を生み出すことができると考えている.

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (15 results)

All 2020 2019 2018 2017 Other

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Open Access: 2 results) Presentation (10 results) (of which Int'l Joint Research: 3 results,  Invited: 6 results) Remarks (1 results)

  • [Journal Article] JAK/STAT guarantees robust neural stem cell differentiation by shutting off biological noise2018

    • Author(s)
      Tanaka Yoshitaro、Yasugi Tetsuo、Nagayama Masaharu、Sato Makoto、Ei Shin-Ichiro
    • Journal Title

      Scientific Reports

      Volume: 8 Issue: 1 Pages: 12484-12484

    • DOI

      10.1038/s41598-018-30929-1

    • NAID

      120006536943

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Reaction diffusion approximation of nonlocal interactions using Jacobi polynomials2018

    • Author(s)
      Ninomiya Hirokazu、Tanaka Yoshitaro、Yamamoto Hiroko
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 印刷中 Issue: 2 Pages: 613-651

    • DOI

      10.1007/s13160-017-0299-z

    • NAID

      210000170293

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Mathematical approach to nonlocal interactions using a reaction-diffusion system2017

    • Author(s)
      Y. Tanaka, H. Yamamoto, and H. Ninomiya
    • Journal Title

      Development, Growth & Differentiation

      Volume: - Issue: 5 Pages: 388-395

    • DOI

      10.1111/dgd.12354

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Reaction, diffusion and non-local interaction2017

    • Author(s)
      H. Ninomiya, Y. Tanaka and H. Yamamoto
    • Journal Title

      Journal of Mathematical Biology

      Volume: 75 Issue: 5 Pages: 1203-1233

    • DOI

      10.1007/s00285-017-1113-x

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] 空間離散モデルに対する細胞の大きさと形状を残す連続化法の提案と応用2020

    • Author(s)
      田中吉太郎
    • Organizer
      反応拡散系と実験の融合3
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Reaction-diffusion approximation for understanding pattern formations through non-local interactions2020

    • Author(s)
      田中吉太郎
    • Organizer
      第16回生物数学の理論とその応用, 京都大学数理解析研究所
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 細胞の大きさと形状を残す空間離散モデルの連続化の提案と応用2019

    • Author(s)
      田中吉太郎
    • Organizer
      日本応用数理学会2019年度年会
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] The continuation method for spatially discretized models with nonlocal interactions2019

    • Author(s)
      田中吉太郎
    • Organizer
      ICIAM2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 円領域上の修正ヘルムホルツ方程式のノイマン境界値問題に対する精度保証付き基本解近似解法2018

    • Author(s)
      田中吉太郎
    • Organizer
      応用数学合同研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] Reaction-diffusion approximation for nonlocal interactions2018

    • Author(s)
      田中吉太郎
    • Organizer
      Society for Mathematical Biology 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The theoretical approach for pattern formations based on the convolution kernels in the networks systems2017

    • Author(s)
      Yoshitaro Tanaka
    • Organizer
      Equadiff 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 分化の波の数理モデルに対する離散構造を保持する連続化の提案2017

    • Author(s)
      田中吉太郎
    • Organizer
      日本応用数理学会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] The theoretical approach for pattern formations based on the convolution kernels in the networks systems2017

    • Author(s)
      Yoshitaro Tanaka
    • Organizer
      日本数理生物学会
    • Related Report
      2017 Research-status Report
  • [Presentation] 分化の波に対する数理モデルの連続化と数理解析2017

    • Author(s)
      田中吉太郎
    • Organizer
      2017年度応用数学合同研究集会
    • Related Report
      2017 Research-status Report
  • [Remarks] Yoshitaro TANAKAのホームページ

    • URL

      https://www.fun.ac.jp/~y-tanaka/

    • Related Report
      2017 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2021-02-19  

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