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Mathematical studies on spatial and temporal patterns in reaction-diffusion systems

Research Project

Project/Area Number 09440075
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionUniversity of Tokyo

Principal Investigator

INABA Hisashi (1998-1999)  Graduate School of Mathematical Sciences, University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (80282531)

柳田 英二 (1997)  東京大学, 大学院・数理科学研究科, 教授 (80174548)

Co-Investigator(Kenkyū-buntansha) TAKAHASHI Katsuo  Graduate School of Mathematical Sciences, University of Tokyo, Research Assistant, 大学院・数理科学研究科, 助手 (90114529)
YAMADA Michio  Graduate School of Mathematical Sciences, University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (90166736)
YANAGIDA Eiji  Graduate School of Mathematical Sciences, University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (80174548)
NINOMIYA Hirokazu  Department of Applied Mathematics and Informatics, Ryukoku University, Assistant Professor, 理工学部, 講師 (90251610)
MIZOGUCHI Noriko  Department of Mathematics, Tokyo Gakugei University, Associate Professor, 教育学部, 助教授 (00251570)
堤 誉志雄  東京大学, 大学院・数理科学研究科, 助教授 (10180027)
三村 昌泰  広島大学, 理学部, 教授 (50068128)
飯田 雅人  岩手大学, 人文社会学部, 講師 (00242264)
稲葉 寿  東京大学, 大学院・数理科学研究科, 教授 (80282531)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥13,500,000 (Direct Cost: ¥13,500,000)
Fiscal Year 1999: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1998: ¥4,900,000 (Direct Cost: ¥4,900,000)
Fiscal Year 1997: ¥5,400,000 (Direct Cost: ¥5,400,000)
Keywordsreaction-diffusion system / nonlinear partial differential equation / stability analysis / activator-inhibitor system / singular limit analysis / blowup of solutions / free boundary / modeling / 興奮-抑制系 / 興奮-柳制系 / 歪勾配系 / 自己相似解 / 有限時間爆発 / バクテリヤコロニーモデル / スペクトル法 / 非線型放物型方程式 / 特異極限 / 爆発
Research Abstract

Concerning the above research project, we obtained the following results.
(1) We carried out the study on skew-gradient systems, which are a generalized activator-inhibitor systems. When a steady state is characterized as a min-max point of an energy functional, we showed that the steady state is stable regardless of parameter values. We also make clear the relation with the so-called Turing instability.
(2) We carried out numerical simulations on a model equation which describes a spatial pattern formation observed In a colony of some bacteria. It is shown that such spatial patterns appear as a history of the process.
(3) Two-species competition system is a main problem in the theory of mathematical ecology. We gave a mathematically rigorous verification for the fact that the system reduces to a Stephen -type free boundary value problem in a singular limit where the competition rate is infinitely large.
(4) The Gierer-Meinhardt system is a mathematical model for the morphogenesis in Mathematical biology. We studied stability of spiky stationary patterns, and gave some criteria for the stability and Instability.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (31 results)

All Other

All Publications (31 results)

  • [Publications] Y. Kabeya, E, Yanagida: "Eigenvalue Problems in the whole space with radially symmetric weight"Comm,PDE. 24. 1121-1166 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] E. Yanagida: "Extinction and blowing of positive radial solutions"Nonlinear Analysis. 39. 365-377 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Mizoguchi: "Slowing decay of solutions in a semilinear dissipative parabolic equation"J. Diflerential Equations. 158. 79-93 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Mizoguchi, H. Ninomiya: "Diffusion -induced blow up in a nonlinear parabolic system"J. Dyn. Diff. Eqs. 10-3. 619-638 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Inaba: "Persistent age-distribution for an age-structured two-sex population model"Mathematical Population Studies. 7(4). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Yamode, Y. Taniganchi: "A statical Property of packet round-trip times in the internet"Advances in Performance Analysis. 2-1. 67-76 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Kabeya and E. Yanagida: "Eigenvalue problem in the whole space with radially symmetric weight"Comm. PDE. 24. 1127-1166 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] E. Yanagida: "Extinction and blowup of positivie radial solutions for a semilinear elliptic equation"Nonlinear Anal.. 39. 365-377 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Mizoguchi: "low decay of solutions in a semilinear dissipative parabolic equation"J. Differential Equations. 158. 79-93 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Mizoguchi, H. Ninomiya, and E. Yanagida: "Diffusion-induced blowup in a nonlinear parabolic system"J. Dynamics and Differential Equations. 10-3. 619-638 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Inaba: "Persistent age-distributions for an age-structured two-sex population model"Mathematical Population Studies. 7(4). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Yamada and Y. Taniguchi: "A statistical property of packet round -trip times in the internet"Advances in Performance Analysis. vol. 2-1. 67-76 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Kabeya, E. Yanagida: "Eigenvalue problems in the whole space with radially symmetric weight"Comm. PDE. 24. 1127-1166 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] E. Yanagida: "Extinction and blowup of positive radial solutions"Nonlinear Analysis. 39. 365-377 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] N. Mizoguchi: "Slow decay of solutions in a semilinear dissipative parabolic equation"J. Differential Equations. 158. 79-93 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. Kawaguchi, M. Mimura: "Collision of travelling waves in a reaction-diffusion system with global coupling"SIAM J. Appl. Math.. 59. 920-941 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. -I. Ei, R. Ikota, M. Mimura: "Segregating partition problems in conpetion-diffusion systems"Interfaces and Free Boundaries. 1. 57-80 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Yamada, Y. Taniguchi: "A statistical property of packet round-trip times in the internet"Advances in Performance Analysis. 2-1. 67-76 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Mizoguchi, Z.Yanagida: "Blownpand life span of solutions for a semilineac parabolic equation" SIAM J.Math.Anal.29. 1434-1446 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Iida, T.Muramatsu, H.Iida, Z.Yanagida: "Diffusion-induced extinctrin of a superior species in a competion system" Japan J.Indust.Appl.Math.15(2). 233-252 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Critical exponents for the below up of solutions with sig changes in a semilinear parabolic equation II" J.Differential Equations. 145(2). 295-331 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Mizoguchi, H.Ninomiya, E.Yanagida: "Critical exponents for the bipolar blowup in a semilinar parabolic equactions" J.Math.Anal.Appl.218(2). 495-518 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Matsushita, J.Wakita, H.Sakaguchi, H.Itoh, I.Ratols, H.Matsuyama, M.Mimura: "Interfacial growth and pattern fomation in bacterial colonies" Physica A. 249. 517-524 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Yagisita, M.Mimura, M.Yamada: "Spiral wave behaviors in an excitable reaction diffusion system on a sphere" Physica D.124. 126-136 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Iida, E.Yanagida: "Dynamics of Interfaces in a Scalar Parabolic Equation" Japan J.Indust.Appl.Math.14. 1-23 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Hirose, E.Yanagida: "Biofurcation of Fapidly Decaying Solutions for H-W Eq." Adv.Math.Sci.Appl.7. 619-636 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Critical Exponents in a Semilinear Parabolic Equation" Math.Annal.307. 663-675 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Mimura, S,-I, Ei: "Singnlar Perturbation Problems to a Combnstion" Studies in Adv.Math.3. 75-84 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Ohta, M.Mimura: "Collision of Propagating Pulses in R-D System" J.Phys.Soc.Japan. 66. 1551-1558 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 稲葉 寿: "数理人工学の発展" 人工学研究. 1997 (21)

    • Related Report
      1997 Annual Research Report
  • [Publications] 稲葉 寿: "数理人工学の発展" 人工学研究. 1997 (21)7-17:

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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