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2005 Fiscal Year Final Research Report Summary

Bifurcation structure of positive stationary solutions for a competition-diffusion system and its numerical verification

Research Project

Project/Area Number 15540124
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionEhime University

Principal Investigator

KAN-ON Yukio  Ehime University, Faculty of Education, Associate Professor, 教育学部, 助教授 (00177776)

Co-Investigator(Kenkyū-buntansha) FANG Qing  Yamagata University, Faculty of Science, Associate Professor, 理学部, 助教授 (10243544)
Project Period (FY) 2003 – 2005
Keywordscompetition-diffusion system / bifurcation theory / comparison principle / numerical verification
Research Abstract

We intend to understand the mechanism of the coexistence by studying the existence and stability of positive stationary solutions for a competition-diffusion system, which describes the dynamics of the population density for a two competing species community. In earlier studies, for the case where the habitat of the community is an interval, we investigate the spatial profile and the distribution of eigenvalues to the positive stationary solution, and then we establish the global bifurcation structure of positive stationary solutions for the system. To do this, we employ mathematical methods such as the bifurcation theory and the comparison principle, and numerical methods such as the numerical computation and the numerical verification. It seems that enough results have not been obtained so far, because the habitat is in general a two- or three-dimensional region. In this research, we assume that the habitat is the inside of a ball, and try to study the bifurcation structure of radially symmetric positive stationary solutions for the system.
It is hard to investigate the property of positive stationary solutions for this case, so that the information on the set of positive stationary solutions is not obtained enough to establish the bifurcation structure. However, it could be shown that the set of monotone positive stationary solutions is represented as the graph of a certain function with respect to the value of the positive stationary solution at the origin of the ball. Moreover, we see from the numerical verification that the secondary bifurcation of saddle-node type occurs. These facts give us a clue to understand the bifurcation structure.
In the future, it will be necessary to solve some open problems such as what spatial profile each positive stationary solution has, what kind of entire positive stationary solution exists, and so on.

  • Research Products

    (12 results)

All 2006 2005 2003

All Journal Article (12 results)

  • [Journal Article] Bifurcation structure of positive stationary solutions for a Lotka-Volterra competition model with diffusion II : Global structure2006

    • Author(s)
      Yukio Kan-on
    • Journal Title

      Discrete Contin. Dyn. Syst. 14・1

      Pages: 135-148

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Bifurcation structure of positive stationary solutions for a Lotka-Volterra Competition model with diffusion II : Global structure2006

    • Author(s)
      Yukio Kan-on
    • Journal Title

      Discrete Contin.Dyn.Syst. 14,no.1

      Pages: 135-148

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the structure of the set of stationary solutions for a system of reaction-diffusion equations with competitive interaction2005

    • Author(s)
      Yukio Kan-on
    • Journal Title

      Japan J. Indust. Appl. Math. 22・3

      Pages: 385-402

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Superconvergence of finite difference methods for initial-boundary value problems of convection-diffusion equations2005

    • Author(s)
      Qing Fang
    • Journal Title

      Information 8・3

      Pages: 359-368

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the structure of the set of stationary solutions for a system of reaction-diffusion equations with competitive interaction2005

    • Author(s)
      Yukio Kan-no
    • Journal Title

      Japan J.Indust.Appl.Math. 22,no.3

      Pages: 385-402

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Superconvergence of finite difference methods for initial-boundary value problems of convection-diffusion equations2005

    • Author(s)
      Qing Fang
    • Journal Title

      Information 8,no.3

      Pages: 359-368

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Global bifurcation structure of positive stationary solutions for a classical Lotka-Volterra competition model with diffusion2003

    • Author(s)
      Yukio Kan-on
    • Journal Title

      Japan J. Indust. Appl. Math. 20・3

      Pages: 285-310

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A note on the condition number of a matrix2003

    • Author(s)
      Qing Fang
    • Journal Title

      J. Comput. Appl. Math. 157・1

      Pages: 231-234

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence of finite difference methods for convection-diffusion problems with singular solutions2003

    • Author(s)
      Qing Fang
    • Journal Title

      J. Comput. Appl. Math. 152・1-2

      Pages: 119-131

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Global bifurcation structure of positive stationary solutions for a classical Lotka-Volterra competition model with diffusion2003

    • Author(s)
      Yukio Kan-no
    • Journal Title

      Japan J.Indust.Appl.Math. 20,no.3

      Pages: 285-310

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A note on the condition number of a matrix2003

    • Author(s)
      Qing Fang
    • Journal Title

      J.Comput.Appl.Math. 157,no.1

      Pages: 231-234

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Convergence of finite difference methods for convection-diffusion problems with singular solutions2003

    • Author(s)
      Qing Fang
    • Journal Title

      J.Comput.Appl.Math. 152,no.1-2

      Pages: 119-131

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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