Skip to main content
SpringerLink
Log in

Stability of monotone travelling waves for competition-diffusion equations

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We consider the Lotka-Volterra competition model with diffusion inR, and establish a theorem on the asymptotic stability of monotone travelling waves relative to the space of bounded uniformly continuous functions with the supremum norm. In consideration of the result of Alexander et al. [1] and Derndinger [4], we shall arrive at a study of the non-negative eigenvalues of the linearized operator around the travelling wave.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math.,410 (1990), 167–212.

    MATH  MathSciNet  Google Scholar 

  2. P. W. Bates and C. K. R. T. Jones, Invariant, manifolds for semilinear partial differential equations. Dynamics Reported Vol.2 (eds. U. Kirchgraber et al.), Wiley Chichester, 1989, 1–38.

  3. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  4. R. Derndinger, Über das Spektrum positiver Generatoren. Math. Z.,172 (1980), 281–293.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach. J. Differential Equations,44 (1982), 343–364.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Gardner and C. K. R. T. Jones, Stability of travelling wave solutions of diffusive predatorprey systems. Trans. Amer. Math. Soc.,327 (1991), 465–524.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math.840, Springer-Verlag, New York-Berlin-Tokyo, 1981.

    MATH  Google Scholar 

  8. Y. Hosono, Singular perturbation analysis of travelling waves of diffusive Lotka-Volterra competition models. Numerical and Applied Mathematics Part II (Paris, 1988) Baltzer, Basel, 1989, 687–692.

    Google Scholar 

  9. Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. Preprint.

  10. Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations. Hiroshima Math. J.23 (1993), 193–221.

    MATH  MathSciNet  Google Scholar 

  11. M. Mimura and P. C. Fife, A 3-component, system of competition and diffusion. Hiroshima Math. J.,16 (1986), 189–207.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ehime University, 790, Matsuyama, Japan

    Yukio Kan-On

  2. Department of Mathematics, Faculty of Science, Ehime University, 790, Matsuyama, Japan

    Qing Fang

Authors
  1. Yukio Kan-On
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research was partially carried out while visiting the Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology.

About this article

Cite this article

Kan-On, Y., Fang, Q. Stability of monotone travelling waves for competition-diffusion equations. Japan J. Indust. Appl. Math. 13, 343–349 (1996). https://doi.org/10.1007/BF03167252

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167252

Key words

Search

Navigation