Journal of Mathematical Biology

, Volume 16, Issue 1, pp 1–24 | Cite as

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

  • Hitoshi Ishii
  • Izumi Takagi


We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.

We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.

Key words

Global stability Nonlinear diffusion equation Self-shading 


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  1. 1.
    Ishii, H.: On a certain estimate of the free boundary in the Stefan problem. J. Differential Equations 42, 106–115 (1981)Google Scholar
  2. 2.
    Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N.: Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monog., Vol. 23. Amer. Math. Soc., Providence, R.I., 1968Google Scholar
  3. 3.
    Protter, M. H., Weinberger, H. F.: Maximum principles in differential equations. Englewood Cliffs, N.J.: Prentice-Hall 1967Google Scholar
  4. 4.
    Shigesada, N., Okubo, A.: Analysis of the self-shading effect on algal vertical distribution in natural waters. J. Math. Biol. 12, 311–326 (1981)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Hitoshi Ishii
    • 1
  • Izumi Takagi
    • 2
  1. 1.Department of Mathematics, Faculty of Science and EngineeringChuo UniversityTokyoJapan
  2. 2.Tokyo Metropolitan College of Aeronautical EngineeringTokyoJapan

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