Abstract
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.
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References
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Shigesada, N., Okubo, A.: Analysis of the self-shading effect on algal vertical distribution in natural waters. J. Math. Biol. 12, 311–326 (1981)
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Ishii, H., Takagi, I. Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics. J. Math. Biology 16, 1–24 (1982). https://doi.org/10.1007/BF00275157
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DOI: https://doi.org/10.1007/BF00275157