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Self- and mutual shading effect on competing algal distribution. (English) Zbl 0760.92023

Shading of light by algae growing in a watercolumn plays an interesting and important role in the dynamics of algae blooms. In natural waters many species of algae compete not only for light but also for nutrients. For a single species of phytoplankton without resource limitation, a nonlinear model for self-shading effects on algal vertical distribution is already studied. The second author [see ibid. 13, No. 8, 969-986 (1989; Zbl 0721.92027)] developed a nonlinear model for two species of algae that compete for light without nutrient limitation.
The presence of a species in water can attenuate light intensity not only on itself through self- and mutual shading effects, but also for other species that are present. The second author analysed a system of two nonlinear integro-partial differential equations for the existence of a unique positive global solution and examined a criterion for the existence of nonnegative stationary solutions in terms of the parameters appearing in the equations.
In this paper the second author’s model of algae populations which compete for nutrients is extended. The existence of a unique positive global mild solution is proved using techniques of semigroup theory. The existence of nonnegative steady-state solutions to the model system is also proved and the criterion for existence of the solutions is identified.

MSC:

92D40 Ecology
45K05 Integro-partial differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
47H20 Semigroups of nonlinear operators

Citations:

Zbl 0721.92027
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References:

[1] Shigesada, N.; Okubo, A., Analysis of the self-shading effect on algal vertical distribution in the natural waters, J. math. Biol., 12, 311-326 (1981) · Zbl 0477.92018
[2] Ishii, H.; Takagi, I., Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. math. Biol., 16, 1-24 (1982) · Zbl 0501.92020
[3] Totaro, S., Mutual shading on algal distribution: a nonlinear problem, Nonlinear Analysis, 13, 968 (1989) · Zbl 0721.92027
[4] Wroblewski, J. S.; O’Brien, J. J., A spatial model of phytoplankton patchiness, Mar. Biol., 35, 161-175 (1976)
[5] Kato, T., Perturbation Theory for Linear Operator (1976), Springer: Springer New York
[6] Belleni-Morante, A., Applied Semigroups and Evolution Equations (1979), Oxford University Press: Oxford University Press Oxford · Zbl 0426.47020
[7] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer New York · Zbl 0516.47023
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