Timm, Uwe; Totaro, Silvia; Okubo, Akira Self- and mutual shading effect on competing algal distribution. (English) Zbl 0760.92023 Nonlinear Anal., Theory Methods Appl. 17, No. 6, 559-576 (1991). 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. Reviewer: Y.Y.Sugai (Chiba-shi) Cited in 4 Documents MSC: 92D40 Ecology 45K05 Integro-partial differential equations 45N05 Abstract integral equations, integral equations in abstract spaces 47H20 Semigroups of nonlinear operators Keywords:competing algal populations; dynamics of algae blooms; self-shading effects; mutual shading effects; nonlinear integro-partial differential equations; unique positive global mild solution; existence of nonnegative steady-state solutions Citations:Zbl 0721.92027 PDFBibTeX XMLCite \textit{U. Timm} et al., Nonlinear Anal., Theory Methods Appl. 17, No. 6, 559--576 (1991; Zbl 0760.92023) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.