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A scaled nonlinear mathematical model for interaction of algae with light: Existence and uniqueness results. (English) Zbl 0923.92026
Summary: We study the mathematical properties of a scaled nonlinear model for interaction of algae with light. The growth dynamics is modeled by a couple of nonlinear equations for the evolution of the concentration of algae and the light intensity, in a column of water. A diffusion model is derived applying formally the modified Chapman-Enskog expansion to the Boltzmann-like evolution system for a small parameter $$\varepsilon$$, proportional to the average time between successive photon-water collisions. Existence and uniqueness of a positive solution are proved, using semigroup perturbation theory.

##### MSC:
 92D40 Ecology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 82C70 Transport processes in time-dependent statistical mechanics 46N60 Applications of functional analysis in biology and other sciences 35B20 Perturbations in context of PDEs
##### Keywords:
interaction; algae; light; diffusion model
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##### References:
 [1] Banasiak J., Diffusion approximation and analysis of initial layer for evolution equations of kinetic type (1994) [2] DOI: 10.1002/mma.1670171306 · Zbl 0805.35090 [3] DOI: 10.1090/S0002-9947-1984-0743736-0 [4] Belleni-Morante A., A concise guide to semigroups and evolution equations (1994) · Zbl 0807.34074 [5] Carslaw H. S., Conduction of heat in solids (1959) · Zbl 0029.37801 [6] Frosali G., A scaled mathematical model for interaction of algae with light: formal derivation of a diffusion-like model (1995) [7] DOI: 10.1002/mma.1670030113 · Zbl 0474.34050 [8] DOI: 10.1080/00411459508205119 · Zbl 0821.45004 [9] DOI: 10.1142/9789812831248 [10] Poupaud F., Asymptotic Analysis 4 pp 293– (1991) · Zbl 0762.35092 [11] Reed M., Methods of modern mathematical physics 1 (1972) [12] DOI: 10.1007/BF00276919 · Zbl 0477.92018 [13] DOI: 10.1016/0362-546X(91)90064-8 · Zbl 0760.92023 [14] DOI: 10.1007/BF00285346 · Zbl 0555.92017 [15] DOI: 10.1016/0362-546X(89)90024-2 · Zbl 0721.92027
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