×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.