A finite difference approximation for a nonlinear size-structured phytoplankton aggregation model

Ackleh, Azmy S.; Ferdinand, Robert R.

doi:10.1090/qam/1704439

Authors: Azmy S. Ackleh and Robert R. Ferdinand
Journal: Quart. Appl. Math. 57 (1999), 501-520
MSC: Primary 92D25; Secondary 65M06, 92C37
DOI: https://doi.org/10.1090/qam/1704439
MathSciNet review: MR1704439
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear model that describes the dynamics of a phytoplankton population with aggregation and competition between individual cells. A finite difference method is developed for approximating the solution of this partial differential equation. The convergence of this approximation to a unique bounded variation solution of the model is proved. Numerical results showing the accuracy of this scheme are presented.

Azmy S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. 28 (1997), no. 5, 837–854. MR 1422189, DOI https://doi.org/10.1016/0362-546X%2895%2900195-2
Azmy S. Ackleh and Ben G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol. 35 (1997), no. 4, 480–502. MR 1478594, DOI https://doi.org/10.1007/s002850050062
Azmy S. Ackleh, Ben G. Fitzpatrick, and Thomas G. Hallam, Approximation and parameter estimation problems for algal aggregation models, Math. Models Methods Appl. Sci. 4 (1994), no. 3, 291–311. MR 1282237, DOI https://doi.org/10.1142/S0218202594000182

A. S. Ackleh, T. G. Hallam, and Helene C. Muller Landau, Estimation of Sticking and Contact Efficiencies in Aggregation of Phytoplankton: The 1993 SIGMA Tank Experiment, Deep-Sea Research II, 42, 185-201 (1995)

A. S. Ackleh, T. G. Hallam, and W. O. Smith, Influences of Aggregation and Grazing on Phytoplankton Dynamics and Fluxes: An Individual Based Modeling Approach, Nonlinear World 1, 473-492 (1994)

Azmy S. Ackleh and Kazufumi Ito, An implicit finite difference scheme for the nonlinear size-structured population model, Numer. Funct. Anal. Optim. 18 (1997), no. 9-10, 865–884. MR 1485984, DOI https://doi.org/10.1080/01630569708816798
Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI https://doi.org/10.1090/S0025-5718-1980-0551288-3

G. A. Jackson, Compared Observed Changes in Particle Size Spectra with Those Predicted Using Coagulation Theory, Deep-Sea Research II, 42, 159-184 (1994)

G. Jackson, A Model of Formation of Marine Algal Flocs by Physical Coagulation Processes, Deep-Sea Research 37, 1197-1211 (1990)

I. N. McCave, Size-Spectra and Aggregation of Suspended Particles in the Deep Ocean, Deep-Sea Research 31, 329-352 (1984)

C. R. O’Melia and K. S. Bowman, Origins and Effects of Coagulation in Lakes, Schweiz. Z. Hydrol. 46, 64-85 (1984)

T. R. Parsons and M. Takahashi, Biological Oceanographic Processes, Pergamon Press, 1984

U. Riebesell and D. A. Wolf-Gladrow, The Relationship between Physical Aggregation of Phytoplankton and Particle Flux: A Numerical Model, Deep-Sea Research 39, 1085-1102 (1992)

Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779

U. Weilenmann, C. R. O’Melia, and W. Stumm, Particle Transport in Lakes: Models and Measurements, Limnology and Oceanography 34, 1-18 (1989)

David M. Young, Iterative solution of large linear systems, Academic Press, New York-London, 1971. MR 0305568