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This paper proposes a mathematical model for replicating a simple dynamics inan aquarium with two components; bacteria and organic matter. The model isbased on a system of partial differential equations (PDEs) with fourcomponents: the drift-diffusion equation, the dynamic boundary condition, thefourth boundary condition, and the prey-predator model. The system of PDEs isstructured to represent typical dynamics, including the increase of organicmatter in the aquarium due to the excretion of organisms ($e.g$. fish), itsadsorption into the filtration filter, and the decomposition action of theorganic matter both on the filtration filter and within the aquarium. In thispaper, we prove the well-posedness of the system and show some results ofnumerical experiments. The numerical experiments provide a validity of themodeling and demonstrate filter clogging phenomena. We compare the feeding ratewith the filtration performance of the filter. The model exhibits convergenceto a bounded steady state when the feed rate is reasonable, and grow up to anunbounded solution when the feeding is excessively high. The latter correspondsto the clogging phenomenon of the filter.

Modeling and Mathematical Analysis of the Clogging Phenomenon in Filtration Filters Installed in Aquaria