<p>This paper proposes a mathematical model for replicating a simple dynamics in an aquarium with two components; bacteria and organic matter. The model is based on a system of partial differential equations (PDEs) with four components: the drift-diffusion equation, the dynamic boundary condition, the fourth boundary condition, and the prey-predator model. The system of PDEs is structured to represent typical dynamics, including the increase of organic matter in the aquarium due to the excretion of organisms (e.g. fish), its adsorption into the filtration filter, and the decomposition action of the organic matter both on the filtration filter and within the aquarium. In this paper, we prove the well-posedness of the system and show some results of numerical experiments. The numerical experiments provide a validity of the modeling and demonstrate filter clogging phenomena. We compare the feeding rate with the filtration performance of the filter. The model exhibits convergence to a bounded steady state when the feed rate is reasonable, and grow up to an unbounded solution when the feeding is excessively high. The latter corresponds to the clogging phenomenon of the filter.</p>