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Box models are simplified versions of complex systems, reducing them to boxes (or reservoirs) linked by fluxes. The boxes are assumed to be mixed homogeneously. Within a given box, the concentration of any chemical species is therefore uniform. However, the abundance of a species within a given box may vary as a function of time due to the input to (or loss from) the box or due to the production, consumption or decay of this species within the box.
Simple box models, i.e. box model with a small number of boxes whose properties (e.g. their volume) do not change with time, are often useful to derive analytical formulas describing the dynamics and steady-state abundance of a species. More complex box models are usually solved using numerical techniques.
Box models are used extensively to model environmental systems or ecosystems and in studies of ocean circulation and the carbon cycle. See Sarmiento et. al in the references.
Box model, Wikipedia
Slingerland, Kump Princeton University Press 2011, Mathematical Modeling of Earth’s Dynamical Systems- a Primer
Sarmiento, J.L.; Toggweiler, J.R. (1984). “A new model for the role of the oceans in determining atmospheric P CO 2”. Nature 308 (5960): 621–24. doi:10.1038/308621a0.