Extensions of the chemostat model with flocculation
In this work a model of the chemostat is studied in the case when the species are presented in two forms, isolated and aggregated individuals, such as attached bacteria or bacteria in flocks. More precisely, general model is considered of a bio-process with three compartments involving the substrate, the planktonic and attached biomass densities, respectively. Each compartment of the biomass is characterized by its own specific growth rate and apparent dilution rate, generalizing previous models of biofilms (with no dilution rate for the attached bacteria) or models of perfect flocks (with no growth rate for aggregated individuals). A class of models with planktonic and structured biomass is analyzed under the assumptions that attachment and detachment processes are fast compared to the biological scale. Assuming that the flocculation and deflocculation dynamics is fast compared to the growth of the species, a reduced chemostat-like model is constructed in which both the growth functions and the apparent dilution rate depend on the density of the species. In the case when the previous assumption is not fulfilled, the analysis of the original three order model must be done to establish its qualitative behavior. The study performed in the paper reveals two main characteristics of this model: the reduced dynamics may exhibit a bi-stable behavior even though each growth function is monotonic. This phenomenon seems to be new and is met in the chemostat but when the growth function is non-monotonic. For bio-processes in which part of the biomass is under a structured form (in flocks or bio films), the macroscopic models (with reduced dynamics involving only the aggregated biomass and the substrate) should include a growth rate and an apparent dilution rate that are both density dependent.
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