Mathematical analysis of a three-tiered model of anaerobic digestion
Résumé
In this paper, we are interested in a mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web. The model is a six-dimensional system of ordinary differential equations. In our study, we take into account the phenol and the hydrogen input concentrations as well as the maintenance terms. Moreover, we consider the case of a large class of growth rates, instead of specific kinetics. In this general case, a recent study shows that the system can have up to eight steady states and their existence conditions were analytically determined. We focus here on the necessary and sufficient conditions of the local stability of the steady states, according to the four operating parameters of the process, which are the dilution rate and the chlorophenol, phenol and hydrogen input concentrations. In previous studies, this stability analysis was performed only numerically. Using the Liénard-Chipart stability criterion, we show that the positive steady state can be unstable and we give numerical evidence for a supercritical Hopf bifurcation with the appearance of a stable periodic orbit. We give two bifurcation diagrams with respect to the dilution rate, first, and then to the chlorophenol input concentration as the bifurcating parameters, showing that the system can present rich behavior including bistability, coexistence and occurrence of limit cycle.
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