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, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. In previous studies in the existing literature, the stability of the steady states was considered only in the particular case without maintenance, where the model can be reduced to a three-dimensional system. Moreover, we consider the case of a large class of growth kinetics, instead of Monod kinetics. According to the four operating parameters of the process, represented by the dilution rate and input concentrations of the chlorophenol, the phenol and the hydrogen, we show that the system can have up to eight steady states and we analytically determine the necessary and sufficient conditions for their existence and their local stability. In previous studies of the case including maintenance, the stability analysis was performed only numerically. 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. Finally, we give a bifurcation diagram with the concentration of influent chlorophenol as the bifurcating parameter, clarifying the findings of a recent study in literature.
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