No abstract

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.

In this paper, we consider a three-tiered food web model in a chemostat, including chlorophenol, phenol, and hydrogen substrates and their degraders. The model takes into account the three substrate inflowing concentrations, as well as maintenance, that is, decay terms of the species. The operating diagrams give the asymptotic behavior of the model with respect to the four operating parameters, which are the dilution rate and the three inflowing concentrations of the substrates. These diagrams were obtained only numerically in the existing literature. Using the mathematical analysis of this model obtained in our previous studies, we construct the operating diagrams, by plotting the curves that separate their various regions. Hence, the regions of the operating diagrams are constructed analytically and there is no requirement for time-consuming algorithms to generate the plots, as in the numerical method. Moreover, our method reveals behaviors that have not been detected in the previous numerical studies. The growth functions are of Monod form with the inclusion of a product inhibition term. However, our method applies for a large class of growth functions. We construct operating diagrams with and without maintenance showing the role of maintenance on the stability of the system.