Mathematical Analysis of a Three-Tiered Model of Anaerobic Digestion

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.

Mortality can produce limit cycles in density-dependent models with a predator-prey relationship

“…However, the accuracy of the results depends on the step used for the discretization of the set of operating parameters. For example, important phenomena were not detected in [53], as the emergence of limit cycles through Hopf bifurcations, which were discovered in the studies using analytical methods [39,38,44].…”

Operating diagrams for a three-tiered microbial food web in the chemostat

“…The organisms involved in the resulting three-tiered model are the chlorophenol-dechlorinating bacterium, the phenol degrader and the hydrogenotrophic methanogen. Several authors studied the three-tiered model, see [12,21,20,26,29,30]. Using specific growth rates, Wade et al [30] have proven that the model can have up to eight steady states.…”

“…The necessary and sufficient conditions of the local stability are performed analytically when maintenance is excluded. Then, these results of local stability are extended in [20] to the case where maintenance is included. Using a large class of growth functions, we have provided rigorous proofs for the results of [30] on the existence and stability of the steady states, which, for the most part, were obtained only numerically and for specific kinetics in [30].…”

“…In this work, we consider the three-tiered model proposed in [30], and we are interested to complete the mathematical analysis of this model by studying theoretically the operating diagrams. Using the existence and stability conditions of all steady states determined analytically in [20,21], our main contribution in this paper is the construction of the operating diagrams, by plotting the curves which separate their different regions, in both cases with and without considering decay terms. The operating diagram shows the dynamic behavior of the system according to the conventional chemostat operating parameters (the dilution rate and the input concentrations of the substrates) such that the values of the biological parameters are fixed.…”

“…The one parameter bifurcation diagrams of the three-tiered model presented in [20,21] were analytically constructed in the cases with and without maintenance. They show that the model has in both cases rich dynamics including bistability, coexistence and occurrence of the limit cycle due to supercritical Hopf bifurcation.…”