Mathematical study of multi-resource models in a chemostat

Résumé

This thesis focuses on the mathematical analysis of an anaerobic digestion model involving three species and three resources in a chemostat. The model includes several specific ecological interactions between these species, such that competition, syntrophy, and inhibition. More precisely, we analyze the mechanistic model describing the anaerobic chlorophenol mineralization in a three-tiered microbial food-web. We investigate the effects of including three inflowing concentrations, the mortality of the species, together with the inhibition of the third substrate on the first species. In this general case, we prove that the system can have up to eight types of steady states and we give a complete analysis by determining the necessary and sufficient conditions for their existence, according to the operating parameters. Then, we study the local stability in the case when the maintenance is ignored, where we can reduce the system to a three-dimensional one. We examine the effect of adding the hydrogen input concentration to the chlorophenol mineralization model and analyze the bifurcation diagrams by varying the input concentration of chlorophenol as a bifurcation parameter. We show that the positive steady state, which corresponds to the coexistence of the three microorganisms, can be unstable and that the system exhibits rich behaviors including bistability, coexistence, and emergence of a limit cycle through a supercritical Hopf bifurcation. When the mortality terms are present in the model, we use the Liénard-Chipart stability criterion to determine explicitly the necessary and sufficient local stability properties, when the eigenvalues of the Jacobian matrix can not be calculated. Subsequently, we give a numerical analysis of the bifurcation diagrams which suggested the presence of a supercritical Hopf bifurcation emerging through the positive steady state with the appearance of a stable periodic solution. Finally, we focus on the study of the operating diagrams which illustrate the existence and stability regions of the steady states. Our analytical study results in the discovery of several interesting regions, namely the existence of an instability region of the positive steady state, a fact that has not been detected, previously, by the numerical study.

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