A Mathematical Model of Anaerobic Digestion with Syntrophic Relationship, Substrate Inhibition, and Distinct Removal Rates

Understanding and exploiting the syntrophic relationship between microbial species is a major challenge in the mathematical theory of the anaerobic digestion process. In this work, we focus on the acetogenesis and hydrogenotrophic methanogenesis phases and include distinct removal rates for the species. Our study gives a quite comprehensive analysis of a syntrophic model by analyzing the joined effects of syntrophy relationship, mortality, substrate inhibition, and input concentrations that were neglected in previous studies. The mathematical analysis of the model involving mortality is a difficult problem since the model is not reduced to a planar system as in the case where the dilution rates of the substrates and the removal rates of microbial species are equal. Using general nonmonotonic growth rates, the necessary and sufficient conditions of existence and local stability of all steady states of the four-dimensional system are determined, according to the operating parameters. This general model exhibits a rich behavior with the coexistence of two microbial species, the bistability, the multiplicity of coexistence steady states, and the existence of two steady states of extinction of the first species. The operating diagram shows how the model behaves by varying the control parameters and illustrates the effect of the substrate inhibition and the new input substrate concentration (hydrogen) on the appearance or the disappearance of coexistence and bistability regions. Similarly to the classical chemostat model, including the substrate inhibition can destabilize a two-tiered microbial "food chain," where the asymptotic behavior of the system depends on the initial condition.