Stability of density-dependent model with indirect feedback and biomass inhibition

In this study, we conduct a mathematical and numerical investigation of a density-dependent model for the anaerobic digestion process, described by a system of four nonlinear ordinary differential equations, featuring an indirect feedback loop. Our analysis focuses on the acetogenesis and hydrogenotrophic methanogenesis phases. The model incorporates two microbial populations, acetogenic bacteria and hydrogenotrophic methanogens, and two substrates, volatile fatty acids (VFA) and hydrogen, with a specific emphasis on the inhibition of acetogen growth by methanogens. Using a broad class of nonmonotonic growth functions, we establish the necessary and sufficient conditions for the existence and stability of the system's steady states through rigorous mathematical analysis. Operating diagrams are constructed as functions of inlet substrate concentrations and the dilution rate. Numerical simulations further reveal the range of dynamic behaviors, highlighting the impact of methanogen-induced inhibition on acetogen dynamics. Contrary to the findings of Di and Yang in (JRSI 16:20180859, 2019), we demonstrate that when inhibition is sufficiently strong and VFA concentrations are high, the microbial community exhibits damped oscillations that converge to a positive steady state. These results illustrate the system's ability to stabilize at a coexistence equilibrium, even under the influence of an indirect feedback loop.