Dynamical analysis of a chemostat model for 4-chlorophenol and sodium salicylate mixture biodegradation

We consider a mathematical continuous-time model for biodegradation of 4-chlorophenol and sodium salicylate mixture by the microbial strain Pseudomonas putida in a chemostat. The model is described by a system of three nonlinear ordinary differential equations and is proposed for the first time in the paper [Y.-H. Lin, B.-H. Ho, Biodegradation kinetics of phenol and 4-chlorophenol in the presence of sodium salicylate in batch and chemostat systems, Processes, 10:694, 2022], where the model is only quantitatively verified. This paper provides a detailed analysis of the system dynamics. Some important basic properties of the model solutions like existence, uniqueness and uniform boundedness of positive solutions are established. Computation of equilibrium points and study of their local asymptotic stability and bifurcations in dependence of the dilution rate as a key model parameter are also presented. Thereby, particular intervals for the dilution rate are found, where one or three interior (with positive components) equilibrium points do exist and possess different types of local asymptotic stability/instability. Hopf bifurcations are detected leading to the occurrence of stable limit cycles around some interior equilibrium points. A transcritical bifurcation also exists and implies stability exchange between an interior and the boundary (washout) equilibrium. The results are illustrated by lots of numerical examples.