A Mathematical Modeling of Stability of Steady- State for Microbial Growth in a Chemostat

Abstract

This thesis investigates bacterial growth dynamics in a chemostat using Monod-type growth kinetics. A mathematical model describing a single microbial species limited by a single nutrient is formulated as a system of coupled ordinary differential equations. The physical configuration of the chemostat is presented, and the equilibrium points of the model are derived and analyzed. Numerical simulations are carried out using MATLAB to examine the influence of key parameters on microbial growth and substrate concentration. Parameter values are adopted from the literature and biologically reasonable assumptions to evaluate the effects of the maximum growth rate, dilution rate, nutrient input, and saturation constant on system dynamics. The main result demonstrates that the stability of the positive steady state is primarily governed by the relationship between the maximum growth rate and the dilution rate. In particular, the system admits a stable non-washout equilibrium when the maximum growth rate exceeds the dilution rate, whereas washout occurs when the dilution rate dominates. Sensitivity analysis further confirms that these two parameters are the most influential factors determining the long term behavior of the system. From a biological perspective, the derived stability conditions characterize the long term behavior of microbial populations in a chemostat. The results reveal that microbial persistence or washout is governed by the balance between nutrient supply, microbial growth, and dilution. Moreover, the local asymptotic stability of the steady state indicates that the microbial population can self-regulate and maintain a stable concentration under small environmental perturbations, highlighting the robustness of continuous bioreactor operation. Overall, this study provides quantitative insight into how operational parameters influence biological sustainability in controlled microbial ecosystems.