Sensitivity, Equilibria, and Lyapunov Stability Analysis in Droop's Nonlinear Differential Equation System for Batch Operation Mode of Microalgae Culture Systems

Microalgae-based biomass has been extensively studied because of its potential to produce several important biochemicals, such as lipids, proteins, carbohydrates, and pigments, for the manufacturing of value-added products, such as vitamins, bioactive compounds, and antioxidants, as well as for its applications in carbon dioxide sequestration, amongst others. There is also increasing interest in microalgae as renewable feedstock for biofuel production, inspiring a new focus on future biorefineries. This paper is dedicated to an in-depth analysis of the equilibria, stability, and sensitivity of a microalgal growth model developed by Droop (1974) for nutrient-limited batch cultivation. Two equilibrium points were found: the long-term biomass production equilibrium was found to be stable, whereas the equilibrium in the absence of biomass was found to be unstable. Simulations of estimated parameters and initial conditions using literature data were performed to relate the found results to a physical context. In conclusion, an examination of the found equilibria showed that the system does not have isolated fixed points but rather has an infinite number of equilibria, depending on the values of the minimal cell quota and initial conditions of the state variables of the model. The numerical solutions of the sensitivity functions indicate that the model outputs were more sensitive, in particular, to variations in the parameters of the half saturation constant and minimal cell quota than to variations in the maximum inorganic nutrient absorption rate and maximum growth rate.