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

Abstract

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.

Control Factors for the Equilibrium Composition of Microbial Communities in Open Systems: Theory and Experiments

“…Note that in this study we left aside the formal model problems of equilibrium stability, and of dynamic properties of model solutions. All these problems are an independent scientific area [62][63][64][65][66][67].…”

Two Approaches to Modeling Phytoplankton Biomass Dynamics Based on the Droop Model

“…, где  0 -максимальная скорость поглощения биогенов фитопланктоном, Q 0минимальная квота биогенного вещества, ниже которой рост биомассы прекращается, Q 1 -максимальная клеточная квота, q -текущее внутриклеточное содержание биогенов, k s -константа полунасыщения по минеральному питанию. Большое количество работ посвящено исследованию модели клеточной квоты и свойств ее решений, в первую очередь, равновесных [15,18,19]. В работе [16] нами рассмотрено существование и устойчивость равновесных решений модели Друпа, выраженной системой (1), а также аттрактора, к которому принадлежат существующие равновесия.…”

“…Было также установлено, что для расширенной модели (2) области устойчивости определяются, главным образом, параметрами Q 0 (минимальная клеточная квота) и α (доля хлорофилла, ассоциированная со световой фазой фотосинтеза). Известно, что в пространстве переменных (p, s, q) для базовой модели существует компактное (асимптотически) притягивающее множество (аттрактор) [19,24]:…”

Dynamical Simulation, Sensitivity, and Productivity Analysis of a Light-Photoacclimation Model for Microalgae-Based Carbohydrate Production in Continuous Photobioreactors

“…Proper control of the environmental variables governing these processes is essential to the successful design and operation of a photobioreactor [40]. As mentioned above, few parameters are nearly constant; as a result, identifying the parameters that have the greatest impact on the model through sensitivity analysis can help determine which parameters are most critical during parameter calibration [41,42]. In this study, a dynamic photoacclimated model for continuous microalgae culture systems was used to study time-varying state variables behavior as well as the first-order effects of solutions to parameter variations under optimal productivity conditions.…”