Predictive models may be considered a tool to ensure food quality as they provide insights that support decision making on the design of processes, such as fermentation. Objective: To formulate a mathematical model that describes the growth of lactic acid bacteria (LAB) in batch fermentation. Methodology: Based on real-life experimental data from eight LAB strains, we formulated a primary model in the form of a third-degree polynomial function that successfully describes the four phases observed in LAB growth, i.e., lag, exponential, stationary, and death. Our cubic mathematical model allows us to understand the fundamental nonlinear dynamics of LAB as well as its time-variant dependencies. Parameters of the model are written in terms of initial biomass, maximum biomass, maximum growth rate, and lag phase duration. Further, a statistical analysis was performed to compare our cubic primary model with the ones proposed by Gompertz, Baranyi, and Vázquez-Murado by computing the coefficient of determination $\left(R^2\right)$, the residual sum of squares $\left(R,S,S\right)$, and the Akaike Information Criterion $\left(A,I,C\right)$. Results: The average statistical results from the cubic model are as follows: $R^2=0.820$ providing a better fit than the other three models, $RSS=0.658$ and $AIC=-6.499$, where both values are lower than the other models considered in this study. Conclusion: The cubic primary model formulated in this work describes the behavior of biomass as it accurately represents the four phases of biomass growth in batch fermentation process.

This paper presents results concerning mechanistic modeling to describe the dynamics and interactions between biomass growth, glucose consumption and ethanol production in batch culture fermentation by Kluyveromyces marxianus (K. marxianus). The mathematical model was formulated based on the biological assumptions underlying each variable and is given by a set of three coupled nonlinear first-order Ordinary Differential Equations. The model has ten parameters, and their values were fitted from the experimental data of 17 K. marxianus strains by means of a computational algorithm design in Matlab. The latter allowed us to determine that seven of these parameters share the same value among all the strains, while three parameters concerning biomass maximum growth rate, and ethanol production due to biomass and glucose had specific values for each strain. These values are presented with their corresponding standard error and $95\%$ confidence interval. The goodness of fit of our system was evaluated both qualitatively by in silico experimentation and quantitative by means of the coefficient of determination and the Akaike Information Criterion. Results regarding the fitting capabilities were compared with the classic model given by the logistic, Pirt, and Luedeking–Piret Equations. Further, nonlinear theories were applied to investigate local and global dynamics of the system, the Localization of Compact Invariant Sets Method was applied to determine the so-called localizing domain, i.e., lower and upper bounds for each variable; whilst Lyapunov’s stability theories allowed to establish sufficient conditions to ensure asymptotic stability in the nonnegative octant, i.e., ${\mathbf{R}}_{+,0}^3$. Finally, the predictive ability of our mechanistic model was explored through several numerical simulations with expected results according to microbiology literature on batch fermentation.

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The preservation of foods such as milk, meat, and vegetables through fermentation results in products like yogurt, cheese, pickles, sausages, and silage with an extended shelf life compared to their natural unprocessed counterparts. This work aims to formulate a mathematical model of first-order ordinary differential equations (ODEs) that accounts for both the physicochemical and microbiological parameters affecting biomass kinetics [B(t)], acidity [A(t)], and viscosity [V(t)] as a function of temperature across different yogurt samples. In order to validate the efficacy of the model in predicting yogurt shelf life, we compared its fitting results with commonly employed systems or equations, including the Weibull model, the Reaction Order model, the Arrhenius Equation, and the Q10 Factor. Our evaluation, based on R-squared (R2) values exceeding 0.95, demonstrates the robustness of the proposed model. Furthermore, all parameters were estimated along with their corresponding 95 % confidence intervals. The mathematical model estimates the dynamic of each of the physicochemical and microbiological parameters which will help to predict the behavior over time of the shelf life of yogurt at different temperatures.