Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System

Fernando Giménez-Palomares, Pedro Fernández de Córdoba, Juan C. Mejuto, Ricardo J. Bendaña-Jácome and Nelson Pérez-Guerra
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Fernando Giménez-Palomares: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Pedro Fernández de Córdoba: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Juan C. Mejuto: Department of Physical Chemistry, Faculty of Science, University of Vigo, 32004 Ourense, Spain
Ricardo J. Bendaña-Jácome: Department of Engineering, Materials, Structural Mechanics and Construction, Faculty of Sciences, University of Vigo, Ourense Campus, As Lagoas, s/n, 32004 Ourense, Spain
Nelson Pérez-Guerra: Department of Analytical and Food Chemistry, Faculty of Sciences, University of Vigo, Ourense Campus, As Lagoas, s/n, 32004 Ourense, Spain

Mathematics, 2022, vol. 10, issue 5, 1-31

Abstract: Nisin, an antibacterial compound produced by Lactococcus lactis strains, has been approved by the US Food and Drug Administration to be used as a safe food additive to control the growth of undesirable pathogenic bacteria. Nisin is commonly described as a pH-dependent primary metabolite since its production depends on growth and culture pH evolution. However, the relationships between bacteriocin synthesis (BT), biomass production (X), culture pH, and the consumption of the limiting nutrient (total nitrogen: TN) have not been described until now. Therefore, this study aims to develop a competitive four-dimensional Lotka–Volterra-like Equation (predator-prey system) to describe these complex relationships in three series of batch fermentations with L. lactis CECT 539 in diluted whey (DW)-based media. The developed four-dimensional predator-prey system accurately described each individual culture, providing a good description of the relationships between pH, TN, X, and BT, higher values for R 2 and F -ratios, lower values (<10%) for the mean relative percentage deviation modulus, with bias and accuracy factor values approximately equal to one. The mathematical analysis of the developed equation showed the existence of one asymptotically stable equilibrium point, and the phase’s diagram obtained did not show the closed elliptic trajectories observed in biological predator-prey systems.

Keywords: nisin; asymptotically stable equilibrium point; batch fermentation; four-dimensional predator-prey system; mathematical analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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