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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{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.

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Design of the Four-Dimensional Equation

- (i)
- The culture pH (pH(t)) decline could be described as the difference between the buffer capacity of the medium [19] and the decrease in pH due to lactic acid production by the biomass as follows:$$\frac{dpH\left(t\right)}{dt}=-pH\left(t\right)\xb7\left(a-b\xb7pH\left(t\right)-c\xb7X\left(t\right)\right)$$
- (ii)
- The total nitrogen (TN(t)), the limiting nutrient in these cultures [5], which is channeled into the cells and nisin molecules, could be considered as a prey that is consumed for both biomass and nisin production, and its consumption depends on the culture pH [5,20], as follows:$$\frac{dTN\left(t\right)}{dt}=-TN\left(t\right)\xb7\left(d-e\xb7TN\left(t\right)-f\xb7X\left(t\right)-g\xb7BT\left(t\right)-h\xb7pH\left(t\right)\right)$$
- (iii)
- Biomass (X(t)) could be considered as one predator that grows logistically competing with nisin for the nitrogen source and depending on the culture pH and the TN source concentration [5], as follows:$$\frac{dX\left(t\right)}{dt}=X\left(t\right)\xb7\left(i-j\xb7X\left(t\right)-k\xb7BT\left(t\right)-l\xb7pH\left(t\right)-m\xb7TN\left(t\right)\right)$$
- (iv)
- Nisin (BT(t)) could be considered as the second predator that is produced by the biomass but competes with it for the nitrogen source and depends on the culture pH and the TN source concentration [5], as follows:$$\frac{dBT\left(t\right)}{dt}=BT\left(t\right)\xb7\left(n-o\xb7BT\left(t\right)-p\xb7X\left(t\right)-q\xb7pH\left(t\right)-r\xb7TN\left(t\right)\right)$$

#### 2.2. Modeling the Batch Nisin Production System in Different Series of Fermentations in DW Media Using a Global Set of Model Parameters

_{pH}

^{2}and F-ratio were relatively low, the RPDM values were almost always higher than 10, and both the B

_{f}and A

_{f}values were generally far from one (Table 3). In addition, the pH, TN, X, and BT trajectories predicted by the global Equations (1)–(4) for the three series of cultures showed a clear deviation from the experimental pH, TN, X, and BT data (dashed lines in Figure 1, Figure 2, Figure 3 and Figure 4).

#### 2.3. Modeling the Batch Nisin Production System in Individual Cultures Corresponding to Each Series of Fermentations

^{2}and F-values considerably higher, and B

_{f}and A

_{f}values ~ 1 (Table 4, Table 5, Table 6 and Table 7).

#### 2.3.1. Series of Fermentation DW-G

_{pH}

^{2}(between 0.9968 and 0.9980), R

_{TN}

^{2}(between 0.9987 and 0.9998), R

_{X}

^{2}(between 0.9990 and 1.0000), and R

_{BT}

^{2}(between 0.9992 and 1.0000) were obtained. In addition, the values of B

_{f}and A

_{f}calculated for Equations (1)–(4) were ~ 1 and the RPDM values were considerably lower than 10% (Table 4). Therefore, it could be considered that the use of the four-dimensional predator-prey system (1)–(4) accurately described the trend observed for the culture pH, TN, X, and BT in the DW-G cultures.

_{0}] (from 0 to 25 g/L) in the media (Figure 1). In addition, the value of c decreased from 0.039 to 0.002 with the increase in [G

_{0}] due to the inhibition that increasing glucose concentration produced on the growth of L. lactis CECT 539 and, consequently, on lactic acid production [16], causing a gradual reduction in the pH drop in the culture media (Figure 1).

_{0}] (Figure 1). Similarly, the values of f, g, and h were almost similar for all cultures because the TN consumption decreased with the increase in [G

_{0}], in parallel with the reduction in biomass production, nisin synthesis, and pH drop (Figure 1).

_{0}] was proportional to that observed in nisin synthesis because this bacteriocin was produced in this series of cultures as a pH-dependent primary metabolite [16].

_{0}] (Figure 1). The constant value obtained for the o constant could be explained by a proportional decrease in the nisin production rate and maximum nisin levels produced by L. lactis CECT 539. The constant value obtained for p is in perfect agreement with the constant value obtained for k in Equation (3), indicating again that the competition between biomass production and nisin synthesis for the nitrogen source was very similar in the different glucose-supplemented cultures.

_{0}] negatively affected the synthesis of nisin. In the unsupplemented culture ([G

_{0}] = 0), the final pH value was 4.73, which is in perfect agreement with the optimum final pH between 4.78 and 4.90 observed in L. lactis cultures in whey [8]. In contrast, a constant value was obtained for r (Table 4), indicating proportional TN consumption for nisin synthesis.

_{0}] on the rates rpH(t), rTN(t), rX(t), and rBT(t)).

#### 2.3.2. Series of Fermentation DW-TS-TP

#### 2.3.3. Series of Fermentation DW-MRS

_{0}) in the medium.

#### 2.4. Mathematical Analysis of the Four-Dimensional Lotka–Volterra Equation

#### 2.4.1. Generalized Four-Dimensional Lotka–Volterra Equation

_{i}is the i-th row of the matrix A.

**Lemma 1**.

**Proof.**

**Theorem 1**.

#### 2.4.2. Numerical Analysis

_{1},p

_{2},p

_{3},p

_{4}) and ε = (ε

_{1},ε

_{2},ε

_{3},ε

_{4}), the approximate solutions x

_{ε}(t), y

_{ε}(t), z

_{ε}(t), and w

_{ε}(t), corresponding to the following initial conditions: x

_{0}= p

_{1}+ ε

_{1}, y

_{0}= p

_{2}+ ε

_{2}, z

_{0}= p

_{3}+ ε

_{3}, w

_{0}= p

_{4}+ ε

_{4}, can be calculated. Figure 8 shows the graphs corresponding to x

_{ε}(t), y

_{ε}(t), z

_{ε}(t), and w

_{ε}(t) for 40 random values of ε in the 4-dimensional sphere with the center at the origin and a radius of 0.01.

_{1}(0.000,0.000,0.000,0.000), P

_{2}(0.000,0.000,1.071,0.000), P

_{3}(0.000,0.000,0.000,103.528), P

_{4}(0.000,0.000,0.019,103.453), P

_{5}(0.000,0.213,0.000,0.000), P

_{6}(0.000,−0.014,0.000,103.491), P

_{7}(0.000,0.134,1.089,0.000), P

_{8}(0.000,−0.015,0.018,103.420), P

_{9}(4.162,0.000,0.000,0.000), P

_{10}(4.162,0.000,0.000,33.349), P

_{11}(4.162,0.369,0.000,0.000), P

_{12}(4.162,0.294,0.000,34.101), P

_{13}(4.861,0.000,0.663,0.000), P

_{14}(4.627,0.000,0.441,23.837), P

_{15}(4.906,0.346,0.705,0.000), and P

_{16}(4.665,0.301,0.477,23.812).

_{16}(4.665,0.301,0.477,23.812) has biological interest since the other equilibrium points contain at least one zero, and this implies that there are no viable cells (X = 0), nitrogen source (TN = 0), or nisin (BT = 0), or that the culture pH reached the value zero. However, in the fermentation analyzed, this was not possible because during the incubation, the culture variables were all greater than zero during the incubation period (Figure 1). At the beginning of fermentation, the values of the culture pH, [TN], [X], and [BT] were 6.230, 0.439 g/L, 0.010 g/L, and 0.430 BU/mL, respectively, and reached the final values of 4.730, 0.319 g/L, 0.480 g/L, 22.897 BU/mL, respectively.

_{16}) with all its components different from zero is as follows:

#### 2.4.3. Study of the Parameter Values for Which There Is an Asymptotically Stable Solution

_{1}and R

_{2}, as large as possible, so that if

_{i}∈ [p

_{i}

_{0}(1-R

_{1}), p

_{i}

_{0}(1+R

_{2})] (if p

_{i}0 ≥ 0)

_{i}∈ [p

_{i}

_{0}(1+R

_{2}), p

_{i}

_{0}(1-R

_{1})] (if p

_{i}0 < 0) being i = 1,2,…,18

_{1}, and the step h = 0.001, and we determined the first value of a natural k such that if R

_{2}=k·h, and 10.000 random values were taken for each one of the intervals $\left[{p}_{i0}\left(1-{R}_{1}\right),{p}_{i0}\left(1+{R}_{2}\right)\right]$ (if p

_{i}0 ≥ 0) and $\left[{p}_{i0}\left(1+{R}_{2}\right),{p}_{i0}\left(1-{R}_{1}\right)\right]$ (if p

_{i}0 < 0), then $\tilde{x}={\mathit{A}}^{-1}\mathit{b}$ has any of its entries nonpositive or the matrix $\mathit{A}\xb7\tilde{x}$ has some eigenvalue with a nonpositive real part.

_{1}> 0.12, there are no asymptotically stable equilibrium points with all their non-zero coordinates. The left and right parts of Figure 11 show the curve R

_{1}→ R

_{2}and the curves R

_{1}→ 1 − R

_{1}and R

_{1}→ 1 + R

_{2}, respectively. As can be observed in these figures, the range of the parameters’ validity is approximately equal to $0.12\left|{p}_{i0}\right|$, i = 1,2,…,18.

## 3. Materials and Methods

#### 3.1. Microorganisms, Culture Media, and Inoculum Preparation

_{2}PO

_{4}to obtain initial total sugars and phosphorous concentrations between 22.61 and 51.35 g/L, and 0.24 and 0.63 g/L, respectively (series of fermentation DW-TS-TP) [17], and (iii) MRS broth nutrients (except glucose and Tween 80) at 25, 50, 75, 100, and 125% (w/v) of their standard concentrations in the complex substrate to produce the DW25, DW50, DW75, DW100, and DW125 media (series of fermentation DW-MRS) [18]. In these three series of fermentation, control cultures in unsupplemented DW medium were performed to obtain data for the comparisons [16,17,18].

^{9}colony-forming units/mL [16,17,18].

#### 3.2. Batch Cultures

#### 3.3. Analytical Methods

#### 3.4. Statistical Significance of the Parameters and Equation

_{f}) and accuracy (A

_{f}) factors [34]:

_{i}is the experimental value and Ypred

_{i}is the value predicted by the equation. Values of R

^{2}≥ 0.95, RPDM < 10% [5], and B

_{f}and A

_{f}close to 1 [34] indicate that the corresponding equation was accurately fitted to the experimental data.

#### 3.5. Mathematical Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

t | Time (h) |

pH(t) | Culture pH value over the time |

pH_{0} | Initial culture pH value |

pH_{f} | Final culture pH |

a | Intrinsic pH drop rate (h^{−1}) |

b | Quotient between the intrinsic pH drop rate and the theoretical minimum pH value for growth (h^{-1}) |

c | Constant that represents the effect of biomass production on pH time course (L/g/h) |

TN(t) | Total nitrogen concentration (g/L) over the time |

TN_{0} | Initial total nitrogen concentration (g/L) |

TN_{f} | Final total nitrogen concentration (g/L) |

d | Intrinsic TN consumption rate (h^{−1}) |

e | Quotient between the intrinsic TN consumption rate and the theoretical maximum TN concentration that biomass can consume (L/g/h) |

f | Intrinsic TN consumption rate (L/g/h) for biomass production |

g | Intrinsic TN consumption rate (mL/BU/h) for nisin production |

h | Constant that represents the effect of pH time course on TN consumption (h^{−1}) |

X(t) | Biomass concentration (g/L) over the time |

X_{0} | Initial biomass concentration (g/L) |

X_{max} | Maximum biomass concentration (g/L) |

i | Intrinsic growth rate (h^{−1}) |

j | Quotient between the intrinsic growth rate and the theoretical maximum biomass concentration that system can support (L/g/h) |

k | Efficiency of TN utilization (mL/BU/h) to be channeled into cells of L. lactis rather than nisin (competition coefficient) |

l | Constant that represents the effect of pH time course on the growth (h^{−1}) |

m | Constant that represents the effect of TN time course on the growth (L/g/h) |

BT(t) | Nisin concentration (BU/mL) over the time |

BT_{max} | Maximum nisin concentration (BU/mL) |

n | Intrinsic nisin production rate (h^{−1}) |

o | Quotient between the intrinsic nisin production rate and the theoretical maximum nisin concentration that biomass can produce (h^{−1}) |

p | Efficiency of TN utilization (mL/BU/h) to be channeled into nisin rather than cells of L. lactis (competition coefficient) |

q | Constant that represents the effect of pH time course on nisin synthesis (h^{−1}) |

r | Constant that represents the effect of TN time course on nisin synthesis (L/g/h) |

A, B, C_{0}, C_{1}, C_{2} | Constants in Equation (5) |

D, E, F | Constants in Equation (6) |

G, H, I | Constants in Equation (7) |

[G]_{0} | Initial glucose concentration (g/L) |

[Nut]_{0} | Initial concentration of MRS broth nutrients (g/L) |

RPDM | Mean relative percentage deviation modulus |

B_{f} | Bias factor |

A_{f} | Accuracy factor |

s | Number of experimental data |

Yexp_{i} | Experimental values of culture pH and concentration of total nitrogen, biomass, and nisin |

Ypred_{i} | Predicted values by the corresponding equation for culture pH and concentration of total nitrogen, biomass, and nisin. |

R^{2} | Correlation coefficient |

## References

- Delves-Broughton, J. Nisin. In Encyclopedia of Food Microbiology, 2nd ed.; Robinson, R., Batt, C.A., Tortorello, M.L., Eds.; Academic Press: London, UK, 2014; pp. 187–193. [Google Scholar]
- Leroy, F.; De Vuyst, L. Growth of the bacteriocin-producing Lactobacillus sakei strain CTC 494 in MRS broth is strongly reduced due to nutrient exhaustion: A nutrient depletion model for the growth of lactic acid bacteria. Appl. Environ. Microbiol.
**2001**, 67, 4407–4413. [Google Scholar] [CrossRef] [PubMed] [Green Version] - van Impe, J.F.; Poschet, F.; Geeraerd, A.H.; Vereecken, K.M. Towards a novel class of predictive microbial growth models. Int. J. Food Microbiol.
**2005**, 100, 97–105. [Google Scholar] [CrossRef] [PubMed] - Wachenheim, D.E.; Patterson, J.A.; Ladisch, M.R. Analysis of the logistic function model: Derivation and applications specific to batch cultured microorganisms. Bioresour. Technol.
**2003**, 86, 157–164. [Google Scholar] [CrossRef] - Guerra, N.P.; Torrado, A.; López, C.; Fajardo, P.; Pastrana, L. Dynamic mathematical models to describe the growth and nisin production by Lactococcus lactis subsp. lactis CECT 539 in both batch and re-alkalized fed-batch cultures. J. Food Eng.
**2007**, 82, 103–113. [Google Scholar] - Vázquez, J.A.; Murado, M.A. Unstructured mathematical model for biomass, lactic acid and bacteriocin production by lactic acid bacteria in batch fermentation. J. Chem. Technol. Biotechnol.
**2008**, 83, 91–96. [Google Scholar] [CrossRef] [Green Version] - Shirsat, N.; Mohd, A.; Whelan, J.; English, N.J.; Glennon, B.; Al-Rubeai, M. Revisiting Verhulst and Monod models: Analysis of batch and fed-batch cultures. Cytotechnology
**2015**, 67, 515–530. [Google Scholar] [CrossRef] [Green Version] - Guerra, N.P. Modeling the batch bacteriocin production system by lactic acid bacteria by using modified three-dimensional Lotka–Volterra equations. Biochem. Eng. J.
**2014**, 88, 115–130. [Google Scholar] [CrossRef] - Callewaert, R.; De Vuyst, L. Bacteriocin production with Lactobacillus amylovorus DCE 471 is improved and stabilized by fed-batch fermentation. Appl. Environ. Microbiol.
**2000**, 66, 606–613. [Google Scholar] [CrossRef] [Green Version] - Wick, L.M.; Weilenmann, H.; Egli, T. The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics. Microbiology
**2002**, 148, 2889–2902. [Google Scholar] [CrossRef] [Green Version] - Luedeking, R.; Piret, E.L. A kinetic study of the lactic acid fermentation. Batch process at controlled pH. J. Biochem. Microbiol. Technol. Eng.
**1959**, 1, 393–412. [Google Scholar] [CrossRef] - Cabo, M.L.; Murado, M.A.; González, M.P.; Pastoriza, L. Effects of aeration and pH gradient on nisin production: A mathematical model. Enzym. Microb. Technol.
**2001**, 29, 264–273. [Google Scholar] [CrossRef] - Constandinides, K.; Damianou, P.A. Lotka–Volterra equations in three dimensions satisfying the Kowalevski-Painlevé property. Regul. Chaotic Dyn.
**2011**, 16, 311–328. [Google Scholar] [CrossRef] - Yang, R.; Ray, B. Factors influencing production of bacteriocins by lactic acid bacteria. Food Microbiol.
**1994**, 11, 281–291. [Google Scholar] [CrossRef] - Wang, R.; Xiao, D. Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system. Nonlinear Dyn.
**2010**, 59, 411–422. [Google Scholar] [CrossRef] - Costas, M.; Alonso, E.; Guerra, N.P. Nisin production in realkalized fed-batch cultures in whey with feeding with lactose- or glucose-containing substrates. Appl. Microbiol. Biotechnol.
**2016**, 100, 7899–7908. [Google Scholar] [CrossRef] - Costas, M.; Alonso, E.; Outeiriño, D.; Guerra, N.P. Production of a highly concentrated probiotic culture of Lactococcus lactis CECT 539 containing high amounts of nisin. 3-Biotech
**2018**, 8, 292. [Google Scholar] - Costas, M.; Alonso, E.; Bazán, D.L.; Bendaña, R.J.; Guerra, N.P. Batch and fed-batch production of probiotic biomass and nisin in nutrient-supplemented whey media. Braz. J. Microbiol.
**2019**, 50, 915–925. [Google Scholar] - Urbansky, E.T.; Schock, M.R. Understanding, deriving, and computing buffer capacity. J. Chem. Educ.
**2000**, 77, 1640–1644. [Google Scholar] [CrossRef] - Poolman, B.; Konings, W.N. Relation of growth of Streptococcus lactis and Streptococcus cremoris to amino acid transport. J. Bacteriol.
**1988**, 170, 700–707. [Google Scholar] [CrossRef] [Green Version] - Abbasiliasi, S.; Tan, J.S.; Ibrahim, T.A.T.; Bashokouh, F.; Ramakrishnan, N.R.; Mustafa, S.; Arif, A.B. Fermentation factors influencing the production of bacteriocins by lactic acid bacteria: A review. RSC Adv.
**2017**, 7, 29395–29420. [Google Scholar] [CrossRef] - Salaün, F.; Mietton, B.; Gaucheron, F. Buffering capacity of dairy products. Int. Dairy J.
**2005**, 15, 95–109. [Google Scholar] [CrossRef] - Parente, E.; Ricciardi, A.; Addario, G. Influence of pH on growth and bacteriocin production by Lactococcus lactis subsp. lactis 14ONWC during batch fermentation. Appl. Microbiol. Biotechnol.
**1994**, 41, 388–394. [Google Scholar] [CrossRef] - van Niel, E.W.J.; Hahn-Hägerdal, B. Nutrient requirements of lactococci in defined growth media. Appl. Microbiol. Biotechnol.
**1999**, 52, 617–627. [Google Scholar] [CrossRef] [Green Version] - Moret-Tatay, C.; Gamermann, D.; Navarro-Pardo, E.; Fernández de Córdoba, P. ExGUtils: A python package for statistical analysis with the ex-gaussian probability density. Front. Psychol.
**2018**, 9, 612. [Google Scholar] [CrossRef] [Green Version] - Hsu, S.-B.; Ruan, S.; Yang, T.-H. Analysis of three species Lotka–Volterra food web models with omnivory. J. Math. Anal. Appl.
**2015**, 426, 659–687. [Google Scholar] [CrossRef] - Adamu, H.A. Mathematical analysis of predator-prey model with two preys and one predator. Int. J. Eng. Appl. Sci.
**2018**, 5, 17–23. [Google Scholar] - Castro-Palacio, J.C.; Isidro, J.M.; Navarro-Pardo, E.; Velázquez-Abad, L.; Fernández-de-Córdoba, P. Monte Carlo Simulation of a modified Chi distribution with unequal variances in the generating gaussians. A Discrete Methodology to Study Collective Response Times. Mathematics
**2021**, 9, 77. [Google Scholar] [CrossRef] - Ortigosa, N.; Orellana-Panchame, M.; Castro-Palacio, J.C.; Fernández de Córdoba, P.; Isidro, J.M. Monte Carlo simulation of a modified Chi distribution considering asymmetry in the generating functions: Application to the Study of Health-Related Variables. Symmetry
**2021**, 13, 924. [Google Scholar] [CrossRef] - Goudar, C.T.; Joeris, K.; Konstantinov, K.B.; Piret, J.M. Logistic equations effectively model mammalian cell batch and fed-batch kinetics by logically constraining the fit. Biotechnol. Prog.
**2005**, 21, 1109–1118. [Google Scholar] [CrossRef] [PubMed] - Tomás Miguel, J.M.; Meléndez Moral, J.C.; Navarro Pardo, E. Factorial confirmatory models of Ryff’s scales in a sample of elderly people. Psicothema
**2008**, 20, 304–310. [Google Scholar] [PubMed] - Kanzow, C.; Yamashita, N.; Fukushima, M. Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math.
**2004**, 172, 375–397. [Google Scholar] [CrossRef] [Green Version] - Behling, R.; Gonçalves, D.S.; Santos, S.A. Local convergence analysis of the Levenberg-Marquardt framework for nonzero-residue nonlinear least-squares problems under an error bound condition. J. Optim. Theory Appl.
**2019**, 183, 1099–1122. [Google Scholar] [CrossRef] - Vazquez, J.A.; Lorenzo, J.M.; Fuciños, P.; Franco, D. Evaluation of non-linear equations to model different animal growths with mono and bisigmoid profiles. J. Theor. Biol.
**2012**, 314, 95–105. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Experimental data (symbols) of culture pH, TN consumption, and X and Nis synthesis by L. lactis CECT 539 in batch fermentations in DW medium supplemented with different glucose levels ([G]

_{0}). Dashed lines drawn through the experimental data are predictions of the global four-dimensional predator-prey system (1)–(4) obtained with the parameters shown in Table 3. Solid lines were obtained by adjusting the four-dimensional predator-prey system to the experimental data corresponding to each individual culture (see parameter values in Table 4). Reproduced with permission from Costas et al. [16], Appl. Microbiol. Biotechnol.; published by Springer Nature, 2016.

**Figure 2.**Experimental data (symbols) of culture pH, TN consumption, and X and Nis production by L. lactis CECT 539 in the first seven batch cultures of the experimental matrix (Table 2), corresponding to the fermentation series DW-TS-TP. Dashed lines drawn through the experimental data are predictions of the global four-dimensional predator-prey systems (1)–(4) obtained with the parameters shown in Table 3. Solid lines were obtained by adjusting the four-dimensional predator-prey system (1)–(4) to the experimental data corresponding to each individual culture (see parameter values in Table 5). Reproduced with permission from Costas et al. [17], 3-Biotech; published by Springer Nature, 2018.

**Figure 3.**Experimental data (symbols) of culture pH, TN consumption, and X and Nis production by L. lactis CECT 539 in the last six batch cultures of the experimental matrix (Table 2) and in the optimum conditions (OC), corresponding to the fermentation series DW-TS-TP. Dashed lines drawn through the experimental data are predictions of the global four-dimensional predator-prey system (1)–(4) obtained with the parameters shown in Table 3. Solid lines were obtained by adjusting the four-dimensional predator-prey system (1)–(4) to the experimental data corresponding to each individual culture (see parameter values in Table 5 and Table 6). Reproduced with permission from Costas et al. [17], 3-Biotech; published by Springer Nature, 2018.

**Figure 4.**Experimental data (symbols) of culture pH, TN consumption, and X and Nis formation by L. lactis CECT 539 in batch cultures in DW medium supplemented with 0, 25, 50, 75, 100, and 125% (w/v) of the standard concentrations of the MRS broth nutrients ([Nut]

_{0}) with the exception of glucose and Tween 80. Dashed lines drawn through the experimental data are predictions of the global four-dimensional predator-prey system (1)–(4) obtained with the parameters shown in Table 3. Solid lines were obtained by adjusting the four-dimensional predator-prey system (1)–(4) to the experimental data corresponding to each individual culture (see parameter values in Table 7).

**Figure 5.**Response surfaces showing the time courses of the experimental rates of pH, TN, X, and Nis, as a function of the initial glucose concentration ([G]

_{0}= 0, 5, 10, 15, 20, and 25 g/L) in the DW medium. The different rates were obtained from the experimental data shown in Figure 1.

**Figure 7.**Response surfaces showing the time courses of the experimental rates of pH, TN, X, and Nis as a function of the initial concentration of MRS both nutrients ([Nut]

_{0}= 0, 25, 50, 75, 100, and 125%) in the DW medium. The different rates were obtained from the experimental data shown in Figure 4.

**Figure 8.**Family of trajectories for the four-dimensional Lotka–Volterra system around the equilibrium point (pH = 4.665, TN = 0.301 g/L, X = 0.477 g/L, BT = 23.812 BU/mL).

**Figure 9.**(

**A**): Family of trajectories for the four-dimensional Lotka–Volterra system for a = −0.154, b = −0.037, c = 0.039, d = −0.194, e = −0.909, f = −0.066, g = −0.002, h = 0.034, i = 0.843, j = 0.787, k = 0.008, l = 0.066, m = −0.104, n = 3.727, o = 0.036, p = 0.137, q = 0.607, and r = −0.092. B: Family of trajectories obtained using the initial conditions (pH

_{0}= 6.230, TN

_{0}= 0.439 g/L, X

_{0}= 0.010 g/L, (

**B**) T

_{0}= 0.430 BU/mL) at the beginning of fermentation, which are in a neighborhood of the equilibrium point (pH = 4.665, TN

_{0}= 0.301 g/L, X = 0.477 g/L, BT = 23.812 BU/mL).

**Figure 11.**Plots of the curves R

_{1}→ R

_{2}(

**left**part), and R

_{1}→ 1 − R

_{1}and R

_{1}→ 1+ R

_{2}(

**right**part).

**Table 1.**Initial concentrations (mean ± standard deviations) of total sugars (TS), nitrogen (TN), phosphorous (TP), and proteins (Pr) in culture media prepared with deproteinized diluted whey (DW) and concentrated mussel-processing wastes (CMPW).

Medium | TS (g/L) | TN (g/L) | TP (g/L) | Pr (g/L) |
---|---|---|---|---|

DW | 20.54 ± 0.514 | 0.45 ± 0.014 | 0.25 ± 0.021 | 2.04 ± 0.083 |

DW25 | 20.96 ± 0.021 | 1.20 ± 0.006 | 0.30 ± 0.001 | 5.39 ± 0.025 |

DW50 | 21.93 ± 0.014 | 1.96 ± 0.013 | 0.36 ± 0.05 | 8.72 ± 0.034 |

DW75 | 22.89 ± 0.016 | 2.72 ± 0.002 | 0.43 ± 0.07 | 12.05 ± 0.019 |

DW100 | 23.85 ± 0.010 | 3.49 ± 0.016 | 0.50 ± 0.017 | 15.38 ± 0.031 |

DW125 | 24.81 ± 0.023 | 4.25 ± 0.004 | 0.57 ± 0.013 | 18.71 ± 0.026 |

CMPW | 101.33 ± 1.314 | 0.54 ± 0.024 | 0.06 ± 0.009 | 3.47 ± 0.046 |

**Table 2.**Initial concentrations (mean ± standard deviations) of TS, TN, TP, and Pr in culture media prepared with DW medium mixed with different volumes of CMPW medium and supplemented with KH

_{2}PO

_{4}to give different initial TS and TP concentrations (DW-TS-TP cultures).

Points | Experiment | TS (g/L) | TP (g/L) | TN (g/L) | Pr (g/L) |
---|---|---|---|---|---|

Factorial | 1 | 48.321 ± 0.001 | 0.589 ± 0.001 | 0.479 ± 0.002 | 2.531 ± 0.018 |

2 | 48.321 ± 0.001 | 0.281 ± 0.001 | 0.479 ± 0.002 | 2.531 ± 0.018 | |

3 | 25.639 ± 0.001 | 0.589 ± 0.001 | 0.453 ± 0.003 | 2.122 ± 0.028 | |

4 | 25.639 ± 0.001 | 0.281 ± 0.001 | 0.453 ± 0.003 | 2.122 ± 0.028 | |

Axial | 5 | 51.352 ± 0.003 | 0.435 ± 0.002 | 0.483 ± 0.001 | 2.583 ± 0.047 |

6 | 22.611 ± 0.001 | 0.435 ± 0.002 | 0.450 ± 0.001 | 2.068 ± 0.012 | |

7 | 36.984 ± 0.002 | 0.631 ± 0.003 | 0.466 ± 0.002 | 2.318 ± 0.026 | |

8 | 36.984 ± 0.002 | 0.240 ± 0.001 | 0.466 ± 0.002 | 2.318 ± 0.026 | |

Center (five replicates) | 9–13 | 36.984 ± 0.002 | 0.435 ± 0.001 | 0.466 ± 0.002 | 2.318 ± 0.026 |

**Table 3.**Parameter values (as estimates ± confidence intervals) calculated with the global set of model parameters of the four-dimensional predator-prey system (1)–(4) to describe the batch nisin production system in the different series of fermentations.

Parameter | DW-G Series | DW-TS-TP Series | DW-MRS Series |
---|---|---|---|

a | −0.155 ± 0.039 (p = 0.0001) | −0.083 ± 0.022 (p = 0.0003) | −0.105 ± 0.022 (p < 0.0001) |

b | −0.036 ± 0.007 (p < 0.0001) | −0.019 ± 0.003 (p < 0.0001) | −0.0229 ± 0.003 (p = 0.0001) |

c | 0.040 ± 0.018 (p = 0.0254) | 0.006 ± 0.009 (p = 0.5015) | −0.018 ± 0.010 (p = 0.0023) |

R_{pH}^{2} | 0.5382 | 0.8603 | 0.6108 |

RPDM | 10.9930 | 10.1454 | 13.8907 |

B_{f} | 0.8972 | 0.9516 | 0.9999 |

A_{f} | 1.1285 | 1.1001 | 1.1483 |

F-ratio | 21.41 | 414.42 | 41.94 |

p-value | <0.0001 | <0.0001 | <0.0001 |

d | −0.087 ± 0.022 (p = 0.0002) | −0.034 ± 0.029 (p = 0.2427) | −0.163 ± 0.060 (p = 0.0079) |

e | −0.636 ± 0.055 (p < 0.0001) | −0.512 ± 0.056 (p < 0.0001) | −0.012 ± 0.002 (p < 0.0001) |

f | −0.081 ± 0.010 (p < 0.0001) | −0.123 ± 0.014 (p < 0.0001) | −0.346 ± 0.052 (p < 0.0001) |

g | −0.0001 ± 0.000 (p = 0.4338) | 0.002 ± 0.000 (p < 0.0001) | 0.004 ± 0.0005 (p < 0.0001) |

h | 0.032 ± 0.001 (p < 0.0001) | 0.029 ± 0.006 (p < 0.0001) | −0.018 ± 0.010 (p = 0.0602) |

R_{TN}^{2} | 0.9650 | 0.6909 | 0.2184 |

RPDM | 1.4121 | 12.4855 | 20.6241 |

B_{f} | 0.9745 | 1.0389 | 1.1310 |

A_{f} | 1.0143 | 1.1077 | 1.1957 |

F-ratio | 349.16 | 66.00 | 22.58 |

p-value | < 0.0001 | < 0.0001 | 1.0000 |

i | 1.800 ± 0.327 (p < 0.0001) | 2.381 ± 0.150 (p < 0.0001) | 1.415 ± 0.221 (p < 0.0001) |

j | 1.376 ± 0.098 (p < 0.0001) | 0.736 ± 0.067 (p < 0.0001) | 0.366 ± 0.078 (p = 0.0001) |

k | −0.007 ± 0.001 (p = 0.0016) | 0.021 ± 0.001 (p < 0.0001) | 0.006 ± 0.002 (p = 0.0023) |

l | 0.404 ± 0.035 (p = 0.0016) | 0.262 ± 0.0290 (p < 0.0001) | 0.227 ± 0.040 (p < 0.0001) |

m | −1.880 ± 0.432 (p < 0.0001) | 0.636 ± 0.166 (p = 0.0002) | −0.104 ± 0.013 (p < 0.0001) |

R_{X}^{2} | 0.9669 | 0.8686 | 0.5670 |

RPDM | 13.3796 | 25.9596 | 29.5432 |

B_{f} | 0.9079 | 0.7488 | 0.8132 |

A_{f} | 1.1626 | 1.4377 | 1.3899 |

F-ratio | 369.40 | 222.04 | 16.46 |

p-value | < 0.0001 | < 0.0001 | < 0.0001 |

n | 3.730 ± 0.335 (p < 0.0001) | 1.406 ± 0.170 (p < 0.0001) | 0.232 ± 0.141 (p = 0.0998) |

o | 2.058 × 10^{−5} ± 0.004(p = 0.9955) | 0.051 ± 0.001 (p < 0.0001) | 0.002 ± 0.001 (p = 0.0089) |

p | 1.788 ± 0.114 (p < 0.0001) | −1.148 ± 0.073 (p < 0.0001) | 0.136 ± 74.506 (p = 0.0020) |

q | 0.532 ± 0.041 (p < 0.0001) | 0.260 ± 0.030 (p < 0.0001) | 0.008 ± 0.025 (p = 0.7806) |

r | 0.984 ± 1.295 (p = 0.4489) | −0.723 ± 0.140 (p < 0.0001) | −0.051 ± 0.008 (p < 0.0001) |

R_{BT}^{2} | 0.9001 | 0.9208 | 0.8890 |

RPDM | 18.6672 | 34.7477 | 32.1298 |

B_{f} | 0.8735 | 0.6200 | 0.7513 |

A_{f} | 1.2311 | 1.9114 | 1.4540 |

F-ratio | 109.89 | 402.53 | 130.96 |

p-value | <0.0001 | <0.0001 | <0.0001 |

**Table 4.**Statistically significant (p < 0.05) parameter values (as estimates ± confidence intervals) calculated with the four-dimensional predator-prey system (1)–(4) for each individual culture of the series of fermentation DW-G.

Initial Glucose Concentrations (g/L) in the DW Medium | ||||||
---|---|---|---|---|---|---|

Parameter | 0 | 5 | 10 | 15 | 20 | 25 |

a | −0.154 ± 0.003 (p < 0.0001) | −0.147 ± 0.001 (p < 0.0001) | −0.137 ± 0.002 (p < 0.0001) | −0.122 ± 0.001 (p < 0.0001) | −0.108 ± 0.004 (p < 0.0001) | −0.090 ± 0.003 (p < 0.0001) |

b | −0.037 ± 0.001 (p < 0.0001) | −0.033 ± 0.001 (p < 0.0001) | −0.030 ± 0.001 (p < 0.0001) | −0.027 ± 0.001 (p < 0.0001) | −0.024 ± 0.001 (p < 0.0001) | −0.020 ± 0.002 (p < 0.0001) |

c | 0.039 ± 0.001 (p < 0.0001) | 0.021 ± 0.001 (p < 0.0001) | 0.014 ± 0.001 (p < 0.0001) | 0.013 ± 0.001 (p < 0.0001) | 0.009 ± 0.001 (p < 0.0001) | 0.002 ± 0.000 (p < 0.0001) |

R_{pH}^{2} | 0.9988 | 0.9980 | 0.9973 | 0.9973 | 0.9973 | 0.9968 |

RPDM | 0.2083 | 0.3362 | 0.4013 | 0.4057 | 0.4107 | 0.4180 |

B_{f} | 0.9997 | 0.9990 | 0.9988 | 0.9989 | 0.9988 | 0.9993 |

A_{f} | 1.0021 | 1.0034 | 1.0040 | 1.0041 | 1.0041 | 1.0042 |

F-ratio | 369.17 | 322.20 | 313.95 | 313.38 | 310.46 | 305.77 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

d | −0.194 ± 0.003 (p < 0.0001) | −0.190 ± 0.001 (p < 0.0001) | −0.184 ± 0.001 (p < 0.0001) | −0.171 ± 0.001 (p < 0.0001) | −0.168 ± 0.002 (p < 0.0001) | −0.164 ± 0.001 (p < 0.0001) |

e | −0.909 ± 0.001 (p < 0.0001) | −0.899 ± 0.001 (p < 0.0001) | −0.884 ± 0.004 (p < 0.0001) | −0.851 ± 0.002 (p < 0.0001) | −0.846 ± 0.001 (p < 0.0001) | −0.836 ± 0.003 (p < 0.0001) |

f | −0.066 ± 0.002 (p < 0.0001) | −0.067 ± 0.005 (p < 0.0001) | −0.067 ± 0.003 (p < 0.0001) | −0.067 ± 0.002 (p < 0.0001) | −0.067 ± 0.001 (p < 0.0001) | −0.066 ± 0.002 (p < 0.0001) |

g | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) |

h | 0.034 ± 0.001 (p < 0.0001) | 0.034 ± 0.001 (p < 0.0001) | 0.033 ± 0.001 (p < 0.0001) | 0.034 ± 0.001 (p < 0.0001) | 0.034 ± 0.001 (p < 0.0001) | 0.034 ± 0.002 (p < 0.0001) |

R_{TN}^{2} | 0.9995 | 0.9993 | 0.9987 | 0.9997 | 0.9988 | 0.9998 |

RPDM | 0.2355 | 0.2260 | 0.3286 | 0.1284 | 0.2986 | 0.0979 |

B_{f} | 1.0001 | 0.9996 | 0.9994 | 0.9999 | 1.0006 | 1.0002 |

A_{f} | 1.0024 | 1.0023 | 1.0033 | 1.0013 | 1.0030 | 1.0010 |

F-ratio | 198.79 | 156.66 | 179.35 | 159.14 | 167.55 | 366.64 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

i | 0.843 ± 0.001 (p < 0.0001) | 0.810 ± 0.004 (p < 0.0001) | 0.740 ± 0.001 (p < 0.0001) | 0.626 ± 0.003 (p < 0.0001) | 0.531 ± 0.003 (p < 0.0001) | 0.513 ± 0.000 (p < 0.0001) |

j | 0.787 ± 0.016 (p < 0.0001) | 0.787 ± 0.009 (p < 0.0001) | 0.787 ± 0.002 (p < 0.0001) | 0.787 ± 0.008 (p < 0.0001) | 0.787 ± 0.008 (p < 0.0001) | 0.787 ± 0.001 (p < 0.0001) |

k | 0.008 ± 0.001 (p < 0.0001) | 0.008 ± 0.000 (p < 0.0001) | 0.008 ± 0.001 (p < 0.0001) | 0.008 ± 0.001 (p < 0.0001) | 0.008 ± 0.001 (p < 0.0001) | 0.008 ± 0.001 (p < 0.0001) |

l | 0.066 ± 0.003 (p < 0.0001) | 0.065 ± 0.001 (p < 0.0001) | 0.056 ± 0.002 (p < 0.0001) | 0.040 ± 0.001 (p < 0.0001) | 0.026 ± 0.000 (p < 0.0001) | 0.026 ± 0.001 (p < 0.0001) |

m | −0.104 ± 0.012 (p < 0.0001) | −0.103 ± 0.007 (p < 0.0001) | −0.102 ± 0.012 (p < 0.0001) | −0.101 ± 0.001 (p < 0.0001) | −0.092 ± 0.001 (p < 0.0001) | −0.079 ± 0.001 (p < 0.0001) |

R_{X}^{2} | 0.9997 | 0.9990 | 0.9991 | 1.0000 | 1.0000 | 1.0000 |

RPDM | 2.6773 | 2.1414 | 1.6724 | 0.9334 | 0.6215 | 0.4064 |

B_{f} | 0.9764 | 0.9952 | 1.0056 | 0.9937 | 0.9962 | 1.0005 |

A_{f} | 1.0282 | 1.0218 | 1.0168 | 1.0095 | 1.0063 | 1.0041 |

F-ratio | 163893.47 | 158803.25 | 165456.52 | 169736.76 | 179962.55 | 18968.82 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

n | 3.727 ± 0.015 (p < 0.0001) | 3.664 ± 0.016 (p < 0.0001) | 2.672 ± 0.033 (p < 0.0001) | 1.939 ± 0.022 (p < 0.0001) | 1.496 ± 0.016 (p < 0.0001) | 0.922 ± 0.005 (p < 0.0001) |

o | 0.036 ± 0.002 (p < 0.0001) | 0.036 ± 0.001 (p < 0.0001) | 0.036 ± 0.004 (p < 0.0001) | 0.036 ± 0.003 (p < 0.0001) | 0.036 ± 0.003 (p < 0.0001) | 0.036 ± 0.002 (p < 0.0001) |

p | 0.137 ± 0.006 (p < 0.0001) | 0.137 ± 0.004 (p < 0.0001) | 0.137 ± 0.022 (p < 0.0001) | 0.137 ± 0.011 (p < 0.0001) | 0.137 ± 0.010 (p < 0.0001) | 0.137 ± 0.002 (p < 0.0001) |

q | 0.607 ± 0.013 (p < 0.0001) | 0.605 ± 0.009 (p < 0.0001) | 0.409 ± 0.003 (p < 0.0001) | 0.276 ± 0.002 (p < 0.0001) | 0.201 ± 0.021 (p < 0.0001) | 0.100 ± 0.001 (p < 0.0001) |

r | −0.092 ± 0.003 (p < 0.0001) | −0.092 ± 0.002 (p < 0.0001) | −0.092 ± 0.002 (p < 0.0001) | −0.092 ± 0.001 (p < 0.0001) | −0.092 ± 0.004 (p < 0.0001) | −0.092 ± 0.001 (p < 0.0001) |

R_{BT}^{2} | 1.0000 | 0.9997 | 0.9998 | 1.0000 | 0.9992 | 0.9997 |

RPDM | 0.7956 | 7.5772 | 2.8486 | 0.5489 | 1.2762 | 4.3632 |

B_{f} | 0.9953 | 0.9232 | 0.9748 | 1.0042 | 0.9997 | 1.0365 |

A_{f} | 1.0082 | 1.0894 | 1.0300 | 1.0054 | 1.0124 | 1.0423 |

F-ratio | 16623.57 | 14125.66 | 14185.31 | 16845.44 | 9845.37 | 14054.20 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

**Table 5.**Statistically significant (p < 0.05) parameter values (as estimates ± confidence intervals) calculated with the four-dimensional predator-prey system (1)–(4) for each individual culture corresponding to the four factorial points and the four axial points of the experimental matrix (Table 2) of the series of fermentation DW-TS-TP.

Factorial Points | Axial Points | |||||||
---|---|---|---|---|---|---|---|---|

Parameter | TS = 48.3 g/L TP = 0.59 g/L | TS = 48.3 g/L TP = 0.28 g/L | TS = 25.6 g/L TP = 0.59 g/L | TS = 25.6 g/L TP = 0.28 g/L | TS = 51.3 g/L TP = 0.43 g/L | TS = 22.6 g/L TP = 0.43 g/L | TS = 37.0 g/L TP = 0.63 g/L | TS = 37.0 g/L TP = 0.24 g/L |

a | −0.009 ± 0.001 (p < 0.0001) | −0.008 ± 0.011 (p < 0.0001) | −0.015 ± 0.002 (p < 0.0001) | −0.013 ± 0.001 (p < 0.0001) | −0.008 ± 0.001 (p < 0.0001) | −0.014 ± 0.001 (p < 0.0001) | −0.012 ± 0.001 (p < 0.0001) | −0.011 ± 0.002 (p < 0.0001) |

b | −0.007 ± 0.002 (p < 0.0001) | −0.007 ± 0.002 (p < 0.0001) | −0.012 ± 0.003 (p < 0.0001) | −0.009 ± 0.001 (p < 0.0001) | −0.007 ± 0.001 (p < 0.0001) | −0.012 ± 0.001 (p < 0.0001) | −0.009 ± 0.002 (p < 0.0001) | −0.008 ± 0.002 (p < 0.0001) |

c | 0.041 ± 0.003 (p < 0.0001) | 0.050 ± 0.011 (p < 0.0001) | 0.045 ± 0.005 (p < 0.0001) | 0.034 ± 0.000 (p < 0.0001) | 0.050 ± 0.000 (p < 0.0001) | 0.057 ± 0.004 (p < 0.0001) | 0.036 ± 0.001 (p < 0.0001) | 0.043 ± 0.002 (p < 0.0001) |

R_{pH}^{2} | 0.9961 | 0.9918 | 0.9941 | 0.9952 | 0.9918 | 0.9971 | 0.9897 | 0.9903 |

RPDM | 0.5322 | 0.6954 | 0.9820 | 0.7802 | 0.6954 | 0.6045 | 1.1740 | 0.9703 |

B_{f} | 0.9997 | 0.9995 | 0.9994 | 0.9994 | 0.9995 | 0.9999 | 0.9988 | 0.9990 |

A_{f} | 1.0053 | 1.0070 | 1.0099 | 1.0078 | 1.0070 | 1.0061 | 1.0118 | 1.0098 |

F-ratio | 1225.62 | 1502.17 | 1414.37 | 1203.42 | 1203.42 | 588.82 | 1308.83 | 698.32 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

d | −0.188 ± 0.023 (p < 0.0001) | −0.209 ± 0.016 (p < 0.0001) | −0.595 ± 0.029 (p < 0.0001) | −0.741 ± 0.026 (p < 0.0001) | −0.453 ± 0.023 (p < 0.0001) | −0.154 ± 0.002 (p < 0.0001) | −0.196 ± 0.007 (p < 0.0001) | −0.170 ± 0.008 (p < 0.0001) |

e | −0.914 ± 0.037 (p < 0.0001) | −0.940 ± 0.022 (p < 0.0001) | −1.897 ± 0.088 (p < 0.0001) | −2.224 ± 0.101 (p = 0.0103) | −1.447 ± 0.066 (p < 0.0001) | −0.903 ± 0.016 (p < 0.0001) | −0.955 ± 0.021 (p < 0.0001) | −0.885 ± 0.013 (p < 0.0001) |

f | −0.058 ± 0.002 (p < 0.0001) | −0.073 ± 0.006 (p < 0.0001) | −0.288 ± 0.013 (p < 0.0001) | −0.368 ± 0.019 (p < 0.0001) | −0.206 ± 0.018 (p < 0.0001) | −0.058 ± 0.002 (p < 0.0001) | −0.073 ± 0.001 (p < 0.0001) | −0.073 ± 0.003 (p < 0.0001) |

g | −0.001 ± 0.000 (p = 0.0012) | −0.001 ± 0.000 (p = 0.0025) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p = 0.0024) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) |

h | 0.034 ± 0.002 (p < 0.0001) | 0.034 ± 0.007 (p < 0.0001) | 0.034 ± 0.004 (p < 0.0001) | 0.034 ± 0.003 (p < 0.0001) | 0.034 ± 0.002 (p < 0.0001) | 0.034 ± 0.004 (p < 0.0001) | 0.034 ± 0.003 (p < 0.0001) | 0.034 ± 0.002 (p < 0.0001) |

R_{TN}^{2} | 0.9998 | 0.9996 | 0.9974 | 0.9947 | 0.9987 | 0.9987 | 0.9995 | 0.9973 |

RPDM | 0.1460 | 0.1608 | 0.5997 | 0.7355 | 0.2837 | 0.6074 | 0.2451 | 0.7085 |

B_{f} | 1.0000 | 1.0002 | 0.9992 | 0.9995 | 1.0002 | 0.9992 | 0.9998 | 0.9989 |

A_{f} | 1.0015 | 1.0016 | 1.0060 | 1.0074 | 1.0028 | 1.0061 | 1.0025 | 1.0071 |

F-ratio | 111.24 | 136.45 | 128.63 | 145.52 | 809.54 | 101.89 | 99.51 | 221.34 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

i | 0.805 ± 0.027 (p < 0.0001) | 0.781 ± 0.011 (p < 0.0001) | 0.923 ± 0.026 (p < 0.0001) | 0.940 ± 0.005 (p < 0.0001) | 0.845 ± 0.019 (p < 0.0001) | 0.978 ± 0.027 (p < 0.0001) | 0.860 ± 0.027 (p < 0.0001) | 0.823 ± 0.016 (p < 0.0001) |

j | 1.172 ± 0.084 (p < 0.0001) | 1.319 ± 0.025 (p < 0.0001) | 0.884 ± 0.010 (p < 0.0001) | 1.127 ± 0.014 (p < 0.0001) | 1.349 ± 0.103 (p < 0.0001) | 0.967 ± 0.010 (p < 0.0001) | 0.973 ± 0.025 (p < 0.0001) | 1.098 ± 0.103 (p < 0.0001) |

k | 0.005 ± 0.001 (p < 0.0001) | 0.005 ± 0.000 (p < 0.0001) | 0.005 ± 0.000 (p < 0.0001) | 0.005 ± 0.000 (p = 0.0012) | 0.005 ± 0.000 (p < 0.0001) | 0.005 ± 0.000 (p < 0.0001) | 0.005 ± 0.000 (p = 0.0012) | 0.005 ± 0.000 (p < 0.0001) |

l | 0.017 ± 0.001 (p < 0.0001) | 0.009 ± 0.001 (p < 0.0001) | 0.033 ± 0.002 (p < 0.0001) | 0.030 ± 0.002 (p < 0.0001) | 0.018 ± 0.001 (p < 0.0001) | 0.044 ± 0.003 (p < 0.0001) | 0.029 ± 0.001 (p < 0.0001) | 0.024 ± 0.002 (p < 0.0001) |

m | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.002 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) |

R_{X}^{2} | 0.9973 | 0.9958 | 0.9939 | 0.9989 | 0.9965 | 0.9992 | 0.9983 | 0.9984 |

RPDM | 3.2062 | 3.5977 | 4.8630 | 2.0830 | 3.3137 | 2.4733 | 3.0536 | 2.8520 |

B_{f} | 0.9797 | 0.9788 | 0.9680 | 0.9802 | 0.9796 | 0.9816 | 0.9778 | 0.9794 |

A_{f} | 1.0336 | 1.0378 | 1.0525 | 1.0215 | 1.0347 | 1.0259 | 1.0321 | 1.0299 |

F-ratio | 2655.11 | 2222.57 | 2246.34 | 18756.55 | 11253.26 | 36008.93 | 5001.64 | 4953.73 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

n | 1.068 ± 0.018 (p < 0.0001) | 1.066 ± 0.009 (p < 0.0001) | 1.314 ± 0.015 (p < 0.0001) | 1.136 ± 0.086 (p < 0.0001) | 1.067 ± 0.043 (p < 0.0001) | 1.226 ± 0.111 (p < 0.0001) | 1.136 ± 0.044 (p < 0.0001) | 1.071 ± 0.003 (p < 0.0001) |

o | 0.035 ± 0.003 (p < 0.0001) | 0.040 ± 0.003 (p < 0.0001) | 0.030 ± 0.001 (p < 0.0001) | 0.033 ± 0.002 (p < 0.0001) | 0.040 ± 0.005 (p < 0.0001) | 0.036 ± 0.001 (p < 0.0001) | 0.036 ± 0.001 (p < 0.0001) | 0.037 ± 0.002 (p < 0.0001) |

p | 0.098 ± 0.010 (p < 0.0001) | 0.098 ± 0.006 (p < 0.0001) | 0.098 ± 0.002 (p < 0.0001) | 0.098 ± 0.006 (p < 0.0001) | 0.098 ± 0.004 (p < 0.0001) | 0.098 ± 0.005 (p < 0.0001) | 0.098 ± 0.006 (p < 0.0001) | 0.098 ± 0.001 (p < 0.0001) |

q | 0.080 ± 0.005 (p < 0.0001) | 0.081 ± 0.002 (p < 0.0001) | 0.117 ± 0.001 (p < 0.0001) | 0.086 ± 0.002 (p < 0.0001) | 0.081 ± 0.002 (p < 0.0001) | 0.073 ± 0.004 (p < 0.0001) | 0.073 ± 0.002 (p < 0.0001) | 0.073 ± 0.006 (p < 0.0001) |

r | −0.103 ± 0.008 (p < 0.0001) | −0.103 ± 0.003 (p < 0.0001) | −0.309 ± 0.018 (p < 0.0001) | −0.103 ± 0.008 (p < 0.0001) | −0.103 ± 0.003 (p < 0.0001) | −0.103 ± 0.008 (p < 0.0001) | −0.103 ± 0.003 (p < 0.0001) | −0.103 ± 0.009 (p < 0.0001) |

R_{BT}^{2} | 0.9988 | 0.9983 | 0.9988 | 0.9971 | 0.9987 | 0.9982 | 0.9983 | 0.9983 |

RPDM | 6.2540 | 7.5243 | 7.4690 | 4.1038 | 7.3319 | 21.2434 | 16.3565 | 14.6312 |

B_{f} | 1.0562 | 1.0597 | 1.0602 | 1.0275 | 1.0653 | 1.1581 | 1.1369 | 1.1155 |

A_{f} | 1.0596 | 1.0714 | 1.0702 | 1.0406 | 1.0691 | 1.1700 | 1.1398 | 1.1272 |

F-ratio | 15333.43 | 15698.19 | 6008.51 | 9253.62 | 57045.82 | 11115.87 | 12489.56 | 13001.43 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

**Table 6.**Statistically significant (p < 0.05) parameter values (as estimates ± confidence intervals) calculated with the four-dimensional predator-prey system (1)–(4) for each individual culture corresponding to the five center points (TS = 37.0 g/L, TP = 0.43 g/L) of the experimental matrix (Table 2) and to the optimum conditions (TS = 22.6 g/L, TP = 0.46 g/L) of the series of fermentation DW-TS-TP.

Center Points | Optimum Conditions | |||||
---|---|---|---|---|---|---|

Parameter | TS = 37.0 g/L TP = 0.43 g/L | TS = 37.0 g/L TP = 0.43 g/L | TS = 37.0 g/L TP = 0.43 g/L | TS = 37.0 g/L TP = 0.43 g/L | TS = 37.0 g/L TP = 0.43 g/L | TS = 22.6 g/L TP = 0.46 g/L |

a | −0.012 ± 0.002 (p < 0.0001) | −0.012 ± 0.001 (p < 0.0001) | −0.012 ± 0.001 (p < 0.0001) | −0.012 ± 0.001 (p < 0.0001) | −0.012 ± 0.002 (p < 0.0001) | −0.016 ± 0.001 (p < 0.0001) |

b | −0.008 ± 0.001 (p < 0.0001) | −0.008 ± 0.003 (p < 0.0001) | −0.009 ± 0.001 (p < 0.0001) | −0.008 ± 0.002 (p < 0.0001) | −0.008 ± 0.001 (p < 0.0001) | −0.013 ± 0.001 (p < 0.0001) |

c | 0.033 ± 0.004 (p < 0.0001) | 0.033 ± 0.002 (p < 0.0001) | 0.035 ± 0.002 (p < 0.0001) | 0.035 ± 0.002 (p < 0.0001) | 0.035 ± 0.002 (p < 0.0001) | 0.048 ± 0.000 (p < 0.0001) |

R_{pH}^{2} | 0.9903 | 0.9905 | 0.9779 | 0.9805 | 0.9794 | 0.9937 |

RPDM | 1.1229 | 1.1156 | 1.7516 | 1.6095 | 1.6444 | 1.0972 |

B_{f} | 0.9988 | 0.9988 | 0.9979 | 0.9980 | 0.9979 | 0.9989 |

A_{f} | 1.0113 | 1.0112 | 1.0177 | 1.0163 | 1.0166 | 1.0110 |

F-ratio | 1442.53 | 1286.83 | 1399.58 | 1306.41 | 1119.82 | 2610.26 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

d | −0.168 ± 0.008 (p < 0.0001) | −0.174 ± 0.010 (p < 0.0001) | −0.196 ± 0.014 (p < 0.0001) | −0.169 ± 0.008 (p < 0.0001) | −0.168 ± 0.016 (p < 0.0001) | −0.472 ± 0.024 (p < 0.0001) |

e | −0.893 ± 0.013 (p < 0.0001) | −0.913 ± 0.004 (p < 0.0001) | −0.955 ± 0.010 (p < 0.0001) | −0.893 ± 0.016 (p < 0.0001) | −0.893 ± 0.055 (p < 0.0001) | −1.605 ± 0.102 (p < 0.0001) |

f | −0.087 ± 0.005 (p < 0.0001) | −0.087 ± 0.002 (p < 0.0001) | −0.073 ± 0.002 (p < 0.0001) | −0.073 ± 0.001 (p < 0.0001) | −0.087 ± 0.004 (p < 0.0001) | −0.367 ± 0.008 (p < 0.0001) |

g | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) |

h | 0.034 ± 0.001 (p < 0.0001) | 0.034 ± 0.002 (p < 0.0001) | 0.034 ± 0.001 (p < 0.0001) | 0.034 ± 0.002 (p < 0.0001) | 0.034 ± 0.001 (p < 0.0001) | 0.036 ± 0.003 (p < 0.0001) |

R_{TN}^{2} | 0.9947 | 0.9933 | 0.9941 | 0.9883 | 0.9941 | 0.9983 |

RPDM | 1.3103 | 1.2626 | 2.0280 | 1.6987 | 1.6361 | 1.0880 |

B_{f} | 0.9968 | 0.9982 | 1.0042 | 0.9970 | 0.9835 | 1.0000 |

A_{f} | 1.0132 | 1.0127 | 1.0205 | 1.0172 | 1.0168 | 1.0109 |

F-ratio | 256.84 | 231.13 | 512.73 | 488.79 | 385.82 | 419.77 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

i | 0.933 ± 0.017 (p < 0.0001) | 0.932 ± 0.011 (p < 0.0001) | 0.937 ± 0.010 (p < 0.0001) | 0.933 ± 0.006 (p < 0.0001) | 0.936 ± 0.008 (p < 0.0001) | 0.995 ± 0.019 (p < 0.0001) |

j | 1.052 ± 0.010 (p < 0.0001) | 1.034 ± 0.00034 (p < 0.0001) | 1.036 ± 0.022 (p < 0.0001) | 1.048 ± 0.013 (p < 0.0001) | 1.057 ± 0.016 (p < 0.0001) | 0.851 ± 0.008 (p < 0.0001) |

k | 0.005 ± 0.001 (p < 0.0001) | 0.005 ± 0.000 (p < 0.0001) | 0.005 ± 0.001 (p < 0.0001) | 0.005 ± 0.001 (p < 0.0001) | 0.005 ± 0.001 (p < 0.0001) | 0.005 ± 0.001 (p < 0.0001) |

l | 0.037 ± 0.001 (p < 0.0001) | 0.039 ± 0.001 (p < 0.0001) | 0.038 ± 0.001 (p < 0.0001) | 0.038 ± 0.001 (p < 0.0001) | 0.037 ± 0.001 (p < 0.0001) | 0.055 ± 0.001 (p < 0.0001) |

m | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.000 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) | −0.002 ± 0.001 (p < 0.0001) |

R_{X}^{2} | 0.9990 | 0.9992 | 0.9993 | 0.9984 | 0.9996 | 0.9993 |

RPDM | 3.3758 | 2.6198 | 2.1861 | 3.9710 | 3.7304 | 2.5775 |

B_{f} | 1.0056 | 0.9919 | 0.9786 | 1.0054 | 1.0251 | 0.9767 |

A_{f} | 1.0344 | 1.0270 | 1.0229 | 1.0406 | 1.0374 | 1.0272 |

F-ratio | 7954.52 | 11136.19 | 24025.14 | 8212.25 | 33154.51 | 27521.83 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

n | 1.136 ± 0.013 (p < 0.0001) | 1.136 ± 0.009 (p < 0.0001) | 1.136 ± 0.010 (p < 0.0001) | 1.136 ± 0.005 (p < 0.0001) | 1.136 ± 0.011 (p < 0.0001) | 1.322 ± 0.007 (p < 0.0001) |

o | 0.036 ± 0.001 (p < 0.0001) | 0.036 ± 0.003 (p < 0.0001) | 0.036 ± 0.001 (p < 0.0001) | 0.037 ± 0.002 (p < 0.0001) | 0.035 ± 0.002 (p < 0.0001) | 0.032 ± 0.001 (p < 0.0001) |

p | 0.098 ± 0.001 (p < 0.0001) | 0.098 ± 0.002 (p < 0.0001) | 0.098 ± 0.001 (p < 0.0001) | 0.098 ± 0.002 (p < 0.0001) | 0.098 ± 0.001 (p < 0.0001) | 0.098 ± 0.002 (p < 0.0001) |

q | 0.073 ± 0.003 (p < 0.0001) | 0.073 ± 0.001 (p < 0.0001) | 0.073 ± 0.004 (p < 0.0001) | 0.073 ± 0.003 (p < 0.0001) | 0.037 ± 0.006 (p < 0.0001) | 0.107 ± 0.001 (p < 0.0001) |

r | −0.103 ± 0.011 (p < 0.0001) | −0.103 ± 0.009 (p < 0.0001) | −0.103 ± 0.015 (p < 0.0001) | −0.103 ± 0.021 (p < 0.0001) | −0.103 ± 0.008 (p < 0.0001) | −0.103 ± 0.013 (p < 0.0001) |

R_{BT}^{2} | 0.9975 | 0.9974 | 0.9985 | 0.9934 | 0.9971 | 0.9988 |

RPDM | 18.0670 | 17.3313 | 16.4117 | 20.0055 | 17.1747 | 15.0306 |

B_{f} | 1.1513 | 1.1488 | 1.1279 | 1.1670 | 1.1209 | 1.1187 |

A_{f} | 1.1542 | 1.1493 | 1.1388 | 1.1758 | 1.1459 | 1.1286 |

F-ratio | 7764.25 | 9873.12 | 8895.19 | 7983.49 | 10895.37 | 3211.61 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

**Table 7.**Statistically significant (p < 0.05) parameter values (as estimates ± confidence intervals) calculated with the four-dimensional predator-prey system (1)–(4) for each individual culture of the series of fermentation DW-MRS.

Initial Nutrient Concentrations (%, w/v) in the DW Medium | ||||||
---|---|---|---|---|---|---|

Parameter | 0 | 25 | 50 | 75 | 100 | 125 |

a | −0.032 ± 0.001 (p < 0.0001) | −0.037 ± 0.001 (p < 0.0001) | −0.043 ± 0.002 (p < 0.0001) | 0.177 ± 0.002 (p < 0.0001) | 0.290 ± 0.015 (p < 0.0001) | 0.282 ± 0.004 (p < 0.0001) |

b | −0.017 ± 0.001 (p < 0.0001) | −0.016 ± 0.003 (p < 0.0001) | −0.014 ± 0.001 (p < 0.0001) | 0.022 ± 0.000 (p < 0.0001) | 0.039 ± 0.002 (p < 0.0001) | 0.038 ± 0.001 (p < 0.0001) |

c | 0.103 ± 0.002 (p < 0.0001) | 0.057 ± 0.001 (p < 0.0001) | 0.022 ± 0.001 (p < 0.0001) | 0.086 ± 0.001 (p < 0.0001) | 0.116 ± 0.010 (p < 0.0001) | 0.115 ± 0.008 (p < 0.0001) |

R_{pH}^{2} | 0.9964 | 0.9950 | 0.9941 | 0.9987 | 0.9994 | 0.9991 |

RPDM | 0.5787 | 0.8501 | 1.0040 | 0.4249 | 0.3100 | 0.3525 |

B_{f} | 0.9997 | 0.9989 | 0.9988 | 1.0001 | 1.0001 | 1.0003 |

A_{f} | 1.0058 | 1.0086 | 1.0101 | 1.0043 | 1.0031 | 1.0035 |

F-ratio | 378.83 | 375.38 | 371.42 | 396.38 | 422.44 | 409.56 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

d | −0.068 ± 0.001 (p < 0.0001) | −0.189 ± 0.002 (p < 0.0001) | −0.355 ± 0.015 (p < 0.0001) | −0.379 ± 0.016 (p < 0.0001) | 0.364 ± 0.005 (p < 0.0001) | 0.416 ± 0.002 (p < 0.0001) |

e | −0.415 ± 0.004 (p < 0.0001) | −0.430 ± 0.006 (p < 0.0001) | −0.382 ± 0.003 (p < 0.0001) | −0.297 ± 0.011 (p < 0.0001) | 0.066 ± 0.001 (p < 0.0001) | 0.068 ± 0.003 (p < 0.0001) |

f | 0.292 ± 0.011 (p < 0.0001) | 0.338 ± 0.007 (p < 0.0001) | 0.392 ± 0.004 (p < 0.0001) | 0.395 ± 0.020 (p < 0.0001) | 0.230 ± 0.016 (p < 0.0001) | 0.148 ± 0.011 (p < 0.0001) |

g | −0.007 ± 0.001 (p < 0.0001) | −0.012 ± 0.002 (p < 0.0001) | −0.010 ± 0.001 (p < 0.0001) | −0.008 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) | −0.001 ± 0.000 (p < 0.0001) |

h | 0.016 ± 0.001 (p < 0.0001) | 0.045 ± 0.001 (p < 0.0001) | 0.057 ± 0.001 (p < 0.0001) | 0.061 ± 0.002 (p < 0.0001) | 0.018 ± 0.001 (p < 0.0001) | 0.018 ± 0.001 (p < 0.0001) |

R_{TN}^{2} | 0.9871 | 0.9982 | 0.9978 | 0.9850 | 0.9992 | 0.9962 |

RPDM | 1.2891 | 0.9744 | 0.5372 | 1.7542 | 0.2335 | 0.2001 |

B_{f} | 1.0013 | 1.0004 | 0.9996 | 0.9975 | 0.9998 | 0.9997 |

A_{f} | 1.0129 | 1.0098 | 1.0054 | 1.0178 | 1.0023 | 1.0020 |

F-ratio | 1212.37 | 1311.19 | 1309.21 | 1125.41 | 1321.28 | 1285.43 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

i | 0.928 ± 0.013 (p < 0.0001) | 1.103 ± 0.016 (p < 0.0001) | 1.106 ± 0.008 (p < 0.0001) | 1.110 ± 0.014 (p < 0.0001) | 1.118 ± 0.024 (p < 0.0001) | 1.243 ± 0.011 (p < 0.0001) |

j | 0.731 ± 0.010 (p < 0.0001) | 0.830 ± 0.019 (p < 0.0001) | 0.542 ± 0.025 (p < 0.0001) | 0.289 ± 0.005 (p < 0.0001) | 0.139 ± 0.003 (p < 0.0001) | 0.021 ± 0.002 (p < 0.0001) |

k | 0.007 ± 0.001 (p < 0.0001) | 0.007 ± 0.001 (p < 0.0001) | 0.007 ± 0.001 (p < 0.0001) | 0.007 ± 0.000 (p < 0.0001) | 0.007 ± 0.000 (p < 0.0001) | 0.007 ± 0.001 (p < 0.0001) |

l | 0.087 ± 0.004 (p < 0.0001) | 0.087 ± 0.003 (p < 0.0001) | 0.087 ± 0.004 (p < 0.0001) | 0.086 ± 0.002 (p < 0.0001) | 0.086 ± 0.001 (p < 0.0001) | 0.091 ± 0.001 (p < 0.0001) |

m | 0.019 ± 0.002 (p < 0.0001) | 0.019 ± 0.001 (p < 0.0001) | 0.080 ± 0.002 (p < 0.0001) | 0.080 ± 0.004 (p < 0.0001) | 0.076 ± 0.003 (p < 0.0001) | 0.090 ± 0.002 (p < 0.0001) |

R_{X}^{2} | 0.9994 | 0.9999 | 0.9994 | 0.9987 | 0.9998 | 0.9991 |

RPDM | 2.6384 | 0.7688 | 2.6386 | 2.0128 | 1.8566 | 4.4775 |

B_{f} | 0.9785 | 0.9946 | 0.9787 | 0.9964 | 0.9862 | 0.9671 |

A_{f} | 1.0279 | 1.0078 | 1.0277 | 1.0204 | 1.0191 | 1.0480 |

F-ratio | 6014.11 | 7051.64 | 6001.32 | 4318.51 | 6886.49 | 4806.29 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

n | 0.546 ± 0.024 (p < 0.0001) | 0.899 ± 0.013 (p < 0.0001) | 0.967 ± 0.019 (p < 0.0001) | 1.134 ± 0.010 (p < 0.0001) | 1.186 ± 0.008 (p < 0.0001) | 1.225 ± 0.015 (p < 0.0001) |

o | 0.015 ± 0.001 (p < 0.0001) | 0.015 ± 0.001 (p < 0.0001) | 0.009 ± 0.001 (p < 0.0001) | 0.007 ± 0.001 (p < 0.0001) | 0.005 ± 0.001 (p < 0.0001) | 0.004 ± 0.001 (p < 0.0001) |

p | 0.385 ± 0.010 (p < 0.0001) | 0.385 ± 0.007 (p < 0.0001) | 0.385 ± 0.013 (p < 0.0001) | 0.385 ± 0.014 (p < 0.0001) | 0.385 ± 0.004 (p < 0.0001) | 0.385 ± 0.018 (p < 0.0001) |

q | 0.003 ± 0.000 (p < 0.0001) | 0.049 ± 0.003 (p < 0.0001) | 0.077 ± 0.002 (p < 0.0001) | 0.106 ± 0.008 (p < 0.0001) | 0.120 ± 0.004 (p < 0.0001) | 0.127 ± 0.0015 (p < 0.0001) |

r | 0.013 ± 0.002 (p < 0.0001) | 0.013 ± 0.003 (p < 0.0001) | 0.013 ± 0.001 (p < 0.0001) | 0.013 ± 0.001 (p < 0.0001) | 0.013 ± 0.001 (p < 0.0001) | 0.013 ± 0.003 (p < 0.0001) |

R_{BT}^{2} | 0.9991 | 0.9984 | 0.9994 | 0.9994 | 0.9986 | 0.9987 |

RPDM | 4.8680 | 5.8719 | 6.4048 | 7.4056 | 11.5425 | 11.6937 |

B_{f} | 0.9541 | 0.9405 | 0.9330 | 0.9237 | 0.8769 | 0.8765 |

A_{f} | 1.0556 | 1.0719 | 1.0781 | 1.0890 | 1.1505 | 1.1503 |

F-ratio | 892.54 | 808.19 | 801.22 | 2315.09 | 1366.93 | 2404.16 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

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**MDPI and ACS Style**

Giménez-Palomares, F.; Fernández de Córdoba, P.; Mejuto, J.C.; Bendaña-Jácome, R.J.; Pérez-Guerra, N.
Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System. *Mathematics* **2022**, *10*, 677.
https://doi.org/10.3390/math10050677

**AMA Style**

Giménez-Palomares F, Fernández de Córdoba P, Mejuto JC, Bendaña-Jácome RJ, Pérez-Guerra N.
Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System. *Mathematics*. 2022; 10(5):677.
https://doi.org/10.3390/math10050677

**Chicago/Turabian Style**

Giménez-Palomares, Fernando, Pedro Fernández de Córdoba, Juan C. Mejuto, Ricardo J. Bendaña-Jácome, and Nelson Pérez-Guerra.
2022. "Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System" *Mathematics* 10, no. 5: 677.
https://doi.org/10.3390/math10050677