Figure 1.
Set $ B: = [0,\gamma]\times[0,\lambda] $
-
Previous Article
Nonclassical diffusion with memory lacking instantaneous damping
- CPAA Home
- This Issue
-
Next Article
Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
April
2020, 19(4): 2015-2034.
doi: 10.3934/cpaa.2020089
Dynamics of fermentation models for the production of dry and sweet wine
| 1. | Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131, Ancona, Italy |
| 2. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, Sevilla 41012, Spain |
In this work we consider two classical mathematical models of wine fermentation. The first model describes the wine-making process that is used to produce dry wine. The second model is obtained by introducing a term in the equation of the dynamics of the yeast. Thanks to this change it will be possible to inhibit the fermentation of the sugar and as a consequence a sweet wine will be obtained. We first prove the existence, uniqueness, positiveness and boundedness of solutions for both models. Then we pass to analyse the the long-time dynamics. For the second model we also provide estimates for the concentration of ethanol, nitrogen and sugar at the end of the process. Moreover, several numerical simulations are provided to support the theoretical results.
Keywords:
Wine fermentation,
dry wine,
sweet wine,
asymptotic dynamics,
qualitative behaviour of solutions.
Mathematics Subject Classification:
Primary: 34A34, 34A40, 34D05; Secondary: 92B05.
Citation:
Renato Colucci, Javier López-de-la-Cruz. Dynamics of fermentation models for the production of dry and sweet wine. Communications on Pure and Applied Analysis,
2020, 19
(4)
: 2015-2034.
doi: 10.3934/cpaa.2020089
![]()
References:
| [1] |
S. Aiba, M. Shoda and M. Nagatani,
Kinetics of product inhibition in alcohol fermentation, Biotechnology and Bioengineering, 10 (1968), 845-864.
|
| [2] |
L. Alba-Lois and C. Segal-Kischinevzky,
Yeast fermentation and the making of beer and wine, Nature Education, 3 (2010), 17.
|
| [3] |
R. Boulton,
The prediction of fermentation behavior by a kinetic model, Am J Enol Vitic, 31 (1980), 40-45.
|
| [4] |
R. Boulton, V. Singleton, L. Bisson and R. Kunkee, Principles and Practices of Winemaking, New York: Chapman & Hall, 1996. |
| [5] |
I. Caro, L. Pérez and D. Cantero,
Development of a kinetic model for the alcoholic fermentation of must, Biotechnology and Bioengineering, 38 (1991), 742-748.
|
| [6] |
M. C. Coleman, R. Fish and D. E. Block,
Temperature-dependent kinetic model for nitrogen-limited wine fermentations, Applied and Environmental Microbiology, 73 (2007), 5875-5884.
|
| [7] |
A. C. Cramer, S. Vlassides and D. E. Block,
Kinetic model for nitrogen-limited wine fermentations, Biotechnology and Bioengineering, 77 (2002), 49-60.
|
| [8] |
A. C. Cramer, S. Vlassides and D. E. Block,
Kinetic model for nitrogen-limited wine fermentations, Biotechnology and Bioengineering, 77 (2001), 49-60.
|
| [9] |
R. David, D. Dochain, J.-R. Mouret, A. V. Wouwer and J.-M. Sablayrolles, Dynamical modeling of alcoholic fermentation and its link with nitrogen consumption, in Proceedings of the 11th International Symposium on Computer Applications in Biotechnology, 2010, 496-501. |
| [10] |
S. Malherbe, V. Fromion, N. Hilgert and J. Sablayrolles, Modeling the effects of assimilable nitrogen and temperature on fermentation kinetcis in enological conditions, Biotechnology and Bioengineering, 83 (2004), 2. |
| [11] |
L. Perko, Differential Equations and Dynamical Systems, 3rd edition Springer, 2008.
doi: 10.1007/978-1-4684-0249-0. |
| [12] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer Berlin Heidelberg, 1996.
doi: 10.1007/978-3-642-61453-8. |
show all references
References:
| [1] |
S. Aiba, M. Shoda and M. Nagatani,
Kinetics of product inhibition in alcohol fermentation, Biotechnology and Bioengineering, 10 (1968), 845-864.
|
| [2] |
L. Alba-Lois and C. Segal-Kischinevzky,
Yeast fermentation and the making of beer and wine, Nature Education, 3 (2010), 17.
|
| [3] |
R. Boulton,
The prediction of fermentation behavior by a kinetic model, Am J Enol Vitic, 31 (1980), 40-45.
|
| [4] |
R. Boulton, V. Singleton, L. Bisson and R. Kunkee, Principles and Practices of Winemaking, New York: Chapman & Hall, 1996. |
| [5] |
I. Caro, L. Pérez and D. Cantero,
Development of a kinetic model for the alcoholic fermentation of must, Biotechnology and Bioengineering, 38 (1991), 742-748.
|
| [6] |
M. C. Coleman, R. Fish and D. E. Block,
Temperature-dependent kinetic model for nitrogen-limited wine fermentations, Applied and Environmental Microbiology, 73 (2007), 5875-5884.
|
| [7] |
A. C. Cramer, S. Vlassides and D. E. Block,
Kinetic model for nitrogen-limited wine fermentations, Biotechnology and Bioengineering, 77 (2002), 49-60.
|
| [8] |
A. C. Cramer, S. Vlassides and D. E. Block,
Kinetic model for nitrogen-limited wine fermentations, Biotechnology and Bioengineering, 77 (2001), 49-60.
|
| [9] |
R. David, D. Dochain, J.-R. Mouret, A. V. Wouwer and J.-M. Sablayrolles, Dynamical modeling of alcoholic fermentation and its link with nitrogen consumption, in Proceedings of the 11th International Symposium on Computer Applications in Biotechnology, 2010, 496-501. |
| [10] |
S. Malherbe, V. Fromion, N. Hilgert and J. Sablayrolles, Modeling the effects of assimilable nitrogen and temperature on fermentation kinetcis in enological conditions, Biotechnology and Bioengineering, 83 (2004), 2. |
| [11] |
L. Perko, Differential Equations and Dynamical Systems, 3rd edition Springer, 2008.
doi: 10.1007/978-1-4684-0249-0. |
| [12] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer Berlin Heidelberg, 1996.
doi: 10.1007/978-3-642-61453-8. |
Figure 3.
The time series of microbial mass, nitrogen, ethanol and sugar concentrations with initial data as in (14) and parameters values as in (15)
Figure 4.
The dynamics of microbial mass, nitrogen, ethanol and sugar concentrations with initial data as in (14) and parameters values as in (16)
Figure 5.
Case $ \lambda>\mu_{\text{max}} $ . In yellow we represent the positive invariant region. The region between the $ e- $ axis, the vertical line $ n = n(0) $ , the horizontal line $ e = \lambda $ and the curve $ e = \frac{ {{1}}}{ {{k}}}\mu(n) $ is also positively invariant. In violet we have represented the function $ e(n) $ (see Theorem 3.5). Note that $ \frac{ {{de}}}{ {{dn}}}<0 $ as observed in the proof of Theorem 3.5
Figure 6.
The vector field of the system (31), (32) with $ k_s = 1 $ , $ k_e = 2 $ , $ k = 2 $ ; $ \rho = 2 $ , $ \lambda = 3 $ and $ \beta_{max} = 1 $ . It is easy to see that solutions starting on the set $ \{(y,e): \quad y>0, e\in[0,\lambda)\} $ converges to a fixed point $ (0,e^*) $ with $ e^*\in(\frac{ {{\rho}}}{ {{k}}},\lambda) $
Figure 7.
The dynamics of Microbial biomass, nitrogen, ethanol and sugar concentrations respectively with values of parameters as in (35) and initial data as in (36) and $ k = 0.05 $
Figure 11.
The solutions of system (1)-(4) are in blue while solutions of system (6)-(9) are in yellow for $ k = 0.05 $ and in orange for $ k = 0.25 $ respectively. The values of parameters and initial data are as in (37) and (38) respectively
Figure 12.
The solutions of system (1)-(4) are in blue while solutions of system (6)-(9) are in yellow for $ k = 0.05 $ and in orange for $ k = 0.25 $ respectively. The values of parameters and initial data are as in (37) and (39) respectively
Figure 13.
Prof. T. Caraballo tasting a good dry wine
| [1] |
Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206 |
| [2] |
Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations and Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1 |
| [3] |
Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088 |
| [4] |
Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks and Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003 |
| [5] |
Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793 |
| [6] |
Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 |
| [7] |
Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439 |
| [8] |
Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927 |
| [9] |
Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 |
| [10] |
Maciej Smołka. Asymptotic behaviour of optimal solutions of control problems governed by inclusions. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 641-652. doi: 10.3934/dcds.1998.4.641 |
| [11] |
Giovanni Colombo, Paolo Gidoni, Emilio Vilches. Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 737-757. doi: 10.3934/dcds.2021135 |
| [12] |
Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019 |
| [13] |
Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks and Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197 |
| [14] |
Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 |
| [15] |
Akisato Kubo, Hiroki Hoshino, Katsutaka Kimura. Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model. Conference Publications, 2015, 2015 (special) : 733-744. doi: 10.3934/proc.2015.0733 |
| [16] |
Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347 |
| [17] |
Brahim Alouini. Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 45-72. doi: 10.3934/dcdsb.2021032 |
| [18] |
Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28 (2) : 1037-1048. doi: 10.3934/era.2020056 |
| [19] |
Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 |
| [20] |
María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure and Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]