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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

*Corresponding author

Decicated to Prof. T. Caraballo for his 60th birthday

Received  April 2019 Revised  October 2019 Published  January 2020

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.

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.
Set $B: = [0,\gamma]\times[0,\lambda]$
Vector field of system (12)-(13) with -$\gamma = 1$ and $\lambda = 3$
The time series of microbial mass, nitrogen, ethanol and sugar concentrations with initial data as in (14) and parameters values as in (15)
The dynamics of microbial mass, nitrogen, ethanol and sugar concentrations with initial data as in (14) and parameters values as in (16)
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
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 $e^*\in(\frac{ {{\rho}}}{ {{k}}},\lambda)$
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 $k = 0.05$
The dynamics of Microbial biomass, nitrogen, ethanol and sugar concentrations respectively with values of parameters and initial data as figure 7 and $k = 0.05$
Same parameters and initial data as in figure 7 and $k = 2.5$
Same parameters and initial data as in figure 7 and $k = 0$
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
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
Prof. T. Caraballo tasting a good dry wine
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