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Forwards dynamics of nonautonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
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 winemaking 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 longtime 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.
References:
[1] 
S. Aiba, M. Shoda and M. Nagatani, Kinetics of product inhibition in alcohol fermentation, Biotechnology and Bioengineering, 10 (1968), 845864. 
[2] 
L. AlbaLois and C. SegalKischinevzky, 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), 4045. 
[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), 742748. 
[6] 
M. C. Coleman, R. Fish and D. E. Block, Temperaturedependent kinetic model for nitrogenlimited wine fermentations, Applied and Environmental Microbiology, 73 (2007), 58755884. 
[7] 
A. C. Cramer, S. Vlassides and D. E. Block, Kinetic model for nitrogenlimited wine fermentations, Biotechnology and Bioengineering, 77 (2002), 4960. 
[8] 
A. C. Cramer, S. Vlassides and D. E. Block, Kinetic model for nitrogenlimited wine fermentations, Biotechnology and Bioengineering, 77 (2001), 4960. 
[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, 496501. 
[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: 10.1007/9781468402490. 
[12] 
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer Berlin Heidelberg, 1996. doi: 10.1007/9783642614538. 
show all references
References:
[1] 
S. Aiba, M. Shoda and M. Nagatani, Kinetics of product inhibition in alcohol fermentation, Biotechnology and Bioengineering, 10 (1968), 845864. 
[2] 
L. AlbaLois and C. SegalKischinevzky, 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), 4045. 
[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), 742748. 
[6] 
M. C. Coleman, R. Fish and D. E. Block, Temperaturedependent kinetic model for nitrogenlimited wine fermentations, Applied and Environmental Microbiology, 73 (2007), 58755884. 
[7] 
A. C. Cramer, S. Vlassides and D. E. Block, Kinetic model for nitrogenlimited wine fermentations, Biotechnology and Bioengineering, 77 (2002), 4960. 
[8] 
A. C. Cramer, S. Vlassides and D. E. Block, Kinetic model for nitrogenlimited wine fermentations, Biotechnology and Bioengineering, 77 (2001), 4960. 
[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, 496501. 
[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: 10.1007/9781468402490. 
[12] 
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer Berlin Heidelberg, 1996. doi: 10.1007/9783642614538. 
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