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# Dynamics of fermentation models for the production of dry and sweet wine

• *Corresponding author

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

• 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.

Mathematics Subject Classification: Primary: 34A34, 34A40, 34D05; Secondary: 92B05.

 Citation: • • Figure 1.  Set $B: = [0,\gamma]\times[0,\lambda]$

Figure 2.  Vector field of system (12)-(13) with -$\gamma = 1$ and $\lambda = 3$

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 8.  The dynamics of Microbial biomass, nitrogen, ethanol and sugar concentrations respectively with values of parameters and initial data as figure 7 and $k = 0.05$

Figure 9.  Same parameters and initial data as in figure 7 and $k = 2.5$

Figure 10.  Same parameters and initial data as in figure 7 and $k = 0$

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

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