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# Dynamics of some stochastic chemostat models with multiplicative noise

Partially supported by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P and Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492.
• In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.

Mathematics Subject Classification: Primary: 34C11, 34F05; Secondary: 60H10.

 Citation: • • Figure 2.  Stochastic chemostat without wall growth. Values of parameters: $S_0=5$, $x_0=10$, $S^0=1$, $D=2$, $a=0.6$, $m=1$, $\alpha=0.2$ (left) and $\alpha= 0.5$ (right)

Figure 1.  Stochastic chemostat without wall growth. Values of parameters: $S_0=5$, $x_0=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $\alpha=0.2$ (left) and $\alpha= 0.5$ (right)

Figure 5.  Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.2$

Figure 6.  Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.5$

Figure 3.  Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.2$

Figure 4.  Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.5$

Table 1.  Internal structure of the random attractor -Random chemostat model with wall growth

 ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A: $b\nu c_\xi-m\geq 0$ (A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$ $\displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$ (A-2) $\,\, \nu+\frac{\alpha^2}{2}c$ $\displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$ (B-2) $\,\, \nu+\frac{\alpha^2}{2} Table 2. Internal structure of the random attractor -Stochastic chemostat model with wall growth  ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A:$ b\nu c_\xi-m\geq 0$(A-1)$\,\,\nu+\frac{\alpha^2}{2}>c \displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$(A-2)$\,\, \nu+\frac{\alpha^2}{2}c \displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$(B-2)$\,\, \nu+\frac{\alpha^2}{2}
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Tables(2)

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