Dynamics of some stochastic chemostat models with multiplicative noise

T. Caraballo; M. J. Garrido-Atienza; J. López-de-la-Cruz

doi:10.3934/cpaa.2017092

Article Contents

2017, Volume 16, Issue 5: 1893-1914. Doi: 10.3934/cpaa.2017092

This issue Previous Article Global dynamics of a microorganism flocculation model with time delay Next Article Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

Dynamics of some stochastic chemostat models with multiplicative noise

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n. Sevilla, 41012, Spain

* Corresponding author: caraball@us.es
* Corresponding author: caraball@us.es

Received: July 31, 2016

Revised: February 28, 2017

Published: October 2017

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.

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Abstract

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.

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

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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)

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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)

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

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

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

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

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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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\sigma(t)$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information

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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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(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 }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-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 }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$S$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information

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