Modeling and analysis of random and stochastic input flows in the chemostat model

Tomás Caraballo; Maria-José Garrido-Atienza; Javier López-de-la-Cruz; Alain Rapaport

doi:10.3934/dcdsb.2018280

Article Contents

2019, Volume 24, Issue 8: 3591-3614. Doi: 10.3934/dcdsb.2018280

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Modeling and analysis of random and stochastic input flows in the chemostat model

1.
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, Sevilla 41012, Spain
2.
MISTEA, Univ. Montpellier, Inra, Montpellier SupAgro, 2, place Pierre Viala, 34060 Montpellier, France

* Corresponding author: Alain Rapaport
* Corresponding author: Alain Rapaport

Received: March 31, 2018

Revised: May 31, 2018

Early access: October 2018

Published: July 2019

Partially supported by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P, Junta de Andalucía under the Proyecto de Excelencia P12-FQM-1492 and Ⅵ Plan Propio de Investigación y Transferencia de la Universidad de Sevilla

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Abstract

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.

Keywords:

Mathematics Subject Classification: Primary: 37N25, 92D25, 34F05, 37H10; Secondary: 37B55, 60H10.

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Figure 1. Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\beta = 2$

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Figure 2. Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\nu = 0.5$

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Figure 3. Realizations of the perturbed dilution rate, $\underline{s}$ and $\bar{s}$

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Figure 4. Attracting set $\widehat{B}_0$

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Figure 5. $\mbox{Absorbing set } B_\varepsilon(\omega)$

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Figure 6. $\mbox{Absorbing set }{\color{blue}{ B_0(\omega)}}$

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Figure 7. Persistence of the species in the random chemostat model

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Figure 8. Extinction of the species in the random chemostat model

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Figure 9. Stochastic chemostat model. Extinction (left) and persistence (right)

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Figure 10. Comparison in case of extinction

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Figure 11. Comparison in case of persistence

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Table 1. Internal structure of the attracting set $\widehat{B}_0$

	$s_{in}>\bar{s}$	$s_{in}=\bar{s}$	$s_{in}< \bar{s}$
$D>\mu(s_{in})$	impossible	impossible	Extinction Proposition 2.2 $\{(s_{in}, 0)\}$
$D=\mu(s_{in})$	Persistence Theorem 2.3 $\underline{s}\leq s\leq \bar{s}$ $s_{in}-\bar{s}\leq x\leq s_{in}-\underline{s}$ $s+x=s_{in}$	(2.14) not fulfilled $\underline{s}\leq s\leq s_{in}$ $0\leq x\leq s_{in}-\underline{s}$	(2.14) not fulfilled $\underline{s}\leq s\leq s_{in}$ $0\leq x\leq s_{in}-\underline{s}$
$D< \mu(s_{in})$	impossible	impossible	(2.14) not fulfilled $\underline{s}\leq s\leq s_{in}$ $0\leq x\leq s_{in}-\underline{s}$

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