# American Institute of Mathematical Sciences

August  2019, 24(8): 3591-3614. doi: 10.3934/dcdsb.2018280

## 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

Received  April 2018 Revised  June 2018 Published  October 2018

Fund Project: 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.

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.

Citation: Tomás Caraballo, Maria-José Garrido-Atienza, Javier López-de-la-Cruz, Alain Rapaport. Modeling and analysis of random and stochastic input flows in the chemostat model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3591-3614. doi: 10.3934/dcdsb.2018280
##### References:

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##### References:
Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\beta = 2$
Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\nu = 0.5$
Realizations of the perturbed dilution rate, $\underline{s}$ and $\bar{s}$
Attracting set $\widehat{B}_0$
$\mbox{Absorbing set } B_\varepsilon(\omega)$
$\mbox{Absorbing set }{\color{blue}{ B_0(\omega)}}$
Persistence of the species in the random chemostat model
Extinction of the species in the random chemostat model
Stochastic chemostat model. Extinction (left) and persistence (right)
Comparison in case of extinction
Comparison in case of persistence
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 ExtinctionProposition 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}$$0\leq x\leq s_{in}-\underline{s}  s_{in}>\bar{s} s_{in}=\bar{s} s_{in}< \bar{s} D>\mu(s_{in}) impossible impossible ExtinctionProposition 2.2\{(s_{in}, 0)\} D=\mu(s_{in}) Persistence Theorem 2.3\underline{s}\leq s\leq \bar{s}$$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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