A general chemostat model with second-order Poisson jumps: asymptotic properties and application to industrial waste-water treatment

Yassine Sabbar; José Luis Diaz Palencia; Mouhcine Tilioua; Abraham Otero; Anwar Zeb; Salih Djilali; Yassine Sabbar; José Luis Diaz Palencia; Mouhcine Tilioua; Abraham Otero; Anwar Zeb; Salih Djilali

doi:10.3934/math.2023656

AIMS Mathematics

2023, Volume 8, Issue 6: 13024-13049. doi: 10.3934/math.2023656

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Research article

A general chemostat model with second-order Poisson jumps: asymptotic properties and application to industrial waste-water treatment

1.
MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P. O. Box 509, Errachidia 52000, Morocco
2.
Department of Information Technology, Escuela Politecnica Superior, Universidad San Pablo-CEU, CEU Universities, Campus Monteprincipe, Boadilla del Monte, Madrid 28668, Spain
3.
Department of Mathematics and Education, Universidad a Distancia de Madrid, Spain
4.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Khyber Pakhtunkhwa, Pakistan
5.
Faculty of Exact Sciences and Informatics, Mathematic Department, Hassiba Benbouali University, Chlef, Algeria

Received: 29 January 2023 Revised: 02 March 2023 Accepted: 06 March 2023 Published: 03 April 2023
MSC : 34D10, 34E10, 34H20, 37A30

A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter $ \mathfrak{C}^{\star}_0 $. If $ \mathfrak {C}^{\star}_0 $ is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If $ \mathfrak {C}^{\star}_0 $ is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction.
- chemostat,
- Poisson jumps,
- dynamics,
- ergodicity,
- waste-water treatment
Citation: Yassine Sabbar, José Luis Diaz Palencia, Mouhcine Tilioua, Abraham Otero, Anwar Zeb, Salih Djilali. A general chemostat model with second-order Poisson jumps: asymptotic properties and application to industrial waste-water treatment[J]. AIMS Mathematics, 2023, 8(6): 13024-13049. doi: 10.3934/math.2023656

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Abstract

A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter $ \mathfrak{C}^{\star}_0 $. If $ \mathfrak {C}^{\star}_0 $ is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If $ \mathfrak {C}^{\star}_0 $ is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction.

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