Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process

Amer Hassan Albargi; Miled El Hajji; Amer Hassan Albargi; Miled El Hajji

doi:10.3934/mbe.2023283

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 6591-6611. doi: 10.3934/mbe.2023283

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Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process

Amer Hassan Albargi ¹,
Miled El Hajji ^{2,3
,
,}

1.
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2.
ENIT-LAMSIN, BP. 37, 1002 Tunis-Belvédère, Tunis El Manar University, Tunisia
3.
Department of Mathematics, Faculty of Science, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia

Academic Editor: Gilberto González-Parra

Received: 06 November 2022 Revised: 17 January 2023 Accepted: 27 January 2023 Published: 02 February 2023

In this research paper, we presented a four-dimensional mathematical system modeling the anaerobic mineralization of phenol in a two-step microbial food-web. The inflowing concentrations of the hydrogen and the phenol are considered in our model. We considered the case of general class of nonlinear growth kinetics, instead of Monod kinetics. Due to some conservative relations, the proposed model was reduced to a two-dimensional system. The stability of the steady states was carried out. Based on the species growth rates and the three main operating parameters of the model, represented by the dilution rate and input concentrations of the phenol and the hydrogen, we showed that the system can have up to four steady states. We gave the necessary and sufficient conditions ensuring the existence and the stability for each feasible equilibrium state. We showed that in specific cases, the positive steady state exists and is stable. We gave numerical simulations validating the obtained results.
- anaerobic digestion,
- chemostat,
- phenol mineralisation,
- food-web,
- local stability,
- global stability,
- optimal strategy
Citation: Amer Hassan Albargi, Miled El Hajji. Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6591-6611. doi: 10.3934/mbe.2023283

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Abstract

In this research paper, we presented a four-dimensional mathematical system modeling the anaerobic mineralization of phenol in a two-step microbial food-web. The inflowing concentrations of the hydrogen and the phenol are considered in our model. We considered the case of general class of nonlinear growth kinetics, instead of Monod kinetics. Due to some conservative relations, the proposed model was reduced to a two-dimensional system. The stability of the steady states was carried out. Based on the species growth rates and the three main operating parameters of the model, represented by the dilution rate and input concentrations of the phenol and the hydrogen, we showed that the system can have up to four steady states. We gave the necessary and sufficient conditions ensuring the existence and the stability for each feasible equilibrium state. We showed that in specific cases, the positive steady state exists and is stable. We gave numerical simulations validating the obtained results.

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