Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants

Tongqian Zhang; Ning Gao; Tengfei Wang; Hongxia Liu; Zhichao Jiang; Tongqian Zhang; Ning Gao; Tengfei Wang; Hongxia Liu; Zhichao Jiang

doi:10.3934/mbe.2020010

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 179-201. doi: 10.3934/mbe.2020010

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Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China
3.
Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang 065000, China

Received: 26 April 2019 Accepted: 16 September 2019 Published: 29 September 2019

In this paper, a mathematical model for microbial treatment in livestock and poultry sewage is proposed and analyzed. We consider periodic addition of microbial flocculants to treat microorganisms such as Escherichia coli in sewage. Different from the traditional models, a class of composite dynamics models composed of impulsive differential equations is established. Our aim is to study the relationship between substrate, microorganisms and flocculants in sewage systems as well as the treatment strategies of microorganisms. Precisely, we first show the process of mathematical modeling by using impulsive differential equations. Then by using the theory of impulsive differential equations, the dynamics of the model is investigated. Our results show that the system has a microorganismsextinction periodic solution which is globally asymptotically stable when a certain threshold value is less than one, and the system is permanent when a certain threshold value is greater than one. Furthermore, the control strategy for microorganisms treatment is discussed. Finally, some numerical simulations are carried out to illustrate the theoretical results.
- sewage treatment,
- control strategy,
- impulsive differential equation,
- globally asymptotical stability,
- permanence
Citation: Tongqian Zhang, Ning Gao, Tengfei Wang, Hongxia Liu, Zhichao Jiang. Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 179-201. doi: 10.3934/mbe.2020010

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Abstract

In this paper, a mathematical model for microbial treatment in livestock and poultry sewage is proposed and analyzed. We consider periodic addition of microbial flocculants to treat microorganisms such as Escherichia coli in sewage. Different from the traditional models, a class of composite dynamics models composed of impulsive differential equations is established. Our aim is to study the relationship between substrate, microorganisms and flocculants in sewage systems as well as the treatment strategies of microorganisms. Precisely, we first show the process of mathematical modeling by using impulsive differential equations. Then by using the theory of impulsive differential equations, the dynamics of the model is investigated. Our results show that the system has a microorganismsextinction periodic solution which is globally asymptotically stable when a certain threshold value is less than one, and the system is permanent when a certain threshold value is greater than one. Furthermore, the control strategy for microorganisms treatment is discussed. Finally, some numerical simulations are carried out to illustrate the theoretical results.

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