An analysis approach to permanence of a delay differential equations model of microorganism flocculation

Songbai Guo; Jing-An Cui; Wanbiao Ma

doi:10.3934/dcdsb.2021208

Article Contents

2022, Volume 27, Issue 7: 3831-3844. Doi: 10.3934/dcdsb.2021208

This issue Previous Article Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control Next Article On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes

An analysis approach to permanence of a delay differential equations model of microorganism flocculation

1.
School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

^* Corresponding author
^* Corresponding author

Received: February 2021

Revised: July 2021

Published: July 2022

This work is supported in part by the National Natural Science Foundation of China (Nos. 11901027, 11871093 and 11971055), the Scientific Research Project of Beijing Municipal Education Commission (No. KM201910016001), the Pyramid Talent Training Project of BUCEA (JDYC20200327) and the Bill & Melinda Gates Foundation (INV-005834)

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Abstract

In this paper, we develop a delay differential equations model of microorganism flocculation with general monotonic functional responses, and then study the permanence of this model, which can ensure the sustainability of the collection of microorganisms. For a general differential system, the existence of a positive equilibrium can be obtained with the help of the persistence theory, whereas we give the existence conditions of a positive equilibrium by using the implicit function theorem. Then to obtain an explicit formula for the ultimate lower bound of microorganism concentration, we propose a general analysis method, which is different from the traditional approaches in persistence theory and also extends the analysis techniques of existing related works.

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Mathematics Subject Classification: Primary: 34K09, 34K25, 70K42; Secondary: 74G55.

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