Stability and persistence analysis of a microorganism flocculation model with infinite delay

Jiaxin Nan; Wanbiao Ma; Jiaxin Nan; Wanbiao Ma

doi:10.3934/mbe.2023480

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 6: 10815-10827. doi: 10.3934/mbe.2023480

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Theory article Special Issues

Stability and persistence analysis of a microorganism flocculation model with infinite delay

Jiaxin Nan ,
Wanbiao Ma ^,

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Academic Editor: Xiang-Sheng Wang

Received: 31 December 2022 Revised: 27 March 2023 Accepted: 31 March 2023 Published: 19 April 2023

In this paper, we study the global stability and persistence of a microorganism flocculation model with infinite delay. First, we make a complete theoretical analysis on the local stability of the boundary equilibrium (microorganism-free equilibrium) and the positive equilibrium (microorganism co-existent equilibrium), and give a sufficient condition for the global stability of the boundary equilibrium (applicable to the forward bifurcation and the backward bifurcation). Then, for the persistence of the model, we present an explicit estimate of the eventual lower bound of any positive solution for which only the parameter threshold $ R_0 > 1 $ is required. The obtained results extend some of the conclusions of the existing literatures on the case of discrete time delay.
- microorganism flocculation,
- infinite delay,
- persistence,
- local stability,
- global stability
Citation: Jiaxin Nan, Wanbiao Ma. Stability and persistence analysis of a microorganism flocculation model with infinite delay[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10815-10827. doi: 10.3934/mbe.2023480

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Abstract

In this paper, we study the global stability and persistence of a microorganism flocculation model with infinite delay. First, we make a complete theoretical analysis on the local stability of the boundary equilibrium (microorganism-free equilibrium) and the positive equilibrium (microorganism co-existent equilibrium), and give a sufficient condition for the global stability of the boundary equilibrium (applicable to the forward bifurcation and the backward bifurcation). Then, for the persistence of the model, we present an explicit estimate of the eventual lower bound of any positive solution for which only the parameter threshold $ R_0 > 1 $ is required. The obtained results extend some of the conclusions of the existing literatures on the case of discrete time delay.

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