Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth

Ningning Ye; Zengyun Hu; Zhidong Teng

doi:10.3934/cpaa.2022022

Article Contents

2022, Volume 21, Issue 4: 1361-1384. Doi: 10.3934/cpaa.2022022

This issue Previous Article The duality method for mean field games systems Next Article Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth

1.
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China
2.
State Key Laboratory of Desert and Oasis Ecology, Xinjiang Institute of Ecology and Geography, Chinese Academy of Sciences, Urumqi 830011, China

^* Corresponding author
^* Corresponding author

Received: August 31, 2021

Revised: November 30, 2021

Early access: January 2022

Published: April 2022

This work was supported by the Natural Science Foundation of China (Grant No. 11771373, 11861065)

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Abstract

In this paper, the periodic solution and extinction in a periodic chemostat model with delay in microorganism growth are investigated. The positivity and ultimate boundedness of solutions are firstly obtained. Next, the necessary and sufficient conditions on the existence of positive $ \omega $-periodic solutions are established by constructing Poincaré map and using the Whyburn Lemma and Leray-Schauder degree theory. Furthermore, according to the implicit function theorem, the uniqueness of the positive periodic solution is obtained when delay $ \tau $ is small enough. Finally, the necessary and sufficient conditions for the extinction of microorganism species are established.

Keywords:

Mathematics Subject Classification: Primary: 92D25, 34K13, 34D20.

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Figure 1. (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solution of model (1.2) with initial function $ (u(\theta),v(\theta)) = (6,7) $ for all $ \theta\in[-0.15,0] $ converges to a positive periodic solution as $ t\rightarrow \infty $

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Figure 2. (a)-(b): the solutions of model (1.2) with the delay $ \tau = 0 $ and initial points $ (3.5,4.3) $, $ (1.5,1.7) $ and $ (0.5,0.7) $ converge to the unique positive periodic solution $ (u_0^*(t),v_0^*(t)) $ as $ t\rightarrow \infty $; (c): the numerical simulation of $ w^*(t) $; (d)-(e): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((2.1,2.5), (1.5,1.7),(1.1,1.3)) $ for all $ \theta\in[-0.2,0] $ converge to the unique positive periodic solution $ (u^*(t),v^*(t)) $ as $ t\rightarrow \infty $

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Figure 3. (a)-(b): the solutions of model (1.2) with the delay $ \tau = 0 $ and initial points $ (1.1,1.3) $, $ (1.5,1.7) $ and $ (0.5,0.8) $ converge to the unique positive periodic solution $ (u_0^*(t),v_0^*(t)) $ as $ t\rightarrow \infty $; (c): the numerical simulation of $ w^*(t) $; (d)-(e): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((1,2),(1.1,1.5),(1.4,1.9)) $ for all $ \theta\in[-0.1,0] $ converge to the unique positive periodic solution $ (u^*(t),v^*(t)) $ as $ t\rightarrow \infty $

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Figure 4. (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((2.8,1.7),(1.4,2.5)) $ for all $ \theta\in[-0.2,0] $ converge to $ (w^*(t),0) $ as $ t\to\infty $

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Figure 5. (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((0.8,1),(0.7,2),(1,0.6)) $ for all $ \theta\in[-0.3,0] $ converge to $ (w^*(t),0) $ as $ t\to\infty $

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