Open AccessPeriodic solution and extinction in a periodic chemostat model with delay in microorganism growth
Author(s) -
Ningning Ye,
Zengyun Hu,
Zhidong Teng
Publication year - 2022
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2022022
Subject(s) - chemostat , lemma (botany) , uniqueness , mathematics , extinction (optical mineralogy) , function (biology) , pure mathematics , combinatorics , mathematical analysis , biology , ecology , bacteria , paleontology , genetics , poaceae , evolutionary biology
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 \begin{document}$ \omega $\end{document} -periodic solutions are established by constructing Poincaré map and using the Whyburn Lemma and Leray-Schauder degree theory. Furthermore, according to the implicit function theorem, the uniqueness of the positive periodic solution is obtained when delay \begin{document}$ \tau $\end{document} is small enough. Finally, the necessary and sufficient conditions for the extinction of microorganism species are established.