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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-periodic solutions are established by constructing Poincaré map and using the Whyburn Lemma and Leray-Schauder degree theory. Furthermore, according to the implicit function theorem, the uniqueness of the positive periodic solution is obtained when delay <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is small enough. Finally, the necessary and sufficient conditions for the extinction of microorganism species are established.</p>