Global dynamics of a delayed air pollution dynamic model with saturated functional response and backward bifurcation

Abstract

<p style='text-indent:20px;'>This paper studies the global dynamics of a delayed air pollution dynamic model with saturated functional response. This model exhibits forward<inline-formula><tex-math id="M1">\begin{document}$ / $\end{document}</tex-math></inline-formula>backward bifurcation. Through some iterative analysis techniques and constructing appropriate Lyapunov functionals, the global stability of the equilibria and the permanence of the model are obtained. For the case of forward bifurcation, it is shown that the boundary equilibrium is globally asymptotically stable (globally attractive) if <inline-formula><tex-math id="M2">\begin{document}$ R_{0}&lt;1 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ R_{0} = 1 $\end{document}</tex-math></inline-formula>), the positive equilibrium is globally asymptotically stable if <inline-formula><tex-math id="M4">\begin{document}$ R_{0}&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ ab\leq c $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M6">\begin{document}$ R_{0}&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ ab&gt;c $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ n&lt;n^{*} $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M9">\begin{document}$ R_{0}&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ ab&gt;c $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ n\geq n^{*} $\end{document}</tex-math></inline-formula> and the delay is small (the delay can change the stability of the positive equilibrium if <inline-formula><tex-math id="M12">\begin{document}$ R_{0}&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ ab&gt;c $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$ n&gt;n^{*} $\end{document}</tex-math></inline-formula>). For the case of backward bifurcation, it is shown that the boundary equilibrium is globally asymptotically stable if <inline-formula><tex-math id="M15">\begin{document}$ R_{0}&lt;\omega $\end{document}</tex-math></inline-formula>, the model has bistable equilibria (the boundary equilibrium and one positive equilibrium are locally asymptotically stable, and the other positive equilibrium is unstable) if <inline-formula><tex-math id="M16">\begin{document}$ \omega&lt;R_{0}&lt;1 $\end{document}</tex-math></inline-formula>, the positive equilibrium is globally asymptotically stable if <inline-formula><tex-math id="M17">\begin{document}$ R_{0}&gt;1 $\end{document}</tex-math></inline-formula>. Our results largely improve existing results.</p>