Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

Abstract

<p style='text-indent:20px;'>We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> of the zooplankton and the time <inline-formula><tex-math id="M2">\begin{document}$ \tau_1 $\end{document}</tex-math></inline-formula> required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay <inline-formula><tex-math id="M3">\begin{document}$ \tau_1 $\end{document}</tex-math></inline-formula>, the gestation delay <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> may lead to stability switches of the positive equilibrium. Then fixed <inline-formula><tex-math id="M5">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> in stable interval, the effect of <inline-formula><tex-math id="M6">\begin{document}$ \tau_1 $\end{document}</tex-math></inline-formula> is investigated and find <inline-formula><tex-math id="M7">\begin{document}$ \tau_1 $\end{document}</tex-math></inline-formula> can also cause the oscillation of system. Specially, when <inline-formula><tex-math id="M8">\begin{document}$ \tau = \tau_1 $\end{document}</tex-math></inline-formula>, under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in <inline-formula><tex-math id="M9">\begin{document}$ (\tau, \tau_1) $\end{document}</tex-math></inline-formula> plane. When choosing <inline-formula><tex-math id="M10">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses.</p>