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

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 \begin{document}$ \tau $\end{document} of the zooplankton and the time \begin{document}$ \tau_1 $\end{document} required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay \begin{document}$ \tau_1 $\end{document} , the gestation delay \begin{document}$ \tau $\end{document} may lead to stability switches of the positive equilibrium. Then fixed \begin{document}$ \tau $\end{document} in stable interval, the effect of \begin{document}$ \tau_1 $\end{document} is investigated and find \begin{document}$ \tau_1 $\end{document} can also cause the oscillation of system. Specially, when \begin{document}$ \tau = \tau_1 $\end{document} , under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in \begin{document}$ (\tau, \tau_1) $\end{document} plane. When choosing \begin{document}$ \tau $\end{document} 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.