Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

In this paper, a delayed phytoplankton-zooplankton system with the coefficient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.