Dynamical behavior of a delay differential equation system on toxin producing phytoplankton and zooplankton interaction

In this paper, a toxin producing phytoplankton–zooplankton system with the delay is investigated. Firstly, the nonnegativity and boundedness of solutions are given. Then the local and global asymptotic stabilities of the boundary equilibrium are investigated, and the existence of local Hopf bifurcations is established as the delay crosses a threshold value at the positive equilibrium. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. In addition, the impacts of the toxic substances are also investigated. At last, an explicit algorithm is derived for the stability and direction of the bifurcating periodic solution by using center manifold theory and normal form method.