Effects of toxin delay on the dynamics of a phytoplankton-zooplankton model

This paper is concerned with a phytoplankton-zooplankton model with toxin delay. It is shown that the positive equilibrium of the system is stable when the toxin delay is not included. However, we find that the incorporation of a discrete toxin delay not only can destabilize the positive equilibrium of the system but also can cause a Hopf bifurcation at the positive equilibrium as it crosses some critical values. In particular, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. To verify the obtained conditions, a special numerical example is also included. These results in this work demonstrate that the toxin delay plays an important role on deriving the complex dynamics.