Hopf bifurcation and bistability of a nutrient–phytoplankton–zooplankton model

In this paper we consider a nutrient–phytoplankton–zooplankton model in aquatic environment and study its global dynamics. The existence and stability of equilibria are analyzed. It is shown that the system is permanent as long as the coexisting equilibrium exists. The discontinuous Hopf and classical Hopf bifurcations of the model are analytically verified. It is shown that phytoplankton bloom may occur even if the input rate of nutrient is low. Numerical simulations reveal the existence of saddle-node bifurcation of nonhyperbolic periodic orbit and subcritical discontinuous Hopf bifurcation, which presents a bistable phenomenon (a stable equilibrium and a stable limit cycle).