A diffusive plankton model with toxin-producing phytoplankton was considered.For the model without diffusion,the existence and uniqueness of the positive equilibrium point were obtained.Furthermore,it was proved that the equilibrum point is globally asymptotically stable.For the model with spatial diffusion,the condition ensuring the persistence of the corresponding model was obtained.By constructing Lyapunov function,a sufficient condition of the global stability of the positive equilibrium was proved.Numerical simulations were presented to verify the theoretical results.