On the permanence and periodic solutions of a plankton system with impulses and diffusion

In this paper, given the sudden changes of external environment and seasonal variations of climate, we investigate the non-toxic phytoplankton, toxin-producing phytoplankton and zooplankton model with periodic impulses and spatial diffusion. The sufficient conditions for ultimate boundedness of solutions and permanence of system are established by using theory of impulsive differential equations, comparison principle, upper-lower solution method and inequality techniques. Moreover, the existence and uniqueness of asymptotically stable periodic solution are studied with the help of auxiliary function. It is shown that the plankton populations will evolve periodically with time, provided that the system is permanent.