Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication

In this paper, we derive and analyze a mathematical model for the interactions between phytoplankton and zooplankton in a periodic environment, in which the growth rate and the intrinsic carrying-capacity of phytoplankton are changing with respect to time and nutrient concentration. A threshold value: "Predator's average growth rate" is introduced and it is proved that the phytoplankton-zooplankton ecosystem is permanent (both populations survive cronically) and possesses a periodic solution if and only if the value is positive. We use TP (Total Phosphorus) concentration to mark the degree of eutrophication. Based on experimental data, we fit the growth rate function and the environmental carrying capacity function with temperature and nutrient concentration as independent variables. Using measured data of temperature on water bodies we fit a periodic temperature function of time, and this leads the growth rate and intrinsic carrying-capacity of phytoplankton to be periodic functions of time. Thus we establish a periodic system with TP concentration as parameter. The simulation results reveal a high diversity of population levels of the ecosystem that are mainly sensitive to TP concentration and the death-rate of zooplankton. It illustrates that the eruption of algal bloom is mainly resulted from the increasing of nutrient concentration while zooplankton only plays a role to alleviate the scale of algal bloom, which might be used to explain the mechanism of algal bloom occurrence in many natural waters. What is more, our results provide a better understanding of the traditional manipulation method.