Dynamic Behavior Of Nutrient-phytoplankton-zooplankton Model

Plankton refers to organisms that are living in water bodies(such as oceans, lakes,rivers and ponds) freely drifting and weakly mobile. Plant forms of plankton community are known as phytoplankton, they serve as the basic food source and occupy the first trophic level of all aquatic food chains. Animals in the plankton community are known as zooplankton. They consume phytoplankton which are their most favourable food source.Phytoplankton are not only the basis for all aquatic food chains, but also they do huge services for our earth by supplying the essential oxygen and absorbing the harmful carbon dioxide which contributes to global warming. In addition to these benefits phytoplankton act as the biological indicators of water quality. During the recent years, the problems of zooplankton-phytoplankton systems have been discussed by many authors. These systems can exhibit rich dynamic behavior, such as stability of equilibria, Hopf bifurcation, global stability and so on. However, the importance of nutrients to the growth of plankton leads to explicit incorporation of nutrients concentrations in the phytoplanktonzooplankton models, therefore, a better understanding of mechanisms that determine the plankton is to consider plankton-nutrient interaction models. The work mainly focused on the study of a phytoplankton-zooplankton interaction model with harvesting and delay, where phytoplankton is affected by an external toxic substance, a one-phytoplankton two-zooplankton model with harvesting and a nutrient-plankton system with delayed nutrient cycling. This article mainly discusses the positivity, boundedness, the stability of the solution.The main contents in this paper can be summarized as follows:The first section is introduction, in which we present research background, purpose and significance of the nutrient-phytoplankton-zooplankton model, given the nutrientphytoplankton-zooplankton model research present situation and the results. Finally the organization of this paper is presented.In Section 2, we mainly discussed the positively?boundedness of the solution and existence of coexistence equilibrium of a toxic-phytoplankton-zooplankton model. Stability criteria of the model is analyzed both from local and global point of view. Then we get existence conditions of Hopf bifurcation.In Section 3, we mainly discussed the positively?boundedness of the solution and existence of nonnegative equilibria of a zooplankton-phytoplankton model. Meanwhile,we study the local stability of the model. By using the method of Lyapunov function, we get the sufficient conditions for the global asymptotic stability of system.In Section 4, we discussed the positively?boundedness of the solution and existence of coexistence equilibrium of a nutrient-plankton system with delay. And, we proved local stability and global stability of nonnegative equilibria by using Routh-Hurwitz criterion and the Lyapunov-LaSalle's invariance principle, respectively. Using time delay as a bifurcation parameter, we obtain that the model undergoes a Hopf bifurcation.In Section 5, we do some discussions and conclusions about we have study on this paper.