Dynamics Of Several Kinds Of Diffusive Plankton Systems With Delay

Plankton is an important element of water ecosystem. Phytoplankton is the primary producer of water ecosystem that generate half of the oxygen and absorb half of the carbon dioxide of the world by photosynthesis. It is evidently important to understand the dynamics of plankton ecosystems. This dissertation mainly analyzes the dynamical properties of several kinds plankton model with delay and diffusion, obtains some dynamical properties of plankton system such as the stability of equilibrium, Hopf bifurcation,Turing instability and persistent, and give some appropriate explanation in the biological sense. The main work is as follows:(I) The dynamics of a plankton system with toxic substances effect is studied. Existence and priori bound of solution for system with instant toxin are shown. For the system with discrete delay in toxin term, sufficient condition to ensure the globally asymptotically stability of the axial equilibrium is obtained, and the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. The properties of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations.For the system with nonlocal delays in the toxin term, sufficient condition to ensure the asymptotically stability of the positive equilibrium is obtained. Finally, some numerical simulations are carried out for illustrating the theoretical results. The results show that the time delay of the effect of the toxin can affect the dynamical pattern of the system.(II) A minimal model for a plankton ecosystem with spatial diffusion of plankton and the delay of the maturation period of zooplankton is considered. Mathematically, the global stability of the boundary equilibrium is proved. This shows that when the nutrient level is sufficient low, the zooplankton population collapses and the phytoplankton population reaches its carrying capacity. Under an eutrophic condition, we prove that the system has a unique coexistent homogeneous equilibrium, and the equilibrium density of phytoplankton increases while the equilibrium density of herbivorous zooplankton decreases as the fish predation rate on herbivorous zooplankton is increasing. A detailed mathematical analysis about the the existence and properties of Hopf bifurcations is carried out, and we find a Hopf bifurcation curve relation between the delay parameter of the maturation period of herbivorous zooplankton and the fish predation rate on zooplankton.This result reveals that the predation of fish on zooplankton can restrain the occurrence of the oscillation of the system. Finally, we give some numerical results which are consistent with the theoretical analysis.(III) A plankton system with delay and quadratic closure term is investigated. The results show that diffusion, delay and the choice of closure term can influence the dynamics of model. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system with quadratic closure term, but the system with a linear closure term does not have persistence. The results also show that the boundary equilibrium of the system with quadratic closure term is always unstable and the positive equilibrium always exists, but the boundary equilibrium of the system with linear closure term is stable under some conditions and the positive equilibrium does not exist in this situation. The conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.