Study On The Dynamic Problems Of Two Classes Of Ecological System With Spatial Diffusion Effect

In this paper,we mainly study on the dynamic problems of two classes of ecological system with spatial diffusion effect.We deeply investigate some nonlinear problems in the hot fields and derive a series of important results.In chapter 1,we mainly introduce the research background and status of ecological system with spatial diffusion effect,and some basic concepts and important lemmas,which would be the theoretical basis in the following chapters,are also introduced.In chapter 2,based on monitoring date of algae blooms and planktonic ecological theory,an ecological dynamical system with inhibitory effect has been established,and some dynamical problems have been investigated.Theoretical studies mainly have considered the positivity and boundedness of solutions,the existence and stability of equilibria,and the critical conditions for the Hopf bifurcation and Turing instability.Numerical simulations mainly have verified the effectiveness and feasibility about the theoretical derivation.The obtained results could be very helpful to deepen and expand the research of nonlinear dynamics of such plankton ecosystem.In chapter 3,we investigate spatiotemporal dynamics of a diffusive plant-wrack system in freshwater marshes,and some dynamical problems have been investigated.Firstly,we establish conditions for the existence of the Turing-Hopf bifurcation through the distribution of eigenvalues.Then,we calculate the normal form near the bifurcation point by the center manifold reduction and normal form theorem.Furthermore,we derive the detailed dynamical classification near the bifurcation point by analyzing the normal form.Finally,a series of numerical simulations are employed to illustrate theoretical results.It is interesting to point that,under appropriate parameters near the bifurcation point,we observe the coexistence of stable spatially inhomogeneous steady states and stable spatially homogeneous periodic solution.These results may be helpful to understand the diversity of spatiotemporal patterns in such plant-wrack model.In chapter 4,we analyse the investigated results,which are concluded and prospected.