The Spatiotemporal Dynamic Research Of Several Reaction-diffusion Systems

This paper mainly studies the spatiotemporal dynamics of several reaction-diffusion systems. The relevant problems in some hot fields are deeply discussed and a series of important results are obtained.In chapter 1, the introduction part briefly introduces the research background and research status of reaction-diffusion system. In addition, some basic definitions and important lemmas, which are frequently used in the following chapters, are also presented.In chapter 2, the dynamics of a diffusive nutrient-algae model based on the Sanyang wetland are investigated. Based on mathematical theories and actual biological significance, an ecological model with diffusion, including nutrient and algae, is established. Mathematical theoretical studies have considered the positivity, boundedness and local stability of the equilibria in the reaction-diffusion model. What’s more, global stability of the unique interior equilibrium has also been discussed. Furthermore, numerical results verify the correctness of theoretical results. How the input rate of nutrients flowing into the water affects the spatiotemporal dynamics of the system is studied based on numerical results. Research results indicate that the input rate of nutrients has an important influence on the density and spatial distribution of algae populations.In chapter 3, the spatiotemporal dynamics of a plankton system are investigated. The stability of equilibria and the existence of limit cycles for the reduced ordinary differential equation system are discussed by qualitative analysis. Moreover, the global existence, the boundedness of solutions and stability of constant steady states are investigated in detail. And the conditions of existence and nonexistence of nonconstant positive steady states have also been obtained. Furthermore, numerical simulations verify the feasibility of theoretical results, which indicates that diffusion can greatly affect dynamic behaviors of plankton system.In chapter 4, the spatiotemporal dynamics and bifurcation of a diffusive plant-wrack system are studied in detail. Firstly, an analytic technique is used to study dissipation, persistence properties, local and global stability of constant steady states, existence and nonexistence of nonconstant positive steady states. Moreover, the upper and lower solution method is applied to prove global stability of the positive constant steady state. Secondly, periodic solutions and Turing patterns of the system are investigated by means of bifurcation analysis. Finally, based on the previous theoretical analysis, there exist periodic solutions and nonconstant steady-state solutions in the system. And numerical simulations illustrate the theoretical results.