Analysis Of Dynamical Properties For Some Reaction Diffusion Systems

As an significant class of partial differential equations, reaction diffusion systems describe the development process and diffusion law of things and have the widespread application value in many fields, such as, physics, chemistry, biology, economics and en-gineering. Dynamical properties of reaction diffusion systems, such as, the equilibrium problems, Hopf bifurcation, Turing instability, describe the change rule of the whole sys-tem with the changes of parameters of the system, and have important theoretical value and practical significance. Therefore, based on the background of biology, we in this paper systematically analyze the dynamical properties of several important reaction dif-fusion systems.We firstly discuss the dynamical properties of spatial2-D Crowley-Martin predator-prey system with diffusion subject to the no-flux boundary condition. After analyzing the eigenvalues distribution of characteristic equations at the unique positive equilibrium, we obtain the conditions of the system undergoing Hopf bifurcation and Turing bifurca-tion. Using the lower and upper solutions methods, we obtain the global stability of the positive equilibrium.We study a Holling-Ⅱ predator-prey diffusive system with the prey harvesting and delay subject to the homogeneous Neumann boundary condition. After analyze the value of the prey harvesting term, we obtain the existence and stability of the non-negative equilibrium. Considering the delay as the branch parameter, we establish the existence of Hopf bifurcation around the positive equilibrium. By applying the normal form theory and the center manifold theory of the partial functional differential equation, we deduce the calculation formula which decides the Hopf bifurcation direction. Using the analysis method from Fister, we establish the optimal control strategy of the prey and the predator harvesting.We discuss the dynamical properties of the Holling-Ⅱ predator-prey diffusive model incorporating a prey refuge subject to the homogeneous Neumann boundary condition. We prove the global stability of the positive equilibrium by constructing the Lyapunov functional. Taking the prey refuge term as the bifurcation parameter and analyzing the corresponding characteristic equation, we obtain the existence of Hopf bifurcation occur-rence. With the help of the center manifold theory and normal form theory, we discuss the bifurcation properties, such as, the bifurcation direction, the stability of the bifurcation periodic solutions and the bifurcation cycle properties.We finally analyze the Hopf bifurcation existence and the bifurcation properties for N-dimensional Lotka-Volterra competition system with diffusion and delay subject to the homogeneous Dirichlet boundary condition. According to the implicit function theorem, we give the existence and the explicit expressions of the non-constant positive steady-states. Using the time delay as the bifurcation parameter, we establish the existence of Hopf bifurcation, the stability of non-constant positive steady-states and the calcu-lation formula of the bifurcation direction. As an application, we give an example of3-dimensional competition system and present existence computational result to illustrate or complement our mathematical findings.