Analysis Of Spatiotemporal Pattern Dynamics For Two Kinds Of Reaction Diffusion Systems

Investigating on dynamics of nonlinear system is a hot and difficult topic in basic and applied subjects all the time.In recent years,spatiotemporal pattern in reaction diffusion system have been paid much attention to and investigated to much extent.Bifurcation theory is important and effctive tools to handle this problem.In this thesis,under the background of a class of activator-inhibitor chemical reaction diffusion system and nonactivator-inhibitor Lotka-Volterra reaction diffusion system with nonlocal competitions,the complex spatiotemporal patterns of these two classes of models are investigated by using linear stability analysis,center manifold theory,the normal form reduction method and the functional and energy method.The main contents are as follows:Firstly,the Lengyel-Epstein system of the CIMA reaction with homogeneous Neumann boundary condition are considered.By taking feeding concentration b and diffusion coefficient ratio c as bifurcation parameters,and by analyzing the distribution of zeros of characteristic equations corresponding to the linearized system,the conditions for the existence of Turing,Hopf and Turing-Hopf bifurcations of original system are derived.Using center manifold theory and normal form method,the normal form restricted on center manifold up to third-order for the original system are obtained and much complex dynamical behaviors,such as spatial inhomogeneous periodic/quasi-periodic solutions and bistable/tristable phenomenon,are rigorously proved near the Turing-Hopf bifurcation point,where the normal form corresponding to the unfolding is degenerate case 2c of case VIIa.Secondly,in the weak competition case,a Lotka-Volterra system with both local and nonlocal intraspecific and interspecific competitions is studied,where nonlocal competitions depend on both spatial and temporal effects.Compared with the classical LotkaVolterra competition system,strengths of nonlocal intraspecific competitions have great effects on the global convergence of two constant semi-trivial equilibria and the coexistence equilibrium;when the attracting region is limited by the environment capacity of some species,the long time asymptotic behavior of the corresponding constant semitrivial equilibrium cannot be affected by the strength of local and nonlocal intraspecific competitions;The coexistence equilibrium becomes Turing unstable when the kernels are chosen as generally distributed delay functions in temporal and homogeneous in spatial and the nonlocal intraspecific competitions are suitably strong.Finally,based on results of the second part,the bifurcation problem of above LotkaVolterra competition model with effects of nonlocal competitions is studied,where the kernels are taken with spatially homogeneous and discrete delay functions.Conditions for the existence of Hopf,Turing,Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs are derived when the strength of nonlocal intraspecific competition is suitably large.By using center manifold theory,the normal form up to third-order near the codimension two Turing-Hopf bifurcation point for the original system is established,and it can be explicitly expressed by the parameters of the original system.By analyzing the third-order normal form,the existence of complex spatiotemporal patterns,such as the spatial homogeneous periodic orbit,a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits,is proved.