Bifurcation And Spatial Pattern Formation With Some Autocatalytic Reaction And Diffusion Systems

Cubic autocatalator reaction model and the higher autocatalator reaction model arc two kinds of the most significant mathmatical models in chemical biology.Autocatalator reaction model is the nuclear contents of pattern dynamics. This model plays a very important role in DNA reproducing and researching chemical oscillation , chemical wave, pattern.In the light of the recent work on the above two kinds of chemical reaction models, we mainly using the theories of linear analysis and nonlinear partial differential equations, especially those of reaction diffusion equations. we have systematically studied the dynamical behavior of different models in two systems, such as stability of the steady states ,bifurcation solutions, pattern. The tools used here include linear theory, weakly nonlinear analysis , Fredholm theory and so on. The main contents and results in this dissertation are as follows :A reaction diffusion system based on the higher autocatalator, with the reaction taking place inside a closed vessel, is considered. Under suitable conditions, we examine the local stability of the steady state via asymptotic approximations and show that only when the diffusion coefficient A is sufficient small, two types of patterns occur, standing-wave patterns arising out of Hopf bifurcation, together with steady-wave patterns arising out of Pitchfork bifurcation, each pattern is shown to be partially stable to small disturbances with small spacial wave number than its own.A reaction diffusion system with different diffusion coefficient based on the highter autocatalytic, within a closed region, is considered. The stability of the steady state (u, v) = (μ*,μ) is discussed first by the linearized theory. It is shown that a necessary condition for the bifurcation of this steady state to stable spatially nonuniform solutions is that the parameterD(=λ_b/λ_a) < ((?)-1)~2/n-1, whereλ_a,λ_b are the diffusion coefficients of chemical species A and autocatalytic B. The nature of the spatially nonuniform solutions close to their bifurcation points is analyzed from a weakly nonlinear analysis.Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations to spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.Spatiotemporal structures arising in two identical cells, each governed by arbitrary orders autocatalator kinetics and coupled via the diffusive interchange of a reactant, are discussed. The stability of two homogeneous steady states is obtained by the linearized theory. By studying the linearized equations, it is found that two steady states, each of the uncoupled and coupled system, may give rise to the possibility of bifurcations to spatially nonuniform pattern forms. Further information about Turing bifurcation solutions close to these bifurcation points are obtained by weakly nonlinear theory. It is seen that the coupling leads to bifurcations not present in the uncoupled system which give rise to locally stable nonuniform pattern forms. Finally the stability of the equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches about small coupled system with 0 <α<< 1 and large coupling forα>> 1.Two dimensional Turing patterns characteristic is considered for a reaction diffusion system based on the cubic autocatalator,A+2B→3B, B→C, with the reaction taking place within a closed two dimensional region. The linear stability of the steady state (a,b) = (1/μ,μ) (where a and b are the dimensionless concentrations of reactant A and autocalayst B andμis a parameter representing the initial concentration of the precursor P), is discussed firstly by the linearized theory. Proceeding, the possibility of occurrence of two dimensional Turing patterns consisting of rhombic arrays of rectangles and hexagonal is deduced by performing the appropriate weakly nonlinear stability analysis of the homogeneous solution and amplitude functions. Finally the Landau constants and the stability of the equilibrium points of the amplitude equation are determined.