Analysis And Numerical Simulation Of Turing Instability Of Diffusive Biochemical Oscillators

Differential equations can be employed to describe the law of the change rate of molecular concentration with time in chemical reaction system.In the real cytoplasm liquid environment,the molecular concentration changes not only with time,but also in space.Turing explained mathematically that in reaction-diffusion system,the equilibrium state of space uniform stability will lose its stability under some conditions,and generate the pattern of space stationary state spontaneously,which is called Turing instability.In this paper,the condition of Turing instability of biochemical oscillators such as the Goodwin oscillator and Brusselator oscillator are investigated,and the results of theoretical analysis are verified by numerical simulation.This thesis consists of four chapters:The first chapter introduces the research background and preliminaries for this thesis,including Turing instability,one-dimensional reaction diffusion equation,two-dimensional reaction diffusion equation,and time fractional reaction diffusion equation.In Chapter 2,we mainly study the the Turing instability in the classic Goodwin reactiondiffusion model.According to the influence of different parameters on the oscillation mechanism of the classic Goodwin model,the system oscillates when the Hill coefficient n>8.The diffusion term is introduced into the Goodwin model.Studies have shown that,when n>8,the system appears to have a spatially uniform periodic oscillation,and the change of the diffusion coefficient will cause the spatially uniform periodic oscillation to become a spatially uneven periodic oscillation.Then We change the degradation mode of the products in the classical Goodwin oscillator,and adopt the Michaelis-Menten kinetics in degradation.With the qualitative analysis and numerical simulation,the Turing-Hopf bifurcation is observed.The third chapter mainly studies Turing instability in the Goodwin reaction-diffusion time-delay model and nonlinear degradation of product protein Goodwin reaction-diffusion time-delay model.The classical explicit finite difference method is mainly used for numerical simulation,and the critical delay of the Goodwin oscillator and the critical delay of the spatio-temporal distribution of the Goodwin reaction-diffusion system are obtained according to the Hopf bifurcation theory.Secondly,on the basis of determining the existence of Hopf bifurcation in the system,the periodicity,directionality,and stability of the Hopf bifurcation are analyzed and critical values are given according to the normal form theory and the central manifold theorem.In the last chapter,we mainly study the Turing instability of two-dimensional spatiotemporal genetic oscillators.Firstly,we study the pattern of predator-prey reaction-diffusion system under linear and nonlinear coupling with different parameters.Secondly,we study the pattern of Brusselator system under different parameters.By changing the coupling strength,a complex pattern is formed.For the two-dimensional space-time model,we use the diference method of alternating direction implicit scheme(ADI)for numerical simulation.By selecting different parameters,we find that after a long time of evolution,an uneven concentration distribution is observed in the uniform grid.In all,we,in this paper,we have proved the existence of Turing instability of the onedimensional Goodwin oscillator.Secondly,we have confirmed that one-dimensional delayed Goodwin oscillator can have Hopf bifurcation and Turing-Hopf bifurcation.Finally,we have studied Turing instability of the two-dimensional Lengyel-Epstein oscillator and the twodimensional Brusselator.The results of numerical simulation show that the system produce new pattern by coupled Lengyel-Epstein oscillators and the coupled Brusselators can produce more abundant patterns.