Dynamical Analysis In Some Chemical Reaction Models With Delay And Diffusion

Chemical reaction is a common phenomenon in nature.Study on the dynamics of chemical reaction models can help us to understand the mechanism and evolution of the reaction process,and to predict the trend in the development of the reactants involved in the reaction.The dissertation mainly investigates the dynamics of chemical reaction systems with delay and diffusion,including the stability of constant steady state solution,Turing instability,steady state bifurcation,the existence as well as the properties of Hopf bifurcation.The main work is as follows:(?)A bimolecular autocatalytic model with delayed feedback and homogeneous Neumann boundary conditions is established,and the influence of diffusion and delayed feedback on the dynamics is investigated.By discussing the distribution of the eigenvalues,the sufficient conditions to guarantee the occurrence of Turing instability are obtained,the existence of Hopf bifurcation induced by delay is proven.Finally,the properties of Hopf bifurcation are investigated by utilizing the center manifold theory and normal form theory,and some numerical simulations are given to illustrate our theoretical results.Our results suggest that Turing instability occurs when the diffusivity of inhibitor is larger than that of activator.In addition,under certain conditions,delayed feedback can break the stability of the constant steady state solution,and induce the occurrence of periodic solutions.Particularly,when the feedback intensity is small,the stability switch occurs by selecting suitable delay.In this case,delayed feedback can stabilize the unstable constant steady state solution.(?)An arbitrary order autocatalysis model with delayed feedback subject to homogeneous Neumann boundary conditions is considered.The influence of delayed feedback on the stability of the constant positive steady state solution is studied in details,the conditions for the existence of Hopf bifurcation are obtained,the direction of Hopf bifurcation and stability of bifurcating periodic solutions is analyzed.Finally,some simulations coincide with the theoretical results are given.Our results show that delayed feedback not only breaks the stability of the constant positive steady state solution,but also induces the occurrence of spital homogeneous and inhomogeneous periodic solutions when delayed feedback varies.Besides,the simulations suggest that delayed feedback could break the stability of the spatial inhomogeneous periodic solutions when the feedback intensity increases.(?)A diffusive photosensitive CDIMA system with delayed feedback subject to homogeneous Neumann boundary conditions is considered.We perform a detailed investigation of the effect of diffusion on the stability of constant positive steady state solution,and derive the sufficient conditions to guarantee the existence of the instability induced by diffusion.The study shows that the concentration of starch and diffusion can break the stability of the constant positive steady state solution,and results in the occurrence of Turing instability.We derive the sufficient conditions for the existence of Hopf bifurcation induced by delayed feedback,and investigate the properties by utilizing the center manifold theory and normal form theory.Our conclusions illustrate that delay can induce the occurrence of Hopf bifurcation by selecting suitable feedback intensity.Particularly,the stability switch may occur when delay reaches some critical values.(?)A detailed investigation of the effect of gene expression delay on the dynamics of a glycolysis model is given.The conclusions illustrate that delay due to gene expression can make the constant positive steady state solution unstable,and result in the existence of Hopf bifurcation.By computing the normal form on the center manifold,we derive an algorithm for determining the properties of the bifurcating periodic solutions.Finally,we give the sufficient conditions for the existence of steady state bifurcation,and carry out some simulations to show our theoretical results.