Dynamic Analysis Of Degn-Harrison Reaction-Diffusion Model With Time Delay

The Degn-Harrison reaction-diffusion model describes a chemical reaction.The chemical reaction is a complex and changeable process,and the time delay is inevitable.The state of the system is not only affected by the current state,but also depends on the past state.The delayed reaction-diffusion model can accurately describe the state of the dynamic system,which has important significance and broad prospects.In this paper,the dynamic behavior of the Degn-Harrison reaction-diffusion model with time delay is studied under the homogeneous Neumann boundary condition.The influence of time delay on the positive equilibrium and Hopf bifurcation of the system is discussed.For the Degn-Harrison system with time delay,firstly,the time delay is used as the bifurcation parameter,by linearizing the system at the positive equilibrium and analyzing the characteristic equation,the conditions for judging the stability of the positive equilibrium and the existence of Hopf bifurcation are given.Whenτ is in a certain range,the positive equilibrium of the Degn-Harrison system is locally asymptotically stable,and when τ passes through a series of critical values,the Degn-Harrison system produces Hopf bifurcation at the positive equilibrium.Using the normal form theory and the center manifold theorem to judge the stability of the periodic solution and the direction of the Hopf bifurcation,the parameters that determine the direction of the Hopf bifurcation and the stability of the periodic solution of the Degn-Harrison system are given.For the Degn-Harrison reaction-diffusion system,the local and global stability of the positive equilibrium of the Degn-Harrison reaction-diffusion system is discussed.Firstly,by discussing the eigenvalue,the condition for judging the local asymptotic stability of the positive equilibrlum of the system is given by taking the parameter c representing the inhibition strength.Secondly,by constructing the invariant rectangular region and the Lyapunov function,the sufficient conditions for the global stability of the system are given.The results show that the parameter c representing the inhibition strength is closely related to the local and global stability of the Degn-Harrison reaction-diffusion system.For the Degn-Harrison reaction-diffusion system with time delay,firstly,by analyzing the eigenvalue problem of the linearization operator and taking time delay τ as the bifurcation parameter,the influence of time delay on the existence of Hopf bifurcation and the stability of positive equilibrium of the Degn-Harrison reaction-diffusion system is discussed.Secondly,by using the normal form theory and the center manifold theorem,the parameters for determining the direction of Hopf bifurcation and the stability of periodic solution of the Degn-Harrison reaction-diffusion system are given.Finally,the theoretical results are presented intuitively by numerical experiments.