Analysis Of Stability And Bifurcation In Lengyel-epstein Diffusion System With Time Delay

This paper considers mainly the dynamic properties of the diffusive delayed LengyelEpstein system subject to the homogeneous Neumann boundary condition.In the absence of the effect of the delay,by analyzing the eigenvalue problem of the linearized system of the original system at the positive constant equilibrium,we investigate the local asymptotic stability of the positive equilibrium of the corresponding ordinary differential equations(ODEs)system and the positive constant equilibrium of the reactiondiffusion system.When the the positive constant equilibrium for the reaction-diffusion system without delay is stable,the effect of the change of delay on the stability of the positive constant equilibrium of the delayed diffusive system is considered.It is shown that the increase of the delay can lead to that the stable positive constant equilibrium becomes unstable through Hopf bifurcation.At the critical values of parameters,by means of the normal form method and the canter manifold theorem for partial functional differential equations(PFDEs),the explicit formula judging the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions is obtained.Finally,in order to verify the correction of the obtained theoretical results,suitable numerical verifications for the considered model are carried out according to MATALAB software package.The content and structure of this paper are as follows:The first chapter summarizes the research background and current situation of LengyelEpstein system,and states the main content of this paper.In the second chapter,firstly,the stability of the positive equilibrium point of the corresponding ODE system is discussed.Secondly,the stability of the positive constant equilibrium of reaction-diffusion system is analyzed.Finally,the stability of the positive constant equilibrium and the existence of Hopf bifurcation for the reaction diffusion system with delay are investigated.The third chapter discusses the direction of Hopf bifurcation obtained in the second and the stability of the bifurcating periodic solution.The forth chapter carries out the suitable numerical simulation to verify the correction of the obtained theoretical results.