Bifurcation Analysis Of Schnakenberg Model With Delay Feedback

Many scholars have extensively studied the dynamic property of Schnakenbergreaction-diffusion model, which describes an autocatalytic reaction in chemical reaction,and gained some important conclusions. Although reaction-diffusion system arousemany scholars’ wide concern, the research of dynamic behavior of reaction-diffusionsystem with time-delay feedback has not been very perfect yet.Time-delay feedback control can make the unstable fixed points and periodicsolutions become stable. Therefore the stability of the equilibrium maybe change whilethe time-delay feedback is considered. Making quantitative or qualitative description toSchnakenberg reaction diffusion model by using the theory of bifurcation can narrowdown parameters selection. Consequently, we can choose a better research plan tocontrol the chemical reaction effectively. For the reasons mentioned above, stability ofthe equilibrium and Hopf bifurcation of the Schnakenberg model with time delayfeedback are studied in this paper.This paper can be divided into four parts. The first part mainly introduces theorigin of the topic, and present situations of Schnakenberg model. Secondly, consideringtime delay as bifurcation parameters, we analysis two Schnakenberg model with orwithout reaction-diffusion item. By analyzing roots distribution of the correspondingcharacteristic equation at the linear part of the model respectively, the sufficientcondition of stability and Hopf bifurcation of the system are obtained. In the third part,direction of Hopf bifurcation in positive equilibrium and stability of the bifurcationperiodic solutions are studied by the theory of center manifold theorem and the normalform theory. Also, the calculating expression are given. The last part gives numericalsimulation using Matlab software to support the theoretical analysis in this paper.