Analytic And Numerical Hopf Bifurcation For A Retarded Chemostat Model

Delay differential equations arise widely in the fields of Physics, Engineering,Biology, Medical Science, Economics and so on. The bifurcation phenomena canoccur in the parameter dependent systems. The type of bifurcation that connects equi-libria with periodic solution is called Hopf bifurcation. The study of Hopf bifurcationincludes determining the bifurcation value, the direction of bifurcation and the stabil-ity of bifurcating periodic solution.This paper deals with the Hopf bifurcation of a Chemostat model when delay istaken as the parameters, including bifurcation parameter value, bifurcation directionand the stability of bifurcating periodic solution. And applied the Euler method tothis equation, the properties of the numerical Hopf bifurcation are investigated as thedelay varies. The main work of the paper is summarized as follows:Firstly, the existence of the Hopf bifurcation of the internal equilibrium E3 ofthe retarded Chemostat model and the Hopf bifurcation valueτ~* are analyzed. Byusing Euler method, the numerical Hopf bifurcation of the Chemostat model whendelayτis taken as the parameters is investigated. It is proved that the numerical Hopfbifurcation exists when the stepsize is small enough.Secondly, the direction and the stability of the Hopf bifurcation of the Chemostatmodel are considered. The parameters which determine the stability of the periodicsolution and the direction of the bifurcation are given.Finally, by using Euler method, the direction and the stability of the numericalHopf bifurcation when delayτis taken as the parameters are discussed. It is shownthat the numerical Hopf bifurcation satisfiesτh =τ~* + O(h) and the bifurcationperiodic solutions of the difference system obtained by using Euler method have thesame bifurcation direction and stability as the original system when the stepsize htends to zero.