The Stability And Bifurcation Of A Class Of Gene Expression System With The Time Delay

A lot of differential equation models with delays are developed in auto-control, physics, biology, economics and many other fields. It is of importance to study dynamical behaviors of these models.Many results about the system of circadian oscillation are obtained recently. But a large part of the results are from numerical methods. In this thesis, a circadian oscillation model is studied from the point of view of theoretical analysis. The circadian oscillation model is analyzed using the theory of delay differential equations. Firstly, the influences of the delay, nonlinearity in the protein production and cooperativity in the negative feedback on circadian oscillation are analyzed. Furthermore, taking the time delay as a parameter, the critical value of the time delay that Hopf bifurcation occurs and the areas in parameter space where the Hopf bifurcation occurs are obtained. Secondly, based on center manifold and normal form, the formulas for determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation are derived. Finally, the matlab programs are used to verify the results. The methods used and the results obtained in this paper are important to future researches.