Hopf bifurcation analysis for a model of single genetic negative feedback autoregulatory system with delay

In this paper, we deal with a model of single genetic negative feedback autoregulatory system with delay. Choosing the transcriptional rate k as the bifurcation parameter, we demonstrate that the Hopf bifurcation would occur when k exceeds a critical value. By using the normal form theory and center manifold argument, we derive the explicit formulas which determine the bifurcation direction and the stability of bifurcated periodic solution. In order to illustrate our theoretical analysis, a numerical simulation is also included in the end.