[PERSON] and [PERSON] [15] proposed that time delay can change the level of Hes1 protein concentration, and pointed out that high steady-state Hes1 is susceptible to time delay, which excites oscillatory behavior. [PERSON] and [PERSON] [16] applied a factorization method to the characteristic polynomial of the bifurcation point of the negative feedback model to obtain the frequency of Hopf bifurcation, and then determined the stability of the periodic solution of Hopf bifurcation. [PERSON] and [PERSON] [17] took the fractional order as the bifurcation parameter and proposed that Hopf bifurcation occurs only when the number of suppressor genes is odd based on the stability analysis. For further understanding, please refer to [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. There are basically two types of genetic network models, namely Boolean (or discrete) models and differential equation models (or continuous models). In this paper, the differential equation model is chosen to describe the concentration of gene products in terms of variables more accurately and to provide more detailed information on the behavior of nonlinear dynamics. Inspired by the above work, this paper studies a gene expression model:  
\[ \begin{array}{l}\dot{M}(t)=\frac{k}{1+(P(t-\tau)/P_{0})^{n}}-\mu M(t)\\ \dot{P}(t)=M(t)-dP(t)\end{array}  \tag{1}\]  
where \(M(t)\) and \(P(t)\) represent the density of mRNA and protein, \(k\) is the initiation rate of transcription in the absence of the associated protein, \(\mu\) and \(d\) denote the degradation rates of mRNA and protein, respectively, \(\tau\) is the node delay caused by transcription, and \(n\) is called Hill coefficients. The model is derived from [23] and can be traced back to [30; 31]. The time dependence of the variables is shown explicitly in [31], but the delay differential equation in the text is not solved analytically, but only numerically. It is only based on some numerical simulations and does not give a rigorous mathematical proof of the Hopf bifurcation, and the direction of the Hopf bifurcation is not discussed in the paper. In addition, the effect of transcriptional parameters on the system was not considered in [23]. In this paper, we discuss the effect of transcription parameters for time delay, which we use as a static delay state feedback control gain. We adopt the \(\tau\)-decomposition method, which can quickly and accurately solve the equilibrium point and critical delay of the system. The results show that the critical value of delay is related to the static state delay feedback control. In the nonlinear part of the system, the center manifold theorem is combined with this paper, different from the nonlinear methods in other papers, such as [PERSON]'s method, perturbation method, or multiple scale method; these methods are more complicated and require a great deal of calculation. This paper encapsulates and simplifies the formulation based on the center manifold theorem. It can make the analysis of the nonlinear part clearer and more concise. The remainder of this article is organized as follows: In Section 2, we make a preliminary analysis of the system and give a brief description of the \(\tau\)-decomposition method. In Section 3, we have two conclusions, one is the linear part, and we use \(\tau\)-decomposition to analyze the distribution of the characteristic equations associated with this model and obtain the existence of local Hopf bifurcations. In addition, the other is the nonlinear part, which uses the center manifold theorem to determine the direction of the Hopf bifurcation and the stability of the bifurcation period. In Section 4 section, numerical examples and simulation plots are given to confirm our theoretical results. Finally, in Section 5, we discuss our findings.