| Summary: | The stability and Hopf bifurcation of gene expression models with a mechanism of delayed state feedback are considered. An effective algorithm for the calculations on the delay stable interval of the equilibrium point, the direction, and stability of the bifurcating periodic solution is also proposed. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>-decomposition strategy is applied to tackle the issue of local stability, and the explicit formula for the delay stable interval is provided. In addition, the asymptotical behaviors of the bifurcation solutions are investigated by the center manifold theorem and normal form theory. The direction and stability of the Hopf bifurcation are determined naturally. In addition, a subtle bilinear form of the adjoint system is proposed to calculate the bifurcation parameters directly. Finally, the correctness and effectiveness of our results and algorithm are verified by typical numerical examples.
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