## 3 Main Results
### Local Stability and Hopf Bifurcation  
In this subsection, we focus on the stability of the system at the equilibrium point and the conditions required for a Hopf bifurcation to occur. The discussion of local stability first requires us to obtain the value of the critical time delay. As described above in the first work on the \(\tau\)-decomposition method, we need to compute the pure imaginary roots. For the pure imaginary roots of the system, we give the following lemma.  
Lemma 1: _Assuming that \(K>\mu d\), the transcendental equation has only one purely imaginary root (PIR), where_  
\[\omega_{0}=(\frac{1}{2}(-\mu^{2}-d^{2}+\sqrt{(\mu^{2}-d^{2})^{2}+4K^{2}}))^{ \frac{1}{2}} \tag{9}\]  
Proof of Lemma 1: The characteristic polynomial of the system (3) is  
\[P(\lambda)=\lambda^{2}+(\mu+d)\lambda+\mu d+Ke^{-\tau\lambda}\]  
Introducing A and C concerning \(\lambda\) to represent the characteristic polynomial,  
\[A(\lambda)=\lambda^{2}+(\mu+d)\lambda+\mu d,C(\lambda)=K\]  
Since conjugated complex roots appear symmetrically, we only need to consider positive pure imaginary roots, set \(\lambda=\omega j\), and we can obtain  
\[A(\omega j)=-\omega^{2}+(\mu+d)\omega j+\mu d,C(\omega j)=K\]  
We use the method to obtain the following equation:  
\[\omega_{0}^{4}+\mu^{2}\omega_{0}^{2}+d^{2}\omega_{0}^{2}+\mu^{2}d^{2}-K^{2}=0\]  
and the solution of this equation is  
\[\omega_{0}=(\frac{1}{2}(-\mu^{2}-d^{2}+\sqrt{(\mu^{2}-d^{2})^{2}+4K^{2}}))^{ \frac{1}{2}}\]  
Thus, we obtain the result in (9).  
On the other hand, the cross direction around \(\omega_{0}j\) is always to the right, which is explicitly mentioned in the exposition of the \(\tau\)-decomposition method. Therefore, as for the local stability of system in (3), it is stated below.  
Theorem 1: The equilibrium point \((M_{e},\ P_{e})\) of system (3) is exponentially stable for \(\tau\in[0,\tau_{0})\), and \(\tau_{0}\) is a Hopf bifurcation point, within  
\[\tau_{0}=\frac{\theta_{0}}{\omega_{0}} \tag{10}\]  
in which \(\theta_{0}=\angle(\omega_{0}^{2}+(\mu+d)\omega_{0}j-\mu d)\), \(\angle\) is arg (argument of a complex number). Moreover, \(\tau_{0}\) is a Hopf bifurcation point for the system in (3).  
Proof of Theorem 1:.: Lemma 1 tells us that there is only one pair of \(j\omega_{0}\) in the characteristic polynomial, which is the root of the characteristic equation, and then we have  
\[-\omega_{0}^{2}+(\mu+d)\omega_{0}j+\mu d+Ke^{-\tau_{0}\omega_{0}j}=0\]  
We can obtain an expression for \(e^{\tau_{0}\omega_{0}j}\)  
\[e^{\tau_{0}\omega_{0}j}=\frac{K}{\omega_{0}^{2}-(\mu+d)\omega_{0}j-\mu d}\]  
Furthermore, the following formula can be obtained:  
\[\tau_{0}\omega_{0}=-\angle(\omega_{0}^{2}-(\mu+d)\omega_{0}j-\mu d)\]  
Therefore,  
\[\tau_{0}=\frac{\angle(\omega_{0}^{2}+(\mu+d)\omega_{0}j-\mu d)}{\omega_{0}}\]  
Thus, we obtain the result in (10). On the other hand, we will explain the reason why \(\tau_{0}\) is a Hopf bifurcation point.  
At \(\tau=\tau_{0}\), it only corresponds to a pair of pure imaginary roots \(\pm\omega_{0}j\). In addition, when \(\tau\to\tau_{0}^{-}\), all roots lie on the left half plane, but when \(\tau=\tau_{0}\), the root lies on the imaginary axis, at this time, all other roots lie on the left half plane, that is, \(\text{Re}(\tau_{0})<0\). Moreover, since there is only one pair of the pure imaginary root, the crossing direction must be to the right, so we think \(\alpha^{{}^{\prime}}(0)>0\).