# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Stability of Interior Equilibria and Hopf Bifurcation
Thus, from the monotonicity of \(x_{2}^{*}\) on \(d\), \(d^{*}=c\) if and only if \(c=h(\beta)\), \(d^{*}>c\) if and only if \(c>h(\beta)\), and \(d^{*}<c\) if and only if \(c<h(\beta)\). It is easy to see that \(h(\beta)<f_{2}(\beta)=\frac{3}{\sqrt{1+3\beta-2}}\) if and only if \(h(\beta):=3\sqrt{\beta}-\sqrt{1+3\beta}-1>0\). Since \(\tilde{h}(1)=0\) and  
\[\tilde{h}^{\prime}(\beta)=\frac{3}{2}\Bigg{(}\frac{1}{\beta}-\frac{1}{\sqrt{1 +3\beta}}\Bigg{)}>0,\quad\beta>1,\]  
we know that for \(\beta>1\), \(\tilde{h}(\beta)>0\). It follows that for \(\beta>1\), \(h(\beta)<f_{2}(\beta)\). So, conclusively, we have  
\[f_{1}(\beta)<h(\beta)<f_{2}(\beta),\quad\beta>1. \tag{22}\]  
Based on (22), we divide the region \(\Lambda_{2b}\) into the following three subregions (see Figure 3):  
\[\Lambda_{2b-1} :=\{(\beta,c)\in\Lambda_{2b}:f_{1}(\beta)<c<h(\beta)\}, \tag{23}\] \[\Lambda_{2b-2} :=\{(\beta,c)\in\Lambda_{2b}:c=h(\beta)\},\] \[\Lambda_{2b-3} :=\{(\beta,c)\in\Lambda_{2b}:h(\beta)<c<f_{2}(\beta)\},\]  
**Theorem 5**. : _If \((\beta,c)\in\Lambda_{1b}\cup\Lambda_{2b}\), then a supercritical and backward Hopf bifurcation occurs at \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) when \(d=d^{*}\). The Hopf bifurcation surface \(d=H(\beta,c)\) is given by (21). In addition, we have the following statements:_  
1. _If_ \((\beta,c)\in\Lambda_{1b}\)_, then_ \(d^{*}<c\)_;_
2. _If_ \((\beta,c)\in\Lambda_{2b-1}\)_, then_ \(d^{*}<c\)_;_
3. _If_ \((\beta,c)\in\Lambda_{2b-2}\)_, then_ \(d^{*}=c\)_;_
4. _If_ \((\beta,c)\in\Lambda_{2b-3}\)_, then_ \(d^{*}>c\)_._  
Proof. : When \(d=d^{*}\), \(x_{2}^{*}=\frac{1}{c\beta}\). Hence, \(\omega_{1}=(c\beta x_{2}^{*}-1)\phi_{2}(x_{2}^{*})=0\) and the characteristic Equation (14) at \(E_{2}^{*}\) has a pair of imaginary eigenvalues \(\kappa_{1,2}=\pm\sqrt{\omega_{2}}\). Let \(\kappa=\varrho_{1}(d)\pm i\sqrt{\varrho_{2}(d)}\) be the root of (14) when \(d\) is near \(d^{*}\). Then, \(\varrho_{1}(d)=\frac{1}{2}\omega_{1}=\frac{1}{2}(c\beta x_{2}^{*}-1)\phi_{2}( x_{2}^{*})\), \(\varrho_{2}(d)=\frac{1}{2}(\omega_{1}^{2}-4\omega_{2})\). Clearly,  
\[\frac{\partial\varrho_{1}(d)}{\partial d}=\frac{1}{2}c\beta\phi_{2}(x_{2}^{*}) \frac{\partial x_{2}^{*}}{\partial d}+\frac{1}{2}(c\beta x_{2}^{*}-1)\phi^{ \prime}(x_{2}^{*})\frac{\partial x_{2}^{*}}{\partial d}.\]  
Noticing that \(x_{2}^{*}\mid_{d=d^{*}}=\frac{1}{c\beta}\) and \(\frac{\partial x_{2}^{*}}{\partial d}<0\) (see Remark 1), we observe that for \((\beta,c)\in\Lambda_{1b}\cup\Lambda_{2b}\),  
\[\frac{\partial\varrho_{1}(d)}{\partial d}\mid_{d=d^{*}}=\left[\frac{1}{2}c\beta \phi_{2}(x_{2}^{*})\frac{\partial x_{2}^{*}}{\partial d}\right]_{d=d^{*}}<0.\]  
From the Poincare-Hopf Bifurcation Theorem [35], it follows that System (3) undergoes a Hopf Bifurcation when \(d=d^{*}\). Below, we calculate the first Lyapunov coefficient at \(d=d^{*}\), which determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits. Let \(X=x-x_{2}^{*}\), \(Y=y-y_{2}^{*}\). For simplicity, we still use the variables \(x\) and \(y\) instead of \(X\) and \(Y\), respectively.