# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Stability of Interior Equilibria and Hopf Bifurcation
Now, we consider the stability of \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) for the case of \((\beta,c)\in\Lambda_{2}\). According to Theorem 2, when \(d\in(0,F(x_{0}))\), the interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) exists. 1. Let \((\beta,c)\in\Lambda_{2a}\), then, similar to the case of \((\beta,c)\in\Lambda_{1a}\), \Let \((\beta,c)\in\Lambda_{2b}\), then \(x_{0}<\frac{1}{c\beta}<x_{max}^{*}\), as indicated by (17) and (18). From the monotonicity of \(x_{2}^{*}\) on \(d\), we know that \(x_{2}^{*}>\frac{1}{c\beta}\) for \(d\in(0,d^{*})\) and \(x_{2}^{*}<\frac{1}{c\beta}\) for \(d\in(d^{*},F(x_{0}))\). It follows from Lemma 1 that \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is a source for \(d\in(0,d^{*})\) and a sink for \(d\in(d^{*},F(x_{0}))\). 3. Let \((\beta,c)\in\Lambda_{2c}\), then \(\frac{1}{c\beta}\leq x_{0}\), as indicated by (18), which implies from the monotonicity of \(x_{2}^{*}\) on \(d\) that for all \(d\in(0,F(x_{0}))\), the corresponding interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) satisfies \(x_{2}^{*}>\frac{1}{c\beta}\). So, \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is a source for all \(d\in(0,F(x_{0}))\). Then, we obtain the following conclusive result on the local stability of the interior equilibria. **Theorem 4**. : _1_\(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\)_, if it exists, is always a saddle._  
_At \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\), we have the following statements:_  
1. _If_ \((\beta,c)\in\Lambda_{1a}\cup\Lambda_{2a}\)_, then_ \(E_{2}^{*}\)_, if it exists, is a sink._
2. _If_ \((\beta,c)\in\Lambda_{1b}\)_, then_ \(E_{2}^{*}\) _is a source for_ \(d\in(0,d^{*})\) _and a sink for_ \(d\in(d^{*},c)\)_._
3. _If_ \((\beta,c)\in\Lambda_{2b}\)_, then_ \(E_{2}^{*}\) _is a source for_ \(d\in(0,d^{*})\) _and a sink for_ \(d\in(d^{*},F(x_{0}))\)_._
4. _If_ \((\beta,c)\in\Lambda_{2c}\)_, then_ \(E_{2}^{*}\) _is a source for all_ \(d\in(0,F(x_{0}))\)_._  
According to Theorem 4, if \((\beta,c)\in\Lambda_{1b}\) or \((\beta,c)\in\Lambda_{2b}\), i.e., \(\beta\leq 1\) and \(c>f_{1}(\beta)\), or \(\beta>1\) and \(f_{1}(\beta)<c<f_{2}(\beta)\), there exists a unique \(d^{*}\) (\(d^{*}<c\) or \(d^{*}<F(x_{0})\)) such that when \(d=d^{*}\), System (3) may undergo the Hopf bifurcation at \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\), where \(d^{*}\) is given by (21), and \(x_{2}^{*}=\frac{1}{c\beta}\). Below, we discuss the Hopf bifurcation of (3) and its directions, together with the stability of bifurcating periodic orbits. We say that a Hopf bifurcation with respect to the parameter \(d\) at \(d=d^{*}\) is backward (respectively, forward) if there is a small amplitude periodic orbit for each \(d\in(d^{*}-c,d^{*})\) (respectively, \(d\in(d^{*},d^{*}+\epsilon)\)), where \(\epsilon>0\) is a small constant. Additionally, we say that a Hopf bifurcation is supercritical (respectively, subcritical) if the bifurcating periodic solutions are orbitally asymptotically stable (respectively, unstable). To compare \(d\) to \(c\), we first determine the location of \(d^{*}\). If \((\beta,c)\in\Lambda_{1b}\), i.e., \(\beta\leq 1\) and \(c>f_{1}(\beta)\), we have \(d^{*}<c\) since the interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) exists only for \(d\in(0,c)\) according to Theorem 4. Let \((\beta,c)\in\Lambda_{2b}\), i.e., \(\beta>1\) and \(f_{1}(\beta)<c<f_{2}(\beta)\). According to Theorem 2, System (3) has the unique interior equilibrium \(E_{2}^{*}\) for \(d\leq c\) and two interior equilibria \(E_{1}^{*}\) and \(E_{2}^{*}\) for \(c<d<F(x_{0})\). From the equation \(F(x)=d\), one can deduce that the first component of the interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) corresponding to \(d=c\) is given by \(x_{2}^{*}=\frac{-1+\sqrt{\beta}}{\beta}\). It is easy to see that  
\[\frac{-1+\sqrt{\beta}}{\beta}<x_{2}^{*}\mid_{d=d^{*}}=\frac{1}{c\beta}\ \ \text{if and only if}\ \ c<h(\beta):=\frac{1}{\sqrt{\beta}-1},\quad\beta>1.\]  
Clearly, for \(\beta>1\), \(f_{1}(\beta)<h(\beta)\).