# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 5 Complete Description of the Dynamic Behaviors
### Global Dynamics of the Case \((\beta,c)\in\Lambda_{2}\)  
In this subsection, we consider the global dynamics for the case \((\beta,c)\in\Lambda_{2}\), i.e., \(\beta>1\) and \(f_{1}(\beta)<c<f_{2}(\beta)\). When \(d\leq c\), System (3) has the unique positive equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) according to Theorem 2, and according to Theorem 2, \(E_{1}(0,1)\) is a saddle or saddle node with its hyperbolic sectors in the first quadrant. When \(c<d<F(x_{0})\), there are two positive equilibria \(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\) and \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\), where \(E_{1}^{*}\) is always a saddle and the stability of \(E_{2}^{*}\) is dependent on parameters and \(E_{1}\) becomes a sink.  
**Lemma 2**.: _Let \((\beta,c)\in\Lambda_{2}\) and \(c<d<F(x_{0})\)._  
1. _One branch of the stable manifolds of_ \(E_{1}^{*}\)_, denoted by_ \(\Gamma_{1}^{s}\)_, enters_ \(E_{1}^{*}\) _from the region_ \(\{(x,y)\in\mathbb{R}_{+}^{2}:y>\phi_{1}(x),y>\phi_{2}(x)\}\)_; another one, denoted by_ \(\Gamma_{2}^{s}\)_, enters_ \(E_{1}^{*}\) _from the region_ \(\{(x,y)\in\mathbb{R}_{+}^{2}:y<\phi_{1}(x),y<\phi_{2}(x)\}\)_._
2. _One branch of the unstable manifolds of_ \(E_{1}^{*}\)_, denoted by_ \(\Gamma_{1}^{u}\)_, is the heteroclinic orbit connecting_ \(E_{1}^{*}\) _and_ \(E_{1}(0,1)\) _entering the region_ \(\{(x,y)\in\mathbb{R}_{+}^{2}:y<\phi_{1}(x),y<\phi_{2}(x)\}\) _and then approaching_ \(E_{1}\)_. Another one, denoted by_ \(\Gamma_{2}^{u}\)_, enters the region_ \(\{(x,y)\in\mathbb{R}_{+}^{2}:y>\phi_{1}(x),y>\phi_{2}(x)\}\)_._  
Figure 8: The dynamics of System (3) as \(d\in(0,c)\) in the case \((\beta,c)\in\Lambda_{1b}\). Here, \(\beta=0.6,c=4>f_{1}(\beta)\approx 2.3699\). The Hopf bifurcation value \(d^{*}=H(\beta,c)\approx 2.3958333\). In (**a**), \(d\in(0,d^{*})\), \(E_{2}^{*}\) is a source and surrounded by a stable limit cycle. In (**b**), \(d=d^{*}\) and a backward and supercritical Hopf bifurcation occurs. In (**c**), \(d\in(d^{*},c)\) and \(E_{2}^{*}\) is globally asymptotically stable.  
Proof.: At \(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\), the eigenvalues of (13) are \(\kappa_{1}=\frac{\omega_{1}-\sqrt{\alpha_{1}^{2}-4\omega_{2}}}{2}<0\), \(\kappa_{2}=\frac{\omega_{1}+\sqrt{\alpha_{1}^{2}-4\omega_{2}}}{2}>0\). The eigenvector corresponding to \(\kappa_{1}\) is \(\left(-\frac{c(1+\beta x_{1}^{*})x_{1}^{*}}{c\beta x_{1}^{*}y_{1}^{*}-\kappa_ {1}},1\right)\). Then, the tangential direction of the stable manifold at \(E_{1}^{*}\) is  
\[k_{1}=-\frac{c\beta x_{1}^{*}y_{1}^{*}-\kappa_{1}}{c(1+\beta x_{1}^{*})x_{1}^{* }}<-\frac{c\beta y_{1}^{*}}{c(1+\beta x_{1}^{*})}=-\frac{d\beta}{c(1+\beta x_{1 }^{*})^{2}}=\phi_{1}^{\prime}(x_{1}^{*})<\phi_{2}^{\prime}(x_{1}^{*})\]  
since \(y_{1}^{*}=\phi_{1}(x_{1}^{*})=\frac{d}{c(1+\beta x_{1}^{*})}\). So, \(\Gamma_{1}^{*}\) enters from the region \(\{(x,y)\in\mathbb{R}_{+}^{2}:y>\phi_{1}(x)\), \(y>\phi_{2}(x)\}\), and \(\Gamma_{2}^{*}\) approaches \(E_{1}^{*}\) from the region \(\{(x,y)\in\mathbb{R}_{+}^{2}:y<\phi_{1}(x),y<\phi_{2}(x)\}\).  
The eigenvector corresponding to \(\kappa_{2}\) is \(\left(-\frac{y_{1}^{*}+\kappa_{2}}{(1+2\beta x_{1}^{*})y_{1}^{*}},1\right)\). Then, the tangential direction of the unstable manifold at \(E_{1}^{*}\) is  
\[k_{2}=-\frac{(1+2\beta x_{1}^{*})y_{1}^{*}}{y_{1}^{*}+\kappa_{2}}>-(1+2\beta x_ {1}^{*})=\phi_{2}^{\prime}(x_{1}^{*})>\phi_{1}^{\prime}(x_{1}^{*}).\]  
So, \(\Gamma_{1}^{u}\) enters the region \(\{(x,y)\in\mathbb{R}_{+}^{2}:y<\phi_{1}(x),y<\phi_{2}(x)\}\), and \(\Gamma_{2}^{u}\) enters the region \(\{(x,y)\in\mathbb{R}_{+}^{2}:y>\phi_{1}(x),y>\phi_{2}(x)\}\). From the direction of the vector field in (3), \(\Gamma_{1}^{u}\) is the heteroclinic orbit connecting \(E_{1}^{*}\) and \(E_{1}(0,1)\).  
Below, we consider the global dynamics for the following scenarios: \((\beta,c)\in\Lambda_{2u}\), \((\beta,c)\in\Lambda_{2b}=\Lambda_{2b-1}\cup\Lambda_{2b-2}\cup\Lambda_{2b-3}\), and \((\beta,c)\in\Lambda_{2c}\). When \(d\in(c,F(x_{0}))\), the dynamics of (3) are complex and are described through numerical simulations.