# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 5 Complete Description of the Dynamic Behaviors
### Global Dynamics for \((\beta,c)\in\Lambda_{1}\)
Thus, we can construct the region \(\mathcal{R}\) with the vertices \(M_{2}\) and \(M_{3}\) as follows: the boundaries of \(\mathcal{R}\) are composed of the orbit \(\gamma^{-}(M_{2})\) above the line \(y=\frac{d}{c+1}\) and the line segment \(y=\frac{d}{c+1}\) between \(M_{3}\) and \(M_{2}\). It is easy to see that \(\mathcal{R}\) is positively invariant. Similar to the above arguments, System (3) has no periodic orbit. Remark 2: _For the case \((\beta,c)\in\Lambda_{1a}\), according to Theorems 7 and 8, the global features of System (3) are described as follows: when the degree of cooperative hunting of the predator species \(x\) and the predator conversion \(c\) are both low such that \(\beta\leq 1\) and \(c\leq f_{1}(\beta)\), the high mortality of the predator population (\(d\geq c\)) drives the predator to extinction (see Figure 6a). On the contrary, if the mortality of the predator population is small such that \(d<c\), two species coexist (see Figure 7)._  
Theorem 4.2: _If \((\beta,c)\in\Lambda_{1b}\), then we can make the following statements:_  
1. _If_ \(d\in(0,d^{*})\)_, then (_3_) has a stable limit cycle (see Figure 8a)._
2. _If_ \(d\in(d^{*},c)\)_, then_ \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) _is globally asymptotically stable (see Figure 8c)._  
Proof: When \(d\in(d^{*},c)\), similar to the proof of Theorem 4.2, one can prove that \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is globally asymptotically stable. Now, let \(d\in(0,d^{*})\). According to Theorem 4.2, \(E_{2}^{*}\) is a source. Let \(M_{1}\) be the intersection point of the lines \(y=1\) and \(x=x_{2}^{*}\), i.e., \(M_{1}=(x_{2}^{*},1)\). We consider the positive half-trajectory \(\gamma^{+}(M_{1})\) passing the point \(M_{1}\). By examining the direction of the vector field in System (3), \(\gamma^{+}(M_{1})\) must intersect the nullcline \(y=\phi_{2}(x)\) at some point \(M_{2}(x_{M_{2}},y_{M_{2}})\) below \(y=\phi_{1}(x)\). Denote \(M_{3}=(x_{M_{2}},0)\). Then, we construct a region \(\mathcal{R}\) with the vertices \(M_{1}\), \(M_{2}\), \(M_{3}\), and \(E_{1}\) as follows: between \(M_{1}\) and \(M_{2}\), the boundary of \(\mathcal{R}\) is the orbit \(\gamma^{+}(M_{1})\), and all other parts of the boundary of \(\mathcal{R}\) are the line segments connecting \(M_{2}\to M_{3}\to E_{1}\to M_{1}\). It is easy to see that \(\mathcal{R}\) is positively invariant. Thus, from the Poincare-Bendixson theorem [36], System (3) has a stable limit cycle since \(E_{2}^{*}\) is unstable. **Remark 3**. : _For the case \((\beta,c)\in\Lambda_{1b}\), Theorems 7 and 9 indicate that although the conversion efficiency \(c\) is high such that \(c>f_{1}(\beta)\), when the degree of cooperative hunting is low (\(\beta\leq 1\) ), the high mortality of the predator population (\(d\geq c\)) still drives the predator to extinction and the prey to survival (see Figure 6a). On the contrary, if the mortality of the predator is small (\(d<c\)), both species coexist. However, unlike the case \((\beta,c)\in\Lambda_{1a}\), the coexistence of the two species in the case \((\beta,c)\in\Lambda_{1b}\) takes two different forms: when \(d\in(0,d^{*})\), the coexistence exhibits an oscillatory mode (there is a stable limit cycle; see Figure 8a); when \(d\in(d^{*},c)\), the coexistence exhibits a steady mode (\(E_{2}^{*}\) is globally asymptotically stable; see Figure 8c)._