# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 5 Complete Description of the Dynamic Behaviors
### Global Dynamics for \((\beta,c)\in\Lambda_{1}\)  
In this subsection, we consider the global dynamics of the case \((\beta,c)\in\Lambda_{1}=\Lambda_{1a}\cup\Lambda_{1b}\). In this case, System (3) has the positive equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) only for \(d<c\), as indicated by Theorem 2. According to Theorems 3 and 5, when \((\beta,c)\in\Lambda_{1a}\), \(E_{2}^{*}\) is a sink; when \((\beta,c)\in\Lambda_{1b}\), there exists a \(d^{*}:d^{*}<c\) such that \(E_{2}^{*}\) is a source for \(d\in(0,d^{*})\), a sink for \(d\in(d^{*},c)\), and a Hopf bifurcation occurs at \(E_{2}^{*}\) for \(d=d^{*}\), where \(d^{*}=H(\beta,c)\) and \(H(\beta,c)\) is given by (21). If \((\beta,c)\in\Lambda_{1a}\) and \(d<c\), then \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is globally asymptotically stable (see Figure 7). Proof. : When \(\lambda\in\Lambda_{1a}\) and \(d<c\), both \(E_{0}(0,0)\) and \(E_{1}(0,1)\) are saddles according to Theorem 3, and \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is the unique interior equilibrium and is locally asymptotically stable, as indicated by Theorem 4. The stable and unstable manifolds of \(E_{0}\) are along the \(x\)-axis and \(y\)-axis, respectively. The stable manifold of \(E_{1}\) is along the \(y\)-axis and its unstable manifold enters the interior of \(\mathbb{R}_{+}^{2}\). If System (3) has no limit cycle, according to the Poincare-Bendixson theorem [36], we can conclude that \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is globally asymptotically stable. On the contrary, we assume that System (3) has a periodic orbit \((x(t),y(t))\) with the period \(T\). Then, by integrating the equations of (3) on \([0,T]\), we have  
\[\int_{0}^{T}((1+\beta x)y-d)dt=0,\quad\int_{0}^{T}(1-y-(1+\beta x)x)dt=0. \tag{29}\]  
Denote the right-hand sides of (3) by \(P_{1}(x,y)\) and \(P_{2}(x,y)\), respectively. Using (29), we obtain the integral of the divergence of (3):  
\[\int_{0}^{T}\Bigl{(}\tfrac{\partial P_{1}}{\partial x}+\tfrac{\partial P_{2}} {\partial y}\Bigr{)}dt=\int_{0}^{T}(d-(c+1)y(t))dt. \tag{30}\]  
Since \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is locally asymptotically stable, we know that \(\omega_{1}=d-(c+1)y_{2}^{*}<0\) (see (15)), that is, \(y_{2}^{*}>\tfrac{d}{c+1}\). Let \(M_{1}(0,\tfrac{d}{c+1})\). The line \(y=\tfrac{d}{c+1}\) must intersect the \(y\)-nullcline \(y=\phi_{2}(x)\) at some point \(M_{2}\) below \(y=\phi_{1}(x)\). We consider the negative half-trajectory \(\gamma^{-}(M_{2})\) passing the point \(M_{2}\). If \(\gamma^{-}(M_{2})\) intersects the line \(y=1\) at some point \(M_{3}\) above the \(y\)-nullcline \(y=\phi_{2}(x)\), we can construct a region \(\mathcal{R}\) with the vertices \(M_{1},M_{2},M_{3}\), and \(E_{1}=(0,1)\) as follows: between \(M_{2}\) and \(M_{3}\), the boundary of \(\mathcal{R}\) is the orbit \(\gamma^{-}(M_{2})\), and all other parts of the boundary of \(\mathcal{R}\) are the line segments connecting \(M_{3}\to E_{1}\to M_{1}\to M_{2}\). It is easy to see that \(\mathcal{R}\) is positively invariant. Since \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is the unique equilibrium point in the first quadrant, any periodic orbit, if it exists, must encircle it and lie wholly in the region \(\mathcal{R}\). Assume that System (3) has a \(T\)-periodic orbit \((x(t),y(t))\). Then, \(y(t)>\tfrac{d}{c+1}\) and \(\int_{0}^{T}\left(\frac{a\rho_{1}}{ax}+\frac{a\rho_{2}}{\beta y}\right)dt<0\). It follows that the periodic orbit \((x(t),y(t))\) is asymptotically stable. However, it is impossible since \(E_{2}^{*}\) is locally asymptotically stable. If \(\gamma^{-}(M_{2})\) does not meet the line \(y=1\), by examining the direction of the vector field in System (3), it enters the region \(\{(x,y)\in\mathbb{R}_{+}^{2}:y<\phi_{2}(x)\}\). After that, if \(\gamma^{-}(M_{2})\) always stays above the line \(y=\frac{d}{c+1}\), its \(\alpha\)-limit set must be a periodic orbit. In (30), the periodic orbit is stable. But it is also impossible since \(E_{2}^{*}\) is locally asymptotically stable. So, \(\gamma^{-}(M_{2})\) intersects the line \(y=\frac{d}{c+1}\) at some point \(M_{3}\).