# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Summary  
Our main results obtained in the previous arguments can be summarized in Figure 5a,b, where the parameter \(\beta\) is chosen as \(\beta=0.6<1\) and \(\beta=4>1\), respectively. The red curve denotes the Hopf bifurcation curve \(d=H(\beta,c)\), and the blue and green straight lines are \(d=c\) and \(d=F(x_{0})=\frac{c(2(3\beta+1)^{\frac{3}{2}}+9\beta+2)}{27\beta}\) (\(x_{0}\approx 0.1337959395\)), respectively. 1. As \(d\) decreases, Figure 5a (\(\beta=0.6<1\)) indicates the following implications: 1. When \(d>c\), there is no interior equilibrium and \(E_{1}(0,1)\) is a sink; when \(d=c\), \(E_{1}\) becomes a half-stable equilibrium (a saddle-node point; see Theorem 3); when \(d\) decreases from \(d=c\), the saddle-node bifurcation occurs, and \(E_{1}\) becomes a saddle and a stable interior equilibrium \(E_{2}^{*}\) appears (see Theorem 4). 2. When \(d\) continues to decrease, according to Theorem 4, if \(c\leq f_{1}(\beta)\) (\(f_{1}(\beta)\approx 2.3699\)), i.e., \((\beta,c)\in\Lambda_{1a}\), \(E_{2}^{*}\) is a sink for all \(d<c\); if \(c>f_{1}(\beta)\) (i.e., \((\beta,c)\in\Lambda_{1b}\)), as \(d\) decreases to the red curve, i.e., \(d=H(\beta,c)\) (\(\beta=0.6\)), a supercritical Hopf bifurcation occurs. Below it, i.e., \(d<H(\beta,c)\), \(E_{2}^{*}\) becomes a source. 2. As \(d\) decreases, Figure 5b (\(\beta=4>1\)) indicates the following implications: 1. According to Theorem 2, when \(d>F(x_{0})\), System (3) has no interior equilibrium and \(E_{1}\) is stable node. According to Theorems 2 and 6, when \(d\) decreases to the green line \(d=F(x_{0})\), System (3) has a unique interior equilibrium \(\tilde{E}^{*}(x_{0},\phi_{1}(x_{0}))\), which is an attracting saddle node when \(c<f_{2}(\beta)\) (\(f_{2}(\beta)\approx 1.868517092\)), a repelling saddle node when \(c>f_{2}(\beta)\), and a Bogdanov-Takens-type cusp when \(c=f_{2}(\beta)\) (see Figure 4). Note that when \(c=f_{2}(\beta)\), the line \(d=F(x_{0})\) is tangent to the Hopf bifurcation curve \(d=H(\beta,c)\). According to Theorems 2 and 4, as \(d\) continues to decrease from \(F(x_{0})\) (\(d<F(x_{0})\) but near \(F(x_{0})\)), two interior equilibria \(E_{1}^{*}\) and \(E_{2}^{*}\) appear, where \(E_{1}^{*}\) is a saddle and \(E_{2}^{*}\) is a sink when \(c<f_{2}(\beta)\) (i.e., \((\beta,c)\in\Lambda_{2a}\cup\Lambda_{2b}\)), whereas it is a source when \(c\geq f_{2}(\beta)\) (i.e., \((\beta,c)\in\Lambda_{2c}\)). When \(d>c\), \(E_{1}\) is a stable node. (3) When \(c\leq f_{1}(\beta)\) (\(f_{1}(\beta)\approx 0.6404\)), i.e., \((\beta,c)\in\Lambda_{2a}\), and \(d\) continues to decrease to \(d=c\), the stability of \(E_{1}\) switches from a stable node to a half-stable equilibrium (a saddle node), \(E_{1}^{*}\) disappears, and System (3) has the unique interior equilibrium \(E_{2}^{*}\). When \(d<c\), \(E_{1}\) becomes a saddle and \(E_{2}^{*}\) remains asymptotically stable. When \(f_{1}(\beta)<c<h(\beta)\) (here, \(h(\beta)=1\)), i.e., \((\beta,c)\in\Lambda_{2b-1}\), and \(d\) continues to decrease to \(d=c\), similar to the case \((\beta,c)\in\Lambda_{2a}\), the stability of \(E_{1}\) switches from a stable node to a saddle node, \(E_{1}^{*}\) disappears, and System (3) has the unique interior equilibrium \(E_{2}^{*}\). When \(d\) decreases from \(d=c\) and passes the Hopf bifurcation curve \(d=H(\beta,c)\) (\(\beta=4\)), \(E_{1}\) becomes a saddle and \(E_{2}^{*}\) goes from a sink to a source. When \(c=h(\beta)=1\), i.e., \((\beta,c)=(4,1)\in\Lambda_{2b-2}\), and \(d\) decreases to \(d=c=1\), \(E_{1}\) becomes a saddle node, \(E_{1}^{*}\) disappears, and System (3) has the unique interior equilibrium \(E_{2}^{*}\). The Hopf bifurcation curve \(d=H(\beta,c)\) intersects the line \(d=c\) when \(c=h(\beta)=1\). Therefore, when \((\beta,c)\in\Lambda_{2b-2}\) and \(d=c\), the Hopf bifurcation and saddle-node bifurcation occur simultaneously. After that, as \(d\) continues to decrease from \(d=c\), \(E_{1}\) becomes a saddle and \(E_{2}^{*}\) becomes a source.