# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Saddle Node and Cusp
Then, when \(0<\rho\ll 1\), we have \(\frac{\mathrm{d}\rho}{\mathrm{d}t}>0\) along the direction \(\theta=\theta_{1}\) and \(\frac{\mathrm{d}\rho}{\mathrm{d}t}<0\) along the directions \(\theta=\theta_{2}\), \(\theta_{3}\), and \(\theta_{4}\), which implies that the half node of \(\tilde{E}^{*}\) is stable, i.e., \(\tilde{E}^{*}\) is an attracting saddle node of codimension one. If \(c>f_{2}(\beta)\), then, when \(0<\rho\ll 1\), we have \(\frac{\mathrm{d}\rho}{\mathrm{d}t}<0\) along the direction \(\theta=\theta_{1}\) and \(\frac{\mathrm{d}\rho}{\mathrm{d}t}>0\) along the directions \(\theta=\theta_{2}\), \(\theta_{3}\), and \(\theta_{4}\), which implies that the half node of \(\tilde{E}^{*}\) is unstable, i.e., \(\tilde{E}^{*}\) is a repelling saddle node of codimension one. Now, we assume that \(c=f_{2}(\beta)\), which implies that \(\omega_{1}=0\). Then, \(\kappa_{1}=\kappa_{2}=0\). In this case, we set \(x=-u-\frac{1}{y_{0}}v\), \(y=(1+2\beta x_{0})u\). Then, System (24) becomes  
\[ \begin{split}\frac{\mathrm{d}u(t)}{\mathrm{d}t}&=v -(1+\beta x_{0})u^{2}-\frac{1}{y_{0}}uv-\frac{\beta}{y_{0}(1+2\beta x_{0})}v^ {2}+O(3),\\ \frac{\mathrm{d}v(t)}{\mathrm{d}t}&=y_{0}(2+3\beta x _{0})u^{2}-(1+c)uv-\frac{\beta(1+c)}{1+2\beta x_{0}}v^{2}.\end{split}  \tag{27}\]  
Let \(u=\varphi_{1}+\frac{1}{2}\left(-\frac{1}{y_{0}}-\frac{\beta(1+c)}{1+2\beta x_ {0}}\right)\varphi_{1}^{2}-\frac{\beta}{y_{0}(1+2\beta x_{0})}\varphi_{1} \varphi_{2}\), \(v=\varphi_{2}+(1+\beta x_{0})\varphi_{1}^{2}-\frac{\beta(c+1)}{1+2\beta x_{0} }\varphi_{1}\varphi_{2}\). Then, System (27) is transformed into  
\[ \begin{split}\frac{\mathrm{d}\varphi_{1}(t)}{\mathrm{d}t}& =\varphi_{2}+o(|(\varphi_{1},\varphi_{2})|^{2}),\\ \frac{\mathrm{d}\varphi_{2}(t)}{\mathrm{d}t}&=\delta _{1}\varphi_{1}^{2}-\delta_{2}\varphi_{1}\varphi_{2}+o(|(\varphi_{1},\varphi_ {2})|^{2}),\end{split}  \tag{28}\]  
where  
\[\delta_{1}=y_{0}(2+3\beta x_{0})>0,\quad\delta_{2}=(1+c)+2(1+\beta x_{0})>0.\]  
So, \(\tilde{E}^{*}\) is a Bogdanov-Takens-type cusp of codimension two (see [34; 37]). The proof is complete. For example, we take \(\beta=4\), then, \(x_{0}\approx 0.1337959395\). We consider the dynamics of System (3) when \((\beta,c)=(4,c)\in\Lambda_{2}\) and \(d=F(x_{0})\). 1. In Figure 4a, we choose \(c=1<f_{2}(\beta)\)\(\approx 1.868517092\). When \(d=F(x_{0})\approx 1.2198549364\), System (3) has the unique interior equilibrium \(\tilde{E}^{*}(x_{0},y_{0})\), which is an attracting saddle node. There are three separatrices, two approaching \(\tilde{E}^{*}\) and one connecting \(\tilde{E}^{*}\) and \(E_{1}(0,1)\) and entering \(E_{1}\). The three separatrices divide \(\mathbb{R}_{+}^{2}\) into three parts, two of which are hyperbolic sectors and one of which is a parabolic sector. The solutions initiated in the hyperbolic sectors converge to \(E_{1}\), whereas the solutions initiated in the parabolic sector tend to \(\tilde{E}^{*}\). 2. In Figure 4b, we choose \(c=3>f_{2}(\beta)\). When \(d=F(x_{0})\approx 3.6595648104\), the unique interior equilibrium \(\tilde{E}^{*}\) is a repelling saddle node. \(\tilde{E}^{*}\) consists of two hyperbolic sectors, one parabolic sector, and three separatrices. Only one of the separatrices approaches \(\tilde{E}^{*}\) and all the other solutions approach \(E_{1}\). 3. In Figure 4c, we choose \(c=f_{2}(\beta)\). When \(d=F(x_{0})\approx 2.2793198\), the unique interior equilibrium \(\tilde{E}^{*}\) is a Bogdanov-Takens-type cusp. Cusp \(\tilde{E}^{*}\) consists of two hyperbolic sectors and two separatrices. Only one of the separatrices approaches \(\tilde{E}^{*}\) and all the other solutions approach \(E_{1}\).