# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Saddle Node and Cusp  
According to Theorem 2, when \((\beta,c)\in\Lambda_{2}\) and \(d=F(x_{0})\), System (3) has the unique interior equilibrium \(\tilde{E}^{*}(x_{0},y_{0})\), where \(x_{0}=\frac{-2+\sqrt{3\beta+1}}{3\beta}\) (see (8)), \(y_{0}=\phi_{1}(x_{0})\). Below, we discuss the dynamic behavior near \(\tilde{E}^{*}(x_{0},y_{0})\). Theorem 4.2: _Let \((\beta,c)\in\Lambda_{2}\) and \(d=F(x_{0})\). (1) If \(c>f_{2}(\beta)\), then \(\tilde{E}^{*}(x_{0},y_{0})\) is a repelling saddle node of codimension one. (2) If \(c<f_{2}(\beta)\), then \(\tilde{E}^{*}(x_{0},y_{0})\) is an attracting saddle node of codimension one. (3) If \(c=f_{2}(\beta)\), then \(\tilde{E}^{*}(x_{0},y_{0})\) is a Bogdanov-Takens-type cusp of codimension two._  
Proof. : At \(\tilde{E}^{*}(x_{0},y_{0})\), it is easy to see that \(\omega_{2}=0\) and the characteristic Equation (14) has two roots: \(\kappa_{1}=0\) and \(\kappa_{2}=\omega_{1}\). Therefore, \(\tilde{E}^{*}\) is a non-hyperbolic equilibrium. We use the center manifold theorem and the blow-up technique [34] to determine the type of \(\tilde{E}^{*}\). Let \(c\
eq f_{2}(\beta)\), which implies that \(\omega_{1}=(c\beta x_{0}-1)y_{0}\
eq 0\). Perform a transformation of \(X=x-x_{0}\), \(Y=y-y_{0}\) to translate \(\tilde{E}^{*}=(x_{0},y_{0})\) to the origin. For simplicity, we still use the variables \(x\) and \(y\) instead of \(X\) and \(Y\), respectively. Then, System (3) becomes  
\[ \begin{split}\frac{\mathrm{d}x(t)}{\mathrm{d}t}&=c \beta x_{0}y_{0}x+c(1+\beta x_{0})x_{0}y+c\beta y_{0}x^{2}+c(1+2\beta x_{0})xy+ c\beta x^{2}y,\\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}&=-(1+2\beta x_{0 })y_{0}x-y_{0}y-\beta y_{0}x^{2}-(1+2\beta x_{0})xy-y^{2}-\beta x^{2}y.\end{split}  \tag{24}\]  
Let \(x=-u+cx_{0}(1+\beta x_{0})v\) and \(y=(1+2\beta x_{0})u-y_{0}v\). Then, System (24) becomes  
\[ \begin{split}\frac{\mathrm{d}u(t)}{\mathrm{d}t}&=a_{ 1}u^{2}+a_{2}uv+a_{3}v^{2}+O(|(u,v)|^{3}),\\ \frac{\mathrm{d}v(t)}{\mathrm{d}t}&=\omega_{1}v+b_{1 }u^{2}+b_{2}uv+b_{3}v^{2}+O(|(u,v)|^{3}).\end{split}  \tag{25}\]  
Here,  
\[ \begin{split} a_{1}&=-\frac{c\beta x_{0}y_{0}}{ \omega_{1}}(2+3\beta x_{0}),\quad a_{2}=-\frac{cy_{0}}{\omega_{1}}[\omega_{1}+3 \beta x_{0}y_{0}+c\beta x_{0}^{2}(1+\beta x_{0})],\\ a_{3}&=\frac{cx_{0}y_{0}(1+\beta x_{0})}{\omega_{ 1}}[c^{2}\beta x_{0}^{2}(1+\beta x_{0})^{2}+\omega_{1}],\\ b_{1}&=\frac{1+2\beta x_{0}}{\omega_{1}}(c\beta x_{ 0}(1+2\beta x_{0})+(1+\beta x_{0})),\\ b_{2}&=-\frac{1}{\omega_{1}}[cy_{0}(1+2\beta x_{ 0})(1+2\beta x_{0}-c\beta x_{0})+y_{0}(c\beta x_{0}+\beta x_{0}+1)],\\ b_{3}&=-\frac{1}{\omega_{1}}[c^{2}\beta x_{0}y_{ 0}(1+2\beta x_{0})(cx_{0}(1+\beta x_{0})-y_{0})-c^{2}\beta x_{0}^{2}y_{0}(1+ \beta x_{0})^{2}-y_{0}\omega_{1}].\end{split} \]  
It is clear that the flow on the center manifold is given by \(\frac{\mathrm{d}u(t)}{\mathrm{d}t}=-\frac{c\beta x_{0}y_{0}}{\omega_{1}}(2+3 \beta x_{0})u^{2}+O(u^{3})\). So, according to the center manifold theorem, \(\tilde{E}^{*}\) is a saddle node, i.e., half of \(\tilde{E}^{*}\) is a node and the other half is a saddle. In order to determine the stability of the half node of \(\tilde{E}^{*}\), below, the blow-up technique [34] is applied to System (25). By applying the polar coordinate transformation \(u=\rho\cos(\theta)\), \(v=\rho\sin(\theta)\), System (25) is transformed into the following system  
\[ \begin{split}\frac{\mathrm{d}\rho}{\mathrm{d}t}&= \rho\omega_{1}\sin^{2}(\theta)+\rho^{2}[\cos(\theta)(a_{1}\cos^{2}(\theta)+a_{2} \cos(\theta)\sin(\theta)+a_{3}\sin^{2}(\theta))\\ &\quad+\sin(\theta)(b_{1}\cos^{2}(\theta)+b_{2}\cos(\theta)\sin( \theta)+b_{3}\sin^{2}(\theta))]+o(\rho^{2}),\\ \frac{\mathrm{d}\theta}{\mathrm{d}t}&=\omega_{1} \cos(\theta)\sin(\theta)+O(\rho).\end{split}  \tag{26}\]On the unit circle \(\{0\}\times\mathbb{S}\), System (26) clearly has four singular points \((0,\theta_{i})\) corresponding to \(\theta_{i}=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\), \(i=1,2,3,4\). If \(c<f_{2}(\beta)\), then \(\omega_{1}<0\) and \(a_{1}>0\).