# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 4 Stability of Equilibria
### Stability of Interior Equilibria and Hopf Bifurcation  
Now, we consider the local stability of the interior equilibria of (3). At an interior equilibria \(E^{*}=(x^{*},y^{*})\), the Jacobian matrix (9) becomes  
\[J|_{E^{*}}= \begin{pmatrix}c\beta x^{*}y^{*}&c(1+\beta x^{*})x^{*}\\ -(1+2\beta x^{*})y^{*}&-y^{*}\end{pmatrix} . \tag{13}\]  
The characteristic equation is given by  
\[\kappa^{2}-\omega_{1}\kappa+\omega_{2}=0. \tag{14}\]  
where \(\omega_{1}=\mathrm{Tr}J|_{E^{*}},\omega_{2}=\mathrm{Det}J|_{E^{*}}\). By using \(y^{*}=\phi_{1}(x^{*})=\frac{d}{c(1+\beta)x^{*}}\) and \(y^{*}=\phi_{2}(x^{*})=-\beta(x^{*})^{2}-x^{*}+1\), we have  
\[\omega_{1}=(c\beta x^{*}-1)y^{*}=d-(c+1)y^{*}=(c\beta x^{*}-1)\phi_{2}(x^{*}), \tag{15}\]and  
\[\omega_{2}=-x^{*}y^{*}(c\beta y^{*}-c(1+\beta x^{*})(1+2\beta x^{*}))=-x^{*}y^{*}F^ {\prime}(x^{*}). \tag{16}\]  
Note that it is very important to observe that the determinant \(\text{Det}|_{E^{*}}\) is expressed by \(F^{\prime}(x^{*})\), that is, \(\text{Det}|_{E^{*}}=-x^{*}y^{*}F^{\prime}(x^{*})\). If \(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\) exists, it is a saddle since \(F^{\prime}(x_{1}^{*})>0\); if \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) exists, its stability is determined by the sign of \(\omega_{1}=\text{Tr}|_{E_{2}^{*}}\) since \(F^{\prime}(x_{2}^{*})<0\) (see Remark 1). Thus, from (15), we can directly obtain the following result. Lemma 1: \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) _is a sink if \(x_{2}^{*}<\frac{1}{c\beta}\) and a source if \(x_{2}^{*}>\frac{1}{c\beta}\)._  
Lemma 1 does not explicitly reveal whether \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is stable or unstable in explicit parameter regions. For this reason, we need to conduct an in-depth analysis of different parameter regions. Obviously, for \(\beta>0\),  
\[\frac{1}{c\beta}<x_{max}^{*}\quad\text{if and only if}\quad c>f_{1}(\beta):= \frac{2}{\sqrt{1+4\beta-1}}, \tag{17}\]  
and for \(\beta>1\),  
\[\frac{1}{c\beta}<x_{0}=\frac{-2+\sqrt{3\beta+1}}{3\beta}\quad\text{if and only if}\quad c>f_{2}(\beta):=\frac{3}{\sqrt{1+3\beta}-2} \tag{18}\]  
Based on (17) and (18), we divide the parameter region \(\Lambda_{1}\) into the following two subregions (see Figure 3):  
\[\Lambda_{1a}=\{(\beta,c)\in\Lambda_{1}:c\leq f_{1}(\beta)\},\quad\Lambda_{1b} =\{(\beta,c)\in\Lambda_{1}:c>f_{1}(\beta)\}. \tag{19}\]  
Also, the parameter region \(\Lambda_{2}\) is divided into the following three subregions (see Figure 3):  
\[ \begin{split}&\Lambda_{2a}=\{(\beta,c)\in\Lambda_{2}:c\leq f_{1}( \beta)\},\quad\Lambda_{2b}=\{(\beta,c)\in\Lambda_{2}:f_{1}(\beta)<c<f_{2}( \beta)\},\\ &\Lambda_{2c}=\{(\beta,c)\in\Lambda_{2}:c\geq f_{2}(\beta)\}. \end{split}  \tag{20}\]  
Now, we apply the monotonicity of \(x_{2}^{*}\) on \(d\) (see Remark 1) to analyze the stable parameter region of the interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\). Define \(d^{*}=F\Big{(}\frac{1}{c\beta}\Big{)}\), that is,  
\[d^{*}=H(\beta,c):=(1+c)\bigg{(}1-\frac{1}{c\beta}-\frac{1}{c^{2}\beta}\bigg{)}. \tag{21}\]  
Note that \(d^{*}>0\) is equivalent to \(c>f_{1}(\beta)\). First, we consider the stability of \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) for the case of \((\beta,c)\in\Lambda_{1}\). According to Theorem 2, System (3) has the unique interior equilibrium \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) when \(d\in(0,c)\). 1. Let \((\beta,c)\in\Lambda_{1a}\), then \(x_{max}^{*}\leq\frac{1}{c\beta}\), as indicated by (17). According to Remark 1, if \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) exists, then \(x_{2}^{*}<x_{max}^{*}\). Hence, for all \(d\in(0,c)\), \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is a sink according to Lemma 1. 2. Let \((\beta,c)\in\Lambda_{1b}\), then \(\frac{1}{c\beta}<x_{max}^{*}\). In addition, (21) implies that when \(d=d^{*}\), \(x_{2}^{*}=\frac{1}{c\beta}\). Then, from the monotonicity of \(x_{2}^{*}\) on \(d\) (see Remark 1), we know that \(x_{2}^{*}>\frac{1}{c\beta}\) for \(d\in(0,d^{*})\) and \(x_{2}^{*}<\frac{1}{c\beta}\) for \(d\in(d^{*},c)\). It follows from Lemma 1 that \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) is a source for \(d\in(0,d^{*})\) and a sink for \(d\in(d^{*},c)\).