# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 3 Equilibria  
In this section, we consider the number of equilibria in System (3). Clearly, System (3) always has two boundary equilibria \(E_{0}(0,0)\) and \(E_{1}(0,1)\). Below, we discuss the existence of interior equilibria. System (3) has the \(x\)-nullcline  
\[y=\phi_{1}(x):=\frac{d}{c(1+\beta x)}\]  
and the \(y\)-nullcline  
\[y=\phi_{2}(x):=-\beta x^{2}-x+1.\]  
It is easy to see that \(y=\phi_{1}(x)\) decreases to zero as \(x\) increases on \([0,\infty)\) with \(\phi_{1}(0)=\frac{d}{c}\). \(y=\phi_{2}(x)\) is a parabola with \(\phi_{2}(0)=1\) and has a unique positive zero, denoted as  
\[x^{*}_{max}=\frac{-1+\sqrt{1+4\beta}}{2\beta}. \tag{5}\]\((x^{*},y^{*})\) is an interior equilibrium point of (3) if and only if it is the intersection point of \(y=\phi_{1}(x)\) and \(y=\phi_{2}(x)\) in the first quadrant of the \((x,y)\)-plane (see Figure 1). Therefore, System (3) has at most two interior equilibria. That is, \((x^{*},y^{*})\) is an interior equilibrium point if and only if \(x^{*}\) is a positive root of \(\phi_{1}(x)-\phi_{2}(x)=0\), i.e., \(x^{*}\) satisfies \(F(x)=d\), and \(y^{*}=\phi_{1}(x^{*})\). Here,  
\[F(x)=c(1+\beta x)\phi_{2}(x)=c(-\beta^{2}x^{3}-2\beta x^{2}+(\beta-1)x+1). \tag{6}\]  
Clearly, \(F(0)=c\). Therefore, we only need to discuss the positive intersection point of the functions \(u=F(x)\) and \(u=d\). When there are two interior equilibria, the larger one of the predator is denoted as \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\), and the other one is denoted as \(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\). In (6), the sign of the coefficient \(\beta-1\) is important for determining the graph of the function \(F(x)\) and the range of the parameter \(d\) such that System (3) has zero, one, and two interior equilibria. So, we choose \(d\) as the bifurcation parameter. Based on the sign of \(\beta-1\), the coefficient of the first-order term of \(F(x)\), we divide the parameter space \(\Lambda\) given in (4) into  
\[\Lambda_{1}=\{(\beta,c)\in\Lambda:\beta\leq 1\},\quad\text{and}\quad\Lambda_{2 }=\{(\beta,c)\in\Lambda:\beta>1\}. \tag{7}\]  
If \((\beta,c)\in\Lambda_{1}\), from  
\[F^{\prime}(x)=c(-3\beta^{2}x^{2}-4\beta x+\beta-1)\]  
we know that \(F(x)\) is decreasing on \([0,\infty)\), and for all \(x>0\), \(F(x)<F(0)=c\) (see Figure 2a). If \((\beta,c)\in\Lambda_{2}\), then at the point \(x=x_{0}\), where  
\[x_{0}=\frac{-2+\sqrt{3\beta+1}}{3\beta}, \tag{8}\]  
\(F(x)\) has a positive local maximum value  
\[F(x_{0})=\frac{c(2(3\beta+1)^{\frac{3}{2}}+9\beta+2)}{27\beta}.\]  
Figure 1: The \(x\)-nullcline and \(y\)-nullcline. When \(x\in(0,x_{0})\), \(F(x)\) increases and \(c=F(0)<F(x)<F(x_{0})\); when \(x>x_{0}\), \(F(x)\) decreases and \(F(x)<F(x_{0})\) (see Figure 2b). Thus, we can obtain the following result. **Theorem 2**. : _System (3) may have zero, one, or two interior equilibria. More precisely:_  
1. _If_ \((\beta,c)\in\Lambda_{1}\)_, then System (_3_) has a unique interior equilibrium_ \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) _when_ \(d<c\)_, and no interior equilibrium when_ \(d\geq c\) _(see Figure_ 2a)._
2. _If_ \((\beta,c)\in\Lambda_{2}\)_, then System (_3_) has two interior equilibria_ \(E_{1}^{*}(x_{1}^{*},y_{1}^{*})\) _and_ \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\)_, with_ \(x_{1}^{*}<x_{0}<x_{2}^{*}\) _when_ \(c<d<F(x_{0})\)_; one interior equilibrium_ \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\) _when_ \(d\leq c\)_; and no interior equilibrium when_ \(d>F(x_{0})\)_. When_ \(d=F(x_{0})\)_,_ \(E_{1}^{*}\) _and_ \(E_{2}^{*}\) _coicide and System (_3_) has a unique interior equilibrium_ \(\tilde{E}^{*}(x_{0},\phi_{1}(x_{0}))\) _(see Figure_ 2b)._  
**Remark 1**. : _Based on the above arguments, we can make the following statements._  
1. _When the cooperative predation rate of the predator is small (_\(\beta\leq 1\)_), (_3_) has at most one coexistence equilibrium, whereas if_ \(\beta\) _is large (_\(\beta>1\)_), there may exist two coexistence equilibria._
2. _If_ \(d<c\)_, then for each_ \(\beta>0\)_, System (_3_) always has a unique interior equilibrium_ \(E_{2}^{*}(x_{2}^{*},y_{2}^{*})\)_._
3.