# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 6 Discussion  
In this paper, the dynamics of the predator-prey model (3) with cooperative hunting are considered. The parameter space \(\Lambda\) of \((\beta,c)\) involved in (3) is divided into different regions: \(\Lambda=\Lambda_{1}\cup\Lambda_{2}\), \(\Lambda_{1}=\Lambda_{1a}\cup\Lambda_{1b}\), \(\Lambda_{2}=\Lambda_{2a}\cup\Lambda_{2b}\cup\Lambda_{2c}\), \(\Lambda_{2b}=\Lambda_{2b-1}\cup\Lambda_{2b-2}\cup\Lambda_{2b-3}\) (see Figure 3). The division is very important for the dynamic analysis of System (3). In each region, as \(d\) changes, the dynamic behaviors of (3) are presented through theoretical analysis and numerical simulations, including the number of equilibria (see Theorem 2 and Figure 2); the stability of equilibria (see Theorems 3 and 4); the bifurcation phenomena, including the Hopf bifurcation (see Theorem 5), saddle-node bifurcation and Bogdanov-Takens bifurcation (see Theorem 6 and Figure 4), and homoclinic bifurcation (see Figure 13); and the existence and non-existence of limit cycles (see Theorems 9-13 and Figures 8 and 10-13). Based on the theoretical analysis and numerical simulations, a complete description of the dynamic behaviors of (3) for \(d\), along with the corresponding biological explanations, is presented for each region. If the predator is without cooperative hunting, then (3) becomes the following well-known Lotka-Volterra model:  
\[\frac{\mathrm{d}x}{\mathrm{d}t}=cxy-dx,\quad\frac{\mathrm{d}y}{\mathrm{d}t}=y(1-y )-xy. \tag{31}\]  
The dynamics of System (31) are quite clear. System (31) has two boundary equilibria: \(E_{0}=(0,0)\), which is always a saddle, and \(E_{1}=(0,1)\), which is locally asymptotically stable if \(d\geq c\) and a saddle if \(d<c\). If \(d<c\), there exists a unique interior equilibrium \(E^{*}=(1-\frac{d}{c},\frac{d}{c})\), which is globally asymptotically stable. If \(d\geq c\), \(E_{1}\) is globally asymptotically stable, which implies that the high death rate of the predator leads it to become extinct. Comparing Model (3) to the Lotka-Volterra model (31), we find that System (3) has exactly the same dynamics as the Lotka-Volterra model (31) only when \((\beta,c)\in\Lambda_{1a}\) (see Theorem 8). When \((\beta,c)\in\Lambda_{2a}\cup\Lambda_{2}\), due to the nonlinearity of the cooperative hunting term \(-\beta x^{2}y\), Model (3) exhibits more complex dynamic behavior than (31), including rich bifurcation phenomena such as the Hopf bifurcation (see Theorem 5), saddle-node bifurcation and Bogdanov-Takens bifurcation (see Theorems 6), and homoclinic bifurcation (see Figure 13). Moreover, we can make the following statements, which answer the questions posed in Section 1. 1. If the mortality of the predator is small such that \(d<c\), then both species coexist regardless of the intensity of hunting cooperation. The coexistence of the two species exhibits either a steady mode or an oscillatory mode. If the Hopf bifurcation occurs, it only changes the coexistence mode. Therefore, compared to the Lotka-Volterra model (31), when \(d<c\), the hunting cooperation term \(-\beta x^{2}y\) does not affect the coexistence of populations but it affects the pattern of coexistence. 2. When \((\beta,c)\in\Lambda_{2a}\cup\Lambda_{2b}\) and \(c<d<F(x_{0})\), or \((\beta,c)\in\Lambda_{2c}\) and \(c<d<d^{\sharp}\), Model (3) has two attractors (bistability): one is the boundary equilibrium \(E_{1}(0,1)\) and the other is the interior equilibrium \(E_{2}^{*}\) or the stable limit cycle surrounding \(E_{2}^{*}\). The separatrices of the attracting basins of these two attractors are the stable manifolds of the interior equilibrium \(E_{1}^{*}\). This indicates that two species may coexist but this depends on the initial values. For the Lotka-Volterra model (31), when \(d>c\), the predator becomes extinct regardless of the initial value. Therefore, hunting cooperation is beneficial to the coexistence of the prey and predator.