# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 1 Introduction  
In an ecosystem, cooperative hunting behavior among predators is an important biological phenomenon [1] that is frequently observed in carnivores [2] such as African wild dogs [3], wolves [4], lions [5; 6; 7], chimpanzees [8], and other organisms such as spiders [9], birds [10; 11], and ants [12]. Cooperative hunting has many advantages for the evolution of predator populations. For example, cooperation increases hunting success, prey mass, and the probability of multiple kills. It also decreases the chase distance [3], increases the probability of capturing large prey [13], enables the population to find food more quickly [14], and allows individuals in large packs to accrue foraging advantages by helping prevent the carcass from being stolen by other predators [15]. In recent years, various predator-prey mathematical models with hunting cooperation have been proposed and studied. [PERSON] et al. [16] proposed a predator-prey model with a functional response for predators that forage in a spatially linear formation and aggregate when they encounter a cluster of prey. [PERSON] [17] derived a predator-prey model with cooperative foraging, which is described by a Holling type II-like functional response, and investigated the impacts of foraging facilitation among predators. Due to this functional response, it was presented in [17] that hunting cooperation has a destabilizing effect on predator-prey dynamics and the populations exhibit oscillatory behavior. [PERSON] et al. [18] proposed a three-species food chain model with hunting cooperation for the predator and analyzed the dynamics of the species under several degrees of cooperative hunting. [PERSON] et al. [19] considered a modified Leslie-Gower predator-prey model, where predators cooperate during hunting and prey populations show anti-predator behavior due to fear of predation risk. It was observed that in the absence of the fear effect, hunting cooperation can induce both supercritical and subcritical Hopf bifurcations. Recently, [PERSON] et al. [20] proposed a predator-prey model with a non-differentiable functional response, where the prey exhibits group defense and the predator exhibits cooperative hunting, and explored the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state, as well as the dynamics of the system. We also refer thereader to [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31] for models of ordinary differential equations, reaction-diffusion equations, and difference equations with hunting cooperation. [PERSON] and [PERSON] [32] studied the following predator-prey model with cooperative hunting in predators  
\[ \begin{split}\frac{\mathrm{d}N}{\mathrm{d}t}&=rN \Big{(}1-\frac{N}{K}\Big{)}-(\lambda+aP)NP,\\ \frac{\mathrm{d}P}{\mathrm{d}t}&=e(\lambda+aP)NP- mP.\end{split}  \tag{1}\]  
where \(N\) and \(P\) represent the population density of the prey and predator, respectively; \(r\) and \(K\) are the intrinsic growth rate and carrying capacity of the prey population, respectively; \(e\) is the conversion coefficient; \(m\) is the mortality of the predator; \(\lambda\) is the attack rate of each predator on the prey; and \(a\) is a parameter describing predator cooperation in hunting. All these parameters are positive. By introducing the dimensionless variables \(n=\frac{e\lambda}{m}N\), \(p=\frac{\lambda}{m}P\), and \(\tau=mt\), and the dimensionless parameters \(\sigma=\frac{r}{m}\), \(k=\frac{e\lambda K}{m}\), and \(\alpha=\frac{am}{\lambda^{2}}\), and still using \(t\) for \(\tau\), Model (1) is written as the following three-parameter system  
\[ \begin{split}\frac{\mathrm{d}n}{\mathrm{d}t}&=\sigma n \Big{(}1-\frac{n}{k}\Big{)}-(1+\alpha p)np,\\ \frac{\mathrm{d}p}{\mathrm{d}t}&=(1+\alpha p)np-p. \end{split}