# Global Dynamics of a Predator-Prey System with Cooperative Hunting
## 1 Introduction
 \tag{2}\]  
The study of [PERSON] and [PERSON] [32] indicates that Model (2) has more complex dynamics than the well-known Lotka-Volterra model (i.e., when \(\alpha=0\)), and hunting cooperation can be beneficial to the predator population by increasing the attack rate brought about by the cooperation. In [32], through numerical simulations, the stability of the equilibria and the various bifurcation behaviors of (2), including the Hopf bifurcation, homoclinic bifurcation, and Bogdanov-Takens bifurcation, were investigated for one-parameter cases (bifurcation parameter is taken as \(\alpha\) and \(\sigma\), respectively) and two-parameter cases (bifurcation parameter is taken as \((\alpha,\sigma)\) and \((k,\alpha)\), respectively). However, in [32], the important parameter conditions that describe the dynamical behavior of (2) were not provided mathematically. For example, the exact parameter regions such that Model (2) has zero, one, and two interior equilibria, the parametric conditions for the stability of equilibria, and the critical parameter value for the Hopf bifurcation were not explicitly presented. Motivated by the works of [PERSON] and [PERSON] [32], in this paper, we also study the dynamics of (1) but apply changes to the variables \(x=\frac{\lambda}{r}P,y=\frac{N}{K}\), and \(\tilde{t}=rt\), and define the constants as \(c=\frac{e\lambda K}{r}\), \(\beta=\frac{ra}{\lambda^{2}}\), and \(d=\frac{m}{r}\). After dropping the \"tilde\" on \(\tilde{t}\), we rewrite Model (1) in the following non-dimensional form  
\[ \begin{split}\frac{\mathrm{d}x}{\mathrm{d}t}&=c(1+ \beta x)xy-dx,\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=y(1-y)-(1+\beta x) xy. \end{split}  \tag{3}\]  
where the population density of the predator and prey are represented by \(x\) and \(y\), respectively. In order to better understand the dynamics of (3), such as the number of interior equilibria, stability of the equilibria, and Hopf bifurcation, we divide the two-parameter space  
\[\Lambda=\{(\beta,c)\in\mathbb{R}_{+}^{2}:c\beta>0\}. \tag{4}\]  
into several mutually exclusive regions. The idea and method of the division are motivated by [33] and are described in this paper, along with an analysis of the dynamic behaviors of (3). In each parameter region, by choosing \(d\) as the bifurcation parameter, the dynamics of (3) are explored in detail. Based on the dynamics of (3), we investigate the following two questions: (i) What are the parameter regions in which both species coexist? (ii) How does hunting cooperation affect the coexistence of the two species and the pattern of coexistence?The rest of the paper is organized as follows. First, we show the positivity and boundedness of the solutions to System (3) in Section 2. In Sections 3 and 4, as \(d\) varies, the existence and stability of equilibria are investigated in each parameter region. The Hopf bifurcation of (3) and its directions, together with the stability of bifurcating periodic orbits, are discussed. It is presented that there is a critical surface \(d=H(\beta,c)\) under which a supercritical and backward Hopf bifurcation occurs. In Section 5, through theoretical analysis and numerical simulations, the global stability and the existence of periodic solutions are considered, and a complete description of the global dynamic behaviors of (3) is presented for each parameter region. Finally, in Section 6, we summarize our main results and compare the dynamics of (3) to those of the well-known Lotka-Volterra model (i.e., when the predator has no cooperative hunting). It turns out that due to cooperative hunting, System (3) has a richer and more complex dynamic behavior than the Lotka-Volterra model, such as various bifurcation phenomena and oscillation behavior. Cooperative hunting by predators is beneficial to the coexistence of the prey and predator. When the mortality of the predator is small (\(d<c\)), the hunting cooperation term \(-\beta x^{2}y\) does not affect the coexistence of populations but it affects the pattern of coexistence.