Impacts of hunting cooperation and prey harvesting in a Leslie-Gower prey-predator system with strong Allee effect

In this paper, we proposed and studied a Leslie-Gower prey-predator system which considered various ecological factors, such as the Allee effect and harvesting effect in prey populations and the hunting cooperation in predator populations. The positivity and boundedness of the system's solutions were determined. The conditions for the uniformly persistence of the system and the extinction of populations have been established. We studied the existence and type of singularities, including primary singularities and higher-order singularities. Using topological equivalent and the blow-up method, we proved that the origin was the attractor, and a defined basin of attraction was given. As the parameters change, the system may experience two saddle-node bifurcations and a Hopf bifurcation. The direction and stability of Hopf bifurcation solutions were established. Numerical simulations were given to validate the primary theoretical findings. In this paper, we found that there existed critical thresholds for Allee threshold, prey harvesting, and hunting cooperation, beyond which both predator and prey populations faced the risk of extinction.