Analyzing effect of harvesting on prey population when prey growth depend on fear-factor and Allee-effect

A mathematical model of predator-prey considering Allee effect, fear factor and harvesting in prey population constructed in this topic. Meanwhile, anti-predation involved in predator population such that prey can counterattack their predators. The second type of Holling type functional response chosen to describe a predator who active searching for prey. We non-dimensionalize our model to reduce the number of parameters, then we analyzed the existence and local stability criteria of all equilibrium points analytically. There are maximum four equilibrium are found, where the extinction of predator population equilibrium might not unique. Our results suggest to consider the rate of harvesting in prey population wisely to guarantee the coexistence of prey and predator in the environment. Our results also show that the level of fear of prey might effect the final size of coexistence equilibrium. Our numerical results show that although the time scale do not effect the size of the coexistence equilibrium, it does change the speed of the system to reach the equilibrium point.