Impact of Michaelis-Menten type harvesting of predators in a predator-prey model with Holling type II functional response and Allee effect on prey

We propose a predator-prey system with Holling type II functional response incorporating both the Allee effect in the growth of the prey population and a nonlinear Michaelis-Menten type harvesting in predator. We provide a detailed mathematical analysis of the proposed model, including, positivity and boundedness of solutions, uniform persistence, existence, and local and global asymptotic stability of equilibria. Detailed bifurcation analysis is carried out and it is observed that the proposed system exhibits very complex dynamics and many local and global bifurcations as transcritical, pitchfork, saddle-node, Hopf, homoclinic, and Bogdanov-Takens have been identified. We observe the bi-stability and tri-stability in the system, so the basins of attraction in all possible cases of the existence of multiple attractors are discussed in detail. The system shows different types of bi-stabilities behavior in the case of strong and weak Allee effect and different types of tri-stability in the case of strong Allee effect. Extensive numerical simulations are performed for supporting evidence of our analytical findings. According to our analysis, the proposed model allows the development of a harvesting policy that can prevent the extinction of predator and prey populations. In the case of weak Allee effect, the maximum threshold for continuous predator harvesting without the extinction risk of both species is obtained. In the case of strong Allee effect, the optimal harvesting threshold has been also determined, but optimal harvesting rate of the predator population can only promote the coexistence of the population whenever the Allee effect is quite low, otherwise, the predator harvesting ceases to have any stabilizing effect.