Bifurcation analysis of a discrete type prey-predator model with Michaelis–Menten harvesting in predator

Abstract

Abstract A discrete predator–prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered in the present work. Three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition. The local stability of the three equilibria has been analyzed. The interior equilibrium point is locally asymptotically stable under a parametric condition. It is examined that a flip and Neimark–Sacker bifurcations have occurred in the system at the axial equilibrium point. The flip and Neimark–Sacker bifurcations have been analyzed by the center manifold theorem and bifurcation theory, considering the harvesting coefficient as the bifurcation parameter. The proposed discrete model with a nonlinear Michaelis–Menten type harvesting effect on the predator population exhibits rich dynamics; for instance, bifurcations, chaos, and more complex dynamical behaviors. The discrete-time model also produced few numerical simulation results that are more accurate than the continuous model. The proposed discrete model will be performed better than the continuous model in populations with non-overlapping generations or smaller densities. The harvesting coefficient’s optimal value has finally been identified, and an optimal harvesting policy has been introduced. To verify the results, further numerical simulations have been performed.