Holling-Tanner model with Beddington-DeAngelis functional response and time delay introducing harvesting

•Holling-Tanner model with Beddington-DeAngelis Functional Response is analyzed.•Harvesting on prey and Gestational delay of predator into the system are introduced.•The direction and stability of Hopf bifurcation are established.•A profit function with the help of bionomic equilibrium is calculated.•Numerical simulations with graphical are included to verify the analytic results.

The paper is formulated with the Holling-Tanner prey-predator model with Beddington-DeAngelis functional response including prey harvesting. Gestational time delay of predator and the dynamic stability of time delay preventing system are incorporated into the system of our paper. The equilibria of the proposed system are determined and the existence of interior equilibrium point for the proposed system is described. Local stability of the system with the magnitude of time delay at the interior equilibrium point is discussed. Thereafter, the direction and the stability of Hopf bifurcation are established with the help of normal theory and center manifold theorem. Furthermore, profit function is calculated with the help of bionomic equilibrium and it is optimized using optimal control. Finally, some numerical simulations are introduced to verify the validity of analytic results of our proposed model.