Stability and Bifurcation Dynamics of a Discrete-Time Predator-Prey Model with Proportional Harvesting

The paper deals with a comprehensive analysis of a discrete-time model of prey-predator with proportional harvesting in the prey population. We show the positivity and boundedness of solutions which is relevant for the biological viability. Rigorous proofs for the existence and uniqueness of the three equilibria—extinction, prey-only, and coexistence—are shown. A detailed sensitivity analysis for the parameters is carried out and local stability conditions are derived with the help of the Jury criterion. The analytical calculation of the first Lyapunov coefficient is carried out to show the supercritical character of the Neimark-Sacker bifurcation. Theoretical results are validated by numerical simulations, such as time-series, phase portraits, bifurcation diagrams and stability maps. We show that the proportional harvesting is a double-edged sword, it both reduces population densities, and yet it can suppress chaotic oscillations at moderate harvesting intensity. Our findings can be used to offer quantitative recommendations for sustainable harvesting in ecological management.