From Instability to Pest Eradication: Linear Harvesting in a Modified Holling–Tanner System

This study analyzes a modified Holling–Tanner predator–prey system with linear harvesting and supplementary food for the predator. The framework examines how harvesting interacts with predation and external resources to determine system dynamics. We derive explicit conditions for the existence and stability of all equilibria and identify a critical predation threshold separating stable coexistence from oscillatory dynamics. Harvesting acts as a control parameter that can suppress oscillations, eliminate interior equilibria, and drive the system toward a prey-free state. We establish sufficient conditions for pest eradication by linking harvesting intensity, predation rate, and the loss of coexistence equilibria. Local bifurcation analysis reveals Hopf and saddle–node bifurcations, marking transitions between steady states and periodic oscillations. For the spatially extended system, diffusion-driven instability is investigated, and conditions for Turing pattern formation are derived from the modified equilibrium structure. Numerical simulations support the analytical results and illustrate transitions between dynamical regimes under varying harvesting levels. The results provide explicit parameter thresholds governing stabilization, oscillation, and eradication in predator–prey systems with external resource support.