We analyze a discrete-time predator–prey model with density-dependent prey harvesting. Through analytical and numerical methods, we establish biologically feasible solution domains, characterize equilibria stability in the \((\mathcal {G}, \mathcal {E}, \alpha )\) -space, and identify codimension-1 and codimension-2 bifurcations. Key findings reveal critical \(1\!:\!k\) resonances ( \(k = 2, 3, 4\) ) where period-doubling and Neimark–Sacker manifolds intersect, acting as organizing centers for chaotic and multistable regimes. A paradoxical stabilization effect is observed: moderate harvesting suppresses chaos via inverse period-doubling cascades. We determine precise thresholds \(\mathcal {E}^{PD}\) and \(\mathcal {E}^{NS}\) that define sustainable harvesting boundaries. Increasing the harvesting effort \(\mathcal {E}\) induces stability-loss bifurcations that lead to chaotic dynamics; however, appropriately managed harvesting can stabilize oscillatory behavior. Population collapse occurs through three distinct pathways: abrupt collapse at 1:2 resonance, patchy extinction near 1:3 resonance, and delayed “echo collapse” associated with 1:4 resonance-each offering quantitative insight into conservation strategy and management.
