Harvesting-driven dynamics in a modified Holling-Tanner model

This paper presents a comprehensive bifurcation analysis of a modified Holling-Tanner model incorporating constant-yield prey harvesting, centered on the five-dimensional parameter space of the rescaled system. Utilizing rigorous analytical methods, we uncover the substantial impacts of harvesting on the dynamical architecture of the classical Holling-Tanner framework, demonstrating that such harvesting induces far richer complex dynamical behaviors and bifurcation phenomena than the unharvested model. We prove the existence of a codimension-4 degenerate Bogdanov-Takens bifurcation and a codimension-at-least-3 Hopf bifurcation, which advances theoretical knowledge of harvesting-induced complexity in predator-prey dynamics. The constant-yield harvesting is identified as a pivotal regulator of the system's qualitative behavior, exerting nontrivial effects on the existence and stability of positive equilibria and the codimension of critical bifurcations, and elevating the system's analytical complexity by transforming bifurcation structures from lower-codimension singularities to high-dimensional degeneracies. Comparative analysis with the unharvested model reveals that the harvested model exhibits higher-codimension cusp, degenerate Bogdanov-Takens and Hopf bifurcations, as well as the novel phenomenon of three coexisting limit cycles in specific parameter regimes, highlighting the synergistic interaction between harvesting and the model's inherent ecological structure. Ecologically, the detected high-codimension degeneracies carry critical implications for predator-prey population stability and resilience. Infinitesimal parameter perturbations near the parametric organizing center can trigger abrupt regime shifts, while nested limit cycles induce bistable dynamics: moderate perturbations are tolerated, but exceeding critical thresholds leads to catastrophic population collapse, aligning with empirical patterns in natural ecosystems.