Dynamics of a Harvested Predator-Prey Model with a Generalized Holling Type~III Response Capturing Group Defense
We investigate a predator-prey model with constant prey harvesting and a generalized Holling type III functional response, focusing on the biological and dynamical consequences of allowing the parameter $b$ to be negative. In this regime, predation efficiency declines at high prey densities, potentially reflecting group defense. This change permits an additional coexistence equilibrium and introduces new bifurcation mechanisms affecting population persistence. We establish the existence of a codimension-three Bogdanov--Takens bifurcation, derive a codimension-three nilpotent-saddle unfolding, and investigate a codimension-two generalized Hopf bifurcation and the emergence of multiple limit cycles. Our analysis show that moderate harvesting weakens the protection conferred by high prey abundance; excessive harvesting leaves too few prey to sustain predator growth; and low harvesting preserves strong group defense, requiring high predator density for population persistence.