Dynamics in a Holling II predator-prey model: interaction between prey refuge and Allee effects

This study examines an extended Leslie–Gower predator–prey model that integrates a Holling II functional response, a fixed proportion of prey refuge, and Allee effects acting on the prey population. Using normal form theory and symbolic computations, we prove the existence of high codimension bifurcations, including cusp type Bogdanov–Takens bifurcations of codimension 4, focus and elliptic type Bogdanov–Takens bifurcations of codimension 3, and a Hopf bifurcation of codimension 3. Our results show that prey refuge size critically mediates population persistence. Under a strong Allee effect or high predation pressure, a small refuge leads to co-extinction, whereas increasing the refuge size promotes coexistence through complex dynamical transitions. In contrast, under a weak Allee effect or low predation pressure, populations persist regardless of refuge size.