Canard cycles in a slow-fast Holling-Tanner predator-prey model incorporating a prey refuge

In this paper, we will study the Holling-Tanner predator-prey model with a prey refuge, considering that the intrinsic growth rate of the predator is much lower than that of the prey. We first discuss the dynamic behavior of the system in the presence of positive equilibrium. Using geometric singular perturbation theory and the slow-fast normal form, we show that the system exhibits a singular Hopf bifurcation near the fold point. This bifurcation is accompanied by a canard explosion, validated through Melnikov integral analysis. Then, we prove the existence of canard limit cycles near the canard point using slow divergence integral methods, and the existence of relaxation oscillations via entry-exit function analysis. Our proof shows that the introduction of the prey refuge significantly enhances the global stability of the system. The refuge can balance the predation pressure through the buffer effect, which plays an important role in regulating the sustainability of the ecosystem.