Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III

Abstract For a classical ratio-dependent predator-prey model with the generalized Holling type III functional response, it was previously investigated in [20] by Hsu and Huang for global stability of an equilibrium, and in [21] by Huang, Ruan and Song for subcritical Hopf and Bogdanov-Takens bifurcations. Here in this model when prey reproduces much faster than predator, by using geometric singular perturbation theory, we achieve much richer new dynamical phenomena than the existing ones, such as the existence of canard cycles, canard explosion and relaxation oscillations, heteroclinic and homoclinic orbits, cyclicity of slow-fast cycles, and the coexistence of the Hopf cycle and the relaxation oscillation. On global stability of the equilibrium we also provide less restricted conditions than the existing ones.

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