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Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III. (English) Zbl 1418.34103

Summary: For a classical ratio-dependent predator-prey model with the generalized Holling type III functional response, it was previously investigated in [S.-B. Hsu and T.-W. Huang, SIAM J. Appl. Math. 55, No. 3, 763–783 (1995; Zbl 0832.34035)] for global stability of an equilibrium, and in [J. Huang et al., J. Differ. Equations 257, No. 6, 1721–1752 (2014; Zbl 1326.34082)] 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.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
92D25 Population dynamics (general)
34E15 Singular perturbations for ordinary differential equations
34E17 Canard solutions to ordinary differential equations
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