Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

Abstract

This paper studies bifurcations of canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Based on geometric singular perturbation theory (GSPT) and canard theory, canard explosion is observed and the associated bifurcation curve is determined. Due to the canard point, a homoclinic orbit with slow and fast segments and homoclinic to a saddle can also exist, in which, the stable and unstable manifolds of the saddle are connected under certain parameter value. By analyzing the slow divergence integral, it is proved that the cyclicity of canard cycles in this model is at most four. Finally, by calculating the entry-exit function explicitly, a unique, orbitally stable canard relaxation oscillation passing through a transcritical bifurcation point is detected. All these theoretical predictions on the birth of canard explosion, canard limit cycles and homoclinic orbits are verified by numerical simulations.