Dynamics of a slow–fast Leslie–Gower predator–prey model with prey harvesting

For the Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov–Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow–fast cycles. Here, we focus on the global dynamics of the model in the slow–fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.