Multiparametric dynamics of a slow–fast modified Leslie–Gower predator–prey system with Beddington–DeAngelis functional response

We reveal the complete multiparametric dynamics of a slow–fast modified Leslie–Gower predator–prey system with Beddington–DeAngelis functional response. More specifically, the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the multiparametric slow–fast system and its location (or their locations) are found, and then the explicit conditions to further completely determine the dynamics of positive equilibria are derived. Besides, through studying the corresponding slow–fast normal form, we are able to classify its rich dynamics completely under corresponding parameters conditions, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits, and global attraction of equilibrium. Moreover, the conditions to guarantee these various rich dynamics are explicitly given, including the explicit necessary and sufficient conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which were not completely and explicitly characterized before. Additionally, we also find that the cyclicity of canard cycles without head is at most one under some explicit conditions. Finally, numerical simulations are employed to illustrate our theoretical analysis.