Super-Explosion and Inverse Canard Explosion in a Piecewise-Smooth Slow-Fast Leslie-Gower Model

In this paper, we study a slow-fast Leslie-type predator-prey model with piecewise-linear functional response. Our approach is based on the geometric singular perturbation theory and the canard theory. When the fold point of the critical curve is lower than the transcritical bifurcation point, theoretical and numerical analyses show that a supercritical super-explosion occurs near the non-smooth corner followed by an inverse canard explosion close to the smooth fold. The critical values of parameters corresponding to these dynamical behaviors are obtained. Moreover, a stable relaxation oscillation is generated by the supercritical super-explosion, which will vanish as the occurrence of the inverse canard explosion.