Dynamics of slow–fast modified Leslie–Gower model with weak Allee effect and fear effect on predator

In this paper, we focus on the dynamics of the modified Leslie–Gower model with a weak Allee effect and fear effect on predators. Assuming that the inherent growth rate of prey is much faster than that of predators, this becomes a singular perturbation problem. Based on the geometric singular perturbation theory, we prove the existence of canard cycles and relaxation oscillation and provide an asymptotic expression of the relaxation oscillation period. Furthermore, it demonstrates that, under certain conditions, the degenerate transcritical bifurcation point is a global attractor using geometric singular perturbation theory, the center manifold theorem and the blow-up method. Numerical simulations verify the theoretical results.