Global dynamics of a singularly perturbed predator–prey system with Allee effect and small predator’s death rate

In this paper, we consider the global dynamics of a predator–prey system with simplified Holling type IV functional response and Allee effect in prey. Based on geometric singular perturbation theory, we focus on the nonstandard slow–fast dynamics of the system with a small death rate of the predator. We provide the existence and stability of the positive equilibrium and boundary equilibria under the fact that prey-isocline has a single interior local extremum in the first quadrant, whether the prey suffers from weak or strong Allee effect. In the case of weak Allee effect, we prove the existence and location of the unique relaxation oscillation type nontrivial stable periodic orbit by entry–exit function. Meanwhile, the global attractor is observed in the system with strong Allee effect. In addition, combined with the analysis of two equilibria at infinity by means of Poincaré transformation, the global phase portraits of predator–prey system are characterized.