Dynamics of the predator-prey model with the Sigmoid functional response

For the predator-prey model with the Sigmoid functional response, the known result is on the global stability of its positive equilibrium when it is locally stable. Here, we characterize existence of particular type of limit cycles using qualitative theory and geometric singular perturbation methods. The main results are as follows. If the positive equilibrium exists and is a weak focus, it is a stable weak focus of order 1. The positive equilibrium is unique, and it is either globally stable or unstable and there exists a limit cycle surrounding it. The limit cycle could be a canard cycle without head, or a canard cycle with head, or a relaxation oscillation. The system could present canard explosions consecutively two times, which first births from one canard point, via relaxation oscillation for a large range of parameter values, and then there exhibits an inverse canard explosion and disappears at another canard point with the parameter variation.

Automated summary

This study investigates the global stability and characteristics of limit cycles in the predator-prey model with a Sigmoid functional response. It is found that when the positive equilibrium is a weak focus, it is a stable weak focus of order 1, and various types of limit cycles may exist, including canard cycles and relaxation oscillations.