Predator–prey model with sigmoid functional response

The sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature.