Global Stability and Canard Explosions of the Predator-Prey Model with the Sigmoid Functional Response

Abstract

For the predator-prey model with the sigmoid functional response whose denominator has no real roots, the dynamics has already been characterized. Whereas, when the denominator has a real root, the model is singular there, and its dynamics has not been classified. This paper focuses on the case that the denominator has a repeated real root and obtains the next results. The positive equilibrium (if exists) is globally stable provided that it is locally stable. If it is a weak focus, it must be of order one and stable. When the positive equilibrium is unstable, the system has always a limit cycle, which could come from a singular Hopf bifurcation, and undergoes two consecutive canard explosions via relaxation oscillations. In addition, numerical simulations reveal that the curvature of the critical curve at the canard point affects the period of the canard cycle.