Single-time and multi-time scales dynamic analysis in a predator–prey system with Michaelis-Menten type predator harvesting

The main focus of this paper is to study the dynamics of a predator–prey system with Holling type II functional response and Michaelis-Menten type predator harvesting. In this article, the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcations of codimension 2 and 3 are investigated under the assumption that both the prey and predator evolve in a single time scale. Furthermore, considering that the prey grows much faster than predator in general, the dynamic behavior of slow-fast system is also studied including the singular Hopf bifurcation, canard explosion, and transient dynamics by using the geometric singular perturbation theory and asymptotic expansion technique. Numerical simulations are provided to illustrate the theoretical results. The study presented in this paper may enable us to gain a better understanding of the complex interactions between predator and prey in both single and multiple time scales.