High codimension bifurcations analysis in a predator-prey system with Michaelis-Menten type predator harvesting

Abstract

The main focus of this paper is to study some high codimension bifurcations of a predator-prey system with Michaelis-Menten type predator harvesting. In this paper, using the center manifold theorem, the cusp bifurcation of codimension two is discussed in detail. Furthermore, the Bautin bifurcation of codimension two and the Bogdanov-Takens bifurcation of codimension three are well studied by computing the second Lyapunov coefficient and using the normal form theory, respectively. Numerical simulations are provided to illustrate the theoretical results. The study of these bifurcations reveals the more complex dynamics of the system and may help us to better understand the complex interactions between predator and prey.