Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model

Abstract

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

Keywords

• discrete predator-prey model
• fold bifurcation
• 1:1 strong resonance bifurcation
• fold-flip bifurcation
• 1:2 strong resonance bifurcation