Complicate bifurcation behaviors of a discrete predator–prey model with group defense and nonlinear harvesting in prey

Abstract

In this paper, some complicate dynamical behaviors are formulated for a discrete predator–prey model with group defense and nonlinear harvesting in prey. After considering the existence and stability for all of its nonnegative fixed points, our main work is to present those conditions for the occurrences of transcritical bifurcation, saddle-node bifurcation and Neimark–Sacker bifurcation, respectively. Numerical simulations not only verify the theoretical results for saddle-node bifurcation and Neimark–Sacker bifurcation but also display more interesting dynamical properties of the model.

Keywords:

• Discrete predator–prey model
• group defense and nonlinear harvesting
• transcritical bifurcation
• saddle-node bifurcation
• Neimark–Sacker bifurcation
• center manifold theorem

2000 Mathematics Subject Classifications:

• 39A28
• 39A30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China [grant number 61473340], Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province [grant number 201805], and National Natural Science Foundation of Zhejiang University of Science and Technology [grant number F701108G14].