Rich dynamical behaviours for predator–prey model with weak Allee effect

Abstract

The weak Allee effect on the predator is introduced into the classic predator–prey model of Lotka–Volterra type. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. It is shown that the weak Allee effect can bring rich and complicated dynamics to the previous simple model, such as the saddle–node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov–Takens bifurcations, implying that weak Allee effect can be one of the simple reasons for many complicated behaviours in the predator–prey communities.

Keywords:

• predator–prey system
• weak Allee effect
• bifurcation
• limit cycle

AMS Subject Classifications::

• 92A15
• 34A47

Acknowledgements

We would like to thank two anonymous referees for their careful reading and very valuable comments, which led to an improvement of our original manuscript. We are very grateful to professors Huaiping Zhu, Wendi Wang, Sze-Bi Hsu and Ying-Hen Hsieh for their helpful comments and constructive suggestions. We also thank Mister Jinliang Wang for his kind help in numeric simulations. This work is supported by the National Natural Science Foundation of China (No. 10601042), Science Research Foundation in Harbin Institute of Technology (HITC200714) and program of excellent Team in Harbin Institute of Technology.