Complex dynamics of a discrete predator–prey model with the prey subject to the Allee effect

Abstract

In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.

Keywords:

• Allee effect
• predator–prey model
• flip bifurcation
• Neimark–Sacker bifurcation
• codimension-two bifurcation
• Marotto’s Chaos

AMS Subject Classifications:

• 92D25
• 39A30

Acknowledgements

Authors would like to thank the anonymous referees for their helpful comments and suggestions which led to an improvement of original manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is partially supported by the National Natural Science Foundation of China [grant number 11571170], [grant number 61174155], [grant number 31570417]; the Natural Science Foundation of Anhui Province of China [grant number 1608085MA14], [grant number 1608085MC63]; the Key Project of Natural Science Research of Anhui Higher Education Institutions of China [grant number KJ2015A152].