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Affiliation of Author(s):理学院

Title of Paper:Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect

Journal:JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS

Key Words:Allee effect predator-prey model flip bifurcation Neimark-Sacker bifurcation codimension-two bifurcation Marotto's Chaos 92D25 39A30

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.

ISSN No.:1023-6198

Translation or Not:no

Date of Publication:2017-01-01

Co-author:Wu, Daiyong

Correspondence Author:zhy