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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 1613709, 12 pages
Research Article

The Bifurcation of Two Invariant Closed Curves in a Discrete Model

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Correspondence should be addressed to Yicang Zhou; nc.ude.utjx@cyuohz

Received 2 March 2018; Accepted 24 April 2018; Published 30 May 2018

Academic Editor: Guang Zhang

Copyright © 2018 Yingying Zhang and Yicang Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.