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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 1613709, 12 pages
https://doi.org/10.1155/2018/1613709
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.

Linked References

  1. Z. He and X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-prey system,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 403–417, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. Z. Jing and J. Yang, “Bifurcation and chaos in discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 27, no. 1, pp. 259–277, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. Fazly and M. Hesaaraki, “Periodic solutions for a discrete time predator–prey system with monotone functional responses,” Comptes Rendus Mathematique, vol. 345, no. 4, pp. 199–202, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. N. Yi, P. Liu, and Q. Zhang, “Bifurcations analysis and tracking control of an epidemic model with nonlinear incidence rate,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 36, no. 4, pp. 1678–1693, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Z. Limin, Z. Chaofeng, and Z. Min, “Dynamic complexities in a discrete predator–prey system with lower critical point for the prey,” Mathematics and Computers in Simulation, vol. 105, pp. 119–C131, 2014. View at Google Scholar
  7. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NewYork, NY, USA, 2nd edition, 1997.
  8. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet
  9. B.-S. Goh, Management and analysis of biological populations, Elsevier, Amsterdam, The Netherlands, 1980.
  10. X. C. Huang and L. Zhu, “Limit cycles in a general Kolmogorov model,” Nonlinear Analysis, vol. 60, pp. 1393–1414, 2005. View at Google Scholar
  11. E. Gonzalez-Olivares and A. Rojas-Palma, “Multiple limit cycles in a Gause type predator-prey model with Holling type {III} functional response and Allee effect on prey,” Bulletin of Mathematical Biology, vol. 73, pp. 1378–1397, 2011. View at Google Scholar
  12. J. D. Flores, J. Mena-Lorca, B. Gonzßlez-Ya, and E. Gonzßlez-Olivares, “Consequences of depensation in a Smiths bioeconomic model for open-access fishery,” in Proceedings of international symposium on mathematical and computational biology, vol. 27, pp. 219–232, 2007.
  13. E. González-Olivares, B. González-Yañez, E. Sáez, and I. Szantó, “On the number of limit cycles in a predator prey model with non-monotonic functional response,” Discrete and Continuous Dynamical Systems, vol. 6, pp. 525–534, 2006. View at Google Scholar
  14. C. Chicone, Ordinary differential equations with applications, Berlin, Germany, 2006.
  15. Z. Hu, Z. Teng, and H. Jiang, “Stability analysis in a class of discrete {SIRS} epidemic models,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2017–2033, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Z. Hu, Z. Teng, and L. Zhang, “Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2356–2377, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. H. Cao, Y. Zhou, and Z. Ma, “Bifurcation analysis of a discrete {SIS} model with bilinear incidence depending on new infection,” Mathematical Biosciences and Engineering, vol. 10, no. 5-6, pp. 1399–1417, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, NY, USA, 1981. View at MathSciNet
  19. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, NY, USA, 1983. View at MathSciNet
  20. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2, Springer, New York, NY, USA, 2003. View at MathSciNet
  21. Y. Zhou, H. Cao, and Y. Xiao, Difference equation and its application, Science Press, Beijing, China, 2014.
  22. Z. Yingying and Z. Yicang, The Existence of Multiple Invariant Closed Curves for Discrete Dynamical Models [Master, thesis], Xian Jiao Tong University, 2015.
  23. Y. A. Kuznetsov, A Tutorial for MatcontM GUI, Utretcht University, Utrecht, The Netherlands, 2013.
  24. V. Hajnová and L. Pribylová, “Two-parameter bifurcations in {LPA} model,” Journal of Mathematical Biology, vol. 75, pp. 1235–1251, 2017. View at Google Scholar
  25. P. Aguirre, E. González-Olivares, and E. Sáez, “Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,” Nonlinear Analysis: Real World Applications, vol. 10, pp. 1401–1416, 2009. View at Google Scholar
  26. H. Zhu, S. A. Campbell, and G. S. Wolkowicz, “Bifurcation analysis of a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 63, no. 2, pp. 636–682, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 2005.