IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v27y2006i1p259-277.html
   My bibliography  Save this article

Bifurcation and chaos in discrete-time predator–prey system

Author

Listed:
  • Jing, Zhujun
  • Yang, Jianping

Abstract

The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3,5,6,7,8,9,10,12,18,20,22,30,39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.

Suggested Citation

  • Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    • Handle: RePEc:eee:chsofr:v:27:y:2006:i:1:p:259-277
    DOI: 10.1016/j.chaos.2005.03.040
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077905003267
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2005.03.040?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhong, Shihong & Xia, Juandi & Liu, Biao, 2021. "Spatiotemporal dynamics analysis of a semi-discrete reaction-diffusion Mussel-Algae system with advection," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Xu, Rui & Ma, Zhien, 2008. "Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 669-684.
    3. Wang, Jinliang & Li, You & Zhong, Shihong & Hou, Xiaojie, 2019. "Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 1-17.
    4. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    5. Huang, Tousheng & Zhang, Huayong, 2016. "Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 92-107.
    6. Zhang, Xue & Zhang, Qing-ling & Zhang, Yue, 2009. "Bifurcations of a class of singular biological economic models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1309-1318.
    7. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Analysis and control of the bifurcation of Hodgkin–Huxley model," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 247-256.
    8. Yijie Li & Zhiming Guo, 2022. "Wolbachia Invasion Dynamics by Integrodifference Equations," Mathematics, MDPI, vol. 10(22), pages 1-13, November.
    9. Chen, Yanguang, 2009. "Spatial interaction creates period-doubling bifurcation and chaos of urbanization," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1316-1325.
    10. Lv, Jian Cheng & Yi, Zhang, 2007. "Stability and chaos of LMSER PCA learning algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1440-1447.
    11. McAllister, A. & McCartney, M. & Glass, D.H., 2023. "Stability, collapse and hyperchaos in a class of tri-trophic predator–prey models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 628(C).
    12. Simas, Fabiano C. & Nobrega, K.Z. & Bazeia, D., 2022. "Bifurcation and chaos in one dimensional chains of small particles," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    13. Lv, Jian Cheng & Yi, Zhang, 2007. "Some chaotic behaviors in a MCA learning algorithm with a constant learning rate," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 1040-1047.
    14. Zhao, Jiantao & Wei, Junjie, 2009. "Stability and bifurcation in a two harmful phytoplankton–zooplankton system," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1395-1409.
    15. Jiang, Yongxin & Yang, Jianping, 2009. "Complex dynamics in a food chain with slow and fast processes," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3160-3168.
    16. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Bifurcation control of the Hodgkin–Huxley equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 217-224.
    17. Sun, Huijing & Cao, Hongjun, 2007. "Bifurcations and chaos of a delayed ecological model," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1383-1393.
    18. Salman, S.M. & Yousef, A.M. & Elsadany, A.A., 2016. "Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 20-31.
    19. Guangye Chen & Zhidong Teng & Zengyun Hu, 2011. "Analysis of stability for a discrete ratio-dependent predator-prey system," Indian Journal of Pure and Applied Mathematics, Springer, vol. 42(1), pages 1-26, February.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:27:y:2006:i:1:p:259-277. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.