IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v91y2016icp92-107.html
   My bibliography  Save this article

Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system

Author

Listed:
  • Huang, Tousheng
  • Zhang, Huayong

Abstract

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is investigated in this research. The conditions for stable homogeneous stationary state of the system are derived via stability analysis. By using center manifold theorem and bifurcation theory, critical parameter values for flip bifurcation, Hopf bifurcation and Turing bifurcation are determined, respectively. Based on the bifurcation analysis, pattern formation conditions are also provided. Numerical simulations are performed not only to illustrate the theoretical results, but also to exhibit new and complex dynamical behaviors, including period-doubling cascade, invariant circles, periodic windows, chaotic dynamics, and pattern formation. Maximum Lyapunov exponents are calculated to distinguish chaos from regular behaviors. In the routes from bifurcation to chaos, flip-Turing instability and Hopf-Turing instability emerge, capturing the formation of diverse complex patterns, such as mosaic, circle, spiral, spatiotemporal chaotic patterns, and so on. The analysis and results in this research contribute to a new understanding on the relationship among bifurcation, chaos and pattern formation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:91:y:2016:i:c:p:92-107
    DOI: 10.1016/j.chaos.2016.05.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077916301771
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2016.05.009?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.

    References listed on IDEAS

    as
    1. Abid, Walid & Yafia, Radouane & Aziz-Alaoui, M.A. & Bouhafa, Habib & Abichou, Azgal, 2015. "Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 292-313.
    2. Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    3. Wang, Caiyun, 2015. "Rich dynamics of a predator–prey model with spatial motion," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 1-9.
    4. Guin, Lakshmi Narayan, 2015. "Spatial patterns through Turing instability in a reaction–diffusion predator–prey model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 109(C), pages 174-185.
    5. Li, Meifeng & Han, Bo & Xu, Li & Zhang, Guang, 2013. "Spiral patterns near Turing instability in a discrete reaction diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 1-6.
    6. Kelly C. De Carvalho & Tânia Tomé, 2006. "Self-Organized Patterns Of Coexistence Out Of A Predator-Prey Cellular Automaton," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(11), pages 1647-1662.
    7. Yongli Cai & Caidi Zhao & Weiming Wang, 2013. "Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-16, December.
    8. Liu, Xiaoli & Xiao, Dongmei, 2007. "Complex dynamic behaviors of a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 80-94.
    9. Rao, Feng & Wang, Weiming & Li, Zhenqing, 2009. "Spatiotemporal complexity of a predator–prey system with the effect of noise and external forcing," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1634-1644.
    10. Zhao, Hongyong & Zhang, Xuebing & Huang, Xuanxuan, 2015. "Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 462-480.
    11. Perc, Matjaž & Grigolini, Paolo, 2013. "Collective behavior and evolutionary games – An introduction," Chaos, Solitons & Fractals, Elsevier, vol. 56(C), pages 1-5.
    12. Chang, Lili & Sun, Gui-Quan & Wang, Zhen & Jin, Zhen, 2015. "Rich dynamics in a spatial predator–prey model with delay," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 540-550.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Zhang, Guang & Zhang, Ruixuan & Yan, Yubin, 2020. "The diffusion-driven instability and complexity for a single-handed discrete Fisher equation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    2. 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).
    3. Zhang, Huayong & Ma, Shengnan & Huang, Tousheng & Cong, Xuebing & Yang, Hongju & Zhang, Feifan, 2018. "A new finding on pattern self-organization along the route to chaos," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 118-130.
    4. 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).
    5. Lu, Guangqing & Smidtaite, Rasa & Howard, Daniel & Ragulskis, Minvydas, 2019. "An image hiding scheme in a 2-dimensional coupled map lattice of matrices," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 78-85.
    6. 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.
    7. Abernethy, Gavin M. & McCartney, Mark & Glass, David H., 2019. "The role of migration in a spatial extension of the Webworld eco-evolutionary model," Ecological Modelling, Elsevier, vol. 397(C), pages 122-140.
    8. Matvey Kulakov & Efim Frisman, 2023. "Clustering Synchronization in a Model of the 2D Spatio-Temporal Dynamics of an Age-Structured Population with Long-Range Interactions," Mathematics, MDPI, vol. 11(9), pages 1-21, April.
    9. Xu, Li & Liu, Jiayi & Zhang, Guang, 2018. "Pattern formation and parameter inversion for a discrete Lotka–Volterra cooperative system," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 226-231.
    10. Tousheng Huang & Huayong Zhang & Xuebing Cong & Ge Pan & Xiumin Zhang & Zhao Liu, 2019. "Exploring Spatiotemporal Complexity of a Predator-Prey System with Migration and Diffusion by a Three-Chain Coupled Map Lattice," Complexity, Hindawi, vol. 2019, pages 1-19, May.
    11. Jialin Chen & Xiaqing He & Fengde Chen, 2021. "The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource," Mathematics, MDPI, vol. 9(8), pages 1-20, April.
    12. Han, Xiaoling & Lei, Ceyu, 2023. "Bifurcation and turing instability analysis for a space- and time-discrete predator–prey system with Smith growth function," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    13. Zhang, Limin & Wang, Tao, 2023. "Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Zhang, Huayong & Ma, Shengnan & Huang, Tousheng & Cong, Xuebing & Yang, Hongju & Zhang, Feifan, 2018. "A new finding on pattern self-organization along the route to chaos," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 118-130.
    3. 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).
    4. Xu, Li & Liu, Jiayi & Zhang, Guang, 2018. "Pattern formation and parameter inversion for a discrete Lotka–Volterra cooperative system," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 226-231.
    5. Gökçe, Aytül, 2021. "A mathematical study for chaotic dynamics of dissolved oxygen- phytoplankton interactions under environmental driving factors and time lag," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    6. Wu, Zeyan & Li, Jianjuan & Liu, Shuying & Zhou, Liuting & Luo, Yang, 2019. "A spatial predator–prey system with non-renewable resources," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 381-391.
    7. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    8. 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.
    9. 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.
    10. Huang, Chengdai & Cao, Jinde & Xiao, Min & Alsaedi, Ahmed & Alsaadi, Fuad E., 2017. "Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 293-310.
    11. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Bifurcation control of the Hodgkin–Huxley equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 217-224.
    12. Mohammed O. Al-Kaff & Ghada AlNemer & Hamdy A. El-Metwally & Abd-Elalim A. Elsadany & Elmetwally M. Elabbasy, 2024. "Dynamic Behavior and Bifurcation Analysis of a Modified Reduced Lorenz Model," Mathematics, MDPI, vol. 12(9), pages 1-20, April.
    13. Chang, Lili & Jin, Zhen, 2018. "Efficient numerical methods for spatially extended population and epidemic models with time delay," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 138-154.
    14. Yousef, A.M. & Rida, S.Z. & Ali, H.M. & Zaki, A.S., 2023. "Stability, co-dimension two bifurcations and chaos control of a host-parasitoid model with mutual interference," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    15. Xiaorong Ma & Qamar Din & Muhammad Rafaqat & Nasir Javaid & Yongliang Feng, 2020. "A Density-Dependent Host-Parasitoid Model with Stability, Bifurcation and Chaos Control," Mathematics, MDPI, vol. 8(4), pages 1-26, April.
    16. Han, Renji & Dai, Binxiang, 2017. "Spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 177-201.
    17. Bozkurt, Fatma & Yousef, Ali & Baleanu, Dumitru & Alzabut, Jehad, 2020. "A mathematical model of the evolution and spread of pathogenic coronaviruses from natural host to human host," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    18. Tang, Biao & Xiao, Yanni, 2015. "Bifurcation analysis of a predator–prey model with anti-predator behaviour," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 58-68.
    19. Mortuja, Md Golam & Chaube, Mithilesh Kumar & Kumar, Santosh, 2021. "Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    20. Zhang, Xue & Zhang, Qing-Ling & Liu, Chao & Xiang, Zhong-Yi, 2009. "Bifurcations of a singular prey–predator economic model with time delay and stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1485-1494.

    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:91:y:2016:i:c:p:92-107. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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.