IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v37y2008i2p476-486.html
   My bibliography  Save this article

Traveling waves in a diffusive predator–prey model with holling type-III functional response

Author

Listed:
  • Li, Wan-Tong
  • Wu, Shi-Liang

Abstract

We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator–prey model with Holling type-III functional response. The analysis is in the three-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator–prey system in the traveling wave variable. The methods used to prove the results are the shooting argument, invariant manifold theory and the Hopf bifurcation theorem.

Suggested Citation

  • Li, Wan-Tong & Wu, Shi-Liang, 2008. "Traveling waves in a diffusive predator–prey model with holling type-III functional response," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 476-486.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:2:p:476-486
    DOI: 10.1016/j.chaos.2006.09.039
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077906008873
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2006.09.039?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. Jiang, Guirong & Lu, Qishao & Qian, Linning, 2007. "Complex dynamics of a Holling type II prey–predator system with state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 448-461.
    2. Lin-Lin Wang & Wan-Tong Li, 2004. "Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response," Discrete Dynamics in Nature and Society, Hindawi, vol. 2004, pages 1-19, January.
    3. Xu, Rui & Chaplain, M.A.J. & Davidson, F.A., 2006. "Travelling wave and convergence in stage-structured reaction–diffusion competitive models with nonlocal delays," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 974-992.
    4. Zhang, Shuwen & Tan, Dejun & Chen, Lansun, 2006. "Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 474-482.
    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. Wu, Chufen & Yang, Yong & Weng, Peixuan, 2013. "Traveling waves in a diffusive predator–prey system of Holling type: Point-to-point and point-to-periodic heteroclinic orbits," Chaos, Solitons & Fractals, Elsevier, vol. 48(C), pages 43-53.
    2. Hu, Guang-Ping & Li, Wan-Tong & Yan, Xiang-Ping, 2009. "Hopf bifurcations in a predator–prey system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1273-1285.
    3. Wu, Chufen & Yang, Yong & Zhao, Qianyi & Tian, Yanling & Xu, Zhiting, 2017. "Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 122-143.
    4. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 584-594.
    5. Wu, Shi-Liang & Li, Wan-Tong, 2009. "Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1229-1239.

    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. Wu, Shi-Liang & Li, Wan-Tong, 2009. "Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1229-1239.
    2. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 584-594.
    3. Wu, Chufen & Yang, Yong & Weng, Peixuan, 2013. "Traveling waves in a diffusive predator–prey system of Holling type: Point-to-point and point-to-periodic heteroclinic orbits," Chaos, Solitons & Fractals, Elsevier, vol. 48(C), pages 43-53.
    4. Tian, Yuan & Li, Chunxue & Liu, Jing, 2023. "Complex dynamics and optimal harvesting strategy of competitive harvesting models with interval-valued imprecise parameters," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    5. Huo, Liang’an & Jiang, Jiehui & Gong, Sixing & He, Bing, 2016. "Dynamical behavior of a rumor transmission model with Holling-type II functional response in emergency event," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 228-240.
    6. Jiang, Guirong & Yang, Qigui, 2009. "Complex dynamics in a linear impulsive system," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2341-2353.
    7. Tian, Yuan & Sun, Kaibiao & Chen, Lansun, 2011. "Modelling and qualitative analysis of a predator–prey system with state-dependent impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 318-331.
    8. Zhao, Min & Lv, Songjuan, 2009. "Chaos in a three-species food chain model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2305-2316.
    9. Jin, Chunhua & Yin, Jingxue & Wang, Yifu, 2009. "Travelling wavefronts of a generalized Fisher equation with spatio-temporal delay," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 599-609.
    10. Elettreby, M.F., 2009. "Two-prey one-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2018-2027.
    11. Li, Wenjie & Zhang, Ying & Huang, Lihong, 2023. "Dynamics analysis of a predator–prey model with nonmonotonic functional response and impulsive control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 529-555.
    12. Nie, Linfei & Teng, Zhidong & Hu, Lin & Peng, Jigen, 2009. "Existence and stability of periodic solution of a predator–prey model with state-dependent impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(7), pages 2122-2134.
    13. Xiao, Qizhen & Dai, Binxiang, 2015. "Dynamics of an impulsive predator–prey logistic population model with state-dependent," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 220-230.
    14. Kang, Li & Tang, Sanyi, 2016. "The reverse effects of random perturbation on discrete systems for single and multiple population models," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 198-209.
    15. Zhao, Zhong & Kong, Yinchang & Chen, Ying, 2016. "Dynamic analysis of the ethanol fermentation with the impulsive state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 274-281.
    16. Tian, Yuan & Li, Huanmeng & Sun, Kaibiao, 2024. "Complex dynamics of a fishery model: Impact of the triple effects of fear, cooperative hunting and intermittent harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 31-48.
    17. Yang, Jin & Tang, Guangyao, 2019. "Piecewise chemostat model with control strategy," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 126-142.
    18. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    19. Sun, Kaibiao & Zhang, Tonghua & Tian, Yuan, 2017. "Dynamics analysis and control optimization of a pest management predator–prey model with an integrated control strategy," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 253-271.
    20. Tian, Yuan & Gao, Yan & Sun, Kaibiao, 2022. "Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

    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:37:y:2008:i:2:p:476-486. 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.