IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v170y2023ics0960077923002989.html

Steady states and spatiotemporal evolution of a diffusive predator–prey model

Author

Listed:
  • Chen, Mengxin
  • Wu, Ranchao

Abstract

The existence of steady states, bifurcations and the spatiotemporal patterns are presented for the diffusive predator–prey model. First, the boundedness and positivity of solutions are justified, respectively. By employing the priori estimates, Poincare inequalities and Leray–Schauder degree, nonexistence and existence of nonconstant steady states are established, respectively. To further explore the pattern dynamics, the Hopf bifurcation and Turing instability are analyzed, the weakly nonlinear analysis is employed to establish the amplitude equations. It is found that the various complex pattern solutions can be identified from amplitude equations. The numerical results are in agreement with the theoretical analysis. We also find that the predator–prey model with Beddington–Deangelis (BD)-type functional response can admit various spatiotemporal patterns, such as labyrinthine-like patterns, the wave patterns near the Hopf–Turing bifurcation threshold, and so on. Such complex spatiotemporal patterns may be useful to help us understand the interaction among species.

Suggested Citation

  • Chen, Mengxin & Wu, Ranchao, 2023. "Steady states and spatiotemporal evolution of a diffusive predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:chsofr:v:170:y:2023:i:c:s0960077923002989
    DOI: 10.1016/j.chaos.2023.113397
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923002989
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.113397?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Li, Yilong & Xiao, Dongmei, 2007. "Bifurcations of a predator–prey system of Holling and Leslie types," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 606-620.
    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. Li, Qiang & Liu, Zhijun & Yuan, Sanling, 2019. "Cross-diffusion induced Turing instability for a competition model with saturation effect," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 64-77.
    4. Chen, Mengxin & Wu, Ranchao & Chen, Liping, 2020. "Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    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. Anita Triska & Agus Yodi Gunawan & Nuning Nuraini, 2023. "The Effects of the Susceptible and Infected Cross-Diffusion Terms on Pattern Formations in an SI Model," Mathematics, MDPI, vol. 11(17), pages 1-18, August.
    2. Gupta, R.P. & Tiwari, Shristi & Kumar, Arun, 2025. "The study of non-constant steady states and pattern formation for an interacting population model in a spatial environment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 652-672.
    3. Owolabi, Kolade M. & Jain, Sonal, 2023. "Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    4. Chen, Mengxin & Zheng, Qianqian, 2023. "Diffusion-driven instability of a predator–prey model with interval biological coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    5. Chen, Mengxin & Ham, Seokjun & Choi, Yongho & Kim, Hyundong & Kim, Junseok, 2023. "Pattern dynamics of a harvested predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    6. Ramasamy, Sivasamy & Banjerdpongchai, David & Park, PooGyeon, 2025. "Stability and Hopf-bifurcation analysis of diffusive Leslie–Gower prey–predator model with the Allee effect and carry-over effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 19-40.

    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. Souna, Fethi & Belabbas, Mustapha & Menacer, Youssaf, 2023. "Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 597-618.
    2. Zhu, Linhe & Tang, Yuxuan & Shen, Shuling, 2023. "Pattern study and parameter identification of a reaction-diffusion rumor propagation system with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Xiaoqing Wen & Yue Chen & Hongwei Yin, 2016. "Positive Solutions of a Diffusive Predator‐Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    4. Ma, Yuanyuan & Dong, Nan & Liu, Na & Xie, Leilei, 2022. "Spatiotemporal and bifurcation characteristics of a nonlinear prey-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Li, Zhenlei & Zhang, Yue, 2024. "Dynamic analysis of a fast slow modified Leslie–Gower predator–prey model with constant harvest and stochastic factor," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 226(C), pages 474-499.
    6. Xia Liu & Jinling Wang, 2013. "Bogdanov‐Takens and Triple Zero Bifurcations of a Delayed Modified Leslie‐Gower Predator Prey System," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    7. Yang, Rui & Yao, Jiaqi & Jiang, Heping, 2024. "Cross-diffusion induced spatial patterns in a chemical self-replication model," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    8. Wang, Fatao & Yang, Ruizhi & Zhang, Xin, 2024. "Turing patterns in a predator–prey model with double Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 170-191.
    9. Yao, Yong, 2025. "Dynamics of a Leslie–Gower type predator–prey system with herd behavior and constant harvesting in prey," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 32-49.
    10. Wenxu Ning & Zhijun Liu & Lianwen Wang & Ronghua Tan, 2021. "Analysis of a Stochastic Competitive Model with Saturation Effect and Distributed Delay," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1435-1459, December.
    11. He, Haoming & Xiao, Min & Cao, Jinde & Zhou, Ying & Park, Ju H. & Rutkowski, Leszek, 2026. "Fractional Gierer-Meinhardt system with cross-diffusion: Pattern analysis in three-dimensional space," Applied Mathematics and Computation, Elsevier, vol. 513(C).
    12. Zhi, Shun & Bai, Dingyong & Su, Youhui, 2026. "Turing patterns induced by cross-diffusion in a predator-prey model with Smith-type prey growth and additive predation effects," Chaos, Solitons & Fractals, Elsevier, vol. 202(P1).
    13. Jiao Jiang & Yongli Song, 2013. "Stability and Bifurcation Analysis of a Delayed Leslie‐Gower Predator‐Prey System with Nonmonotonic Functional Response," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    14. Han, Bingtao & Jiang, Daqing, 2022. "Stationary distribution, extinction and density function of a stochastic prey-predator system with general anti-predator behavior and fear effect," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    15. Sarfaraz, Wakil & Yigit, Gulsemay & Barreira, Raquel & Remaki, Lakhdar & Alhazmi, Muflih & Madzvamuse, Anotida, 2024. "Understanding the dual effects of linear cross-diffusion and geometry on reaction–diffusion systems for pattern formation," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    16. 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.
    17. Yin, Hongwei & Zhou, Jiaxing & Xiao, Xiaoyong & Wen, Xiaoqing, 2014. "Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 51-61.
    18. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    19. Yong Ye & Hua Liu & Yu-mei Wei & Ming Ma & Kai Zhang, 2019. "Dynamic Study of a Predator‐Prey Model with Weak Allee Effect and Delay," Advances in Mathematical Physics, John Wiley & Sons, vol. 2019(1).
    20. Guo, Zhifei & Yang, Haiying, 2025. "Bifurcations of a Leslie-type model with herd behavior and predator harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    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:170:y:2023:i:c:s0960077923002989. 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.