IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v192y2025ics0960077925000566.html
   My bibliography  Save this article

Spatial dynamics and pattern formation in fragmented habitats: A study using a diffusive Bazykin model with Allee effect

Author

Listed:
  • Pal, Pallav Jyoti
  • Biswas, Debabrata
  • Saha, Tapan

Abstract

This article explores the dynamics of a Bazykin-type prey–predator model enhanced with an additive Allee effect in the prey population growth. The model reveals complex dynamics, such as bistability, global asymptotic stability, and a variety of local and global bifurcations, including saddle–node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, Generalized-Hopf, and saddle–node bifurcations of limit cycles (SNLC). By varying two key parameters, the parametric space is partitioned into ten distinct subregions, wherein the study provides an in-depth understanding of the behaviour of the system within each region. The influence of both strong and weak Allee effects on system dynamics is discussed, particularly focusing on the potential for total extinction of both populations under a strong Allee effect. Quantitative analysis and necessary numerical simulations validate the analytical findings and demonstrate the applicability of the temporal model. In addition, the evolution of stationary and non-stationary patterns is explored in a fragmented habitat with a Z-like structure, which presents greater challenges for numerical simulations due to its complex boundaries. The simulations reveal how the geometry of the fragmented habitats influence spatiotemporal pattern formation, emphasizing the impact of domain geometry on transient patterns and their duration. Analytical conditions are derived to guarantee the existence of stationary patterns, and extensive numerical simulations are performed to investigate the effect of the spatial domain structure on pattern formation.

Suggested Citation

  • Pal, Pallav Jyoti & Biswas, Debabrata & Saha, Tapan, 2025. "Spatial dynamics and pattern formation in fragmented habitats: A study using a diffusive Bazykin model with Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000566
    DOI: 10.1016/j.chaos.2025.116043
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925000566
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116043?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. Djilali, Salih & Cattani, Carlo, 2021. "Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. 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).
    3. Pal, Pallav Jyoti & Mandal, Gourav & Guin, Lakshmi Narayan & Saha, Tapan, 2024. "Allee effect and hunting-induced bifurcation inquisition and pattern formation in a modified Leslie–Gower interacting species system," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    4. Banerjee, Malay & Pal, Swadesh & Roy Chowdhury, Pranali, 2022. "Stationary and non-stationary pattern formation over fragmented habitat," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    5. Gui-Quan Sun & Li Li & Zhen Jin & Zi-Ke Zhang & Tao Zhou, 2013. "Pattern Dynamics in a Spatial Predator-Prey System with Allee Effect," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, September.
    6. Armsworth, Paul R. & Kendall, Bruce E. & Davis, Frank W., 2004. "An introduction to biodiversity concepts for environmental economists," Resource and Energy Economics, Elsevier, vol. 26(2), pages 115-136, June.
    7. Matthew Scott Luskin & Wido Rizki Albert & Mathias W. Tobler, 2017. "Sumatran tiger survival threatened by deforestation despite increasing densities in parks," Nature Communications, Nature, vol. 8(1), pages 1-9, December.
    8. Gui-Quan Sun & Li Li & Zhen Jin & Zi-Ke Zhang & Tao Zhou, 2013. "Pattern Dynamics in a Spatial Predator‐Prey System with Allee Effect," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    9. Chen, Mengxin & Zheng, Qianqian, 2023. "Diffusion-driven instability of a predator–prey model with interval biological coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    10. Pal, Pallav Jyoti & Saha, Tapan, 2015. "Qualitative analysis of a predator–prey system with double Allee effect in prey," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 36-63.
    11. Bhunia, Bidhan & Ghorai, Santu & Kar, Tapan Kumar & Biswas, Samir & Bhutia, Lakpa Thendup & Debnath, Papiya, 2023. "A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    Full references (including those not matched with items on IDEAS)

    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. Boli Xie & Zhijun Wang & Yakui Xue & Zhenmin Zhang, 2015. "The Dynamics of a Delayed Predator-Prey Model with Double Allee Effect," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-8, October.
    2. Pal, Pallav Jyoti & Mandal, Gourav & Guin, Lakshmi Narayan & Saha, Tapan, 2024. "Allee effect and hunting-induced bifurcation inquisition and pattern formation in a modified Leslie–Gower interacting species system," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    3. Hanley, Nick & Tinch, Dugald & Angelopoulos, Konstantinos & Davies, Althea & Barbier, Edward B. & Watson, Fiona, 2009. "What drives long-run biodiversity change? New insights from combining economics, palaeoecology and environmental history," Journal of Environmental Economics and Management, Elsevier, vol. 57(1), pages 5-20, January.
    4. Mark Brady & Konrad Kellermann & Christoph Sahrbacher & Ladislav Jelinek, 2009. "Impacts of Decoupled Agricultural Support on Farm Structure, Biodiversity and Landscape Mosaic: Some EU Results," Journal of Agricultural Economics, Wiley Blackwell, vol. 60(3), pages 563-585, September.
    5. Mondal, Argha & Hens, Chittaranjan & Mondal, Arnab & Antonopoulos, Chris G., 2021. "Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    6. Carrión-Flores, Carmen E. & Flores-Lagunes, Alfonso & Guci, Ledia, 2018. "An estimator for discrete-choice models with spatial lag dependence using large samples, with an application to land-use conversions," Regional Science and Urban Economics, Elsevier, vol. 69(C), pages 77-93.
    7. Chakraborty, Priya & Jolly, Mohit Kumar & Roy, Ushasi & Ghosh, Sayantari, 2023. "Spatio-temporal pattern formation due to host-circuit interplay in gene expression dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    8. Eppink, Florian V. & van den Bergh, Jeroen C.J.M., 2007. "Ecological theories and indicators in economic models of biodiversity loss and conservation: A critical review," Ecological Economics, Elsevier, vol. 61(2-3), pages 284-293, March.
    9. Nehring, Klaus & Puppe, Clemens, 2004. "Modelling phylogenetic diversity," Resource and Energy Economics, Elsevier, vol. 26(2), pages 205-235, June.
    10. Rauscher, Michael & Barbier, Edward B., 2010. "Biodiversity and geography," Resource and Energy Economics, Elsevier, vol. 32(2), pages 241-260, April.
    11. 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).
    12. Gourav Mandal & Alejandro Rojas-Palma & Eduardo González-Olivares & Santabrata Chakravarty & Lakshmi Narayan Guin, 2025. "Allee-induced periodicity and bifurcations in a Gause-type model with interference phenomena," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 98(4), pages 1-33, April.
    13. Hua Liu & Yong Ye & Yumei Wei & Weiyuan Ma & Ming Ma & Kai Zhang, 2019. "Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay," Complexity, Hindawi, vol. 2019, pages 1-14, November.
    14. Yu, Xingwang & Wang, Shengdan & Yang, Yanhua & Ma, Yuanlin & Liu, Tiantian & Wei, Yifan, 2025. "P-bifurcation and bistability arising from cross-correlated sine-Wiener bounded noises: A stochastic single-species model incorporating double Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 193(C).
    15. Ding, Linglong & Zhang, Xuebing & Lv, Guangying, 2024. "Dynamics of a plankton community with delay and herd-taxis," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    16. Horsch, Eric J. & Lewis, David J., 2008. "The Effects of Aquatic Invasive Species on Property Values: Evidence from a Quasi-Random Experiment," Staff Papers 92216, University of Wisconsin-Madison, Department of Agricultural and Applied Economics.
    17. 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 e," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 597-618.
    18. Sipilainen, Timo & Marklund, Per-Olov & Huhtala, Anni, 2008. "Efficiency In Agricultural Production Of Biodiversity: Organic Vs. Conventional Practices," 107th Seminar, January 30-February 1, 2008, Sevilla, Spain 6478, European Association of Agricultural Economists.
    19. Li, Xiaoshuang & Pang, Danfeng & Wallhead, Philip & Bellerby, Richard Garth James, 2023. "Dynamics of an aquatic diffusive predator–prey model with double Allee effect and pH-dependent capture rate," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    20. Lewis, David J., 2010. "An economic framework for forecasting land-use and ecosystem change," Resource and Energy Economics, Elsevier, vol. 32(2), pages 98-116, April.

    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:192:y:2025:i:c:s0960077925000566. 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.