IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i9p1500-d807045.html
   My bibliography  Save this article

Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect

Author

Listed:
  • Yangyang Shao

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

  • Yan Meng

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

  • Xinyue Xu

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

Abstract

The Allee effect is widespread among endangered plants and animals in ecosystems, suggesting that a minimum population density or size is necessary for population survival. This paper investigates the stability and pattern formation of a predator–prey model with nonlinear reactive cross-diffusion under Neumann boundary conditions, which introduces the Allee effect. Firstly, the ODE system is asymptotically stable for its positive equilibrium solution. In a reaction system with self-diffusion, the Allee effect can destabilize the system. Then, in a reaction system with cross-diffusion, through a linear stability analysis, the cross-diffusion coefficient is used as a bifurcation parameter, and instability conditions driven by the cross-diffusion are obtained. Furthermore, we show that the system (5) has at least one inhomogeneous stationary solution. Finally, our theoretical results are illustrated with numerical simulations.

Suggested Citation

  • Yangyang Shao & Yan Meng & Xinyue Xu, 2022. "Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1500-:d:807045
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/9/1500/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/9/1500/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Naveed Iqbal & Yeliz Karaca, 2021. "Complex Fractional-Order Hiv Diffusion Model Based On Amplitude Equations With Turing Patterns And Turing Instability," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-16, August.
    2. Ghorai, Santu & Poria, Swarup, 2016. "Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 421-429.
    3. Yao, Shao-Wen & Ma, Zhan-Ping & Cheng, Zhi-Bo, 2019. "Pattern formation of a diffusive predator–prey model with strong Allee effect and nonconstant death rate," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 527(C).
    4. Mansouri, Djamel & Abdelmalek, Salem & Bendoukha, Samir, 2020. "Bifurcations and pattern formation in a generalized Lengyel–Epstein reaction–diffusion model," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    5. Capone, F. & Carfora, M.F. & De Luca, R. & Torcicollo, I., 2019. "Turing patterns in a reaction–diffusion system modeling hunting cooperation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 172-180.
    6. Peng, Yahong & Ling, Heyang, 2018. "Pattern formation in a ratio-dependent predator-prey model with cross-diffusion," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 307-318.
    7. Yao, Shao-Wen & Ma, Zhan-Ping & Yue, Jia-Long, 2018. "Bistability and Turing pattern induced by cross fraction diffusion in a predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 982-988.
    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. Ishtiaq Ali & Maliha Tehseen Saleem, 2023. "Spatiotemporal Dynamics of Reaction–Diffusion System and Its Application to Turing Pattern Formation in a Gray–Scott Model," Mathematics, MDPI, vol. 11(6), pages 1-17, March.

    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 e," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 597-618.
    2. Maria Francesca Carfora & Isabella Torcicollo, 2022. "Traveling Band Solutions in a System Modeling Hunting Cooperation," Mathematics, MDPI, vol. 10(13), pages 1-11, July.
    3. Meng Zhu & Jing Li & Xinze Lian, 2022. "Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation," Mathematics, MDPI, vol. 10(17), pages 1-20, September.
    4. 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.
    5. Ghorai, Santu & Bairagi, Nandadulal, 2022. "Instabilities in hyperbolic reaction–diffusion system with cross diffusion and species-dependent inertia," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    6. Hu, Junlang & Zhu, Linhe, 2021. "Turing pattern analysis of a reaction-diffusion rumor propagation system with time delay in both network and non-network environments," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    7. Maria Francesca Carfora & Isabella Torcicollo, 2020. "Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense," Mathematics, MDPI, vol. 8(8), pages 1-20, July.
    8. Rana, Sourav & Bhattacharya, Sabyasachi & Samanta, Sudip, 2022. "Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 32-49.
    9. Bi, Zhimin & Liu, Shutang & Ouyang, Miao, 2022. "Three-dimensional pattern dynamics of a fractional predator-prey model with cross-diffusion and herd behavior," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    10. Monica De Angelis, 2022. "Transport Phenomena in Excitable Systems: Existence of Bounded Solutions and Absorbing Sets," Mathematics, MDPI, vol. 10(12), pages 1-11, June.
    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).
    12. Djilali, Salih & Cattani, Carlo, 2021. "Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    13. Mohan, Nishith & Kumari, Nitu, 2021. "Positive steady states of a SI epidemic model with cross diffusion," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    14. Yang, Ruizhi & Ma, Jian, 2018. "Analysis of a diffusive predator-prey system with anti-predator behaviour and maturation delay," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 128-139.
    15. Song, Mingrui & Gao, Shupeng & Liu, Chen & Bai, Yue & Zhang, Lei & Xie, Beilong & Chang, Lili, 2023. "Cross-diffusion induced Turing patterns on multiplex networks of a predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    16. Ghorai, Santu & Chakraborty, Bhaskar & Bairagi, Nandadulal, 2021. "Preferential selection of zooplankton and emergence of spatiotemporal patterns in plankton population," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    17. Peng, Yahong & Zhang, Guoying, 2020. "Dynamics analysis of a predator–prey model with herd behavior and nonlocal prey competition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 170(C), pages 366-378.
    18. Kumari, Sarita & Tiwari, Satish Kumar & Upadhyay, Ranjit Kumar, 2022. "Cross diffusion induced spatiotemporal pattern in diffusive nutrient–plankton model with nutrient recycling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 246-272.
    19. Yasmin, Humaira, 2022. "Effect of vaccination on non-integer dynamics of pneumococcal pneumonia infection," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    20. Currò, C. & Grifò, G. & Valenti, G., 2023. "Turing patterns in hyperbolic reaction-transport vegetation models with cross-diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

    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:gam:jmathe:v:10:y:2022:i:9:p:1500-:d:807045. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.