IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i7p1026-d1617500.html
   My bibliography  Save this article

Effects of Diffusion and Delays on the Dynamic Behavior of a Competition and Cooperation Model

Author

Listed:
  • Hassan Y. Alfifi

    (Department of General Courses, College of Applied Studies and Community Service, Imam Abdulrahman Bin Faisal University, Dammam 34211, Saudi Arabia)

Abstract

This study investigates a model of competition and cooperation between two enterprises with reaction, diffusion, and delays. The stability and Hopf bifurcation for variants with two, one, and no delays are considered by examining a system of delay ODE equations analytically and numerically, applying the Galerkin method. A condition is obtained that helps characterize the existence of Hopf bifurcation points. Full maps of stability analysis are discussed in detail. With bifurcation diagrams, three different cases of delay are shown to determine the stable and unstable regions. It is found that when τ i > 0 , there are two different stability regions, and that without a delay ( τ i = 0 ), there is only one stable region. Furthermore, the effects of delays and diffusion parameters on all other free rates in the system are considered; these can significantly affect the stability areas, with important economic consequences for the development of enterprises. Moreover, the relationship between the diffusion and delay parameters is discussed in more detail: it is found that the value of the time delay at the Hopf point increases exponentially with the diffusion coefficient. An increase in the diffusion coefficient can also lead to an increase in the Hopf-point values of the intrinsic growth rates. Finally, bifurcation diagrams are used to identify specific instances of limit cycles, and 2-D phase portraits for both systems are presented to validate all theoretical results discussed in this work.

Suggested Citation

  • Hassan Y. Alfifi, 2025. "Effects of Diffusion and Delays on the Dynamic Behavior of a Competition and Cooperation Model," Mathematics, MDPI, vol. 13(7), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1026-:d:1617500
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/7/1026/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/7/1026/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Fang, Yinhai & Xu, Haiyan & Perc, Matjaž & Tan, Qingmei, 2019. "Dynamic evolution of economic networks under the influence of mergers and divestitures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 89-99.
    2. Alfifi, H.Y., 2022. "Stability analysis for Schnakenberg reaction-diffusion model with gene expression time delay," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    3. Al Noufaey, K.S. & Marchant, T.R., 2014. "Semi-analytical solutions for the reversible Selkov model with feedback delay," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 49-59.
    4. Alfifi, H.Y., 2023. "Effects of diffusion and delayed immune response on dynamic behavior in a viral model," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    5. Yanxia Zhang & Long Li & Antonio Di Crescenzo, 2022. "Dynamic Analysis on a Diffusive Two-Enterprise Interaction Model with Two Delays," Journal of Mathematics, Hindawi, vol. 2022, pages 1-20, February.
    6. Alfifi, H.Y., 2021. "Stability and Hopf bifurcation analysis for the diffusive delay logistic population model with spatially heterogeneous environment," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    7. Hassan Y. Alfifi & Saad M. Almuaddi, 2024. "Stability Analysis and Hopf Bifurcation for the Brusselator Reaction–Diffusion System with Gene Expression Time Delay," Mathematics, MDPI, vol. 12(8), pages 1-19, April.
    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. Alfifi, H.Y., 2024. "Stability analysis and Hopf bifurcation for two-species reaction-diffusion-advection competition systems with two time delays," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    2. Alfifi, H.Y., 2023. "Effects of diffusion and delayed immune response on dynamic behavior in a viral model," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    3. Li, Yanqiu & Zhou, Yibo, 2023. "Turing–Hopf bifurcation in a general Selkov–Schnakenberg reaction–diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    4. Cheng, Yuan & Chang, Meng & Xue, Yanbo, 2020. "A computational study of promotion dynamics and organizational efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 560(C).
    5. Wang, Xianjia & Chen, Wenman, 2020. "Evolutionary dynamics in spatial threshold public goods game with the asymmetric return rate mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    6. 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).
    7. Alfifi, H.Y., 2022. "Stability analysis for Schnakenberg reaction-diffusion model with gene expression time delay," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    8. Alfifi, H.Y., 2021. "Stability and Hopf bifurcation analysis for the diffusive delay logistic population model with spatially heterogeneous environment," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    9. Hassan Y. Alfifi & Saad M. Almuaddi, 2024. "Stability Analysis and Hopf Bifurcation for the Brusselator Reaction–Diffusion System with Gene Expression Time Delay," Mathematics, MDPI, vol. 12(8), pages 1-19, April.
    10. Artés, Joan Carles & Llibre, Jaume & Valls, Claudià, 2018. "Dynamics of the Higgins–Selkov and Selkov systems," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 145-150.

    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:13:y:2025:i:7:p:1026-:d:1617500. 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.