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Stability Analysis and Hopf Bifurcation for the Brusselator Reaction–Diffusion System with Gene Expression Time Delay

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  • Hassan Y. Alfifi

    (Department of General Courses, College of Applied Studies and Community Service, Imam Abdulrahman Bin Faisal University, Dammam 34211, Saudi Arabia
    Basic & Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia)

  • Saad M. Almuaddi

    (Basic & Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia
    Mathematics Department, College of Science, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia)

Abstract

This paper investigates the effect of a gene expression time delay on the Brusselator model with reaction and diffusion terms in one dimension. We obtain ODE systems analytically by using the Galerkin method. We determine a condition that assists in showing the existence of theoretical results. Full maps of the Hopf bifurcation regions of the stability analysis are studied numerically and theoretically. The influences of two different sources of diffusion coefficients and gene expression time delay parameters on the bifurcation diagram are examined and plotted. In addition, the effect of delay and diffusion values on all other free parameters in this system is shown. They can significantly affect the stability regions for both control parameter concentrations through the reaction process. As a result, as the gene expression time delay increases, both control concentration values increase, while the Hopf points for both diffusion coefficient parameters decrease. These values can impact solutions in the bifurcation regions, causing the region of instability to grow. In addition, the Hopf bifurcation points for the diffusive and non-diffusive cases as well as delay and non-delay cases are studied for both control parameter concentrations. Finally, various examples and bifurcation diagrams, periodic oscillations, and 2D phase planes are provided. There is close agreement between the theoretical and numerical solutions in all cases.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:8:p:1170-:d:1375131
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    References listed on IDEAS

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    1. 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.
    2. Lv, Yehu & Liu, Zhihua, 2021. "Turing-Hopf bifurcation analysis and normal form of a diffusive Brusselator model with gene expression time delay," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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