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<p>This paper investigates the stability and Hopf bifurcation of a fractional-order tumor-immune interaction model that incorporates time delay and PD controllers. First, the influence of the fractional order on the stability of equilibrium points and the occurrence of Hopf bifurcation is determined by analyzing the characteristic equation without controllers and using time delay as the bifurcation parameter. Second, the stability conditions for the equilibrium points and the conditions for Hopf bifurcation are determined by analyzing the characteristic equation with the controller and using the time delay as the bifurcation parameter. Theoretical analyses and numerical simulations reveal that time delay and PD control parameters significantly impact system dynamics. Further research reveals that fractional-order parameters differentially regulate the stability of PD control parameters, providing a theoretical basis for elucidating the dynamic mechanisms of tumor-immune interactions.</p>

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Dynamics analysis of a fractional-order tumor-immune system with time delay and PD control: stability and Hopf bifurcation

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Mengyan Liu,
  • Xiaozhou Feng,
  • Jiao Zhang,
  • Chaobo Chen,
  • Changtong Li,
  • Fei Xie

摘要

This paper investigates the stability and Hopf bifurcation of a fractional-order tumor-immune interaction model that incorporates time delay and PD controllers. First, the influence of the fractional order on the stability of equilibrium points and the occurrence of Hopf bifurcation is determined by analyzing the characteristic equation without controllers and using time delay as the bifurcation parameter. Second, the stability conditions for the equilibrium points and the conditions for Hopf bifurcation are determined by analyzing the characteristic equation with the controller and using the time delay as the bifurcation parameter. Theoretical analyses and numerical simulations reveal that time delay and PD control parameters significantly impact system dynamics. Further research reveals that fractional-order parameters differentially regulate the stability of PD control parameters, providing a theoretical basis for elucidating the dynamic mechanisms of tumor-immune interactions.