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<p>This article examines the effects of time delay on two dynamical systems: (1) a fractional-order three-species prey-predator model and (2) a time-fractional reaction-diffusion system. A three-species food chain model incorporating fractional-order dynamics and time delay is analyzed in the absence of diffusion. The existence, uniqueness, and boundedness of solutions are rigorously established. Furthermore, the local asymptotic stability of equilibrium points is investigated. For the time-fractional reaction-diffusion system, the stability properties of the positive equilibrium are analyzed. Both systems exhibit Hopf bifurcation when the delay parameter is varied. Numerical simulations are conducted to validate the analytical results, demonstrating how the fractional-order derivative influences complex dynamical behaviors, including limit cycle oscillations and chaotic dynamics.</p>

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Exploring the Impact of Time Delay with Fractional Derivative and Diffusion in a Three-Species Food Chain Model

  • 期刊论文

  • 发表日期：

    2025年10月6日

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  • Vikas Kumar,
  • Nitu Kumari

摘要

This article examines the effects of time delay on two dynamical systems: (1) a fractional-order three-species prey-predator model and (2) a time-fractional reaction-diffusion system. A three-species food chain model incorporating fractional-order dynamics and time delay is analyzed in the absence of diffusion. The existence, uniqueness, and boundedness of solutions are rigorously established. Furthermore, the local asymptotic stability of equilibrium points is investigated. For the time-fractional reaction-diffusion system, the stability properties of the positive equilibrium are analyzed. Both systems exhibit Hopf bifurcation when the delay parameter is varied. Numerical simulations are conducted to validate the analytical results, demonstrating how the fractional-order derivative influences complex dynamical behaviors, including limit cycle oscillations and chaotic dynamics.