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<p>Nonlocal interactions constitute a fundamental mechanism in biological and ecological systems, spanning processes from intracellular cell dynamics to predator–prey interactions at the population level. In many natural habitats, the detection and response to other individuals or species occur through spatially distributed signals, implying that interactions are inherently nonlocal rather than purely pointwise. This recognition has stimulated increasing research interest in the formulation and analysis of nonlocal advection models, as well as biological and ecological systems governed by nonlocal diffusion–advection equations. In the present study, we develop a new class of nonlocal fractal diffusion–advection models with variable coefficients, motivated by the significant role of fractal dimensions in describing irregular and scale-dependent structures across science and engineering, particularly in biological and ecological contexts. The proposed models extend classical continuous nonlocal diffusion–advection formulations with variable coefficients by incorporating both nonlocal time-delay effects and kernel-based interaction mechanisms. This generalization enables a more realistic representation of systems in which spatial heterogeneity, memory dependence, and long-range interactions coexist. From a theoretical standpoint, we establish the uniqueness of solutions, including the time-dependent component governed by a Volterra-type differential equation. Stability properties are rigorously investigated through the construction of appropriate Lyapunov functionals, and the emergence of periodic dynamics is analyzed via Hopf bifurcation theory. The results demonstrate that the qualitative behavior of the system strongly depends on the range of key parameters, including those associated with the fractal dimension. In particular, the Hopf bifurcation analysis reveals the existence of both stable and unstable equilibrium states, together with transitions toward oscillatory regimes. Additional dynamical features arising from the interplay between nonlocality, delay, and fractal geometry are also identified and discussed.</p>

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Nonlocal Fractal Diffusion-Advection Models with Variable Coefficients and Nonlocal Time Delay: Existence of Solutions, Lyapunov Function, Hopf Bifurcation, and Stability

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Rami Ahmad El-Nabulsi,
  • Waranont Anukool,
  • Raja Valarmathi,
  • Chinnasamy Thangaraj

摘要

Nonlocal interactions constitute a fundamental mechanism in biological and ecological systems, spanning processes from intracellular cell dynamics to predator–prey interactions at the population level. In many natural habitats, the detection and response to other individuals or species occur through spatially distributed signals, implying that interactions are inherently nonlocal rather than purely pointwise. This recognition has stimulated increasing research interest in the formulation and analysis of nonlocal advection models, as well as biological and ecological systems governed by nonlocal diffusion–advection equations. In the present study, we develop a new class of nonlocal fractal diffusion–advection models with variable coefficients, motivated by the significant role of fractal dimensions in describing irregular and scale-dependent structures across science and engineering, particularly in biological and ecological contexts. The proposed models extend classical continuous nonlocal diffusion–advection formulations with variable coefficients by incorporating both nonlocal time-delay effects and kernel-based interaction mechanisms. This generalization enables a more realistic representation of systems in which spatial heterogeneity, memory dependence, and long-range interactions coexist. From a theoretical standpoint, we establish the uniqueness of solutions, including the time-dependent component governed by a Volterra-type differential equation. Stability properties are rigorously investigated through the construction of appropriate Lyapunov functionals, and the emergence of periodic dynamics is analyzed via Hopf bifurcation theory. The results demonstrate that the qualitative behavior of the system strongly depends on the range of key parameters, including those associated with the fractal dimension. In particular, the Hopf bifurcation analysis reveals the existence of both stable and unstable equilibrium states, together with transitions toward oscillatory regimes. Additional dynamical features arising from the interplay between nonlocality, delay, and fractal geometry are also identified and discussed.