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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Nonlinear Dynamics

Background:

  • Turing (wave) instabilities traditionally require multi-component reaction-diffusion systems.
  • Time delays can alter the effective dimensionality and stability of dynamical systems.

Purpose of the Study:

  • To investigate the emergence of diffusion-driven instabilities in single-species reaction-diffusion systems with time delays.
  • To analyze the impact of different delay types (discrete, distributed, variable) on system stability.

Main Methods:

  • Systematic stability analysis of one-component reaction-diffusion systems with various delay types.
  • Analytical arguments and numerical simulations to demonstrate wave instability.
  • Examination of diffusion-induced instabilities from periodic orbits in systems with variable delay.

Main Results:

  • Wave instabilities can arise from equilibrium in single-species systems with fluctuating or distributed delays.
  • Fast asymmetric delay fluctuations or distributed delays can induce wave phenomena.
  • Standing waves with periods matching the variable delay can emerge from homogeneous periodic orbits when diffusion is introduced.

Conclusions:

  • Time delays, particularly fluctuating or distributed ones, can enable Turing-like instabilities in systems previously thought incapable of exhibiting them.
  • This research expands the conditions under which wave patterns can form in reaction-diffusion systems.
  • The findings have implications for understanding pattern formation in biological and chemical systems with inherent time delays.