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Turing's Diffusive Threshold in Random Reaction-Diffusion Systems.

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Increasing the number of diffusing species (N) in reaction-diffusion systems makes Turing instabilities more likely. This finding opens possibilities for true Turing instabilities in many-species systems, overcoming physical limitations in simpler models.

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Area of Science:

  • Chemical kinetics
  • Mathematical biology
  • Nonlinear dynamics

Background:

  • Turing instabilities require significant differences in diffusion rates.
  • This requirement is often unphysical for systems with two diffusing species (N=2).
  • Existing experimental methods rely on fluctuations or non-diffusing species.

Purpose of the Study:

  • To investigate if increasing the number of diffusing species (N>2) lowers the threshold for Turing instabilities.
  • To determine if higher N allows for "true" Turing instabilities in reaction-diffusion systems.
  • To analyze the probability distribution of the diffusive threshold in many-species systems.

Main Methods:

  • Analysis of reaction-diffusion systems using random matrices.
  • Linearized dynamics near a homogeneous fixed point.
  • Inspired by May's stability analysis of random ecological communities.
  • Numerical analysis for systems with N up to 6.

Main Results:

  • The probability of a physical diffusive threshold increases with N.
  • Higher N makes Turing instabilities more likely to occur.
  • Many-species instabilities are often irreducible to simpler, fewer-species models.

Conclusions:

  • Increasing the number of diffusing species is a viable strategy to achieve "true" Turing instabilities.
  • This approach overcomes the physical limitations of low-species systems.
  • Reduced models are insufficient for describing instabilities in many-species reaction-diffusion systems.