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Updated: May 4, 2026

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Stochastic amplification of spatial modes in a system with one diffusing species.

Laura Cantini1, Claudia Cianci, Duccio Fanelli

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Spatially organized patterns can form in reaction-diffusion models, even when only one species diffuses. This occurs through stochastic fluctuations, not classical Turing instability, enabling pattern development.

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Area of Science:

  • Theoretical Biology
  • Mathematical Biology
  • Chemical Ecology

Background:

  • Reaction-diffusion models are crucial for understanding biological pattern formation.
  • Classical Turing instability requires diffusion of multiple species, limiting its applicability.
  • Investigating pattern formation under restricted diffusion is essential for broader biological relevance.

Purpose of the Study:

  • To explore pattern formation in two-species reaction-diffusion systems where only one species diffuses.
  • To determine if spatial patterns can emerge without classical Turing instability.
  • To establish conditions for stochastic pattern formation in such systems.

Main Methods:

  • Studied a generic two-species reaction-diffusion model with asymmetric diffusion.
  • Introduced a stochastic formulation to account for finite-size effects.
  • Derived general conditions for the occurrence of stochastic spatial patterns.

Main Results:

  • Demonstrated that classical Turing instability is not possible when only one species diffuses.
  • Showed that spatially organized patterns can develop via stochastic fluctuations.
  • Identified general conditions under which these stochastic patterns emerge.

Conclusions:

  • Stochasticity can drive pattern formation in reaction-diffusion systems lacking classical Turing instability.
  • Finite-size corrections are key to seeding pattern development in asymmetric diffusion models.
  • The findings provide a framework for understanding pattern formation in a wider range of biological systems.