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Reaction, diffusion and non-local interaction.

Hirokazu Ninomiya1, Yoshitaro Tanaka2, Hiroko Yamamoto3

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Summary
This summary is machine-generated.

Non-local interactions, common in science, can be realized by reaction-diffusion systems. This study links non-local interaction instability to diffusion-driven instability, showing reaction-diffusion systems can approximate these interactions.

Keywords:
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Area of Science:

  • Mathematical modeling
  • Theoretical physics
  • Computational biology

Background:

  • Non-local interactions are increasingly used in diverse scientific fields.
  • A common model involves short-range activation and long-range inhibition within a convolution kernel.
  • Understanding the interplay between non-local interactions and reaction-diffusion systems is crucial.

Purpose of the Study:

  • To investigate the relationship between non-local interactions and reaction-diffusion systems.
  • To analyze the connection between instability from non-local interactions and diffusion-driven instability.
  • To demonstrate how reaction-diffusion systems can realize non-local interactions.

Main Methods:

  • Analysis of instability induced by non-local interactions, specifically using a Mexican-hat kernel.
  • Comparison of non-local interaction instability with diffusion-driven instability phenomena.
  • Construction of reaction-diffusion systems to approximate non-local interaction systems.

Main Results:

  • The Mexican-hat kernel's non-local interaction destabilizes a stable homogeneous state.
  • This induced instability is demonstrated to be related to diffusion-driven instability.
  • The existence of a reaction-diffusion system that approximates non-local interactions with any even convolution kernel is proven.

Conclusions:

  • Reaction-diffusion systems offer a framework for realizing and studying non-local interactions.
  • Diffusion-driven instability is a key mechanism underlying the destabilizing effects of certain non-local interactions.
  • This work provides a theoretical bridge between non-local interaction models and reaction-diffusion dynamics.