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Reaction, diffusion, and nonlocal interaction in high-dimensional space.

Hiroshi Ishii1, Yoshitaro Tanaka2

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Journal of Mathematical Biology
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Summary
This summary is machine-generated.

This study links nonlocal interactions to reaction-diffusion systems. We show how diffusive substances can mathematically approximate nonlocal behaviors in high dimensions.

Keywords:
ApproximationNonlocal evolution equationNonlocal interactionReaction-diffusion system

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Area of Science:

  • Mathematical Physics
  • Dynamical Systems
  • Partial Differential Equations

Background:

  • Nonlocal evolution equations can mimic patterns seen in reaction-diffusion systems.
  • Understanding these connections is crucial for modeling complex phenomena.

Purpose of the Study:

  • To mathematically connect nonlocal convolution-type interactions with multi-component reaction-diffusion systems.
  • To approximate nonlocal interactions using solutions from reaction-diffusion systems.

Main Methods:

  • Approximating radial kernels with Green functions of elliptic partial differential equations.
  • Demonstrating that auxiliary diffusive substances can model nonlocal interactions.
  • Explicitly determining reaction-diffusion parameters for spatial dimensions up to three.

Main Results:

  • A broad class of nonlocal convolution-type interactions can be approximated by linear sums of diffusive substances.
  • The parameters for the reaction-diffusion system are explicitly determined by the kernel shape in dimensions up to three.
  • Established a direct mathematical link between nonlocal interactions and diffusive chemical reactions.

Conclusions:

  • Nonlocal interactions can be effectively represented by reaction-diffusion systems.
  • This work bridges the gap between nonlocal modeling and chemical kinetics in dynamical systems.
  • Provides a framework for analyzing nonlocal phenomena through the lens of reaction-diffusion theory.