Reaction-Diffusion Model

A Reaction-Diffusion Model is a mathematical model that ...

• See: Chemical Reaction, Diffusion, Parabolic Partial Differential Equation, Diffusion Coefficient, Travelling Wave.

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References

2023

• (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Reaction–diffusion_system Retrieved:2023-5-6.
  • Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.

    Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form : [math]\displaystyle{ \partial_t \boldsymbol{q} = \underline{\underline{\boldsymbol{D}}} \,\nabla^2 \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}), }[/math] where q(x, t) represents the unknown vector function, is a diagonal matrix of diffusion coefficients, and R accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons. Such patterns have been dubbed “Turing patterns". Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.

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