# The Azimuth Project Reaction diffusion equation (changes)

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# Contents

## Idea

Reaction diffusion equation is used to model how concentration for example chemical concentrations changes over time and space. According to Wikipedia:

They are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: 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. This description implies that 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 and ecology. Mathematically, reaction–diffusion systems take the form of semi-linearparabolic partial differential equations. They can be represented in the general form

This description implies that 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 and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form

$\partial_t\vec q = D\nabla^2\vec q+R(\vec q)$

where each component of the vector $q(x,t)$ represents the concentration of one substance, $D$ 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.

where each component of the vector $q(x,t)$ represents the concentration of one substance, $D$ 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.