Reaction–diffusion system
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Reaction–diffusion systems are mathematical models that describe how the concentration of one or more substances change under the influence of two processes: chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space.
As this description implies, reaction–diffusion systems are naturally applied in chemistry. However, the reaction process can also refer to other interactions besides chemical reactions. It may for instance stand for competition between different biological species; the resulting models are used in ecology. Reaction–diffusion systems also occur in biology, geology and physics
Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They display a wide range of behaviours, including travelling waves and more complicated wave-like phenomena and the formation of patterns with stripes, hexagons or more intricate structure.
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[edit] Reaction–diffusion equations in one dimension
The simplest reaction–diffusion equation concerns the concentration u of one substance in one spatial dimension.
The first term on the right describes the diffusion process and includes the parameter D. This is a positive number called the diffusivity. The second term is the reaction term.
If the reaction term vanishes, then this is a pure diffusion process. The corresponding equation is the heat equation. The choice f(u) = u(1−u) yields Fisher's equation, which models how an advantageous gene spreads through a population. The variable u varies between zero (no animals possess the advantageous gene) and one. The diffusion term describes the movements of the animals and the reaction term describes the inheritance and selection processes.
If there is more than one substance, then u becomes a vector of concentrations. The form of the equation is the same, but now D is a diagonal matrix containing the diffusion coefficients. For example, the FitzHugh–Nagumo equation
describes how an action potential travels through a nerve. Here, δ, σ and γ are positive constants, and a ∈ (0,1).
[edit] Travelling waves
Many reaction–diffusion systems support travelling wave solutions: solutions that move with constant speed without changing their shape. These solutions are of the form
where c is the speed of the travelling wave.
[edit] Applications
First described by Alan Turing, reaction–diffusion systems are used to model biological and other pattern formations.
Reaction–diffusion systems are a type of continuous-valued cellular automata. They have been used in visual effects to generate patterns for animal coats and skin pigmentation.
[edit] Experimentation
Simple reaction–diffusion-like patterns can be generated in many image editing programs by repeated applications of sharpening and blurring.
[edit] References
- Grindrod, Peter (1991), Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations, Oxford: Clarendon Press, ISBN 0-19-859676-6 (hardback), ISBN 0-19-859692-8 (paperback).
[edit] See also
[edit] External links
- Java applet showing a reaction–diffusion simulation
- Another applet showing RD.
