Reaction-diffusion
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
where q(x, t) represents the unknown vector function, 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. Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.
The simplest reaction–diffusion equation is in one spatial dimension in plane geometry,
is also referred to as the KPP (Kolmogorov-Petrovsky-Piskounov) equation. If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is Fick's second law. The choice R(u) = u(1 − u) yields Fisher's equation that was originally used to describe the spreading of biological populations, the Newell-Whitehead-Segel equation with R(u) = u(1 − u2) to describe Rayleigh-Benard convection, the more general Zeldovich equation with R(u) = u(1 − u)(u − α) and 0 < α < 1 that arises in combustion theory, and its particular degenerate case with R(u) = u2 − u3 that is sometimes referred to as the Zeldovich equation as well.
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