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 Kolmogorov–Petrovsky–Piskunov 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–Bénard 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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