Summer 2011
UCUR Project on Reaction Diffusion Reaction
Diffusion: Turing Systems In 1952, Alan Turing published a paper titled The Chemical Basis of Morphogenesis.
Typically, diffusion is an averaging process that
smoothes differences over space and time. In his paper, Turing presented a
very novel idea by describing a mechanism in which diffusion amplifies small differences
over space and time. A Turing system has two properties. First, the homogeneous solution (in
which there are no differences over space and time) is stable to small
perturbations when there are reaction terms but no diffusion. Second, the homogeneous
solution is not stable to small perturbation when the reaction terms are
coupled with diffusion. In other words, in a Turing system we can expect a pattern if there
small random fluctuations (property 2) and this pattern is attributable to
diffusion by property 1. Note: A reaction diffusion system is a system
described by reaction and diffusion terms in the evolution of the
concentration of some number of variables, and a Turing system is a special
type of reaction diffusion system. Homogeneous Solution Stable To Perturbation Without
Diffusion, But Unstable With Diffusion
Movies of Patterns For Different Initial Conditions
On a Growing Domain Movies
corresponding to one of three initial conditions: 1. a center
point of initial concentration MOVIE 2. a
random field of initial concentration MOVIE 3. center
line of concentration MOVIE Last updated July 28, 2011. |