A Cellular Automaton as an Example for the Propagation of Localized Structures in Three-Component Reaction-Diffusion Systems

The Mechanism, which leads to the propagation of localized structures in a three-component reaction-diffusion system, can be explained using a simple cellular automaton. The applet below realizes such an automata. Every location can take values between 0 and 8. Locally these "activator"-values decreases by one after every iteration. If somewhere the value is 0, it has to be checked, wether there can occur an ignition-process, which sets the value back to 8. The condition which leads to an ignition is the following: All values in an environment of about 3 are added and the sum is divided by a value A. The maximum value of the neighbouring cells (above,below,right and left) will be denoted by M. The ignition-condition is

 M - sum/A > 2      ==>    Ignition (set the value back to 8)

The sum is carried out over the following environment of a point:

    0011100
    0111110
    1111111
    1111111
    1111111
    0111110
    0011100
 
 

Description  The value A can be adjusted in this applet. A higher value A leads to more frequent ignition processes. The simulation speed can be adjusted, too. Using the mouse, it is possible to activate arbitrary points in the simulation domain. The simulation can be paused and restarted again. The following effects can be observed, depending on A: A<7: All structures vanish A=7,8: It is possible to observe localized moving particles, if the initial conditions are chosen well. A>8:  The particles start to divide. A>>8: No particle-like solution are possible. Target-patterns and spiral-dynamics can result.

What is the Connection between this Cellular Automaton and a Reaction-Diffusion-System?

The local values can be interpreted as the activator-concentration u. The activity can spread due to diffusion to the neighbouring cells. The local decrease of the activator reflect the growing of a slow inhibitor which suppresses the activator after some time. So far, this type of dynamics is called an excitable media (A>>8). It is possible to observe spiral-waves in this range of parameters.

To observe spatially localized structures in an excitable media, a second fast diffusing inhibitor has to be present. Such an inhibitor is simulated by the summation over the environment of a point. If there is a very high overall amout of activator, ignition processes and thus the propagation of the structure is not possible. This mechanism limits the possible size of a propagating front and thus enables the possiblity for the existence of localized solutions.
 
 

Author: C. P. Schenk (22.06.1999)

Last change by Andreas W. Liehr (01.06.2000)