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Gray Scott Model of Reaction Diffusion
Abelson, Adams, Coore, Hanson, Nagpal, Sussman
Reaction and diffusion of chemical species can produce a variety of<br>patterns, reminiscent of those often seen in nature. The Gray Scott<br>equations model such a reaction. For more information on this<br>chemical system see the articles "Complex Patterns in a Simple<br>System," by John E. Pearson and "Pattern Formation by Interacting<br>Chemical Fronts," by K.J. Lee, W.D. McCormick, Qi Ouyang, and<br>H.L. Swinney. These articles appeared in Science, Volume 261,<br>9 July 1993.
U and<br>V and<br>P are chemical species.
u and<br>v represent their concentrations.
ru and<br>rv are their diffusion rates.
k represents the rate of conversion of V to P.
f represents the rate of the process that feeds U<br>and drains U,V and P.
The partial differential equations modeling this process may be<br>simulated with a variety of numerical techniques. One may expect good<br>results with even such crude methods as forward Euler integration of<br>the finite-difference equations that one obtains by spatial<br>discretization of the Laplacian.
One may also simulate the underlying<br>process, which is stoichiometrically conservative (accounting for the<br>source of U and the drain of U, V, and P.)<br>The fact that all the interactions are local makes this a good<br>candidate for a parallel implementation. Diffusion may be modeled by<br>an explicitly conservative exchange process among neighbors where<br>the reactions are locally modeled at each locus by a simulated processor.
Xmorphia shows a<br>beautiful presentation of a simulation of the Gray-Scott reaction<br>diffusion mechanism using a uniform-grid finite-difference model<br>running on an Intel Paragon supercomputer. We are interested in<br>being able to make such simulations with an amorphous computer where<br>the precise positions of the individual processing elements is uncertain.
The physical world appears to be homogeneous and isotropic at the<br>scale of atoms and molecules, so the precise positions of the reaction<br>sites should not be essential to the character of the patterns<br>produced by the chemistry. We can test this by comparing a simulation<br>of similar situations using a crystalline array or an amorphous<br>computer. The simulator presented here allows us to conveniently make<br>this comparison.
You can run a Java version of our simulator to see for yourself how the patterns are formed
Gray Scott Simulator Demo
The following figures, made with a simulator written in<br>Scheme,<br>illustrate the kind of patterns that can be obtained by varying the<br>grid.<br>In these simulations each locus of<br>reaction is represented by a simulated processor. The processors are<br>asychronous: each one runs its simulation steps at randomly chosen<br>times. Each simulation step is stoichiometrically conservative.<br>In each of these figures we have 4000 loci of reaction. The<br>parameters are: f = 0.04, k = 0.06, r_u = 0.01, r_v = 0.005. The<br>neighborhood radius is 0.04. (So each simulated processor has<br>approximately 20 neighbors.)
The colors show the concentration of U at each locus. Red labels the<br>maximum concentration and Blue labels the minimum concentration. The<br>patterns that you see are stable -- no significant change will occur,<br>regardless of how long we run the process. Each pattern starts off<br>from the same slightly randomized initial condition. The differences<br>among the patterns is due to either the differences of the grids or<br>the differences in the initial condition.
Amorphous layout (random, smoothed)
Hexagonal close packed layout
Square layout
Another Amorphous layout
Although the details differ, we can see that the general character of<br>the patterns developed are similar, independent of the grid. The<br>features have similar sizes and shapes. They are somewhat aligned<br>with the boundaries.
If we decrease the radius of the neighborhood to 0.03, thereby<br>decreasing the average number of neighbors to about 11<br>(4000*3.14*0.03*0.03), we get correspondingly more delicate features.
Amorphous layout (random, smoothed)
Hexagonal close packed layout
Interestingly, as the amorphous layout shows, the alignment of the<br>features with the boundary is less sure as the features become small<br>by comparison to the figure.
If you have a fast network connection, you can view some MPEG movies<br>we've made with the asychronous Scheme simulator that show how the<br>process evolves. The first three movies show evolution with the same<br>set of parameters, f = 0.025, k = 0.06, r_u = 0.01, r_v = 0.005. In<br>each movie the reaction occurs at 4000 sites with a neighborhood<br>radius of 0.04. The entire image is of unit size in both the<br>horizontal and vertical direction. In these movies the reaction is<br>occuring asynchronously in that at each instant a random site is<br>activated to do one conservative reaction step. We consider 4000 such<br>individual reaction steps (one for each locus) to be "equivalent" to<br>one full step of a synchronous simulation. Each movie spans 6400 full<br>steps. The movie is made by...