Evolution in a Cellular-Automaton Model of Gray-Scott Reaction-Diffusion System
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This Demonstration shows the spatiotemporal evolution of a common spot pattern emerging from the numerical simulation of a simple reaction-diffusion system described by the Gray–Scott model.[more]
The Gray–Scott model is defined by the following two irreversible reactions between two substances that give an inert final product. Let and be the concentration fields of the two substances and that of the final product.
The system of coupled reaction-diffusion equations is given by
Here and are the respective diffusion coefficients in dimensionless units, is the dimensionless feed rate, and is the dimensionless rate constant entering the second equation (see ).
This Demonstration is an implementation of a cellular automaton simulation of the forward Euler integration of the coupled partial differential equations after stencil-type discretization of the Laplacian appearing in the diffusion operators. Von Neumann boundary conditions are implemented according to . At the beginning of the simulation, the entire system is placed in the trivial state with .
A square mesh grid is used with the center of the grid perturbed to . Hence, the lattice for the field appears as a central red square on a blue background, according to the chosen color legend. The grid values are perturbed with random noise in order to break the square symmetry of the lattice. The system is then integrated for several time steps until a spatial pattern emerges for the field.
With the parameter set used in the initialization code of this simulation
you can observe the time evolution of the central spot, which initially increases in size, growing radially outward, and then divides into two spots. Then these two spots get further apart, elongate and further split in a way visually similar to cell division. The process goes on until the field is filled with spots.
This spot multiplication occurs in response to the finite-amplitude perturbations applied to a nonlinear system. It is an example of Turing instability patterns, which often occur in natural systems such as animal coats (e.g. zebra stripes). Some of these can be found in [1, Figure 2], which may require hundreds of thousands of time integration steps to be observed.
In order to see the spot patterns in this Demonstration, first choose the field size of the cellular automaton and a pattern (spot type I or II) by selecting the appropriate control at the top of the Manipulate panel. Then click the Play button to start running the simulation. You can visualize either the or field during the animation, but changes to the field size or pattern type are disabled.
To restart the simulation or change pattern type or field size, first stop the running simulation with the Pause control and check the box to reset the system to the initial state configuration.[less]
Snapshot 1: initial state for the field
Snapshot 2: setting for spot of type II for the field
Snapshot 3: setting for spot of type II for the field with a field size of
 J. E. Pearson, "Complex Patterns in a Simple System," Science, 261(5118), 1993 pp. 189–192. doi:10.1126/science.261.5118.189.
 R. J. Gaylord and K. Nishidate, Modeling Nature: Cellular Automata Simulations with Mathematica, Santa Clara, CA: TELOS, 1996.