Ilya

A Simple Reaction Diffusion Simulator

Hardware requirements: Windows 95, at least 256 colour mode, 133Mhz Pentium, good video card.

Note you will need a decent browser and video mode to get the most out of this page.

Ilya is a small program for Windows 95 which simulates a reaction-diffusion model based on the Brusselator reaction scheme. A reaction-diffusion model is defined in terms of partial differential equations and describes the evolution of the system in both space and time. Ilya simulates a 2-dimensional rectangular plane with opposite edges connected so that the plane actually forms a torus. The plane is divided up into many small area elements (one area element per screen pixel). The simulation involves computing the rates of change of two chemical species labeled X and Y as a result of both chemical transformations (Brusselator scheme) and diffusion of X and Y to and from adjacent area elements. The Brusselator scheme is a well known chemical model and is described by the reactions:

A --> X

B + X --> Y + waste

2X + Y --> 3X (net effect: dx/dt = + X² Y)

X --> waste

The species X and Y are free floating variables whose evolution is determined by the system. The species A and B are fixed boundary conditions and provide the thermodynamic gradient necessary to drive the system away from equilibrium. The system is temporally unstable for certain ratios of A and B, in particular when B > A² + 1. All rate constants are assumed to have values of unity.

The set of reactions defined by the Brusselator scheme can occur in each area element of the simulation and the concentrations of A and B are fixed and present in each area element. The diffusion of X and Y between adjacent elements is governed by Fick's second law.

But what's the point of all this? Classical thermodynamics and generally one's own experience of life, suggests that the 'natural' trend is for physical systems to disperse and all concentration gradients and dynamic structures to dissipate into featureless forms. For example pouring ink into water is an example where an initial gradient or concentration eventually dissipates throughout the water, there are many other such examples. To find a system which appears to go against this general trend is very interesting. Reaction-diffusion models show such behaviour and behave as if the ink, once spread out were to come back together just after it was as it dropped into the water.

When you run Ilya the program starts with a random distribution of species X and Y in space. Once the simulation begins you will see a remarkable effect where the chemical species, initially randomly distributed, appear to gather themselves up to form concentration gradients and what appear to look like walls. The formation of these structures occurs at the expense of the driving thermodynamic forces A and B. If these forces are removed (simply set their concentrations to zero) the structures will disappear and the system will decay to a featureless field. Because these structures are maintained by the dissipation of free energy, these systems are often called Dissipative Systems. Ilya also allows the user to change the various parameters of the model and investigate their effect on the behaviour of the system. To show that the reactions of the Brusselator are essential in this process you can switch the reactions off and just allow diffusion to take place. If you do this you will find that any structures present will decay. Note you can change all the settings as the system evolves ther is no need to stop the simulation first.

The images below where generated using Ilya. The first image shows the state of the system near the start of the simulation. The coloured areas indicate high concentrations of species X and can be seen to be distributed in a random fashion. The second image shows the same system after about 2000 time iterations (about 10 to 15 minutes on a 133Mhz Pentium). Note how X becomes concentrated in specific regions forming structures throughout the space. Do bear in mind that there are no physical barriers, such as walls, preventing the gradients from dissipating. These structures are produced as a result of self-organsiation.

The following images were taken from the anther simulation over a period of about 10 minutes and show the sorts of patterns and structures that can self-organise. On a Pentium 133 MHz computer the evolution can be easily tracked in real-time.

Sequence 1:

Sequence 2:

Ilya is freely available for educational, private study and research purposes. Commercial exploitation is prohibited without prior permission from the author (HSauro@fssc.demon.co.uk).

Download Ilya now (about 360K)

This page was last modified on July 15, 1997
This page is supported by Future Skill Software