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Pattern formation
On this page we present some preliminary results for the pattern formation model.
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Consider a system of identical particles in a 2D space. There are two types events that can occur:
- a death of a particle - the particle is removed from the system,
- a birth of a particle - one of the existing particles causes the creation (birth) of the new particle which is added to the system.
The death has a constant rate d - the probability that the particle dies is the given moment is constant. This means that particles' lifetime is has an exponential distribution with some expected lifetime 1/d.
The probability that given particle gives birth to another one is dependent on other particles, especially on those in the proximity of the both parent and child particles. A particle can not give birth if within the radius R around the particle there are more than C other particles. Above this threshold the particle might give a birth with the constant intensity b. To have a non-vanishing configuration (with high probability) one must set the parameters so the relation b > d is satisified.
The placement of the offspring particle was drawn from the 2D Gaussian distribution centered around the parent particle and standard deviation r in every direction.
Using our software the simulation for the following set of parameters
- b = 10
- d = 1
- R = 10
- r = 1
- C = 314
was run for 90 units of time. The starting distribution was one particle in the origin point. The produced results can be viewed as an animation (7.1 MB).
On the video one can clearly see the hexagonal pattern being formed which is a rather surprising phenomenon for a such simple system. The cut-off function in the birth condition is not the direct reason for the emergence of pattern since other steep functions like atan or tanh produce similar results.
A slightly more complex system had been studied simplification of which lead to the system described above. By making offspring placement dependent on density the particles were forced to not end up to close to each other which resembles the real systems where two objects can not occupy the same spot due to their physical properties.
The simulation over first 90 units of time was run on our software and is available as an animation(23 MB).
One can see that the pattern is blurred, but still present. It's worth noticing that the speed of the expansion (front wave propagation) is higher then in the previous results.
Because the simulation of the simplest model produced interesting results, further investigation was carried out. A mesoscopic equation was obtained from the microscopic model in a straightforward way and - instead of working on individual points it describes the density of the points and all the constants had to be rescaled.
The equation was solved numerically by the Euler method in a simple script in GNU Octave. The animation(925 KB) presents the result. It stays with great correspondence with the microscopic model.