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Pattern formation
On this page we present some preliminary results for the pattern formation model.
Note: on some browsers animations might not work. If this is the case, please use a different browser or try to download the file (right click→save as...) and open the file in a good regular video player.
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 time from 90 units of time. The starting distribution was one particle in the origin point. The results can be viewed in this animation file [7.1 MB] .