Turing Patterns

from Red Blob Games
11 Jan 2022

I last played with this in 2019 and wanted to revisit it now that I’m learning from Phillip Compeau’s online course[1]. Going through this course’s material helped me understand this topic much better than previous references I had read. I took some notes from the chapter on Turing Patterns. They probably won’t make sense unless you’re also following the course.

 1  Random walks#

Let a particle move randomly each time tick. It will follow a random walk. Here’s an example:

Random walk of one particle

Each time the particle will trace out a different random path:

Random walk varies each time

If the particle moves distance 1 each step, after N steps the particle will be on average distance √N away from the origin. This is the Random Walk Theorem.

Random walk of many particles

See the course page[2] for more, and random-walk.js for my source code.

Related: Levy flight random walks[3] are a different distribution instead of each step being 1. See this tweet from Andrei Kashcha[4].

 2  Reaction-diffusion#

If there are two types of particles, “predator” and “prey”, we can set up some simulation rules:

This isn’t exactly the same as the Lotka-Volterra predator-prey model. I read the section but didn’t write a simulation. Watch the videos on the course page[5].

 3  Gray-Scott#

In fluid simulation there are both particle methods which simulate each amount of fluid in a variable amount of space and grid methods which simulate each amount of space with a variable amount of fluid. The previous section was a particle method but we can compute more efficiently by using a grid method, especially on a GPU where we can use 2d arrays (textures) to store the grid and shaders to calculate the simulation.

Instead of storing each particle and its exact location, we store the number of particles in each approximate location. Earlier versions of SimCity did this, storing traffic, pollution, etc. per location, whereas SimCity 5 and Cities Skylines use the particle approach, storing each individual car with exact location. The particle approach is “cooler” but slower and harder to understand at a high level.

My notes on this section of the course:

diffusion
this is a “smoothing” or “blurring” step that spreads out a value to neighboring cells. This corresponds to the random walk effect where particles are moving randomly to nearby locations. There is a parameter here to control how quickly the smoothing happens. If the parameter is too high we can get oscillation. Not covered in the course, but once you think about it as image blurring, you can move values further than just the immediate neighbors. Here are different diffusion speeds:
Different diffusion speeds
dt
I figured this was a standard “delta time” value, and initially set it to the change in timestamp. But Karl Sims’s page[6] always sets it to 1, the course implementation always sets it to 1, Ken Voskuil’s page[7] always sets it to 1, and Pablo Márquez-Neila’s page[8] always sets it to 1. So I’ll set it to 1 too.
colors
in previous projects I’ve visualized the A, B densities in separate color channels but that’s often hard to understand so in this course they’re visualizing A/(A+B) using a diverging colormap[9]. I think I’ll explore color schemes in my next reaction-diffusion project. In this one my priority is learning the math.
Different diffusion colorings
parameters
Gray-Scott is a “fine-tuned” model meaning its behavior is very sensitive to the value of the parameters. I’ve seen it change drastically between feed=0.062 and kill=0.063. See Robert Munafo’s page about a very interesting set of parameters[10], and his clickable map[11] of different parts of the parameter space. There are also “robust” models that are not very sensitive to the value of parameters.
Slightly different parameters, feed=0.026, kill=0.052 to 0.058
simulation
in Karl Sims’s formulas, he uses feed to increase A, and kill+feed to decrease B. Other pages do this too. But in the course notes, kill alone decreases B. I think it works either way but if you are using parameter values from the literature with the course source code you will need to set your kill value to the literature kill+feed value.

Here’s a version where you can change the feed and kill parameters. I decided to use the Karl Sims version (kill+feed) because I wanted to be able to put in numbers from other blog posts I read, so I needed to be consistent with what most people use.

feed=
kill=

See the course page[12] for more, and reaction-diffusion.js for my source code.

 4  Future#

In a future project I want to explore some things Karl Sims mentioned:

  1. A “parameter map” that shows all the different behaviors.
  2. Directionally biased diffusion.
  3. Kill, feed parameters varying across space or time.
  4. Diffusion rate varying across space or time.
  5. Output values feeding back into input parameters.

I also want to try out different coloring and contrast functions. And I’d like to analyze the behavior somehow to see if I can predict which pattern I will get.

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