The Gray-Scott equations can produce most of the types of patterns that are possible in reaction-diffusion systems. post 
For a complete description of Gray-Scott patterns, with many examples and videos, go to my Xmorphia Gray-Scott page. post
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The main parameters are diffusion rate, diffusion offset, feed rate, and kill rate. I can vary them over x, y, or time. But x and y can also be turned into distance and angle for polar coordinates. That’s a lot of combinations to try. Plus, there are color schemes to display A, B, dA, dB, A/B, A/(A+B), etc. post
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. course
Reaction-Diffusion Tutorial:
A simulation of two virtual chemicals reacting and diffusing on a 2D grid using the Gray-Scott model. page
Karl Sims is a digital media artist and visual effects software developer. Karl studied computer graphics at the MIT Media Lab, and Life Sciences as an undergraduate at MIT. site