Reaction-Diffusion by the Gray-Scott Model: Pearson's Parametrization

My first paper was on reaction diffusion equations. They're very dear to my heart -- and they are also very expressive. So expressive, in fact, that one senior academic I knew made a career out of applying them in interesting and novel ways. He was close to retirement (when I was an undergrad) and had published nearly 500 papers, pretty much all math bio. People told him a problem. He writes down equations. They're right. That's kinda his business model. [You might like some of it -- e.g. TGF-Beta and wound repair]

His main passion in life, however, was (European) football. Specifically, playing Championship Manager 1999 [an old football management sim, with wonderfully terrible graphics]. He was _really_ good at it, and did very well in all the fantasy football sweepstakes that are popular -- one of his former students won the Premiership Fantasy Football league a few years ago, for example. [https://www.ox.ac.uk/event/joshua-bull-can-maths-tell-us-how...].

Now, I aged about 20 got the opportunity to go into this professor's office, to talk briefly about parameter space and stability and all that jazz in a set of reaction diffusion equations that I wrote down. I looked around. It was absolutely covered in framed letters. They're not "you're brilliant" or "here's your doctorate" letters. Every time Manchester United, Arsenal, Madrid or any other seriously good club has an opening for a manager, he had sent in his full academic CV – all 500 papers, ten books, the whole shebang. And he writes a covering letter explaining how he'd be perfect for the role -- 'coz he's really good at Championship Manager 1999. And the clubs inevitably write back, saying "Dear Professor X, Thank you very much for your application to be the next Manager of Manchester United. As you can imagine, we have had a lot of interest in this role, and are unable to enter into individual conversations with every applicant. If you do not hear from us again after $DATE, please assume that unfortunately your application has been unsuccessful". And that's it. That's what he frames and puts on his walls. Hundreds of rejection letters from premiership football clubs. That's the kind of brain that becomes a world-leading authority on reaction diffusion equations, and made loads of seminal contributions in the field. (I still love them).

worth to note that apart from the 'typical' patterns you can get some crazy results by tweaking the values. I use it in processing all the time. also it can create a simple but incredible 'cinema' style blur (the stuff you only see on movie credits) on type.

see some extreme examples here: https://ello.co/zssz/post/9skesiflcmsycfhvkdtchw and a mold like structure here: https://www.instagram.com/p/CKxaqKFBF5c/

Highly recommend Ready by the authors of Golly, for experimenting with reaction-diffusion en silico:

https://github.com/GollyGang/ready

I'd be fascinated to know how flexible these are as neural ODE ansatze.

Looks to me they could implement arbitrary logic, but could we prove it.