I liked the final quote: "Nature is more complicated than we thought" .. which I translate into "there's lots to explore!" And I utterly love Turing and the whole field of Reaction-Diffusion and I think people should try 'GollyGang/Ready' and have a play with some of the formulas that it comes-with as there is a really literally infinite world of dynamic possibilities there that has hardly been explored at all so far really! (Speaking as someone who has been on a few adventures in this realm now, see: https://youtube.com/watch?v=Naj_J8aznyk&si=9Z6yvgdVrG69gMbl for a rather strange example!)
In mammals is this process mostly happens in utero by gene activation which lasts for the life of those cells and any cell divisions that happen from them. The spots and stripes can move a bit and fade or strengthen as the animal grows and ages.
But it’s not always just the genes for hair color that activate. You can get spotted cats and dogs that have both pink and black spotted noses and lips. An if your cat ever needs surgery you may discover his skin is also pigmented in the same pattern as the fur. Apparently this is true of tigers.
Fascinating topic. But as far as I know the patterns in cats are not related to skin. I think they are related to genes. After all what we see as patterns are individual hairs of different colors. Some animals like zebras have the same stripes through generations. This is not very interesting. In cats I'm sure there is a random process in effect. But this random process is also "aware" of the symmetry of the body, like having the same black patch on each ear. Or having a black tail in otherwise white cat.