Reaction Diffusion - diffusion in the grid

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I have a problem, which i unfortunately have very little idea how to solve.
I want to write a script that describes the evolution of the state variables u and v for each point over an integration grid.
I attached a image of the equations I would like to use. The yellow marked variables are all constant and the blue underlined ones are the diffusive terms.
What I've found so far is this question where they basically solved one part of my problem. They calculeted the solution for the equations without diffusion.
Now i want to extend the code so that it calculates this shift in concentration for ervery cells in a grid (or rather a few cells that surround the cell, that did the shift).
If you have any ideas or thoughts I would be very happy if you could share them with me.
Thanks a lot

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