Hi!

I would suggest you FEniCS: https://fenicsproject.org/

or MOOSE: http://www.mooseframework.org/

I think that you can find for both libraries an example of reaction diffusion in the documentation.

something like this : https://fr.mathworks.com/matlabcentral/fileexchange/38088-diffusion-in-1d-and-2d?s_tid=gn_loc_drop

Might also take a look at Overture library:

http://www.overtureframework.org/