Mathematicians and biologists have a hammer, so everything looks like a nail.
Then the diffusive part says
du(x, t)/dt = \nabla_x u(x, t).
The \nabla term is the laplacian: a multivariate form of the second derivative.
The equation says that a short time from now, u(x, t) will change in proportion to the average value of u, calculated over a small ball surrounding the point x, minus the value of u at the point x itself.
If there's less "stuff" in the points that neighbour x than at x itself, the function will decrease over time. Similarly if there's more stuff at the neighbours of x, u(x, t) will increase. This is the basis of diffusive behaviour.
(Edit: I think the equation in the article is wrong, unless I've misunderstood something: they have a delta (first derivative) when they should have a nabla (laplacian))
From what I understand, just \nabla(f) describes the gradient of a function f -- meaning all the first order partial derivatives (of a certain point) in a vector.
\nabla * f describes the divergence of the function f -- meaning a scalar field of the quantity of the vector field's sources at each point.
\nabla * \nabla(f), divergence of gradient, then describes the Laplace operator. Also written as \nabla^2(f) or \Delta(f) -- note that it's an uppercase delta.
Maybe there are add-ons that detect valid LaTeX math symbols and convert them whenever possible but as-is you can't get it to render within the comments.
1) "Diffussion" is species vs time equals species spatial laplacian.
2) The "reaction" equations are non-painfully derived from Baez stochastic Petri nets/chemical reaction networks in [1] (species vs time = multivariate polynomial in species, "space dependant rate equation")
So Reaction-Diffusion is just adding up. Species vs time = species spatial laplacian plus multivariate polynomial in species. One more for the toolbox!
[1] https://arxiv.org/abs/1209.3632