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7. Reaction-Diffusion Equations –
Homogeneous and Heterogeneous

Environments

Many physical, chemical, biological, environmental and even sociological pro-
cesses are driven by two different mechanisms: on one hand there is diffusion,
a random particle (= chemical molecule, biological cell or biological specimen)
1
movement microscopically described by Brownian motion , and on the other
hand there are chemical, biological or sociological reactions representing in-
stantaneous interactions, which depend on the state variables themselves and
possibly also explicitely on the particles’ position. Typical examples are flame
propagation, movement of biological cells in plants and animals (see Chap. 4
on chemotaxis and biological pattern formation), spread of biological species
in homogeneous or in heterogeneous environments (for example in the three
dimensionally terraced rice paddies in the southern Chinese Guanxi province
depicted in the Images 7.1–7.3) etc.
For the mathematical modeling, let u = u(x, t)bethe d-dimensional con-
n
centration vector of the interacting particle species, where x in R denotes the
position variable (typically n = 1, 2or3)and t> 0 time. Then the diffusion part
of the motion is (in a quasilinear context) described by the parabolic evolution
equation:

u t = div(D grad u),
where D is a positive definite symmetric diffusion matrix, which may depend
on x describing inhomogeneous diffusion, on t or/and even on the unknown
vector u itself. Note that in the vector-valued case grad u denotes the Jacobi
matrix of the vector field u.If D is a positive scalar valued function, then the
direction of the diffusive flux is parallel to the gradient of the concentration
function u, pointing in the direction of smaller values of u.
In the reaction-diffusion framework the reaction process is modeled by
a ‘local’ dynamical system of the form

u t = F(x, t, u).

F is independent of the position variable x,ifthe processoccursinanun-
structured (homogeneous) environment and x-dependent if spatial structure
interacts instantaneously with the reaction. The t-dependence can be used to
account for external time dependent driving forces.

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see, e.g., the excellent out-of-print book downloadable from
http://www.math.princeton.edu/∼nelson/books/bmotion.pdf