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7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
113
Fig. 7.2. Terraced Rice Paddies in Guanxi Province, China

with boundary S. Clearly, the (temporal) rate of change of the mass of the
particles in V is equal to the mass created in V plus the net flow of material into
V through S. In mathematical terms this law of local mass balance reads:

d
u(x, t)dx = − J(x, t)ds(x)+ F x, t, u(x, t) dx ,
dt
V S V

where ds(x) denotes the (n − 1)-dimensional surface element. The first term on
the right hand side stands for the incoming flux through the boundary S,with
flux density J, and the second term denotes the local mass production in V,with
production per unit volume F(x, t, u). The divergence theorem can be applied to
the boundary integral and we obtain:

(u t +div J − F)dx = 0,
V
such that, since V is arbitrary in G, we conclude that the integrand is zero,
assuming its continuity:

u t = −div J(x, t)+ F(x, t, u), x ∈ G, t> 0.

Assuming a Fick-type law, connecting the flux density with the concentration
vector:

J(x, t) = −D(x, t, u)grad u(x, t)

with a symmetric, positive definite diffusion matrix D, we obtain the reaction-
diffusion equation already derived above. Note that the minus sign in the def-
inition of the flux density accounts for the equilibrating tendency of diffusion,
creating a flow from high densities to low ones.
2
Reaction-diffusion systems have been introduced by Fisher in the year
3
1937 [4] and at the same time by Kolmogorov et al. [5].
Classical examples of reaction-diffusion systems are the so called predator-
5
4
prey models, often referred to as Lotka –Volterra equations, see [8]. Assuming
that there is one species of prey, whose concentration is denoted by u,and one
2 http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Fisher.html
3 http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Kolmogorov.html
4
http://users.pandora.be/ronald.rousseau/html/lotka.html
5
http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Volterra.html