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7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
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Fig. 7.1. Terraced Rice Paddies in Guanxi Province, China

A classical example for F in the scalar case (one species only) is the so-called
Fisher logistic nonlinearity:

F = u(1 − u).

The exact solution of the initial value problem for the so called logistic
ordinary differential equation (ODE)

u t = u(1 − u), u(0) = u 0 ≥ 0

is easily calculated:

u(t) = u 0 .

exp(−t)+ u 0 1−exp(−t)
Note that there are two stationary states u = 0 (total extinction of the species,
unstable) and u = 1 (total saturation, exponentially stable).
In order to describe the interaction of both types of processes, namely dif-
fusion and reaction, we can imagine that on small time intervals the diffusion
process and the reaction process happen consecutively. Then, when the lengths
of the considered time intervals tend to zero, at least on a formal level, this
time-splitting scheme turns into the so-called reaction-diffusion system

u t = div(D(x, t, u)grad u)+ F(x, t, u).

If the process occurs in a spatially confined domain G, then boundary con-
ditions have to be imposed, e.g. the Dirichlet condition

u = 0on ∂G (zero density outside G)

or the Neumann condition

grad u · n = 0on ∂G (no outflow through the boundary) .

Clearly, inhomogeneous boundary conditions may occur, as well as linear com-
binations of Dirichlet and Neumann conditions. Note that diffusion per se does
not change the total number of particles (unless the boundary conditions in-
terfere, as is the case, e.g., with homogeneous Dirichlet conditions) while the
reaction term describes local generation and annihilation of particles of the
considered species.
A different way of deriving reaction-diffusion equations proceeds by local
mass balance. Therefore, denote by V an arbitrary subdomain of the domain G,