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7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
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Fig. 7.6. Namibian savanna

cent papers of Henri Berestycki (and coworkers), which (among others) can be
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downloaded from his webpage .
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Also, we point out the work of S.A. Levin , in particular the review paper [6].
As an important and mathematically interesting example for the interaction
of heterogeneity and reaction-diffusion we mention homogenisation problems.
Consider a periodically fragmented environment, with a periodicity scale which
is small compared to the total characteristic dimension of the environment and
denote by the small positive parameter ε the dimensionless ratio of these two
length scales, i.e. the microscopic-macroscopic ratio.
Then it is reasonable to assume that the diffusion matrix and the reaction
nonlinearity F are periodic in the position variable, with periodicity of the
order ε. In precise terms, let
D = D(y), F = F(y, u),

where D and F are periodic with respect to an n-dimensional lattice L (i.e. the
set of n-vectors with integer components) in y, define the fast scale
x
y =
ε
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http://www.ehess.fr/centres/cams/person/berestycki/
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http://www.eeb.princeton.edu/∼slevin/