  A new method to derive an essential integral kernel from any given
reaction-diffusion network is proposed. Any network describing metabolites or
signals with arbitrary many factors can be reduced to a single or a simpler
system of integro-differential equations called "effective equation" including
the reduced integral kernel (called "effective kernel" ) in the convolution
type. As one typical example, the Mexican hat shaped kernel is theoretically
derived from two component activator-inhibitor systems. It is also shown that a
three component system with quite different appearance from activator-inhibitor
systems is reduced to an effective equation with the Mexican hat shaped kernel.
It means that the two different systems have essentially the same effective
equations and that they exhibit essentially the same spatial and temporal
patterns. Thus, we can identify two different systems with the understanding in
unified concept through the reduced effective kernels. Other two applications
of this method are also given: Applications to pigment patterns on skins (two
factors network with long range interaction) and waves of differentiation
(called proneural waves) in visual systems on brains (four factors network with
long range interaction). In the applications, we observe the reproduction of
the same spatial and temporal patterns as those appearing in pre-existing
models through the numerical simulations of the effective equations.
