  Asymptotic analysis has become a common approach in investigations of
reaction-diffusion equations and pattern formation, especially when considering
generalizations to the original model, such as spatial heterogeneity, where
finding an analytic solution even to the linearized equations is generally not
possible. The WKBJ method, one of the more robust asymptotic approaches for
investigating dissipative phenomena captured by linear equations, has recently
been applied to the Turing model in a heterogeneous environment. It
demonstrated the anticipated modifications to the results obtained in a
homogeneous setting, such as localized patterns and local Turing conditions. In
this context, we attempt a generalization of the scalar WKBJ theory to
multicomponent systems. Our broader mathematical approach results in general
approximation theorems for systems of ODEs. We discuss the cases of exponential
and oscillatory behaviour first before treating the general case. Subsequently,
we demonstrate the spectral properties utilized in the approximation theorems
for a typical Turing system, hence suggesting that such an approximation is
reasonable. Note that our line of approach is via showing that the solution is
close (using suitable weight functions for measuring the error) to a linear
combination of Airy functions.
