Pattern formation from homogeneity is well-studied, but less is known
concerning symmetry-breaking instabilities in heterogeneous media. It is
nontrivial to separate observed spatial patterning due to inherent spatial
heterogeneity from emergent patterning due to nonlinear instability. We employ
WKBJ asymptotics to investigate Turing instabilities for a spatially
heterogeneous reaction-diffusion system, and derive conditions for instability
which are local versions of the classical Turing conditions We find that the
structure of unstable modes differs substantially from the typical
trigonometric functions seen in the spatially homogeneous setting. Modes of
different growth rates are localized to different spatial regions. This
localization helps explain common amplitude modulations observed in simulations
of Turing systems in heterogeneous settings. We numerically demonstrate this
theory, giving an illustrative example of the emergent instabilities and the
striking complexity arising from spatially heterogeneous reaction-diffusion
systems. Our results give insight both into systems driven by exogenous
heterogeneity, as well as successive pattern forming processes, noting that
most scenarios in biology do not involve symmetry breaking from homogeneity,
but instead consist of sequential evolutions of heterogeneous states. The
instability mechanism reported here precisely captures such evolution, and
extends Turing's original thesis to a far wider and more realistic class of
systems.Comment: 23 pages, 7 Figure