| Resumo: | Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional
reaction-cross-diffusion system in the presence of spatial heterogeneity in both
transport and reaction terms. Under a suitable asymptotic assumption that
the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state
approximated by the system in the absence of transport. Using a WKB ansatz,
we find that this steady state can undergo a Turing-type instability in subsets
of the domain, leading to the formation of localized patterns. The boundaries
of the pattern-forming regions are given asymptotically by ‘local’ Turing conditions corresponding to a spatially homogeneous analysis parameterized by
the spatial variable. We developed a general open-source code which is freely
available, and show numerical examples of this localized pattern formation in
a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and
the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We
numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately
within the asymptotically predicted localized regions. This theory can elegantly
differentiate between spatial structure due to background heterogeneity, from
spatial patterns emergent from Turing-type instabilities.
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