Spatial heterogeneity localizes turing patterns in reaction-cross-diffusion systems

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