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

Gaffney, Eamonn A.; Krause, Andrew L.; Maini, Philip K.; Wang, Chenyuan

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Eamonn A. Gaffney

Philip K. Maini

Chenyuan Wang


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.


Gaffney, E. A., Krause, A. L., Maini, P. K., & Wang, C. (2023). Spatial heterogeneity localizes turing patterns in reaction-cross-diffusion systems. Discrete and Continuous Dynamical Systems - Series B, 28(12), 6092-6125.

Journal Article Type Article
Acceptance Date Mar 5, 2023
Publication Date 2023
Deposit Date Apr 14, 2023
Publicly Available Date Apr 17, 2023
Journal Discrete and Continuous Dynamical Systems - B
Print ISSN 1531-3492
Electronic ISSN 1553-524X
Publisher American Institute of Mathematical Sciences (AIMS)
Peer Reviewed Peer Reviewed
Volume 28
Issue 12
Pages 6092-6125
Public URL
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Accepted Journal Article (4.2 Mb)

Copyright Statement
This article has been accepted for publication in a revised form in Discrete and Continuous Dynamical Systems - B, This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative works.

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