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

  • *Corresponding author: Andrew.Krause@durham.ac.uk

    *Corresponding author: Andrew.Krause@durham.ac.uk 
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  • 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.

    Mathematics Subject Classification: Primary: 35B36, 35K57; Secondary: 92C15.

    Citation:

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  • Figure 1.  Plots of $ u $ in blue curves and $ u^* $ in dashed black curves from solutions of the Schnakenberg model, Eq. (66), for various values of $ \varepsilon $ at $ t = 5,000 $, which are essentially at steady state. The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ a(x) = 0.8-12x^2(x-1)^2 $, $ b = 1 $, $ D_{11} = D_{22} = D_{12} = 1 $, $ D_{21} = 3((x-0.5)^2-1) $. Note that in panel (D), $ N = 5\times 10^4 $ grid points were used to accurately represent the solution

    Figure 2.  Plots of $ u $ in blue curves and $ u^* $ in dashed black curves from solutions of the Schnakenberg model, Eq. (66), for various values of $ \varepsilon $ at $ t = 5,000 $ in (A), (C), (E), and kymographs of $ u $ in (B), (D), (F). The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ a(x) = 0.8-12x^2(x-1)^2 $, $ b = 1 $, $ D_{11} = D_{22} = 1 $, $ D_{12} = 0.5+0.8x $, $ D_{21} = 3((x-0.5)^2-1) $

    Figure 3.  Plots of $ u $ in blue curves and $ u^* $ in dashed black curves from solutions of the Schnakenberg model, Eq. (66), for various values of $ \varepsilon $ at $ t = 5000 $ in (A), (C), (E), and kymographs of $ u $ in (B), (D), (F). The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ a(x) = 0.01+0.19(1+\cos(10 x\pi)) $, $ b = 0.9+0.3(1-\cos(6 x\pi)) $, $ D_{11} = D_{22} = 1 $, $ D_{12} = 1+\sin(3x\pi) $, $ D_{21} = 2(x-1) $

    Figure 4.  Plots of $ u $ in blue curves and $ u^* $ in dashed black curves from solutions of the Keller-Segel model, Eq. (67), for various values of $ \varepsilon $ at $ t = 50,000 $. The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ K(x) = 1.2-0.5\cos(2\pi x) $, $ h(x) = (1-0.5\cos(\pi x)) $, $ D_{11} = D_{22} = 1 $, and $ \chi(x) = 3.05-0.1x $

    Figure 5.  Plots of $ u $ from solutions of the Keller-Segel model, Eq. (67), for various values of $ \varepsilon $ at $ t = 50,000 $. The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ K(x) = 1.2-0.5\cos(2\pi x) $ and $ D_{11} = D_{22} = 1 $

    Figure 6.  Plots of $ u $ in blue curves and $ u^* $ in dashed black curves from solutions of the SKT model, Eq. (68), for various values of $ \varepsilon $ at $ t = 50,000 $ in (A), (B), (E), and kymographs of $ u $ in (B), (D), (F). The red vertical lines show the boundary of $ \mathcal{T}_0 $ computed from the conditions in Theorem 2.1. The parameters are taken as $ r_1(x) = 1 $, $ r_2(x) = 2 $, $ a_1(x) = 0.9+0.2\cos(3\pi x) $, $ a_2(x) = 0.9+0.2\cos(4\pi x) $, $ b_1(x) = 0.6 $, $ b_s(x) = 0.2 $, $ d_1(x) = d_2(x) = 1 $, $ d_{21}(x) = 200x $, $ d_{11}(x) = d_{12}(x) = d_{22}(x) = 0 $

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