Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Liu, Ruo-Qi; Jia, Mengmeng; Fan, Weili; He, Yafeng; Liu, Fucheng

doi:10.7498/aps.71.20221294

Cited by 4 publications

(3 citation statements)

References 37 publications

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“…1952年, Turing [1] 发现在反应扩散系统中, 在没有扩散的情况下对小时间扰动稳定的齐次稳态在有扩散项的情况下变得不稳定, 这种现象被称作Turing不稳定, 这种不稳定在空间域中产生了一种称为Turing斑图的模式. 在这一发现之后, 许多研究者受到启发, 将Turing斑图应用于物理、化学、生物和生态过程等各个领域 [2][3][4][5][6][7][8] . 反应扩散系统是空间运动建模的重要工具之一, 已广泛应用于化学反应 [9] 、谣言传播 [10] 、图像处理 [11] 、流行病传播 [12] 在流行病学领域, 研究人员提出了反应扩散模型, Wang等 [13] 研究了包含人口与流行病过程的反应扩散模型以捕捉环境的空间异质性和个体的移动对疾病动态的影响.…”

Section: 引言unclassified

Spatial pattern of a class of SI models driven by cross diffusion

Lu,

Xiao,

Wan

et al. 2024

Acta Phys. Sin.

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Currently, most research on the spatial patterns of the SI model, both domestically and internationally, focuses on the impacts of self-diffusion and system parameters on pattern formation. However, there is a little study on how cross-diffusion influences the evolution of spatial patterns. In this paper, a spatial epidemic model with both self-diffusion and cross-diffusion is established. The study investigates the effects of cross-diffusion on the stability, rate of stability, and pattern structure of the SI model, while considering scenarios where self-diffusion either induces system instability or it does not. The stability of the non-diffusive system is analyzed, and the conditions for Turing instability in the presence of diffusion terms are elucidated. It is found that when the system is stable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the system's local stability, and can produce Turing patterns. Furthermore, different cross-diffusion coefficients can generate patterns with different structures. When the system is unstable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the pattern structure. Specifically, when the cross-diffusion coefficient $D_1$ for the susceptible individuals is negative, the pattern structure transitions from spot-stripe patterns to spot patterns, and when positive, it transitions from spot-stripe patterns to labyrinthine patterns, and eventually to a uniform solid color distribution. When the cross-diffusion coefficient $D_2$ for the infected individuals is positive, the pattern transformation is similar to when the cross-diffusion coefficient $D_1$ for susceptible individuals is negative, gradually changing to spot patterns, while when $D_2$ is negative, the pattern structure exhibits a porous structure, eventually transitioning to a uniform solid color distribution. Regarding the rate of stability of the SI model, in the case of a stable self-diffusion system, the introduction of cross-diffusion may alter the rate of system stability, and the larger the cross-diffusion coefficient $D_1$ for the susceptible individuals, the faster the system stabilizes. When the self-diffusion-driven system is unstable, cross-diffusion causes the system to transition from an unstable state to a locally stable state, and the smaller the susceptible individuals' cross-diffusion coefficient, the slower the rate of system stabilization. Therefore, cross-diffusion has a significant impact on the stability, rate of stability, and pattern structure of the SI model.

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Section: 引言unclassified

Spatial pattern of a class of SI models driven by cross diffusion

Lu,

Xiao,

Wan

et al. 2024

Acta Phys. Sin.

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“…There have been studies of the Turing instability and Turing patterns in the presence of static anisotropic diffusion [13][14][15][16][17][18], and the analysis for static anisotropic diffusion follows similarly to the standard Turing analysis [10]. However, more care is needed when studying time-varying anisotropic diffusion.…”

Section: Turing Instability Under Time-varying Anisotropic Diffusionmentioning

confidence: 99%

“…Das [15] has extended this approach to anisotropic diffusion parameters that depend on the concentrations of species. Further work has been done on simulating anisotropic reaction-diffusion systems using numerical techniques, including the work of Bose & Chanduri [16], Koibuchi et al [17] and Liu et al [18]. Aside from choosing an anisotropic diffusion tensor, note that an isotropic diffusion tensor defined on a strongly curved manifold can result in an effective anisotropy in Turing patterns; see, for instance, §5a and fig.…”

Section: Introductionmentioning

confidence: 99%

Modification of Turing patterns through the use of time-varying anisotropic diffusion

Önder,

Van Gorder

2023

Proc. R. Soc. A.

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The Turing mechanism underpinning pattern formation in reaction–diffusion relies on the interplay between diffusion parameters and reaction kinetics. Diffusion is typically assumed isotropic, however anisotropic diffusion is known to arise in both nature and laboratory conditions. We study how the Turing instability and resulting Turing patterns are modified when the underlying diffusion tensor is both anisotropic and time-dependent, modelling, for instance, chemical species reacting and diffusing through time-varying anisotropic media. We show that the set of unstable wavenumber vectors corresponding to Turing modes evolve in a spatially biased manner under this anisotropy, thereby modifying the spatial scale and structure of any resulting Turing patterns differently along each spatial coordinate. We employ this spatial bias to develop control strategies to modify the shape or even structure of Turing patterns over time. We are able to make minor changes to the aspect ratio of Turing patterns, such as morphing circular Schnakenberg spots into elliptical spots, as well as more major changes to the structure of patterns, for instance converting Gierer–Meinhardt spots into stripes or FitzHugh–Nagumo labyrinthine patterns into target patterns. Our results suggest that time-varying anisotropic media may be used as a tool by which to modify and even control Turing patterns.

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Cross-diffusion-induced transitions between Turing patterns in reaction-diffusion systems

Meng,

Liu,

et al. 2023

Acta Phys. Sin.

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Cross-diffusion is one of the most important factors affecting the formation and transition of Turing patterns in reaction diffusion systems. In this paper, cross-diffusion was introduced into a reaction diffusion Brusselator model to investigate the effects of the directivity and density-dependence of cross-diffusion on Turing pattern transition. Turing space is obtained by the standard linear stability analysis and the amplitude equations are derived based on weakly nonlinear method, by which Turing pattern selection can be theoretically determined. It is found that the degree of deviation from the primary Turing bifurcation point plays an important role in the process of pattern selection in the Turing region. As the degree of deviation from onset is increased, the system exhibits a sequence of pattern transition from homogenous state, to honeycomb hexagonal pattern, to stripe pattern, and to hexagonal spot pattern. In the case of one-way cross-diffusion, the direction of cross-diffusion determines the order of Turing pattern transition. The cross-diffusion from the inhibitor to the activator enhances the Turing mode and drives the system far away from the primary bifurcation point, resulting in the forward order of Turing pattern transition. On the contrary, the cross-diffusion from the activator to the inhibitor suppresses the Turing mode and forces the pattern transition in a reverse order. In the case of two-way cross-diffusion, the cross-diffusion effect from inhibitors to activators is stronger than that from activators to inhibitors with the same diffusion coefficient. Essentially, the cross-diffusion coefficient is depended not only on the local concentration of species itself, but also on the concentrations of other species due to their interaction. It is shown that the dependence of cross-diffusion terms on local densities of other species can also affect the direction of Turing pattern transition. As the linear adjusting parameter β of local concentration depended diffusion is increased, there is a forward order and a reverse order of Turing pattern transition in the case of cross-diffusion coefficient Duv linearly depended on the concentration of the inhibitor and in the case of the cross-diffusion coefficient Dvu linearly depended on the concentration of the activator, respectively. The numerical simulation results are consistent with the theoretical analysis.

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Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Cited by 4 publications

References 37 publications

Spatial pattern of a class of SI models driven by cross diffusion

Spatial pattern of a class of SI models driven by cross diffusion

Modification of Turing patterns through the use of time-varying anisotropic diffusion

Cross-diffusion-induced transitions between Turing patterns in reaction-diffusion systems

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