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Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Liu Ruo-Qi Jia Meng-Meng Fan Wei-Li He Ya-Feng Liu Fu-Cheng

Liu Ruo-Qi, Jia Meng-Meng, Fan Wei-Li, He Ya-Feng, Liu Fu-Cheng. Effects of anisotropic diffusion on Turing patterns in heterogeneous environment. Acta Phys. Sin., 2022, 71(24): 248201. doi: 10.7498/aps.71.20221294
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Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Liu Ruo-Qi, Jia Meng-Meng, Fan Wei-Li, He Ya-Feng, Liu Fu-Cheng
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  • Diffusion plays a crucial role in the forming and evolving of Turing patterns. Generally, the diffusion processes in complex systems do not comply to the complete random walk theory, which means that the diffusion is abnormal rather than normal, such as super-diffusion, sub-diffusion and anisotropic diffusion. However, most of previous studies focused on the pattern formation mechanism under the normal diffusion. In this paper, a two-component reaction-diffusion model with anisotropic diffusion is used to study the effect of anisotropic diffusion on Turing patterns in heterogeneous environments. Three different types of anisotropic diffusions are utilized. It is shown that the system gives rise to stripe patterns when the degree of anisotropic diffusion is high. The directions of stripes are determined by the degree of the diffusion coefficient deviating from the bifurcation point. In a low degree of anisotropic diffusion, the pattern type is the same as the counterpart in a low degree of the isotropic diffusion. When the diffusion coefficient grows linearly in the space, different types of patterns compete with each other and survive in different regions under the influence of spatial heterogeneity. When the diffusion coefficient is modulated by a one-dimensional periodic function, both type and wavelength of the pattern are determined by the modulated wavelength and the intrinsic wavelength. The system can exhibit alternating two-scale mixed patterns of different types when the modulated wavelength is larger than the intrinsic wavelength. Note that each of the diffusion coefficients of some special anisotropic media is a tensor, which can be expressed as a matrix in two-dimensional cases. We also study the influence of off-diagonal diffusion coefficient D on Turing pattern. It is found that the Turing pattern induced by off-diagonal diffusion coefficient always selects the oblique stripe pattern. The off-diagonal diffusion coefficient D not only affects the pattern selection mechanism, but also expands the parameter range of Turing space. The critical diffusion coefficient $ {D_{\text{c}}} $ increases linearly with the diagonal diffusion coefficient $ {D_u} $ increasing. The intrinsic wavelength of the oblique stripe pattern decreases as the off-diagonal diffusion coefficient D increases. It is interesting to note that the critical wavelength corresponding to the critical diffusion coefficient $ {D_{\text{c}}} $ is independent of the diagonal diffusion coefficient $ {D_u} $. These results not only provide a new insight into the formation mechanism of Turing patterns, but also increase the range and complexity of possible patterns.
      PACS:
      82.40.Ck(Pattern formation in reactions with diffusion, flow and heat transfer)
      05.45.-a(Nonlinear dynamics and chaos)
      05.65.+b(Self-organized systems)
      45.70.Qj(Pattern formation)
      Corresponding author: Fan Wei-Li, fanweili@hbu.edu.cn ; Liu Fu-Cheng, hdlfc@hbu.cn
    • Funds: Project supported by National Natural Science Foundation of China (Grant Nos. 12275064, 11975089, 11875014), the Natural Science Foundation of Hebei Province, China (Grant Nos. A2021201010, A2021201003), and the Interdisciplinary Research Program of Natural Science of Hebei University, China (Grant Nos. DXK202010, DXK202108).

    自组织斑图是在空间和时间上具有某些规律性的非均匀宏观结构, 其普遍存在于自然界以及各种非线性系统[1-4]. 图灵机制是理解斑图形成的重要理论框架之一, 该机制不仅对于理解各个领域内的自组织斑图现象具有重要作用[5], 还在很多领域内具有潜在的实际应用价值, 例如新型材料设计[1]、粒子运输辅助[6]、水质净化[7]以及蝗虫集体运动预测[8]等.

    图灵不稳定性本质上是由于扩散所引起的不稳定性, 因而扩散过程是影响图灵机制的一个重要因素[9]. 自20世纪90年代初期欧阳颀等[10]首次在实验上验证图灵斑图以来, 研究者们对于正常扩散下图灵斑图的形成机理已经有了较为全面的认识[9-12], 在正常扩散下, 粒子迁移满足完全随机行走理论, 扩散过程与方向无关, 呈现各向同性的性质. 实验和理论研究表明, 现实中许多非线性物理系统和生物系统的扩散过程大多不满足完全随机行走理论, 其扩散具有方向性, 如分子在多孔介质[13]、胶体[14]以及器官组织[15]中均表现为各向异性扩散. 各向异性扩散作为调节斑图形态的内在因素之一, 具有丰富的研究内涵[16-18]. Bar等[19]发现各向异性双稳态反应扩散介质中存在一种新的动力学状态—分层时空混沌. He和Liu[20]研究了三维各向异性双稳态介质中化学波前的动力学, 发现足够强的各向异性会引起动力学不稳定性并导致波前破裂. 最近, Cui等[21]在粗粒化模型中应用化学波动力学性质成功地实现了纳米颗粒的分离. 在存在随机性和各向异性的情况下, 图灵模式形成机制具有较强的鲁棒性[22], 并且单一扩散方向满足图灵不稳定性的发生条件时, 图灵模式仍然存在[23]. 图灵模式中条纹的取向由物质各向异性扩散的程度所决定, 且各向异性提高条纹的稳定性[17,24-26]. Gao等[18]发现当各向异性扩散在与局部反应耦合过程中起主导作用时可以产生一种新的图灵分岔并导致超晶格斑图的形成.

    此外, 现实斑图形成系统一般都具有非均匀性(异质性)的特点. 研究表明, 空间异质性是造成时空斑图多样性的主要原因之一[27-29]. 非均匀扩散系数是反应扩散系统空间异质性最为常见的一种表现形式. 2005年, Page等[30,31]发现扩散系数的非均匀性可以扩展图灵空间的参数范围, 增加斑图的复杂度以及稳定性. 胡文勇等[32,33]分析了在局域浓度扩散系数调控下图灵斑图的形成机理, 发现通过改变调控系数可实现六角斑点和条纹之间的转化, 且点条比率强烈依赖于调控系数. Chen等[34]在图灵系统中施加一个圆偏振电场, 也实现了六角斑点和条纹之间的转化. 最近, Van Gorder[27]详细阐述了反应扩散系统中空间异质性对图灵斑图的作用机理. 在上述研究基础上, 本文采用双变量反应扩散模型, 系统研究了异质环境下各向异性扩散对图灵斑图的影响.

    将各向异性扩散引入到最为常用的一种反应扩散模型—Brusselator模型[35], 在无量纲的情况下, 该模型具有以下形式:

    $$ \begin{split} &\frac{\partial u}{\partial t}=a-(b+1)u+{u}^{2}v+{D}_{u}^{(x)}\frac{{\partial }^{2}u}{\partial {x}^{2}}+{D}_{u}^{(y)}\frac{{\partial }^{2}u}{\partial {y}^{2}}\text{, }\\ &\frac{\partial v}{\partial t}=bu-{u}^{2}v+{D}_{v}^{(x)}\frac{{\partial }^{2}v}{\partial {x}^{2}}+{D}_{v}^{(y)}\frac{{\partial }^{2}v}{\partial {y}^{2}}\text{, }\\[-15pt]\end{split} $$ (1)

    其中, $u$$ v $分别为活化子和阻滞子浓度; $a$$ b $为系统局部动力学的控制参数; $D_i^{(x)}$$D_i^{(y)}$分别为物质$i$沿$x$方向和$y$方向的扩散系数, $i$代表活化子$u$和阻滞子$ v $, 当${D_u} \equiv D_u^{\left( x \right)} = D_u^{\left( y \right)}$, ${D_v} \equiv D_v^{\left( x \right)} = D_v^{\left( y \right)}$时, 系统为各向同性扩散. 由于反应与扩散的耦合, 在参数空间的不同区域内系统可以经历不同的分岔, 从而产生不同的时空斑图. 均匀定态解$ ({u}_{0}, {v}_{0})=(a, b/a) $经历图灵不稳定的条件是阻滞子的扩散系数大于活化子的扩散系数, 相应的图灵斑图具有本征波数${k_{\text{T}}} = \sqrt[{}]{{\dfrac{{(b - 1){D_v} - {a^2}{D_u}}}{{2{D_u}{D_v}}}}}$. 固定参数$ a\; = \;{\text{3 }} $, $ {\text{ }}b\; = \;{\text{9}} $, ${D_v} = 10$, 选择${D_u}$作为控制参数, 当其小于临界值${D_u} \leqslant {D_{\text{c}}} = 4.44$时, 满足图灵分岔条件. 图1(a)为各向同性扩散中不同扩散系数下的色散关系曲线, 虚线为特征值的实部$ {{\rm{Re}}} (\lambda ) $, 代表了图灵模的线性增长率, $ {{\rm{Re}}} (\lambda ) $越大, 图灵模增长越快; 实线为特征值的虚部$ {{\rm{Im}}} (\lambda ) $, 对应模式的振荡频率, 由于不同扩散系数下特征值的虚部$ {{\rm{Im}}} (\lambda ) $相差非常小, 因此图中仅给出了$ {D_u} = 4 $时的情况作为参考. 随着扩散系数$ {D_u} $的减小, 系统逐渐远离图灵分岔点, 图灵模波数及其对应的$ {{\rm{Re}}} (\lambda ) $也都随之增大.

    图 1 各向同性扩散下的色散曲线图及其相应的图灵斑图 (a)色散曲线; (b)点状六边形斑图, $ {D_u} = 2 $; (c)条纹斑图, $ {D_u} = 3 $; (d)蜂窝状六边形斑图, $ {D_u} = 4 $\r\nFig. 1. Dispersion curves under isotropic diffusion and their corresponding Turing patterns: (a) Dispersion curves; (b) hexagonal spot pattern, $ {D_u} = 2 $; (c) stripe pattern, $ {D_u} = 3 $; (d) honeycomb hexagonal pattern, $ {D_u} = 4 $.
    图 1  各向同性扩散下的色散曲线图及其相应的图灵斑图 (a)色散曲线; (b)点状六边形斑图, $ {D_u} = 2 $; (c)条纹斑图, $ {D_u} = 3 $; (d)蜂窝状六边形斑图, $ {D_u} = 4 $
    Fig. 1.  Dispersion curves under isotropic diffusion and their corresponding Turing patterns: (a) Dispersion curves; (b) hexagonal spot pattern, $ {D_u} = 2 $; (c) stripe pattern, $ {D_u} = 3 $; (d) honeycomb hexagonal pattern, $ {D_u} = 4 $.

    所有模拟采用欧拉向前差分的方法进行积分, 模拟区域包含$128 \times 128$个空间格点, 扩散项在数值计算中选择五点差分格式, 在显式格式下方程(1)的离散形式为

    $$ \begin{split} u_{(i,j)}^{t + 1} = \;&u_{(i,j)}^t + \frac{{\Delta t}}{{{{\left( {\Delta x} \right)}^2}}}\Big[ D_{u\left( {i - 1,j} \right)}^{(x)}\left( {u_{(i - 1,j)}^t - u_{(i,j)}^t} \right) \\ &+ D_{u\left( {i + 1,j} \right)}^{(x)}\left( {u_{(i + 1,j)}^t - u_{(i,j)}^t} \right) \\ & + D_{u\left( {i,j - 1} \right)}^{(y)}\left( {u_{(i,j - 1)}^t - u_{(i,j)}^t} \right) \\ &+ D_{u\left( {i,j + 1} \right)}^{(y)}\left( {u_{(i,j + 1)}^t - u_{(i,j)}^t} \right) \Big] \\ &+ \Delta tf(u_{(i,j)}^t,v_{(i,j)}^t), \\ v_{(i,j)}^{t + 1} =\;& v_{(i,j)}^t + \frac{{\Delta t}}{{{{\left( {\Delta x} \right)}^2}}}\Big[ D_{v\left( {i - 1,j} \right)}^{(x)}\left( {v_{(i - 1,j)}^t - v_{(i,j)}^t} \right) \\ &+ D_{v\left( {i + 1,j} \right)}^{(x)}\left( {v_{(i + 1,j)}^t - v_{(i,j)}^t} \right) \\ & +D_{v\left( {i,j - 1} \right)}^{(y)}\left( {v_{(i,j - 1)}^t - v_{(i,j)}^t} \right)\\ &+ D_{v\left( {i,j + 1} \right)}^{(y)}\left( {v_{(i,j + 1)}^t - v_{(i,j)}^t} \right) \Big] \\ &+ \Delta tg(u_{(i,j)}^t,v_{(i,j)}^t). \\[-10pt] \end{split} $$ (2)

    取时间步长$\Delta t = 0.01$个时间单位, 空间步长$\Delta x = \Delta y = 1.0$个空间单位. 采用更小的空间步长和时间步长获得了相同的模拟结果. 采用均匀定态附近的随机扰动为初始条件, 边界条件选用周期性边界条件.

    图1(b)(d)给出了各向同性扩散下获得的3种图灵斑图. 偏离分岔点的程度是模式选择过程中的一个关键因素[36]. 当$ {D_u} = 2 $时, 图灵模的$ {{\rm{Re}}} (\lambda ) $足够高, 系统远离主分岔点, 产生一种总相位为零的六边形斑图—点状六边形斑图(图1(b)), 其波长$ \lambda $约为7.3. 随着参数$ {D_u} $的增大, 系统接近主分岔点, 产生波长$ \lambda = 7.6 $的条纹斑图(图1(c)), 而不是点状六边形斑图. 当$ {D_u} = 4 $时, 在图灵分岔点附近获得总相位为$ {\text{π }} $的六边形图案(图1(d)), 表示为蜂窝六边形图案, 此斑图波长$ \lambda = 10.7 $.

    各向异性扩散对图灵斑图具有影响. 研究发现活化子$u$和阻滞子$ v $的各向异性扩散对斑图的影响规律一致, 为简便起见, 本文只给出了活化子的情况, 即只考虑了活化子为各向异性扩散的情况. 保持阻滞子的扩散系数${D_v} = 10$不变, 通过改变活化子在不同方向上的扩散系数$ D_u^{\left( x \right)} $$ D_u^{\left( y \right)} $的大小来研究各向异性扩散对图灵斑图的影响. 为了系统研究各向异性扩散系数对图灵斑图的影响, 分3种情况来讨论: 1)空间均匀分布; 2)空间线性增长; 3)空间周期性变化.

    图2给出了均匀环境中两种不同各向异性扩散情况下获得的条纹斑图, 其中图2(a)为扩散系数$ D_u^{\left( x \right)} $满足图灵分岔条件, 而$ D_u^{\left( y \right)} $不满足时的图灵条纹斑图. 当水平$ x $方向上$ D_u^{\left( x \right)} $满足图灵分岔条件, 而竖直$ y $方向$ D_u^{\left( y \right)} $不满足图灵分岔条件时, 系统仅在$ x $方向上激发图灵模, 因而产生竖直取向的条纹斑图(图2(a)), 其本征波长$ \lambda = 8 $, 略大于各向同性扩散情况下的本征波长(图1(b)). 图2(b)为水平和竖直两个方向均满足图灵分岔时的条纹斑图, 由于$ D_u^{\left( x \right)} < D_u^{\left( y \right)} $, 即水平方向上偏离图灵分岔点较远, 相应的图灵模的增长率较大, 因此图灵模的波数沿着$ x $方向, 即条纹取向为竖直方向. 值得说明的是, 此时条纹的本征波长$ \lambda = 6.4 $, 与图2(a)相比其波长较短, 这意味着$ y $方向上的扩散系数也会影响本征波数的大小. 总而言之, 条纹斑图的取向取决于扩散系数在不同方向上偏离分岔点的程度, 扩散系数越小偏离程度越高, 则此方向即为条纹斑图的取向方向.

    图 2 各向异性扩散下的条纹斑图 (a)竖直取向条纹, $D_u^{\left( x \right)}  =    2$, $D_u^{\left( y \right)}  =  5$; (b)竖直取向条纹, $D_u^{\left( x \right)}  =  2$, $D_u^{\left( y \right)}  =    4$\r\nFig. 2. Stripe patterns under anisotropic diffusion: (a) Vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 5 $; (b) vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 4 $.
    图 2  各向异性扩散下的条纹斑图 (a)竖直取向条纹, $D_u^{\left( x \right)} = 2$, $D_u^{\left( y \right)} = 5$; (b)竖直取向条纹, $D_u^{\left( x \right)} = 2$, $D_u^{\left( y \right)} = 4$
    Fig. 2.  Stripe patterns under anisotropic diffusion: (a) Vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 5 $; (b) vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 4 $.

    在现实世界中, 绝大多数非线性系统都处于非均匀的环境中, 斑图形成过程受到空间异质性的影响. 因此异质性对图灵斑图的影响机理一直是人们研究的热点之一[27]. 图3给出了扩散系数$ D_u^{\left( x \right)} $$ x $方向单调增加时的混合斑图. 随着$ x $的不断增加, $ D_u^{\left( x \right)} $从低于$ D_u^{\left( y \right)} $逐步高于$ D_u^{\left( y \right)} $. 在图3(a)的中间区域$ D_u^{\left( x \right)} \approx D_u^{\left( y \right)} $, 当$ D_u^{\left( x \right)} $在1.89—2.02范围时表现为各向同性情况下的点状六边形, 2.02—2.13范围内为斑点与条纹的过渡区; 左侧区域$ D_u^{\left( x \right)} < D_u^{\left( y \right)} $, 表现为竖直取向条纹; 右侧区域$ D_u^{\left( x \right)} > D_u^{\left( y \right)} $, 表现为水平取向条纹. 由于图3(a)左右两侧$ D_u^{\left( x \right)} $大小不同, 所以造成了左右两侧条纹的波长不同. 在图3(b)中, $ D_u^{\left( x \right)} $在3.97—4.17范围时表现为各向同性情况下的蜂窝状六边形, 左侧区域$ D_u^{\left( x \right)} < D_u^{\left( y \right)} $为竖直条纹, 右侧区域$ D_u^{\left( x \right)} > D_u^{\left( y \right)} $为水平条纹. 图3结果表明, 由于空间异质性的影响, 不同区域可以选择不同的斑图类型. 在每个区域内, 斑图由满足图灵条件的图灵模式之间相互作用而成. 斑图的类型由各向异性扩散系数的相对大小决定, 在各向异性程度较小的区域内呈现六边形斑图, 而在各向异性程度较大的区域内呈现条纹斑图.

    图 3 扩散系数在$ x $方向单调增加时的混合斑图 (a)点状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b)蜂窝状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $\r\nFig. 3. Mixed patterns with a monotonic increase in the $ x $ direction: (a) Pattern of dot-like hexagons mixed with stripes, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b) honeycomb hexagonal and stripe mixed pattern, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $.
    图 3  扩散系数在$ x $方向单调增加时的混合斑图 (a)点状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b)蜂窝状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $
    Fig. 3.  Mixed patterns with a monotonic increase in the $ x $ direction: (a) Pattern of dot-like hexagons mixed with stripes, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b) honeycomb hexagonal and stripe mixed pattern, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $.

    周期性变化的各向异性扩散系数是调控图灵斑图的一种重要方式. 引入周期性调控函数后活化子在$ x $方向上扩散系数变为$D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}x} / {{\lambda _{\text{F}}}}}) + {D_{u0}}$, 其中$ \gamma $为周期性调制函数的振幅, 调控波长$ {\lambda _{\text{F}}} = {L \mathord{\left/ {\vphantom {L n}} \right. } n} $, $ n $为整数, $ {D_{u0}} $为无周期调控时的扩散系数. 当$ D_u^{\left( x \right)} = {D_{u0}} $时, 系统斑图的本征波长为$ {\lambda _0} $($D_u^{\left( y \right)} = 3$$ {\lambda _0} = 8 $). 图4给出了周期性调制扩散系数$ D_u^{\left( x \right)} $在不同调制参数下的图灵斑图. 其中图4(a)(c)研究了调制波长对斑图的影响. 当$ n = 16 $时, 此时调控波长($ {\lambda _{\text{F}}} = 8 $)与系统的本征波长相同, 在周期函数调控下, $ D_u^{\left( x \right)} $在不同位置的大小不同, 即各位置的各向异性程度不同, 但由于调控函数周期较短, 各向异性程度较小的区域太窄, 因此只表现出浓度相间的条纹斑图, 如图4(a)所示. 当$ n = 8 $时, 调控函数的波长增大, 各向异性程度不同的区域随之变宽, 在各向异性程度较小的区域呈现点状斑图, 而在各向异性程度较大的区域呈现条纹斑图, 因此得到了双尺度点线相间斑图, 如图4(b)所示, 此时$ {\lambda _{\text{a}}} = 8.4 $($ {\lambda _{\text{a}}} $为点状斑图波长), $ {\lambda _{\text{b}}} = 16 $($ {\lambda _{\text{b}}} $为条纹斑图波长), 其中$ {\lambda _{\text{b}}} = {\lambda _{\text{F}}} = 2{\lambda _0} $. 图4(b)与地松鼠表皮上的点线相间斑图十分相似, 因此空间周期异质性可能是动物体表点线斑图的形成原因[31]. 当$ n = 4 $时, 此时$ {\lambda _{\text{F}}} = 4{\lambda _0} $, 调控函数的周期相较于图4(b)进一步增大, 各向异性程度不同的区域进一步变宽, 在各向异性程度较小的区域呈现出横向拉长的点状斑图, 而在各向异性程度较大的区域呈现浓度不同的条纹斑图, 得到了图4(c)中的点线相间斑图. 保持图4(a)中的其他条件不变, 图4(d)(f)分别给出了参数$ {D_{u0}} $, $ \gamma $$ D_u^{\left( y \right)} $对图灵斑图的影响. 在图4(d)$ {D_{u0}} $的增大使得$ D_u^{\left( x \right)} $不满足图灵条件, 系统仅在$ y $方向上激发图灵模, 此时调制不起作用, 系统产生了水平取向的条纹斑图. 在图4(e)中调制幅度$ \gamma $的增大扩大了$ D_u^{\left( x \right)} $的参数范围, 但是周期大小并没有改变, 因此表现出另外一种浓度周期变化的条纹斑图. 在图4(f)中, 改变$ D_u^{\left( y \right)} $使其不满足图灵条件, 系统仅在$ x $方向上激发图灵模, 因而产生竖直取向条纹斑图.

    图 4 扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/  {\vphantom {{2{\text{π }}nx} L}} \right.  } L}) + {D_{u0}} $周期性变化时的图灵斑图 (a)调制条纹斑图, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b)双尺度点线相间斑图, $ \gamma  = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c)点线相间斑图, $ \gamma  = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d)水平取向条纹, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e)调制条纹斑图, $ \gamma  = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f)调制条纹斑图, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $\r\nFig. 4. Turing patterns with periodically modulated diffusion coefficient $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/  {\vphantom {{2{\text{π }}nx} L}} \right.  } L}) + {D_{u0}} $ in the $ x $ direction: (a) Modulated stripe pattern, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b) two-scale spots-stripes pattern, $ \gamma  = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c) spots-stripes pattern, $ \gamma  = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d) horizontally oriented stripes, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e) modulated stripe pattern, $ \gamma  = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f) modulated stripe pattern, $ \gamma  = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $.
    图 4  扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $周期性变化时的图灵斑图 (a)调制条纹斑图, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b)双尺度点线相间斑图, $ \gamma = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c)点线相间斑图, $ \gamma = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d)水平取向条纹, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e)调制条纹斑图, $ \gamma = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f)调制条纹斑图, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $
    Fig. 4.  Turing patterns with periodically modulated diffusion coefficient $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $ in the $ x $ direction: (a) Modulated stripe pattern, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b) two-scale spots-stripes pattern, $ \gamma = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c) spots-stripes pattern, $ \gamma = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d) horizontally oriented stripes, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e) modulated stripe pattern, $ \gamma = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f) modulated stripe pattern, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $.

    图5$ D_u^{\left( y \right)} $不满足图灵条件, $ D_u^{\left( x \right)} $为周期性函数时的复杂条纹斑图. 当$ n = 4 $时, 虽然调控函数周期较大, 各向异性程度不同的区域较宽, 但由于$ D_u^{\left( y \right)} $不满足图灵条件, 系统仅在$ x $方向上激发图灵模, 因此产生浓度不同的竖直取向条纹斑图, 如图5(a)所示. 当$ n = 8 $时, 调控函数周期变小, 各向异性程度不同的区域变窄, 周期数变多, 因而得到了浓度相间的周期性条纹斑图, 如图5(b)所示. 当$ n = 16 $时, 调控函数周期较短, 因此只能得到周期内为单一浓度的条纹斑图, 如图5(c)所示.

    图 5 扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/  {\vphantom {{2{\text{π }}nx} L}} \right.  } L}) + {D_{u0}} $周期性变化时的复杂条纹斑图($ \gamma  = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $) (a)$n = $$  4$; (b)$ n = 8 $; (c)$ n = 16 $\r\nFig. 5. Complex stripe patterns with periodically modulated diffusion coefficient  $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/  {\vphantom {{2{\text{π }}nx} L}} \right.  } L}) + {D_{u0}} $ in the $ x $ direction, where $ \gamma  = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $: (a) $ n = 4 $; (b) $ n = 8 $; (c) $ n = 16 $.
    图 5  扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $周期性变化时的复杂条纹斑图($ \gamma = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $) (a)$n = $$ 4$; (b)$ n = 8 $; (c)$ n = 16 $
    Fig. 5.  Complex stripe patterns with periodically modulated diffusion coefficient $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $ in the $ x $ direction, where $ \gamma = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $: (a) $ n = 4 $; (b) $ n = 8 $; (c) $ n = 16 $.

    在一些特殊的各向异性介质中, 例如特殊形状的多孔介质[13]、大脑皮层[15]、磁场中的等离子体[37]、生物种群[38]等, 物质的扩散系数为张量, 在二维情况下其形式可以表示为矩阵形式. 同样为简便起见这里只考虑活化子的各向异性扩散情况, 即$ {{\boldsymbol D}_u} = \left[ {\begin{array}{*{20}{c}} {D_u^{\left( x \right)}}&{D_u^{\left( {xy} \right)}} \\ {D_u^{\left( {yx} \right)}}&{D_u^{\left( y \right)}} \end{array}} \right] $. 一般来说, 扩散矩阵中的非对角元素都是相等的, 令$ D_u^{\left( {xy} \right)} = D_u^{\left( {yx} \right)} = D $. 为便于研究非对角元素$ D $对图灵斑图的影响, 在$ x $方向和$ y $方向上取各向同性扩散系数$ {D_u} = D_u^{\left( x \right)} = D_u^{\left( y \right)} $, 即研究各向同性扩散下的情况. 图6(a)图6(b)是相同$ D $值、不同$ {D_u} $值得到的斜条纹斑图. 与无非对角元素情况相比, 当非对角扩散系数$ D = 1 $时, 系统选择了斜条纹斑图(图6(a))而非点状六边形斑图(图1(b)), 这意味着非对角扩散系数$ D $影响了斑图的选择机理. 在各向同性扩散中, 当$ D_u^{} = 5 $时, 此时不满足图灵分岔条件, 系统为均匀态, 但令$ D = 1 $后, 系统由均匀态转变为斜条纹斑图, 如图6(b)所示, 表明各向异性扩散扩展了图灵空间的参数范围. 在各向同性扩散基础上, 当非对角扩散系数$ D $超过一个临界值$ {D_{\text{c}}} $时, 系统会经历图灵分岔点. 图6(c)给出了临界扩散系数$ {D_{\text{c}}} $与各向同性扩散系数$ {D_u} $的变化. 随着$ {D_u} $的增大, $ {D_{\text{c}}} $随之线性增大. 值得说明的是, 非对角扩散引发的图灵斑图总是选择斜条纹斑图, 其本征波长随着非对角扩散系数$ D $的增大而不断减小, 但是临界扩散系数$ {D_{\text{c}}} $所对应的本征波长与$ {D_u} $无关, 如图6(d)所示.

    图 6 非对角扩散系数$ D $对斑图的影响 (a)斜条纹, $ D_u^{} = 2 $, $ D = 1 $; (b)斜条纹, $ D_u^{} = 5 $, $ D = 1 $; (c)扩散系数$ {D_u} $与临界扩散系数$ {D_{\text{c}}} $的关系曲线图; (d)扩散系数$ D $与波长$ \lambda $的关系曲线图\r\nFig. 6. Effect of off-diagonal diffusion coefficient $ D $ on the pattern: (a) Oblique stripes, $ D_u^{} = 2 $, $ D = 1 $; (b) oblique stripes, $ D_u^{} = 5 $, $ D = 1 $; (c) the relationship between the diffusion coefficient $ {D_u} $ and the critical diffusion coefficient $ {D_{\text{c}}} $; (d) the relationship between the diffusion coefficient $ D $ and the wavelength $ \lambda $.
    图 6  非对角扩散系数$ D $对斑图的影响 (a)斜条纹, $ D_u^{} = 2 $, $ D = 1 $; (b)斜条纹, $ D_u^{} = 5 $, $ D = 1 $; (c)扩散系数$ {D_u} $与临界扩散系数$ {D_{\text{c}}} $的关系曲线图; (d)扩散系数$ D $与波长$ \lambda $的关系曲线图
    Fig. 6.  Effect of off-diagonal diffusion coefficient $ D $ on the pattern: (a) Oblique stripes, $ D_u^{} = 2 $, $ D = 1 $; (b) oblique stripes, $ D_u^{} = 5 $, $ D = 1 $; (c) the relationship between the diffusion coefficient $ {D_u} $ and the critical diffusion coefficient $ {D_{\text{c}}} $; (d) the relationship between the diffusion coefficient $ D $ and the wavelength $ \lambda $.

    本文通过改变活化子在不同方向上的扩散系数$ D_u^{\left( x \right)} $$ D_u^{\left( y \right)} $研究了各向异性扩散对图灵斑图的影响. 模拟结果表明, 当单方向上的扩散系数满足图灵条件即可产生图灵条纹斑图, 其空间取向取决于扩散系数在不同方向上偏离分岔点的程度, 扩散系数越小偏离程度越高, 则此方向即为条纹斑图的取向方向. 当扩散系数线性增长时, 由于空间异质性的影响, 不同区域可以选择不同的斑图类型. 在每个区域内, 斑图由满足图灵条件的图灵模式之间相互作用而成. 当扩散系数周期性变化时, 可形成周期性复杂斑图. 若调控函数的周期较小, 各向异性程度较小的区域太窄, 则仅呈现出条纹斑图而不会呈现各向同性扩散时的斑图类型. 非对角扩散系数$ D $不仅可以影响斑图的选择机理而且扩展了图灵空间的参数范围. 随着扩散系数$ {D_u} $的增大, 临界扩散系数$ {D_{\text{c}}} $线性增大. 斜条纹斑图的本征波长均随着非对角扩散系数$ D $的增大而不断减小.

    [1]

    Fuseya Y, Katsuno H, Behnia K, Kapitulnik A 2021 Nat. Phys. 17 1031Google Scholar

    [2]

    Halatek J, Frey E 2018 Nat. Phys. 14 507Google Scholar

    [3]

    Schweisguth F, Corson F 2019 Dev. Cell 49 659Google Scholar

    [4]

    Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar

    [5]

    Turing A M 1952 Philos. Trans. R. Soc. London, Ser. B 237 37Google Scholar

    [6]

    Chen J X, Chen Y G, Kapral R 2018 Adv. Sci. 5 1800028Google Scholar

    [7]

    Tan Z, Chen S F, Peng X S, Zhang L, Gao C J 2018 Science 360 518Google Scholar

    [8]

    Buhl J, Sumpter D J T, Couzin I D, Hale J J, Despland E, Miller E R, Simpson S J 2006 Science 312 1402Google Scholar

    [9]

    Haas P A, Goldstein R E 2021 Phys. Rev. Lett. 126 238101Google Scholar

    [10]

    Ouyang Q, Swinney H L 1991 Nature 352 610Google Scholar

    [11]

    Callahan T K, Knobloch E 1999 Physica D 132 339Google Scholar

    [12]

    Landge A N, Jordan B M, Diego X, Muller P 2020 Dev. Biol. 460 2Google Scholar

    [13]

    Si L L, Zhang H T, Wei J P, Li B, Han H K 2021 Fuel 284 118887Google Scholar

    [14]

    Lin F J, Liao J J, Wu J C, Ai B Q 2022 Chin. Phys. B 31 036401Google Scholar

    [15]

    Zhou F L, McHugh D J, Li Z X, Gough J E, Williams G R, Parker G J M 2021 Bioinspir. Biomim. 16 046016Google Scholar

    [16]

    Chen C N, Ei S I, Lin Y P, Kung S Y 2011 Comm. Part. Diff. Eq. 36 998Google Scholar

    [17]

    Kolokolnikov T, Ward M, Tzou J, Wei J C 2018 Phil. Trans. R. Soc. A 376 20180110Google Scholar

    [18]

    Gao Y P, Zhang Y F, Schwen D, Jiang C, Gan J 2019 Sci. Rep. 9 7835Google Scholar

    [19]

    Bar M, Hagberg A, Meron E, Thiele U 1999 Phys. Rev. Lett. 83 2664Google Scholar

    [20]

    He Z Z, Liu J 2010 Ann. Phys. 325 359Google Scholar

    [21]

    Cui R F, Chen Q H, Chen J X 2020 Nanoscale 12 12275Google Scholar

    [22]

    Bose I, Chaudhuri I 1997 Phys. Rev. E 55 5291Google Scholar

    [23]

    Busiello D M, Planchon G, Asllani M, Carletti T, Fanelli D 2015 Eur. Phys. J. B 88 222Google Scholar

    [24]

    Shoji H, Iwasa Y, Mochizuki A, Kondo S 2002 J. Theor. Biol. 214 549Google Scholar

    [25]

    Shoji H, Mochizuki A, Iwasa Y, Hirata M, Watanabe T, Hioki S, Kondo S 2003 Dev. Dyn. 226 627Google Scholar

    [26]

    Hiscock T W, Megason S G 2015 Cell Syst. 1 408Google Scholar

    [27]

    Van Gorder R A 2021 Phil. Trans. R. Soc. A 379 20210001Google Scholar

    [28]

    Kozak M, Gaffney E A, Klika V 2019 Phys. Rev. E 100 042220Google Scholar

    [29]

    Glimm T, Zhang J Y, Shen Y Q 2009 Nonlinearity 22 2541Google Scholar

    [30]

    Page K, Maini P K, Monk N A M 2003 Physica D 181 80Google Scholar

    [31]

    Page K M, Maini P K, Monk N A M 2005 Physica D 202 95Google Scholar

    [32]

    Li W S, Hu W Y, Pang Y C, Liu T R, Zhong W R, Shao Y Z 2012 Phys. Rev. E 85 066132Google Scholar

    [33]

    胡文勇, 邵元智 2014 物理学报 63 238202Google Scholar

    Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202Google Scholar

    [34]

    Chen W Q, Zhang H, Ying H P, Li B W, Chen J X 2007 J. Chem. Phys. 127 154708Google Scholar

    [35]

    Prigogine I, Lefever R 1968 J. Chem. Phys. 48 1695Google Scholar

    [36]

    Pena B, Perez-Garcia C 2001 Phys. Rev. E 64 056213Google Scholar

    [37]

    Rajkovic M, Milovanovic M, Skoric M M 2017 Phys. Plasmas 24 052303Google Scholar

    [38]

    Hastings A, Abbott K C, Cuddington K, Francis T, Gellner G, Lai Y C, Morozov A, Petrovskii S, Scranton K, Zeeman M L 2018 Science 361 eaat6412Google Scholar

    期刊类型引用(1)

    1. 陆源杉,肖敏,万佑红,丁洁,蒋海军. 交叉扩散驱动的SI模型空间斑图. 物理学报. 2024(08): 7-16 . 百度学术

    其他类型引用(1)

  • 图 1  各向同性扩散下的色散曲线图及其相应的图灵斑图 (a)色散曲线; (b)点状六边形斑图, $ {D_u} = 2 $; (c)条纹斑图, $ {D_u} = 3 $; (d)蜂窝状六边形斑图, $ {D_u} = 4 $

    Figure 1.  Dispersion curves under isotropic diffusion and their corresponding Turing patterns: (a) Dispersion curves; (b) hexagonal spot pattern, $ {D_u} = 2 $; (c) stripe pattern, $ {D_u} = 3 $; (d) honeycomb hexagonal pattern, $ {D_u} = 4 $.

    图 2  各向异性扩散下的条纹斑图 (a)竖直取向条纹, $D_u^{\left( x \right)} = 2$, $D_u^{\left( y \right)} = 5$; (b)竖直取向条纹, $D_u^{\left( x \right)} = 2$, $D_u^{\left( y \right)} = 4$

    Figure 2.  Stripe patterns under anisotropic diffusion: (a) Vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 5 $; (b) vertically oriented stripes, $ D_u^{\left( x \right)} = 2 $, $ D_u^{\left( y \right)} = 4 $.

    图 3  扩散系数在$ x $方向单调增加时的混合斑图 (a)点状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b)蜂窝状六边形与条纹混合斑图, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $

    Figure 3.  Mixed patterns with a monotonic increase in the $ x $ direction: (a) Pattern of dot-like hexagons mixed with stripes, $ D_u^{\left( x \right)} = 1.5 + 0.01 x $, $ D_u^{\left( y \right)} = 2 $; (b) honeycomb hexagonal and stripe mixed pattern, $ D_u^{\left( x \right)} = 3.5 + 0.01 x $, $ D_u^{\left( y \right)} = 4 $.

    图 4  扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $周期性变化时的图灵斑图 (a)调制条纹斑图, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b)双尺度点线相间斑图, $ \gamma = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c)点线相间斑图, $ \gamma = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d)水平取向条纹, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e)调制条纹斑图, $ \gamma = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f)调制条纹斑图, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $

    Figure 4.  Turing patterns with periodically modulated diffusion coefficient $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $ in the $ x $ direction: (a) Modulated stripe pattern, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (b) two-scale spots-stripes pattern, $ \gamma = 1.5 $, $ n = 8 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (c) spots-stripes pattern, $ \gamma = 1.5 $, $ n = 4 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (d) horizontally oriented stripes, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 7 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (e) modulated stripe pattern, $ \gamma = 2.3 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)}{\text{ = 3}} $; (f) modulated stripe pattern, $ \gamma = 1.5 $, $ n = 16 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $.

    图 5  扩散系数在$ x $方向按$ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $周期性变化时的复杂条纹斑图($ \gamma = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $) (a)$n = $$ 4$; (b)$ n = 8 $; (c)$ n = 16 $

    Figure 5.  Complex stripe patterns with periodically modulated diffusion coefficient $ D_u^{\left( x \right)} = \gamma \cos ({{2{\text{π }}nx} \mathord{\left/ {\vphantom {{2{\text{π }}nx} L}} \right. } L}) + {D_{u0}} $ in the $ x $ direction, where $ \gamma = 1.5 $, $ {D_{u0}} = 2.5 $, $ D_u^{\left( y \right)} = 5 $: (a) $ n = 4 $; (b) $ n = 8 $; (c) $ n = 16 $.

    图 6  非对角扩散系数$ D $对斑图的影响 (a)斜条纹, $ D_u^{} = 2 $, $ D = 1 $; (b)斜条纹, $ D_u^{} = 5 $, $ D = 1 $; (c)扩散系数$ {D_u} $与临界扩散系数$ {D_{\text{c}}} $的关系曲线图; (d)扩散系数$ D $与波长$ \lambda $的关系曲线图

    Figure 6.  Effect of off-diagonal diffusion coefficient $ D $ on the pattern: (a) Oblique stripes, $ D_u^{} = 2 $, $ D = 1 $; (b) oblique stripes, $ D_u^{} = 5 $, $ D = 1 $; (c) the relationship between the diffusion coefficient $ {D_u} $ and the critical diffusion coefficient $ {D_{\text{c}}} $; (d) the relationship between the diffusion coefficient $ D $ and the wavelength $ \lambda $.

  • [1]

    Fuseya Y, Katsuno H, Behnia K, Kapitulnik A 2021 Nat. Phys. 17 1031Google Scholar

    [2]

    Halatek J, Frey E 2018 Nat. Phys. 14 507Google Scholar

    [3]

    Schweisguth F, Corson F 2019 Dev. Cell 49 659Google Scholar

    [4]

    Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar

    [5]

    Turing A M 1952 Philos. Trans. R. Soc. London, Ser. B 237 37Google Scholar

    [6]

    Chen J X, Chen Y G, Kapral R 2018 Adv. Sci. 5 1800028Google Scholar

    [7]

    Tan Z, Chen S F, Peng X S, Zhang L, Gao C J 2018 Science 360 518Google Scholar

    [8]

    Buhl J, Sumpter D J T, Couzin I D, Hale J J, Despland E, Miller E R, Simpson S J 2006 Science 312 1402Google Scholar

    [9]

    Haas P A, Goldstein R E 2021 Phys. Rev. Lett. 126 238101Google Scholar

    [10]

    Ouyang Q, Swinney H L 1991 Nature 352 610Google Scholar

    [11]

    Callahan T K, Knobloch E 1999 Physica D 132 339Google Scholar

    [12]

    Landge A N, Jordan B M, Diego X, Muller P 2020 Dev. Biol. 460 2Google Scholar

    [13]

    Si L L, Zhang H T, Wei J P, Li B, Han H K 2021 Fuel 284 118887Google Scholar

    [14]

    Lin F J, Liao J J, Wu J C, Ai B Q 2022 Chin. Phys. B 31 036401Google Scholar

    [15]

    Zhou F L, McHugh D J, Li Z X, Gough J E, Williams G R, Parker G J M 2021 Bioinspir. Biomim. 16 046016Google Scholar

    [16]

    Chen C N, Ei S I, Lin Y P, Kung S Y 2011 Comm. Part. Diff. Eq. 36 998Google Scholar

    [17]

    Kolokolnikov T, Ward M, Tzou J, Wei J C 2018 Phil. Trans. R. Soc. A 376 20180110Google Scholar

    [18]

    Gao Y P, Zhang Y F, Schwen D, Jiang C, Gan J 2019 Sci. Rep. 9 7835Google Scholar

    [19]

    Bar M, Hagberg A, Meron E, Thiele U 1999 Phys. Rev. Lett. 83 2664Google Scholar

    [20]

    He Z Z, Liu J 2010 Ann. Phys. 325 359Google Scholar

    [21]

    Cui R F, Chen Q H, Chen J X 2020 Nanoscale 12 12275Google Scholar

    [22]

    Bose I, Chaudhuri I 1997 Phys. Rev. E 55 5291Google Scholar

    [23]

    Busiello D M, Planchon G, Asllani M, Carletti T, Fanelli D 2015 Eur. Phys. J. B 88 222Google Scholar

    [24]

    Shoji H, Iwasa Y, Mochizuki A, Kondo S 2002 J. Theor. Biol. 214 549Google Scholar

    [25]

    Shoji H, Mochizuki A, Iwasa Y, Hirata M, Watanabe T, Hioki S, Kondo S 2003 Dev. Dyn. 226 627Google Scholar

    [26]

    Hiscock T W, Megason S G 2015 Cell Syst. 1 408Google Scholar

    [27]

    Van Gorder R A 2021 Phil. Trans. R. Soc. A 379 20210001Google Scholar

    [28]

    Kozak M, Gaffney E A, Klika V 2019 Phys. Rev. E 100 042220Google Scholar

    [29]

    Glimm T, Zhang J Y, Shen Y Q 2009 Nonlinearity 22 2541Google Scholar

    [30]

    Page K, Maini P K, Monk N A M 2003 Physica D 181 80Google Scholar

    [31]

    Page K M, Maini P K, Monk N A M 2005 Physica D 202 95Google Scholar

    [32]

    Li W S, Hu W Y, Pang Y C, Liu T R, Zhong W R, Shao Y Z 2012 Phys. Rev. E 85 066132Google Scholar

    [33]

    胡文勇, 邵元智 2014 物理学报 63 238202Google Scholar

    Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202Google Scholar

    [34]

    Chen W Q, Zhang H, Ying H P, Li B W, Chen J X 2007 J. Chem. Phys. 127 154708Google Scholar

    [35]

    Prigogine I, Lefever R 1968 J. Chem. Phys. 48 1695Google Scholar

    [36]

    Pena B, Perez-Garcia C 2001 Phys. Rev. E 64 056213Google Scholar

    [37]

    Rajkovic M, Milovanovic M, Skoric M M 2017 Phys. Plasmas 24 052303Google Scholar

    [38]

    Hastings A, Abbott K C, Cuddington K, Francis T, Gellner G, Lai Y C, Morozov A, Petrovskii S, Scranton K, Zeeman M L 2018 Science 361 eaat6412Google Scholar

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  • 期刊类型引用(1)

    1. 陆源杉,肖敏,万佑红,丁洁,蒋海军. 交叉扩散驱动的SI模型空间斑图. 物理学报. 2024(08): 7-16 . 百度学术

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Publishing process
  • Received Date:  30 June 2022
  • Accepted Date:  30 August 2022
  • Available Online:  10 December 2022
  • Published Online:  24 December 2022

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