“斑图生成”的版本间的差异

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A '''Belousov–Zhabotinsky reaction''', or '''BZ reaction''', is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological<sup>[''clarification needed'']</sup> phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.
 
A '''Belousov–Zhabotinsky reaction''', or '''BZ reaction''', is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological<sup>[''clarification needed'']</sup> phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.
  
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别洛乌索夫-扎波茨基反应(BZ 反应)
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别洛乌索夫-扎波茨基反应(BZ 反应) 是一类作为非平衡热力学经典例子的反应，会促成非线性化学振荡器的建立。（这些振子中唯一共同的元素是溴和酸。）这类反应对于理论化学很重要，因为它们表明化学反应不必受平衡热力学行为的支配。这些反应远未达到平衡，并在相当长的一段时间内保持非平衡状态，并无序地发展。从这个意义上说，它们为非平衡的生物学现象(需要澄清？)提供了一个有价值的化学模型；因此，BZ反应本身的数学模型和模拟具有理论意义，揭示某类现象作为噪声驱动的秩序的性质。
  
 
An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quiescent medium. Some clock reactions such as Briggs–Rauscher and BZ using the tris(bipyridine)ruthenium(II) chloride as catalyst can be excited into self-organising activity through the influence of light.
 
An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quiescent medium. Some clock reactions such as Briggs–Rauscher and BZ using the tris(bipyridine)ruthenium(II) chloride as catalyst can be excited into self-organising activity through the influence of light.
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* [[Liesegang rings]]
 
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'''Liesegang rings''' (/ˈliːzəɡɑːŋ/) are a phenomenon seen in many, if not most, chemical systems undergoing a precipitation reaction under certain conditions of concentration and in the absence of convection. Rings are formed when weakly soluble salts are produced from reaction of two soluble substances, one of which is dissolved in a gel medium. The phenomenon is most commonly seen as rings in a Petri dish or bands in a test tube; however, more complex patterns have been observed, such as dislocations of the ring structure in a Petri dish, helices, and "Saturn rings" in a test tube. Despite continuous investigation since rediscovery of the rings in 1896, the mechanism for the formation of Liesegang rings is still unclear.
 
  
 
===Physics===
 
===Physics===

2021年7月24日 (六) 11:27的版本

本词条由11初步翻译，由CecileLi初步审校 https://wiki.swarma.org/index.php?title=%E5%B9%B3%E8%A1%A1%E7%90%86%E8%AE%BA#:~:text=%E6%9C%AC%E8%AF%8D%E6%9D%A1%E7%94%B1,11%E5%88%9D%E6%AD%A5%E7%BF%BB%E8%AF%91由栗子CUGB继续完善

Complex systems organizational map.jpg
Complex systems
Topics
文件:Self-organizing-Mechanism-for-Development-of-Space-filling-Neuronal-Dendrites-pcbi.0030212.sv003.ogv
Pattern formation in a computational model of dendrite growth.
文件:用于开发填充神经网络树突神经的自发性组织机制-pcbi.0030212.sv003.ogv
Pattern formation in a computational model of dendrite growth.

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.

斑图生成学关注自组织中可见的、统计有序的结果，以及自然界中相似斑图背后的共同原理。

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. The role of genes in pattern formation is an aspect of morphogenesis, the creation of diverse anatomies from similar genes, now being explored in the science of evolutionary developmental biology or evo-devo. The mechanisms involved are well seen in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (a fruit fly), one of the first organisms to have its morphogenesis studied and in the eyespots of butterflies, whose development is a variant of the standard (fruit fly) mechanism.

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. The role of genes in pattern formation is an aspect of morphogenesis, the creation of diverse anatomies from similar genes, now being explored in the science of evolutionary developmental biology or evo-devo. The mechanisms involved are well seen in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (a fruit fly), one of the first organisms to have its morphogenesis studied and in the eyespots of butterflies, whose development is a variant of the standard (fruit fly) mechanism.

在 发育生物学 Developmental Biology 中，斑图生成指的是细胞命运中复杂组织的产生过程。斑图生成由基因控制的。基因在斑图生成中所起的作用属于形态发生的一个方面，即由相似的基因演化出不同的生命结构，当下也属于演化发育生物学所探究的问题。模式生物黑腹果蝇（第一个应用到形态发生研究的物种）的胚胎的斑图生成过程和蝴蝶的眼点清楚地体现了这一机制，而后者的发育过程是标准（果蝇）机制的一种变体。

Examples

Examples 实例

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Examples of pattern formation can be found in biology, chemistry, physics, and mathematics,[1] and can readily be simulated with computer graphics, as described in turn below.

Examples of pattern formation can be found in biology, chemistry, physics, and mathematics, and can readily be simulated with computer graphics, as described in turn below.

在生物、化学、物理和数学中都有斑图生成的实例，并且我们可以用 计算机图形学 Computer Graphics 轻松地来模拟，下面依次进行介绍。

Biology

生物学

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Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.[2]

Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.

动物标记、动物分节和叶序等生物学斑图是以不同的方式形成的。

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions.[3] Embryogenesis, such as of the fruit fly Drosophila, involves coordinated control of cell fates.[4][5][6] Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the French flag model in the 1960s.[7][8] More generally, the morphology of organisms is patterned by the mechanisms of evolutionary developmental biology, such as changing the timing and positioning of specific developmental events in the embryo.[9]

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions. Embryogenesis, such as of the fruit fly Drosophila, involves coordinated control of cell fates. Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the French flag model in the 1960s. More generally, the morphology of organisms is patterned by the mechanisms of evolutionary developmental biology, such as changing the timing and positioning of specific developmental events in the embryo.

在发育生物学中，斑图生成描述了胚胎组织发育中最初相同的细胞逐步呈现出复杂形态和功能的机制。以果蝇为例，胚胎发育 Embryogenesis 涉及到细胞命运的协调控制。斑图生成是由遗传基因所控制，通常涉及一个场中的每个细胞沿着形态发生素梯度感知和响应其位置，然后通过细胞信号通路进行短距离的细胞间通信以完善初始斑图。在此背景下，细胞场 A Field of Cells 是指通过响应同一组位置信息线索而影响其细胞命运的一组细胞。这个概念模型最早在20世纪60年代被描述为法旗模型。更一般来说，生物体的形态是由进化发育生物学的机制，如改变胚胎中特定发育事件的时间和位置所决定的。

Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing[10] and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.[11][12]

Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.

生物系统中可能存在的斑图生成机制包括： 阿兰 · 图灵 Alan Turing 提出的经典反应—扩散模型和最近发现的弹性不稳定机制，后者被认为是高等动物大脑皮层上褶皱模式等形成的原因。

Growth of colonies

菌落生长

Bacterial colonies show a large variety of patterns formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns.[13] Other organisms such as slime moulds display remarkable patterns caused by the dynamics of chemical signaling.[14]

Bacterial colonies show a large variety of patterns formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns. Other organisms such as slime moulds display remarkable patterns caused by the dynamics of chemical signaling.

细菌菌落在生长过程中形成的斑图种类繁多，其形状取决于生长环境。特别是包括培养基环境的严酷程度、营养物质的缺乏等在内的各种外部压力，会增强了所产生斑图的复杂性。对于其他生物，如粘液霉菌，则显示出由化学信号动力学驱动的非同寻常的斑图。

Vegetation patterns

植被斑图

Main article: patterned vegetation

Vegetation patterns such as tiger bush[15] and fir waves[16] form for different reasons. Tiger bush consists of stripes of bushes on arid slopes in countries such as Niger where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation absorbs rainwater from the bare zone immediately above it.[15] In contrast, fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow, protected by the wind shadow of the remaining tall trees[16]. In flat terrains additional pattern morphologies appear besides stripes - hexagonal gap patterns and hexagonal spot patterns. Pattern formation in this case is driven by positive feedback loops between local vegetation growth and water transport towards the growth location[17][18].

虎皮纹灌木和冷杉波纹等植被斑图形成的原因不同。在尼日尔等干旱国家，植物生长受到降雨的限制，虎皮纹灌木由干旱斜坡上的带状灌木组成。每一条大致水平的植被带都可以从紧靠其上方的裸露地带吸收雨水。相比之下，在受到大风摧残后的植被再生过程中，山坡上的森林会出现冷杉波纹。当较外层树木倒下时，曾经受到它们庇护的内层树木会暴露在外，进而这部分植被更容易受到破坏。因此，树林空隙往往向背风面扩大。同时在迎风坡，幼树在生长过程中则受到存活下来的高大树木自然形成的风影区保护。因此在平坦的地形中，植被除了条带状之外，还出现了其他的斑图形态—例如六边形缝隙斑图和六边形点状斑图。在这种情况下，植被斑图是在当地植被生长和朝向生长位置的水分输送之间的正反馈回路驱动下形成的。

Chemistry

化学

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模板:FurtherPattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.[19] The Brusselator model developed by Ilya Prigogine and collaborators is one such example that exhibits Turing instability.[20] Pattern formation in chemical systems often involve oscillatory chemical kinetics or autocatalytic reactions[21] such as Belousov–Zhabotinsky reaction or Briggs–Rauscher reaction. In industrial applications such as chemical reactors, pattern formation can lead to temperature hot spots which can reduce the yield or create hazardous safety problems such as a thermal runaway.[22][19] The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying reaction-diffusion system.[19][21]

Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns. The Brusselator model developed by Ilya Prigogine and collaborators is one such example that exhibits Turing instability. Pattern formation in chemical systems often involve oscillatory chemical kinetics or autocatalytic reactions such as Belousov–Zhabotinsky reaction or Briggs–Rauscher reaction. In industrial applications such as chemical reactors, pattern formation can lead to temperature hot spots which can reduce the yield or create hazardous safety problems such as a thermal runaway. The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying reaction-diffusion system.

在化学和化学工程领域，斑图生成的研究进展良好，其中包括温度和浓度斑图。由伊利亚·普利高津 Ilya Prigogine 和其合作者开发的布鲁塞尔器模型就是一个展示出图灵不稳定性的例子。化学体系中的斑图生成通常涉及 振荡化学动力学 Oscillatory Chemical Kinetics 或 自催化反应 Autocatalytic Reactions，如别洛乌索夫-扎波茨基反应Belousov–Zhabotinsky Reaction 或 Briggs–Rauscher Reaction 布里格斯-劳舍反应。在工业应用中，如化学反应堆，斑图生成可能导致温度热点，进而会降低产量或造成灾害性安全问题，如热失控。斑图生成的出现可以用底层的反应—扩散系统的数学建模与模拟来研究。

A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological[clarification needed] phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.

别洛乌索夫-扎波茨基反应(BZ 反应) 是一类作为非平衡热力学经典例子的反应，会促成非线性化学振荡器的建立。（这些振子中唯一共同的元素是溴和酸。）这类反应对于理论化学很重要，因为它们表明化学反应不必受平衡热力学行为的支配。这些反应远未达到平衡，并在相当长的一段时间内保持非平衡状态，并无序地发展。从这个意义上说，它们为非平衡的生物学现象(需要澄清？)提供了一个有价值的化学模型；因此，BZ反应本身的数学模型和模拟具有理论意义，揭示某类现象作为噪声驱动的秩序的性质。

An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quiescent medium. Some clock reactions such as Briggs–Rauscher and BZ using the tris(bipyridine)ruthenium(II) chloride as catalyst can be excited into self-organising activity through the influence of light.

Physics

物理学

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In the 1980s Lugiato and Lefever developed a model of light propagation in an optical cavity that results in pattern formation by the exploitation of nonlinear effects.

In the 1980s Lugiato and Lefever developed a model of light propagation in an optical cavity that results in pattern formation by the exploitation of nonlinear effects.

20世纪80年代，卢贾托Lugiato和勒弗Lefever开发了一个光在光学谐振腔中传播的模型，该模型通过利用非线性效应形成斑图。

Bénard cells, laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. Dendrites in solidification, liquid crystals. Solitons.

Bénard cells, laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. Dendrites in solidification, liquid crystals. Solitons.

如贝纳尔细胞、激光、条状云或卷状云、冰柱上的涟漪、泥路上的洗衣板等图案以及凝固中的树枝状、液晶、孤子等。

Rayleigh-Bénard convection is a type of natural convection, occurring in a planar horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems.

Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of lesser density fluid from the heated bottom layer. This upwelling spontaneously organizes into a regular pattern of cells.

In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

Mathematics

数学

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Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.

Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.

球形填充和覆盖物。数学是其他斑图生成机制的基础。

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.

A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.

For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.

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Computer graphics

计算机图形学

Pattern resembling a reaction–diffusion model, produced using sharpen and blur

Pattern resembling a reaction–diffusion model, produced using sharpen and blur

类似反应-扩散模型，使用锐化和模糊技术制作的图案。

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Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects.[23][24]

Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects.

一些类型的自动机已经被用来生成看起来有机的纹理，以便对三维物体进行更真实的阴影处理。

A popular Photoshop plugin, KPT 6, included a filter called 'KPT reaction'. Reaction produced reaction–diffusion style patterns based on the supplied seed image.

A popular Photoshop plugin, KPT 6, included a filter called 'KPT reaction'. Reaction produced reaction–diffusion style patterns based on the supplied seed image.

一个流行的Photoshop插件，KPT6，包括一个名为“KPT反应”的过滤器。反应根据提供的种子图像产生反应-扩散风格的图案。

A similar effect to the 'KPT reaction' can be achieved with convolution functions in digital image processing, with a little patience, by repeatedly sharpening and blurring an image in a graphics editor. If other filters are used, such as emboss or edge detection, different types of effects can be achieved.

A similar effect to the 'KPT reaction' can be achieved with convolution functions in digital image processing, with a little patience, by repeatedly sharpening and blurring an image in a graphics editor. If other filters are used, such as emboss or edge detection, different types of effects can be achieved.

利用数字图像处理中的 卷积函数convolution functions，只要稍有耐心，在图形编辑器中反复对图像进行锐化和模糊处理，就能达到类似 "KPT反应 "的效果。如果使用其他滤镜，如浮雕或边缘检测，可以实现不同类型的效果。

Computers are often used to simulate the biological, physical or chemical processes that lead to pattern formation, and they can display the results in a realistic way. Calculations using models like reaction–diffusion or MClone are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

Computers are often used to simulate the biological, physical or chemical processes that lead to pattern formation, and they can display the results in a realistic way. Calculations using models like reaction–diffusion or MClone are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

因为计算机能以真实的方式显示结果，所以我们常常使用它来模拟导致斑图生成的生物、物理或化学过程。根据科学家设计的实际数学方程，通常使用反应-扩散或MClone等模型进行计算，以此来模拟所研究的现象。

References

References

参考资料

  1. ↑ Ball, 2009.
  2. ↑ Ball, 2009. Shapes, pp. 231–252.
  3. ↑ Ball, 2009. Shapes, pp. 261–290.
  4. ↑ Eric C. Lai (March 2004). "Notch signaling: control of cell communication and cell fate". Development. 131 (5): 965–73. doi:10.1242/dev.01074. PMID 14973298.
  5. ↑ Melinda J. Tyler, David A. Cameron (2007). "Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition". Vision Research. 47 (4): 501–511. doi:10.1016/j.visres.2006.08.025. PMID 17034830.{{cite journal}}: CS1 maint: uses authors parameter (link)
  6. ↑ Hans Meinhard (2001-10-26). "Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation". Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany.
  7. ↑ Wolpert L (October 1969). "Positional information and the spatial pattern of cellular differentiation". J. Theor. Biol. 25 (1): 1–47. doi:10.1016/S0022-5193(69)80016-0. PMID 4390734.
  8. ↑ Wolpert, Lewis (2007). Principles of development (3rd ed.). Oxford [Oxfordshire]: Oxford University Press. ISBN 978-0-19-927536-6. 
  9. ↑ Hall, B. K. (2003). "Evo-Devo: evolutionary developmental mechanisms". International Journal of Developmental Biology. 47 (7–8): 491–495. PMID 14756324.
  10. ↑ S. Kondo, T. Miura, "Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation", Science 24 Sep 2010: Vol. 329, Issue 5999, pp. 1616-1620 DOI: 10.1126/science.1179047
  11. ↑ Mercker, M; Brinkmann, F; Marciniak-Czochra, A; Richter, T (4 May 2016). "Beyond Turing: mechanochemical pattern formation in biological tissues". Biology Direct. 11: 22. doi:10.1186/s13062-016-0124-7. PMC 4857296. PMID 27145826.
  12. ↑ Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632
  13. ↑ Ball, 2009. Branches, pp. 52–59.
  14. ↑ Ball, 2009. Shapes, pp. 149–151.
  15. ↑ 15.0 15.1 Tongway, D.J., Valentin, C. & Seghieri, J. (2001). Banded vegetation patterning in arid and semiarid environments. New York: Springer-Verlag. ISBN 978-1461265597. 
  16. ↑ 16.0 16.1 D'Avanzo, C. (22 February 2004). "Fir Waves: Regeneration in New England Conifer Forests". TIEE. Retrieved 26 May 2012.
  17. ↑ Meron, E (2019). "Vegetation pattern formation: the mechanisms behind the forms". Physics Today. 72 (11): 30–36. doi:10.1063/PT.3.4340.
  18. ↑ Meron, E (2018). "From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study". Annual Review of Condensed Matter Physics. 9: 79–103. doi:10.1146/annurev-conmatphys-033117-053959.
  19. ↑ 19.0 19.1 19.2 Gupta, Ankur; Chakraborty, Saikat (January 2009). "Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors". Chemical Engineering Journal. 145 (3): 399–411. doi:10.1016/j.cej.2008.08.025. ISSN 1385-8947.
  20. ↑ Prigogine, I.; Nicolis, G. (1985), Hazewinkel, M.; Jurkovich, R.; Paelinck, J. H. P. (eds.), "Self-Organisation in Nonequilibrium Systems: Towards A Dynamics of Complexity", Bifurcation Analysis: Principles, Applications and Synthesis, Springer Netherlands, pp. 3–12, doi:10.1007/978-94-009-6239-2_1, ISBN 9789400962392
  21. ↑ 21.0 21.1 Gupta, Ankur; Chakraborty, Saikat (2008-01-19). "Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions". Chemical Product and Process Modeling. 3 (2). doi:10.2202/1934-2659.1135. ISSN 1934-2659.
  22. ↑ Marwaha, Bharat; Sundarram, Sandhya; Luss, Dan (September 2004). "Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†". The Journal of Physical Chemistry B. 108 (38): 14470–14476. doi:10.1021/jp049803p. ISSN 1520-6106.
  23. ↑ Greg Turk, Reaction–Diffusion
  24. ↑ Andrew Witkin; Michael Kassy (1991). "Reaction–Diffusion Textures" (PDF). Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques: 299–308. doi:10.1145/122718.122750. ISBN 0897914368.




Bibliography

Bibliography

参考书目




External links

External links

外部链接

  • SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.


  • '15-line Matlab code', A simple 15-line Matlab program to simulate 2D pattern formation for reaction-diffusion model.




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Category:Developmental biology

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Category:Pattern formation

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