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{{进一步|自然界中的模式}}

{{进一步|自然界中的模式}}

Examples of pattern formation can be found in biology, chemistry, physics, and mathematics,<ref>Ball, 2009.</ref> 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,<ref name=":2">Ball, 2009.</ref> 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.

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'''<font color="#ff8000"> 计算机图形学 Computer Graphics</font>''' 轻松地来模拟，下面依次进行介绍。

在生物、化学、物理和数学<ref name=":2" /> 中都有斑图生成的实例，并且我们可以用'''<font color="#ff8000"> 计算机图形学 Computer Graphics</font>''' 轻松地来模拟，下面依次进行介绍。

===Biology 生物学===

===Biology 生物学===

{{深入研究|进化发育生物学|形态发生学领域}}

{{深入研究|进化发育生物学|形态发生学领域}}

Biological patterns such as [[animal markings]], the segmentation of animals, and [[phyllotaxis]] are formed in different ways.<ref>Ball, 2009. ''Shapes'', pp. 231–252.</ref>

Biological patterns such as [[animal markings]], the segmentation of animals, and [[phyllotaxis]] are formed in different ways.<ref name=":3">Ball, 2009. ''Shapes'', pp. 231–252.</ref>

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

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。

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。<ref name=":3" />

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.<ref>Ball, 2009. Shapes, pp. 261–290.</ref> [[Embryogenesis]], such as [[Drosophila embryogenesis|of the fruit fly ''Drosophila'']], involves coordinated [[cell fate determination|control of cell fates]].<ref name="Lai">{{cite journal |author=Eric C. Lai |title=Notch signaling: control of cell communication and cell fate |doi=10.1242/dev.01074 |pmid=14973298 |volume=131 |issue=5 |date=March 2004 |pages=965–73 |journal=Development|doi-access=free }}</ref><ref name="Tyler">{{cite journal |title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition |authors=Melinda J. Tyler, David A. Cameron|journal=Vision Research |volume=47 |issue=4 |pages=501–511 |year=2007 |doi=10.1016/j.visres.2006.08.025 |pmid=17034830}}</ref><ref name="Meinhard">{{cite web|title=Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation |author=Hans Meinhard |agency= Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany |url=http://www1.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm |date=2001-10-26 }}</ref> 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.<ref>{{cite journal |doi=10.1016/S0022-5193(69)80016-0 |author=Wolpert L |title=Positional information and the spatial pattern of cellular differentiation |journal=J. Theor. Biol. |volume=25 |issue=1 |pages=1–47 |date=October 1969 |pmid=4390734 }}</ref><ref>{{cite book |author=Wolpert, Lewis |title=Principles of development |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2007 |isbn=978-0-19-927536-6 |edition=3rd |display-authors=etal}}</ref> More generally, the morphology of organisms is patterned by the mechanisms of [[evolutionary developmental biology]], such as [[heterochrony|changing the timing]] and positioning of specific developmental events in the embryo.<ref>{{cite journal |last1=Hall |first1=B. K. |title=Evo-Devo: evolutionary developmental mechanisms |journal=International Journal of Developmental Biology |date=2003 |volume=47 |issue=7–8 |pages=491–495 |pmid=14756324}}</ref>

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.<ref name=":4">Ball, 2009. Shapes, pp. 261–290.</ref> [[Embryogenesis]], such as [[Drosophila embryogenesis|of the fruit fly ''Drosophila'']], involves coordinated [[cell fate determination|control of cell fates]].<ref name="Lai">{{cite journal |author=Eric C. Lai |title=Notch signaling: control of cell communication and cell fate |doi=10.1242/dev.01074 |pmid=14973298 |volume=131 |issue=5 |date=March 2004 |pages=965–73 |journal=Development|doi-access=free }}</ref><ref name="Tyler">{{cite journal |title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition |authors=Melinda J. Tyler, David A. Cameron|journal=Vision Research |volume=47 |issue=4 |pages=501–511 |year=2007 |doi=10.1016/j.visres.2006.08.025 |pmid=17034830}}</ref><ref name="Meinhard">{{cite web|title=Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation |author=Hans Meinhard |agency= Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany |url=http://www1.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm |date=2001-10-26 }}</ref> 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.<ref name=":5">{{cite journal |doi=10.1016/S0022-5193(69)80016-0 |author=Wolpert L |title=Positional information and the spatial pattern of cellular differentiation |journal=J. Theor. Biol. |volume=25 |issue=1 |pages=1–47 |date=October 1969 |pmid=4390734 }}</ref><ref name=":6">{{cite book |author=Wolpert, Lewis |title=Principles of development |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2007 |isbn=978-0-19-927536-6 |edition=3rd |display-authors=etal}}</ref> More generally, the morphology of organisms is patterned by the mechanisms of [[evolutionary developmental biology]], such as [[heterochrony|changing the timing]] and positioning of specific developmental events in the embryo.<ref>{{cite journal |last1=Hall |first1=B. K. |title=Evo-Devo: evolutionary developmental mechanisms |journal=International Journal of Developmental Biology |date=2003 |volume=47 |issue=7–8 |pages=491–495 |pmid=14756324}}</ref>

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。<ref name=":4" /> 以果蝇为例，'''<font color="#ff8000">胚胎发育 Embryogenesis</font>''' 涉及到细胞命运的协调控制。<ref name="Lai" /><ref name="Tyler" /><ref name="Meinhard" /> 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，'''<font color="#32CD32">细胞场 A Field of Cells</font>''' 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 French Flag Model'''。<ref name=":5" /><ref name=":6" /> 更一般来说，生物体的形态是由'''进化发育生物学 Evolutionary Developmental Biology'''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]]<ref name=":7">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</ref> 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.<ref name="Mercker">{{cite journal |last1=Mercker |first1=M |last2=Brinkmann |first2=F |last3=Marciniak-Czochra |first3=A |last4=Richter |first4=T |title=Beyond Turing: mechanochemical pattern formation in biological tissues. |journal=Biology Direct |date=4 May 2016 |volume=11 |pages=22 |doi=10.1186/s13062-016-0124-7 |pmid=27145826|pmc=4857296 }}</ref><ref name=":8">Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632</ref>

 

Possible mechanisms of pattern formation in biological systems include the classical [[reaction–diffusion]] model proposed by [[Alan Turing]]<ref>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</ref> 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.<ref name="Mercker">{{cite journal |last1=Mercker |first1=M |last2=Brinkmann |first2=F |last3=Marciniak-Czochra |first3=A |last4=Richter |first4=T |title=Beyond Turing: mechanochemical pattern formation in biological tissues. |journal=Biology Direct |date=4 May 2016 |volume=11 |pages=22 |doi=10.1186/s13062-016-0124-7 |pmid=27145826|pmc=4857296 }}</ref><ref>Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632</ref>

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.

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。

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 Reaction–Diffusion'''模型<ref name=":7" /> 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。<ref name="Mercker" /><ref name=":8" />

====Growth of colonies 菌落生长====

====Growth of colonies 菌落生长====

Bacterial colonies show a [[bacterial patterns|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.<ref>Ball, 2009. ''Branches'', pp. 52–59.</ref> Other organisms such as [[slime mould]]s display remarkable patterns caused by the dynamics of chemical signaling.<ref>Ball, 2009. ''Shapes'', pp. 149–151.</ref>

Bacterial colonies show a [[bacterial patterns|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.<ref name=":9">Ball, 2009. ''Branches'', pp. 52–59.</ref> Other organisms such as [[slime mould]]s display remarkable patterns caused by the dynamics of chemical signaling.<ref name=":10">Ball, 2009. ''Shapes'', pp. 149–151.</ref> Cellular embodiment (elongation and adhesion) can also have an impact on the developing patterns.<ref name=":11">Duran-Nebreda, Salva; Pla, Jordi; Vidiella, Blai; Piñero, Jordi; Conde-Pueyo, Nuria; Solé, Ricard (2021-01-15). "Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies". ''ACS Synthetic Biology''. '''10''' (2): 277–285. doi:10.1021/acssynbio.0c00318. ISSN 2161-5063.</ref>

 

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.

====Vegetation patterns 植被斑图====

====Vegetation patterns 植被斑图====

{{Main|patterned vegetation}}

{{Main|patterned vegetation}}

[[patterned vegetation|Vegetation patterns]] such as [[tiger bush]]<ref name="TigerBush">{{cite book | title=Banded vegetation patterning in arid and semiarid environments | publisher=Springer-Verlag | author=Tongway, D.J., Valentin, C. & Seghieri, J. | year=2001 | location=New York|isbn=978-1461265597}}</ref> and [[fir wave]]s<ref name="FirWave">{{cite web | url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html | title=Fir Waves: Regeneration in New England Conifer Forests | publisher=TIEE | date=22 February 2004 | accessdate=26 May 2012 | author=D'Avanzo, C.}}</ref> 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.<ref name="TigerBush" /> 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<ref name="FirWave" />. 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<ref>{{cite journal |author=Meron, E |title=Vegetation pattern formation: the mechanisms behind the forms |journal=Physics Today |volume=72 |issue=11 | pages=30-36 |year=2019 |doi=10.1063/PT.3.4340}}</ref><ref>{{cite journal |author=Meron, E |title=From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study|journal=Annual Review of Condensed Matter Physics  |volume=9 | pages=79-103  |year=2018 |doi=10.1146/annurev-conmatphys-033117-053959}}</ref>.

[[patterned vegetation|Vegetation patterns]] such as [[tiger bush]]<ref name="TigerBush">{{cite book | title=Banded vegetation patterning in arid and semiarid environments | publisher=Springer-Verlag | author=Tongway, D.J., Valentin, C. & Seghieri, J. | year=2001 | location=New York|isbn=978-1461265597}}</ref> and [[fir wave]]s<ref name="FirWave">{{cite web | url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html | title=Fir Waves: Regeneration in New England Conifer Forests | publisher=TIEE | date=22 February 2004 | accessdate=26 May 2012 | author=D'Avanzo, C.}}</ref> 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.<ref name="TigerBush" /> 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<ref name="FirWave" />. 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<ref name=":12">{{cite journal |author=Meron, E |title=Vegetation pattern formation: the mechanisms behind the forms |journal=Physics Today |volume=72 |issue=11 | pages=30-36 |year=2019 |doi=10.1063/PT.3.4340}}</ref><ref name=":13">{{cite journal |author=Meron, E |title=From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study|journal=Annual Review of Condensed Matter Physics  |volume=9 | pages=79-103  |year=2018 |doi=10.1146/annurev-conmatphys-033117-053959}}</ref>.

[[文件:Tiger Bush Niger Corona 1965-12-31.jpg|缩略图|Tiger bush is a vegetation pattern that forms in arid conditions.虎皮纹灌木是一种在干旱条件下形成的植被模式。]]

[[文件:Tiger Bush Niger Corona 1965-12-31.jpg|缩略图|Tiger bush is a vegetation pattern that forms in arid conditions.虎皮纹灌木是一种在干旱条件下形成的植被模式。]]

===Chemistry 化学===

===Chemistry 化学===

{{expand section|date=March 2013}}

{{expand section|date=March 2013}}

{{further|reaction–diffusion system|Turing patterns}}Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.<ref name=":0">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> The [[Brusselator]] model developed by [[Ilya Prigogine]] and collaborators is one such example that exhibits [[Turing instability]].<ref>{{Citation|last=Prigogine|first=I.|title=Self-Organisation in Nonequilibrium Systems: Towards A Dynamics of Complexity|date=1985|work=Bifurcation Analysis: Principles, Applications and Synthesis|pages=3–12|editor-last=Hazewinkel|editor-first=M.|publisher=Springer Netherlands|doi=10.1007/978-94-009-6239-2_1|isbn=9789400962392|last2=Nicolis|first2=G.|editor2-last=Jurkovich|editor2-first=R.|editor3-last=Paelinck|editor3-first=J. H. P.}}</ref> Pattern formation in chemical systems often involve [[Chemical oscillator|oscillatory chemical kinetics]] or [[Autocatalysis|autocatalytic reactions]]<ref name=":1">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|issn=1934-2659}}</ref> 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]].<ref>{{Cite journal|last=Marwaha|first=Bharat|last2=Sundarram|first2=Sandhya|last3=Luss|first3=Dan|date=September 2004|title=Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†|journal=The Journal of Physical Chemistry B|volume=108|issue=38|pages=14470–14476|doi=10.1021/jp049803p|issn=1520-6106}}</ref><ref name=":0" /> The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying [[Reaction–diffusion system|reaction-diffusion system]].<ref name=":0" /><ref name=":1" />

{{further|reaction–diffusion system|Turing patterns}}Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.<ref name=":0">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> The [[Brusselator]] model developed by [[Ilya Prigogine]] and collaborators is one such example that exhibits [[Turing instability]].<ref name=":14">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</ref> Pattern formation in chemical systems often involve [[Chemical oscillator|oscillatory chemical kinetics]] or [[Autocatalysis|autocatalytic reactions]]<ref name=":1">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|issn=1934-2659}}</ref> 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]].<ref name=":15">{{Cite journal|last=Marwaha|first=Bharat|last2=Sundarram|first2=Sandhya|last3=Luss|first3=Dan|date=September 2004|title=Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†|journal=The Journal of Physical Chemistry B|volume=108|issue=38|pages=14470–14476|doi=10.1021/jp049803p|issn=1520-6106}}</ref><ref name=":0" /> The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying [[Reaction–diffusion system|reaction-diffusion system]].<ref name=":0" /><ref name=":1" />

 

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'''<font color="#ff8000"> 振荡化学动力学 Oscillatory Chemical Kinetics</font>''' 或'''<font color="#ff8000"> 自催化反应 Autocatalytic Reactions</font>'''，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。

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。<ref name=":0" />由'''伊利亚·普利高津 Ilya Prigogine'''和其合作者开发的'''布鲁塞尔器 Brusselator'''模型就是一个展示出'''图灵不稳定性 Turing Instability'''的例子。<ref name=":14" /> 化学体系中的斑图生成通常涉及'''<font color="#ff8000">振荡化学动力学 Oscillatory Chemical Kinetics</font>''' 或'''<font color="#ff8000"> 自催化反应 Autocatalytic Reactions</font>'''，<ref name=":1" /> 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。<ref name=":0" /><ref name=":15" /> 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。<ref name=":0" /><ref name=":1" />

* [[Belousov–Zhabotinsky reaction]]

* [[Belousov–Zhabotinsky reaction]]

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.

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.

===Computer graphics 计算机图形学===

===Computer graphics 计算机图形学===

[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[reaction–diffusion]] model, produced using sharpen and blur|链接=Special:FilePath/Homebrew_reaction_diffusion_example_512iter.jpg]]

[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[reaction–diffusion]] model, produced using sharpen and blur

 

利用图像锐化和虚化产生的斑图近似反应-扩散模型|链接=Special:FilePath/Homebrew_reaction_diffusion_example_512iter.jpg]]