• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Turing/Turing-like patterns: Products of random aggregation of spatial components

    2023-09-05 08:47:50JianGao高見XinWang王欣XinshuangLiu劉心爽andChuanshengShen申傳勝
    Chinese Physics B 2023年7期
    關(guān)鍵詞:王欣

    Jian Gao(高見), Xin Wang(王欣), Xinshuang Liu(劉心爽), and Chuansheng Shen(申傳勝)

    1International Joint Research Center of Simulation and Control for Population Ecology of Yangtze River in Anhui,Anqing Normal University,Anqing 246011,China

    2School of Mathematics and Physics,Anqing Normal University,Anqing 246011,China

    Keywords: Turing-like pattern,collective behavior,random aggregation,pattern formation,multi-particle system

    1.Introduction

    Ordered spatiotemporal patterns arising out of randomness are typical and interesting phenomena during morphogenesis.[1]Turing,[2],in his groundbreaking work,has put forward a type of explanation for a variety of patterns observed in nature.[3–6]He believes that emergences of these spatially periodical patterns are induced by interaction between a long-range inhibitor and a short-range activator.Turing’s theory of reaction diffusion (RD) has been proved to be greatly influential across many research fields.[7]For instance, patterns in chlorite iodide malonic acid chemical reactions,[8–12]stripes on tropical fishes,[3,13]and pigment patterns on sea shells[14,15]have been investigated as Turing patterns via the RD theory.Moreover, Turing patterns have the property of cross scale.The typical length scale of Turing patterns on the surface of animals ranges from millimetres to centimetres,and that in chemical reactions is about sub-millimetres.Recent studies[16–18]have found that the scale of Turing patterns can be observed at the nanoscale.

    In addition to RD systems, Turing/Turing-like patterns can also be observed in fluid systems,[1,19–23]gas discharges,[24–26]spatial distribution of bacterial population,[27,28]and even distribution of human settlements.[29]In this article, we call the patterns, which are similar to Turing patterns but have different or ambiguous generation mechanisms, the Turing-like patterns.Generally,Turing/Turing-like patterns are studied in systems driven far from thermodynamic equilibrium.[1]Turing’s RD mechanism containing a long-range inhibiting agent and a local catalytic agent only appears in this type of nonequilibrium systems.Specifically, most of the experimental studies of Turing/Turing-like pattern are devoted to finding the two elements supporting the Turing mechanism, i.e., a long-range inhibition and a short-range self-activation, which exist in systems driven far from thermodynamic equilibrium.[3,5,30,31]

    In fact, a large number of studies[3,8–29]based on different systems show that, the formation mechanism of Turing/Turing-like patterns across different systems should be of a spatial problem, not of local dynamics.Therefore, Turing/Turing-like patterns could exist in systems near thermodynamic equilibrium.The conjecture is encouraging.Studying Turing/Turing-like patterns in near-equilibrium thermodynamic systems can avoid complex dynamics problems in dissipative systems far from thermodynamic equilibrium,thus greatly simplifying the related problems of the mechanism for Turing/Turing-like patterns.Consequently,it is more meaningful and important to study Turing/Turing-like patterns in near-equilibrium systems for understanding the formation mechanism of Turing/Turing-like patterns.

    We accidentally observed a type of Turing-like pattern with a‘labyrinthine’stripe structure in a system near thermodynamic equilibrium, which can hardly be explained by the classical Turing mechanism(see Fig.1).This pattern appears during the slow cooling process of the starch solution.In particular,the pattern still exists when the system finally reaches thermodynamic equilibrium,which is quite different from the Turing/Turing-like patterns in dissipative systems.For instance,Turing/Turing-like patterns in chemical reactions,[8–12]thermal convection[1,19–23]and gas discharges[24–26]can only appear in the process of violent reactions,and disappear when the systems reach thermodynamic equilibrium.

    In this article, by investigating this phenomenon, we put forward new ideas on the formation mechanism of Turing-like patterns.We find that the random aggregation of spatial components leads to the formation of Turing-like patterns,and the proportion of spatial components determines the pattern structures.The rest of the article is organized as follows: Section 2 introduces the materials and methods.Section 3 shows the results, including the experimental results and numerical results.Finally, conclusions and discussions are presented in Section 4.

    Fig.1.(a)A Turing-like pattern with a‘labyrinthine’stripe structure observed in a rice porridge(the area in the white circle).Structure description: Bulges and depressions appear on the original flat surface,namely peaks and valleys.The peaks and valleys are long strips, which appear alternately in space to form a ‘labyrinthine’ stripe structure.The white lumps in the figure are expanded rice grains,e.g.,the one indicated by the white arrow.(b)Schematic diagram of the longitudinal section along the white dotted line of the Turinglike pattern in panel (a).The distance between adjacent stripes λ is about 0.27 cm.

    2.Materials and methods

    2.1.Experiment 1

    We believe that the Turing-like pattern in Fig.1 is a selforganized structure formed by the starch solution in rice porridge with a certain concentration.To reproduce the Turinglike pattern in Fig.1,we designed a group of experiments with starch solution, and took the concentration of starch solutionρ ≡ms/Vas the control parameter,withmsandVindicating the quality of starch and the volume of starch solution,respectively.

    Materials:Starch(rice powder)and water.

    Experimental procedure:

    Step 1: Dissolvemsg of starch in water.

    Step 2: Keep the starch solution at 90?C for three hours.

    Step 3: Place the starch solution in a horizontal container keeping a 3 mm thin layer.For easy observation, we use a black container to hold the starch solution.

    Step 4: Cool naturally at 10?C.

    2.2.Experiment 2

    Studies have shown that thermal convection can lead to the appearance of Turing-like patterns.[1,21–23]A natural conjecture is that,during the cooling process of the starch solution in Experiment 1,thermal convection was generated due to the lower temperature of the upper surface,resulting in the Turinglike patterns in Experiment 1.To examine whether the Turinglike patterns obtained in Fig.1 and Experiment 1 are caused by thermal convection, we designed Experiment 2.Figure 2 shows the experimental design of Experiment 2.This design would reverse the direction of the temperature gradient and eliminate thermal convection.In this group of experiments,we also took the concentration of starch solution as the control parameter.

    Fig.2.Design for Experiment 2.(a)Container structure.Containers 1 and 2 are used to hold starch solution and hot water, respectively.In order to obtain a high rate heat transfer, the containers are made of metal(stainless steel).(b) Placement of equipments.Place the integrated container on a trivet so that no objects prevent heat from escaping from the bottom of the container into air.Then cover the side and top of the container with a heat preservation cover.(c)Front profile of the integrated container.The red arrows in panel(c)indicate the direction of heat flow.The heat flow direction is downward,thus avoiding the occurrence of thermal convection.

    Materials:Starch(rice powder)and water.

    Experimental procedure:

    Step 1: Dissolvemsg of starch inmwg of water.

    Step 2: Keep the starch solution at 90?C for three hours.

    Step 3: As shown in Fig.2,place the starch solution and hot water (90?C) in Containers 1 and 2, respectively.Keep the starch solution in a 3-mm-thick layer.Then place the integrated container containing starch solution and hot water on a trivet and keep it warm with a heat preservation cover.

    Step 4: Cool naturally at 10?C.

    2.3.Model 1

    Starch is a polymer compound with molecular formula(C6H10O5)x(xis an uncertain positive integer).The results of Experiment 2 proved that the patterns observed in experiments are not products of thermal convection.Accordingly,we believe that the Turing-like patterns in experiments are induced by random aggregation of starch macromolecules, and the random aggregation is attributed to the interactions between starch macromolecules.Here, we propose a model to verify the conjecture.

    We regard the thin liquid layer in the experiment as anL×Ltwo-dimensional (2D) system filled with a large number of particles(starch macromolecules),which are evenly and randomly distributed in the system.(There is no interaction between the system boundary and particles.)The position vector of the particlei(i=1,2,3,...,N)can be expressed aspi.Consider repulsive,attractive,and viscous forces between particles, denoted asfr,fa, andfv, respectively.Generally, the magnitude of force between the particlesiandjis negatively correlated with the distancerbetween them,i.e.,

    wherenis a positive integer.In addition, when the distance is less than the equilibrium distance, the repulsive force increases faster with the decreasing distance.When the distance is greater than the equilibrium distance (within the action range), the repulsive force decreases faster with the increasing distance.This feature of the forces prevents the particles from being infinitely close or being isolated from each other.Therefore, the parameterncorresponding to the repulsive force should be greater than that corresponding to attractive force,i.e.,nr>na.Specifically,the forces applied by the particlejto the particleican be expressed as

    with

    We presume that particleimoves along the direction of the resultant forceFiat a constant velocityυ, ifFiis greater than the thresholdFthr,i.e.,

    whereFirepresents the resultant force on the particlei;αrepresents a unit vector with a random direction,WGrepresents a Gaussian white noise with with a mean of 0 and a variance of 1,andσis a parameter that controls the degree of noise.The parameters are fixed:L=50,nr=3,na=2,kr=1,ka=1,kv=5,zr=1,za=3,zv1=1,zv2=1.2,Fthr=0.05,υ=1,andσ=0.2.The spatial density of the particlesρ ≡N/L2has similar physical meaning to the concentration of starch solutionρin the experiments, and is set as the control parameter.The system was carried out by the Euler method with time steps of?t=0.02.Note that, the same results can be obtained when the parametersnrandnaare equal to 2 and 1,respectively(see Fig.A1 in Appendix A).We also got the same results when considering the interaction between the system boundary and particles(see Fig.A2).MATLAB codes to calculate the system are available from the authors on request.

    2.4.Model 2

    Model 1 is based on the specific system of starch solution, so this model has no generalisability.In order to verify our general conclusion (Turing-like patterns are induced by the random aggregation of spatial components),we propose a general model in the following.

    N(N=1,2,3,...,n2)small balls are randomly dispersed in ann×ngrids (each grid can hold only one small ball and the side length of each grid ish=1).Select one small ball randomly and let it walk randomly(in the continuous 2D space)until it meets another small ball(the selected small ball can only walk in the empty grid and cannot cross other small balls).The random walk follows the equations

    whereptrepresents the position vector of the selected small ball at timet;vtis the change of the position vectorpt+1;θtindicates the direction angle ofvt;andrepresents a Gaussian white noise with a mean of 0 and a variance of 0.15.This process is recorded as one operation.Then repeat the above operation.The total number of steps taken by all selected balls is recorded asT.The parameters are fixed:n=300,υ=1,andσ=π/4.The initial valuesp0andθ0are taken randomly in their ranges.The fill ratio of the gridsR ≡N/n2,which has similar physical meaning toρa(bǔ)ndρ, is set as the control parameter.Codes for this model are available from the authors on request.

    3.Results

    3.1.Results of experiment 1

    We examined the effects ofρon the structure of patterns.The results show that Turing-like patterns would not appear on the surface when the concentration of starch solution is too small or too large.Turing-like patterns can appear only when the concentrationρis in a certain range from 0.04 g/cm3to 0.10 g/cm3, and the structural characteristic of the patterns changes with the change of the concentrationρ.The bulges(peaks) in the pattern show circular spots or strips whenρis 0.06 g/cm3[see Fig.3(a)].The peaks(or valleys)in the pattern show a ‘labyrinthine’ stripe structure whenρis 0.07 g/cm3[see Fig.3(b)].The peaks are intertwined to form a network structure,and the valleys in the pattern are scattered spots(inverse spot structure),whenρis 0.08 g/cm3[see Fig.3(c)].

    Fig.3.Two-dimensional patterns observed in Experiment 1 with different values of ρ: (a)ρ =0.06 g/cm3,(b)ρ =0.07 g/cm3,(c)ρ =0.08 g/cm3.

    3.2.Results of experiment 2

    The results of Experiment 2 are consistent with those of Experiment 1.As shown in Fig.4, the same patterns as those in Fig.3 can be obtained in the concentration interval [0.04 g/cm3, 0.10 g/cm3].Specifically, with the increase of the concentrationρ, the structure of the pattern changes from spots to inverse spots (reticular structure)through‘labyrinthine’stripe.

    Fig.4.Two-dimensional patterns observed in Experiment 2 with different values of ρ: (a)ρ =0.06 g/cm3,(b)ρ =0.07 g/cm3,(c)ρ =0.08 g/cm3.

    Findings from Experiment 2 suggest that the Turing-like patterns observed in the experiments in this article are not caused by thermal convection,but may be caused by an undiscovered mechanism.The results also indicate directions for the establishment of models.

    3.3.Numerical results of model 1

    To reproduce the results in our experiments,we examined the effects ofρon the patterns’structure.By the gradual modulation of the parameterρin the system, various 2D patterns can be produced,including spot patterns,‘labyrinthine’stripe patterns and inverse spot patterns(see Fig.5).

    When the particle density is small, the particles can self organize into spots[see Fig.5(a)].With the increase of the particle density, spots closed together can form long strip structures[see Figs.5(b)–5(d)].When the particle density reaches a certain degree,the particles may organize into‘labyrinthine’stripe structures[Figs.5(e)and 5(f)].When the particle density is larger, the white regions will dominate and black strip structures will appear [see Figs.5(g)–5(i)].As the particle density continues to increase, black spots will appear [see Fig.5(j)].

    We also found that the ‘labyrinthine’ stripe patterns appear during the evolution of the spot structure [see Fig.6(b)and 6(c)].However,the temporary‘labyrinthine’stripe structures eventually evolve into spot patterns because of the instability.The result shows that the random aggregation of particles always tends to generate the‘labyrinthine’stripe structure, but the final pattern structure is determined by specific conditions(particle density)of the system.

    Fig.5.Two-dimensional patterns generated by numerical simulations.Gradual modulation of the value the parameter ρ (from 1 to 10 indicated below each figure)can induce pattern changes from spots to inverse spots.Intermediate values appear as labyrinthine stripe patterns.In each figure,the white areas are formed by the accumulation of particles in a stationary distribution.

    Fig.6.The evolution process of the pattern for parameter ρ =1 at different times [(a)–(d)].The ‘labyrinthine’ stripe patterns can appear during the evolution process[(b),(c)].(d)The pattern reached a stationary stable.See Movie 1 in the supplementary materials for the evolution process.

    3.4.Numerical results of model 2

    The results show that the random aggregation of spatial components can generate Turing-like patterns,and the proportion of spatial components can control the specific structures.

    Specifically,we examined the effects of the fill ratioRon the structure of patterns.By the gradual modulation of the fill ratioR,various 2D patterns can be reproduced,including spot patterns, ‘labyrinthine’ stripe patterns, and inverse spot patterns,as shown in Fig.7.When the fill ratio is small,the small balls can self organize into spots[see Fig.7(a)].With the increase of the fill ratio,spots closed together can form long strip structures[see Fig.7(b)].When the fill ratio reaches a certain degree,the small balls may organize into‘labyrinthine’stripe structures[Fig.7(c)].When the fill ratio is larger,the white regions will dominate and black strip structures will appear[see Fig.7(d)].As the fill ratio continues to increase, white spots(inverse spots)will appear[see Fig.7(e)].

    Generally, the spatial scale of Turing-like patterns is related to the effective action distance of the force in the system.Since there is no specific force in Model 2,the generated Turing-like patterns have no fixed spatial scale.Taking the pattern withR=50% as an example, we studied the effect of the time length of random aggregation on the pattern.We found that the spatial scale of the pattern(e.g.,the width of the stripes in patterns)increases with the evolution of the pattern,however, the structural characteristic of the pattern does not change(see Fig.8).This result shows that the process of random aggregation can continuously increase the spatial scale of the Turing-like patterns.One can obtain the same conclusion when the parameterRis equal to other values.

    Fig.7.Two-dimensional patterns generated by numerical simulations at T =1×106.Gradual modulation of the value the parameter R(from 15%to 85%)can induce pattern changes from spots to inverse spots.Intermediate values appear as labyrinthine stripe patterns.In each figure,the white areas are formed by the accumulation of small balls.

    Fig.8.Two-dimensional patterns generated by numerical simulations with R=50%at different T.In each figure,the white areas are formed by the accumulation of small balls.See Movie 2 in the supplementary materials for the evolution process.

    3.5.Quantification of patterns

    It is necessary to quantitatively describe the different structures of Turing/Turing-like patterns.Here, we propose a quantitative method.The spots and inverse spots in Turing/Turing-like patterns are relative and can be considered to have the same structure.We introduce the parameter

    where ?Srepresents the area difference of two spatial components, andSis the total area.WhenΦis in the intervals [0,0.1), [0.1,0.6] and (0.6,1), the pattern is labyrinthine stripes[see Figs.5(a)and 5(j)],irregular strips[see Figs.5(d)and 5(g)[and spots[see Fig.5(e)].

    4.Discussion and conclusion

    We have observed Turing-like patterns in starch solution and suggest that they are caused by random aggregation of starch macromolecules.Through Experiment 1,we study the effect of concentration of starch solution on the patterns, and obtain the corresponding structures of Turing-like patterns,namely,spots,labyrinthine stripes,and inverse spots.To prove that these patterns are not caused by thermal convection, we design and carry out Experiment 2, and obtain the same results.Based on this assumption, we establish Model 1 and reproduce the experimental results perfectly.The mechanism can be generalized:If there are two components in a 2D space,and one of them has the ability to aggregate, the random aggregation of spatial components can generate Turing-like patterns,and the proportion of components determines the structure.To validate this mechanism,we propose Model 2 and obtain the expected results.Our mathematical analysis explains the formation of such Turing-like patterns and the choice of specific structures (see Appendix B).In summary, we have studied the formation of Turing-like patterns in a system near thermodynamic equilibrium experimentally and theoretically,and put forward new ideas on the formation mechanism of Turing-like patterns.

    The experiments and models in this paper belong to nearequilibrium thermodynamic systems.Non-equilibrium systems can be divided into two types according to the degree of leaving the thermodynamic equilibrium,i.e.,near-equilibrium and far-from-equilibrium thermodynamic systems.Nearequilibrium thermodynamic systems, described in an earlier paper,[32]are not far from the thermodynamic equilibrium.They follow the Onsager reciprocal relations and the principle of minimum entropy production.Near-equilibrium thermodynamics is a branch of classical equilibrium thermodynamics which has been used to evaluate many central reactions in intermediary metabolism.It involves the study of systems that are not in their own state of equilibrium,instead have changes in variables that are not infinitely slow.[33]This includes studies on thermodynamics and kinetics near equilibrium,as well as those beyond local equilibrium.Far-from-equilibrium thermodynamic systems are such that they are not in or near a state of thermodynamic equilibrium.Thermodynamics far from equilibrium can be studied using the principles of maximum entropy production[34]and global validity for effectively onevariable, irreversible chemical systems with multiple steady states.[35]For example, the Bernard convection,[1]when the temperature difference between the upper and lower parts is 0,the system is in thermodynamic equilibrium.When the temperature difference is small, the system is near equilibrium and is still laminar flow.As the temperature difference increases,the degree of the system leaving the equilibrium state increases.When the temperature difference exceeds a certain critical value, non-equilibrium phase transition will occur.The original laminar flow loses its stability and forms the Bernard convection, which is a dissipative structure.The slowly cooled, uniform liquid without chemical reaction follows the Onsager reciprocal relations and the principle of minimum entropy production,which is a near-equilibrium thermodynamic system.[1,32]Therefore, the experiments in this paper are near-equilibrium thermodynamic systems.The uniformly distributed particles in Model 1 are only affected by the thermal noise,and there is no self-driving force and quantity change of particles, so the particle system is in a nearequilibrium state.Model 2 is a simplification of Model 1,which is to gradually apply the thermal noise to a single particle.

    The findings of this paper are of great significance to pattern formation.In the 1970s, the dissipative structure theory proposed by Prigogine was widely accepted.[36]According to this theory, dissipative systems can generally self-organize into spatiotemporal patterns.Therefore, patterns have been studied in dissipative systems,for example, Bernard hexagon patterns in thermal convection, distribution patterns of bacterial populations and Turing patterns in chemical reactions.These systems are far from thermodynamic equilibrium and have very complex dynamic behavior.Such complex dynamic behavior may mask the real mechanism of dissipative structure.The generation of ordered structures is a matter of space,so the spatial problem should not depend on the complex dynamic behavior of dissipative systems.We believe that complex dynamic behavior in dissipative systems produces different spatial components, and the spatial components produce Turing/Turing-like patterns due to random aggregation.For example, the ascending heat flow and descending cold flow in the Bernard hexagon pattern correspond to different spatial components, and the viscosity of fluid corresponds to the ability of aggregation.The area of ascending heat flow is consistent with the spot structure of Turing patterns,and the edge of the spots shows hexagonal structures.Also,the long-range inhibitor in the Turing mechanism corresponds to the low concentration spatial component with the ability of aggregation.Specifically, the inhibitor corresponds to the spatial component with low concentration,and the long-range property corresponds to the ability of aggregation.Therefore,our mechanism includes the Turing mechanism.The mechanism of pattern formation across different systems can only be grasped by leaving aside dynamic behavior of specific systems.Consequently, our results provide a new perspective for pattern formation,which may explain the formation of Turing/Turinglike patterns in various systems (including systems far from thermodynamic equilibrium and near-equilibrium systems),and provide a unified mechanism for generation of patterns in different systems.

    Miyazawaet al.[31]pointed out that crossing between animals having inverted spot patterns(for example,light spots on a dark background and dark spots on a light background)will necessarily result in hybrid offspring that have camouflaged labyrinthine patterns as ‘blended’ intermediate phenotypes.For instance, as shown in Fig.7, the ‘blending’ of the spot patterns in (a) and (e) will produce the ‘labyrinthine’ stripe structures in (c).In addition, when Turing patterns exist in a system, the transition from spot patterns to inverse spot patterns can generally be observed,during which the labyrinthine stripes are experienced.[31,37,38]

    In previous studies,the description of different structures of Turing/Turing-like patterns is only qualitative, lacking objective quantitative criteria.Representative examples are spots and labyrinthine stripes.Here,based on our spatial component mechanism,we propose a quantitative description method for Turing/Turing-like patterns.

    Appendix A: Supplementary numerical results of model 1

    We studied the patterns that appeared in the system when the parametersnrandnatake as 1 and 2, respectively.The results show that the pattern structure appearing in the system does not depend on the values ofnrandna(see Fig.A1).

    Fig.A1.Two-dimensional patterns generated by numerical simulations with different ρ: (a)ρ=2,(b)ρ=4,(c)ρ=6,(d)ρ=8,(e)ρ=10.In gradual modulation of the value the parameter ρ can induce pattern changes from spots to inverse spots.Intermediate values appear as labyrinthine stripe patterns.In each figure,the white areas are formed by the accumulation of particles reaching a stationary distribution.

    In other words,each set ofnrandnavalues can describe the general characteristics of attraction and repulsion between particles.

    We examined the influence of the interaction between particles and system boundary on the pattern structure.The results show that the boundary conditions of the system do not change the pattern structure(see Fig.A2).

    Appendix B:Supplementary analytical results

    Based on our spatial component mechanism, we explained the formation of different structures in Turing/Turinglike patterns by a simple plane geometry method.

    The 2D plane was discretized into regular hexagonal meshes of uniform size to meet the particle aggregation towards the center of each regular hexagon.The spot pattern is a common structure in Turing/Turing-like patterns(see Fig.B1).We took two adjacent hexagons as an example to explain the choice of pattern structure.As shown in Fig.B1,his the characteristic scale.The characteristic scalehis divided into entity-scalehfand empty-scalehe.Because the space filled by particles is completely equivalent to the space not filled,hfcannot be greater thanhewhen the particle fill ratio is less than 50%.We considered the case wherehfequalshe,and the minimum fill ratio of particles is

    whereScandSare the areas of circular spots and hexagon,respectively.

    Fig.B1.Discretization of 2D plane.The red and blue circles represent the aggregated particles,and only 7 circles within the hexagon are drawn in the figure.

    Since the space filled by particles is completely equivalent to the space not filled, it is necessary only to analyze the case that the fill ratio is less than or equal to 50%.When the fill ratio of particles is less than 23%,the pattern in the system shows a spot structure,and the diameter of spot increases with the increase of the fill ratio.When the filling rate of particles is greater than 23%and less than 50%, in order to preventhffrom being greater thanhe,the diameter of spot will no longer increase.Instead, some adjacent spots will be connected to each other [as shown in Fig.2(b)].With the increase of fill ratio,the number of particles connected increases.

    When the fill ratio of particles is about 50%,the circular spots no longer exist.The filling of discrete spatial elements(regular hexagon)can be divided into six types(see Fig.B2).

    The random combination of the six types of filling elements on the 2D plane can form patterns similar to the‘labyrinthine’stripe patterns(see Fig.B3).

    We smoothed the pattern in Fig.B3 to eliminate sharp corners.(Codes are available from the authors on request.)The final pattern is the ‘labyrinthine’ stripe pattern (see Fig.B4).

    In summary,we have explained the choice of the structure for Turing-like patterns through the method of plane geometry.That is,we explained how the fill ratio of particles determines the structure of Turing-like pattern.

    Fig.B2.Six types of filling elements.

    Fig.B3.A‘labyrinthine’stripe pattern formed by random combination of the six filling elements.

    Fig.B4.The smoothing process of the ‘labyrinthine’ stripe pattern in Fig.B3.Also see Movie 3 in the supplementary materials for the evolution process.

    Appendix C:Supplementary material

    See the supplementary materials for Movies 1, 2, and 3.Movie 1 shows the evolution process of the pattern when the parameterρis equal to 1.Movie 2 shows the evolution process of the pattern when the parameterRis equal to 50%.Movie 3 shows the smoothing process of the ‘labyrinthine’ stripe pattern in Fig.B4.

    Acknowledgements

    Project supported by the National Natural Science Foundation of China(Grant Nos.12205006 and 11975025),the Excellent Youth Scientific Research Project of Anhui Province(Grant No.2022AH030107), the Natural Science Foundation of Anhui Higher Education Institutions of China (Grant No.KJ2020A0504),and the International Joint Research Center of Simulation and Control for Population Ecology of Yangtze River in Anhui(Grant No.12011530158).

    猜你喜歡
    王欣
    Photonic-plasmonic hybrid microcavities: Physics and applications*
    黑洞
    太空探索(2021年2期)2021-02-27 07:59:34
    風(fēng)
    Universal Pseudo-PT-Antisymmetry on One-Dimensional Atomic Optical Lattices?
    小保安晉身“都教授”有多難?
    小保安晉身“都教授”有多難?
    小保安晉身“都教授”有多難?
    Analysis of Means of Strength in "Letter From a Birmingham Jail"
    還我腎來,我才同意和你離婚
    女性天地(2009年8期)2009-11-23 06:19:50
    捐腎,索腎,這對“血脈相連”的夫妻怎么了
    男女无遮挡免费网站观看| 成年人免费黄色播放视频 | 成人毛片60女人毛片免费| 国产日韩欧美视频二区| 亚洲国产欧美在线一区| 亚洲欧美日韩东京热| 纯流量卡能插随身wifi吗| 国产视频首页在线观看| 蜜臀久久99精品久久宅男| 国产欧美日韩精品一区二区| av一本久久久久| 久久久久国产网址| 免费黄色在线免费观看| 最近中文字幕高清免费大全6| 亚洲三级黄色毛片| 国产熟女午夜一区二区三区 | 国精品久久久久久国模美| 成人二区视频| 日本av免费视频播放| 尾随美女入室| 亚洲精品成人av观看孕妇| 久久ye,这里只有精品| 日本色播在线视频| 日日摸夜夜添夜夜添av毛片| 老司机亚洲免费影院| 搡老乐熟女国产| 如日韩欧美国产精品一区二区三区 | 美女脱内裤让男人舔精品视频| 一级毛片黄色毛片免费观看视频| 七月丁香在线播放| 午夜福利视频精品| 亚洲av在线观看美女高潮| 伊人久久精品亚洲午夜| 最黄视频免费看| 中文欧美无线码| 有码 亚洲区| 婷婷色综合www| 女的被弄到高潮叫床怎么办| 精品酒店卫生间| 亚洲精品乱码久久久久久按摩| 亚洲精品色激情综合| 91久久精品国产一区二区成人| 欧美丝袜亚洲另类| 国产一区二区三区综合在线观看 | 欧美激情极品国产一区二区三区 | 国产精品国产av在线观看| 国产又色又爽无遮挡免| 久久免费观看电影| 婷婷色综合大香蕉| 色视频在线一区二区三区| 国产中年淑女户外野战色| 99热这里只有是精品50| 两个人的视频大全免费| 免费av中文字幕在线| 久久久久网色| a级一级毛片免费在线观看| 人妻夜夜爽99麻豆av| 久久久久视频综合| 男人爽女人下面视频在线观看| 国产日韩欧美亚洲二区| 99视频精品全部免费 在线| 精品久久久久久电影网| 婷婷色综合www| 丁香六月天网| 久久99一区二区三区| 七月丁香在线播放| 麻豆成人午夜福利视频| 80岁老熟妇乱子伦牲交| 日产精品乱码卡一卡2卡三| 夫妻午夜视频| 三级国产精品片| 国产又色又爽无遮挡免| 国产精品成人在线| 久久 成人 亚洲| 特大巨黑吊av在线直播| 波野结衣二区三区在线| 久久久久视频综合| 欧美xxxx性猛交bbbb| 99热这里只有是精品50| 91午夜精品亚洲一区二区三区| 九色成人免费人妻av| 99九九在线精品视频 | 2018国产大陆天天弄谢| 亚洲欧美一区二区三区国产| 国产伦理片在线播放av一区| 精品久久久久久久久亚洲| 国产日韩一区二区三区精品不卡 | 久久精品国产a三级三级三级| 欧美3d第一页| 九色成人免费人妻av| 久久久久精品性色| av有码第一页| 亚洲精品一区蜜桃| 人人澡人人妻人| www.av在线官网国产| 中文字幕制服av| 成人二区视频| 一级黄片播放器| 国产精品99久久99久久久不卡 | 天堂中文最新版在线下载| 美女脱内裤让男人舔精品视频| av天堂中文字幕网| 一级毛片黄色毛片免费观看视频| 亚洲久久久国产精品| 人妻一区二区av| 亚洲性久久影院| 精品国产露脸久久av麻豆| 亚洲欧美日韩东京热| 日韩在线高清观看一区二区三区| 午夜视频国产福利| 中文欧美无线码| 日韩一本色道免费dvd| 亚洲精品aⅴ在线观看| 永久免费av网站大全| 欧美日本中文国产一区发布| 国产精品国产av在线观看| 亚洲精华国产精华液的使用体验| 免费少妇av软件| av在线app专区| 欧美成人精品欧美一级黄| 极品教师在线视频| 三上悠亚av全集在线观看 | 国产成人精品久久久久久| 午夜福利在线观看免费完整高清在| av在线app专区| 欧美97在线视频| 国产免费福利视频在线观看| 大片电影免费在线观看免费| 免费看av在线观看网站| 99视频精品全部免费 在线| 大码成人一级视频| 日韩一区二区三区影片| 国产探花极品一区二区| 一本—道久久a久久精品蜜桃钙片| 久久国内精品自在自线图片| 中文资源天堂在线| 亚洲欧洲国产日韩| 成年人午夜在线观看视频| 一本一本综合久久| 色哟哟·www| 好男人视频免费观看在线| 日韩大片免费观看网站| 最近最新中文字幕免费大全7| 久久ye,这里只有精品| 国产亚洲91精品色在线| 永久免费av网站大全| 欧美高清成人免费视频www| 99热这里只有是精品50| 嫩草影院新地址| 一级毛片电影观看| 一级,二级,三级黄色视频| 精品人妻熟女毛片av久久网站| 91精品一卡2卡3卡4卡| 欧美人与善性xxx| 免费av不卡在线播放| 亚洲一区二区三区欧美精品| 精品国产一区二区久久| av国产精品久久久久影院| 日韩亚洲欧美综合| 久久精品夜色国产| 免费黄频网站在线观看国产| 日韩精品免费视频一区二区三区 | 黄片无遮挡物在线观看| 男女边吃奶边做爰视频| 日韩欧美精品免费久久| 极品教师在线视频| 日韩一区二区视频免费看| 制服丝袜香蕉在线| 在线观看人妻少妇| av在线播放精品| 纯流量卡能插随身wifi吗| 免费人成在线观看视频色| 精品亚洲成a人片在线观看| 久久鲁丝午夜福利片| 老司机影院毛片| 日本wwww免费看| 久久影院123| 春色校园在线视频观看| 最近的中文字幕免费完整| 国产精品国产av在线观看| 一级,二级,三级黄色视频| 国产亚洲欧美精品永久| 一二三四中文在线观看免费高清| 亚洲成人一二三区av| 国产探花极品一区二区| 亚洲国产av新网站| 亚洲天堂av无毛| 色婷婷av一区二区三区视频| 99久久精品国产国产毛片| 午夜影院在线不卡| 蜜桃在线观看..| 欧美精品一区二区大全| 国产精品一区二区在线不卡| 一级毛片我不卡| 熟妇人妻不卡中文字幕| 日韩熟女老妇一区二区性免费视频| 日韩精品有码人妻一区| 天堂8中文在线网| 亚洲精品中文字幕在线视频 | 久久精品国产亚洲av天美| 大又大粗又爽又黄少妇毛片口| 人人妻人人爽人人添夜夜欢视频 | 欧美日韩视频高清一区二区三区二| 国模一区二区三区四区视频| 亚洲av综合色区一区| 91精品一卡2卡3卡4卡| 久久热精品热| 日韩制服骚丝袜av| 黄色怎么调成土黄色| 国产 精品1| 观看免费一级毛片| 一级毛片黄色毛片免费观看视频| 极品人妻少妇av视频| 国内精品宾馆在线| 伦理电影大哥的女人| 99re6热这里在线精品视频| 黑丝袜美女国产一区| 搡女人真爽免费视频火全软件| 一级毛片电影观看| 日本欧美国产在线视频| 91成人精品电影| 精品国产露脸久久av麻豆| 日本wwww免费看| 久久韩国三级中文字幕| a级一级毛片免费在线观看| 亚洲美女视频黄频| 91精品国产九色| 制服丝袜香蕉在线| 好男人视频免费观看在线| 亚洲欧美成人精品一区二区| 国产成人精品婷婷| 新久久久久国产一级毛片| 在线观看免费日韩欧美大片 | 久久人人爽人人片av| 精品酒店卫生间| 欧美另类一区| 又爽又黄a免费视频| 亚洲av国产av综合av卡| 中文乱码字字幕精品一区二区三区| 久久久久国产网址| 我要看日韩黄色一级片| 日韩电影二区| 精品久久久久久电影网| 欧美丝袜亚洲另类| 亚洲av国产av综合av卡| a 毛片基地| 成人亚洲精品一区在线观看| 一级二级三级毛片免费看| 国产高清三级在线| 亚洲综合色惰| 亚洲美女搞黄在线观看| 成人黄色视频免费在线看| 国产精品嫩草影院av在线观看| 日日摸夜夜添夜夜添av毛片| 激情五月婷婷亚洲| 日韩电影二区| 美女国产视频在线观看| 国产精品伦人一区二区| 久久久午夜欧美精品| 亚洲av免费高清在线观看| 国产在线视频一区二区| 国产日韩一区二区三区精品不卡 | 日韩电影二区| 亚洲伊人久久精品综合| 国产无遮挡羞羞视频在线观看| 天天操日日干夜夜撸| 嫩草影院入口| 久久久欧美国产精品| 爱豆传媒免费全集在线观看| 中文字幕免费在线视频6| 欧美亚洲 丝袜 人妻 在线| 午夜91福利影院| 国产伦在线观看视频一区| av又黄又爽大尺度在线免费看| 欧美日韩亚洲高清精品| 如日韩欧美国产精品一区二区三区 | 91久久精品电影网| 婷婷色麻豆天堂久久| 乱系列少妇在线播放| 亚洲国产精品成人久久小说| 美女视频免费永久观看网站| 免费大片黄手机在线观看| 男人舔奶头视频| 久久久精品94久久精品| 亚洲精品成人av观看孕妇| 日日摸夜夜添夜夜添av毛片| 国产 一区精品| 精品久久久精品久久久| 少妇精品久久久久久久| 欧美精品一区二区大全| 校园人妻丝袜中文字幕| videos熟女内射| av天堂中文字幕网| a级毛片免费高清观看在线播放| 麻豆成人午夜福利视频| 欧美+日韩+精品| 欧美bdsm另类| 高清欧美精品videossex| 国产日韩欧美在线精品| 国产极品天堂在线| 国精品久久久久久国模美| 午夜影院在线不卡| 2022亚洲国产成人精品| 91精品国产国语对白视频| tube8黄色片| 久久久午夜欧美精品| 18禁在线无遮挡免费观看视频| 日韩在线高清观看一区二区三区| 国产亚洲最大av| 日本欧美视频一区| 黑人高潮一二区| 亚洲美女搞黄在线观看| 99热国产这里只有精品6| 国产美女午夜福利| 91久久精品电影网| 熟女人妻精品中文字幕| 欧美精品人与动牲交sv欧美| 插阴视频在线观看视频| 国产精品久久久久久久电影| 午夜福利影视在线免费观看| 亚洲电影在线观看av| 美女内射精品一级片tv| 成人亚洲精品一区在线观看| 精品少妇内射三级| 女性生殖器流出的白浆| 91精品伊人久久大香线蕉| 欧美日本中文国产一区发布| 国产高清有码在线观看视频| 伊人久久精品亚洲午夜| 亚洲欧美日韩卡通动漫| 日韩熟女老妇一区二区性免费视频| 日本爱情动作片www.在线观看| 久久av网站| 久久ye,这里只有精品| 人妻 亚洲 视频| 啦啦啦在线观看免费高清www| videos熟女内射| 精品一区二区免费观看| 国产在线一区二区三区精| 成人国产麻豆网| 狠狠精品人妻久久久久久综合| 欧美精品人与动牲交sv欧美| 精品国产国语对白av| 久热这里只有精品99| 九九爱精品视频在线观看| 久久亚洲国产成人精品v| 黄片无遮挡物在线观看| 国产成人精品婷婷| 人妻少妇偷人精品九色| 国产毛片在线视频| 我的女老师完整版在线观看| 色视频www国产| 人妻夜夜爽99麻豆av| 中文字幕av电影在线播放| 久久久a久久爽久久v久久| 日韩av在线免费看完整版不卡| 午夜激情久久久久久久| 国产精品久久久久久精品电影小说| 欧美+日韩+精品| 日韩三级伦理在线观看| 三级国产精品片| 大陆偷拍与自拍| 国产成人精品婷婷| 青春草国产在线视频| 精品人妻偷拍中文字幕| 我要看黄色一级片免费的| 亚洲中文av在线| 少妇高潮的动态图| 九色成人免费人妻av| 亚洲不卡免费看| 亚洲国产精品一区二区三区在线| 久久6这里有精品| 亚洲国产精品999| 亚洲内射少妇av| 精品亚洲乱码少妇综合久久| 一级,二级,三级黄色视频| 亚洲精品久久午夜乱码| 国产午夜精品一二区理论片| 最新的欧美精品一区二区| 热re99久久国产66热| 国产日韩欧美在线精品| 国产亚洲5aaaaa淫片| 国产高清国产精品国产三级| 亚洲精品国产av蜜桃| 成人毛片60女人毛片免费| 欧美国产精品一级二级三级 | 99精国产麻豆久久婷婷| 国产精品女同一区二区软件| 一级毛片aaaaaa免费看小| 国产男女内射视频| 亚洲欧美成人精品一区二区| 亚洲精品久久午夜乱码| 国产精品一区二区在线观看99| 久久久久久久精品精品| 国产白丝娇喘喷水9色精品| 亚洲婷婷狠狠爱综合网| 欧美xxⅹ黑人| 国产美女午夜福利| 黄片无遮挡物在线观看| 免费黄网站久久成人精品| 国产成人91sexporn| 91成人精品电影| av又黄又爽大尺度在线免费看| 亚洲三级黄色毛片| 久久午夜综合久久蜜桃| 少妇的逼水好多| 日韩中文字幕视频在线看片| 亚洲精品日本国产第一区| 人人妻人人看人人澡| 亚洲欧洲国产日韩| 国产视频内射| 这个男人来自地球电影免费观看 | 观看美女的网站| 日韩一区二区视频免费看| av又黄又爽大尺度在线免费看| 中文字幕人妻熟人妻熟丝袜美| 日韩中字成人| 亚洲成人一二三区av| 亚洲欧美精品专区久久| 亚洲欧美一区二区三区国产| 天堂中文最新版在线下载| 性色avwww在线观看| 人人妻人人澡人人爽人人夜夜| 久久韩国三级中文字幕| 人人妻人人爽人人添夜夜欢视频 | 午夜老司机福利剧场| 国产午夜精品一二区理论片| 亚洲一级一片aⅴ在线观看| 高清在线视频一区二区三区| 欧美日韩精品成人综合77777| 亚洲成人一二三区av| 男女边摸边吃奶| 色婷婷久久久亚洲欧美| 夫妻性生交免费视频一级片| 国产av国产精品国产| www.av在线官网国产| 黄色怎么调成土黄色| 亚洲丝袜综合中文字幕| 国产午夜精品久久久久久一区二区三区| 狂野欧美白嫩少妇大欣赏| 中文字幕人妻丝袜制服| 毛片一级片免费看久久久久| 伦精品一区二区三区| 有码 亚洲区| 国产日韩欧美在线精品| 中国三级夫妇交换| 亚洲精品乱码久久久v下载方式| 国产一区二区三区av在线| 下体分泌物呈黄色| 日本与韩国留学比较| 久久久久久久国产电影| 国产精品人妻久久久影院| av视频免费观看在线观看| 观看美女的网站| 精品少妇黑人巨大在线播放| 国产男女内射视频| av有码第一页| 成人影院久久| 人人妻人人看人人澡| 午夜激情久久久久久久| 国产欧美日韩一区二区三区在线 | 91精品一卡2卡3卡4卡| 高清av免费在线| 国产精品麻豆人妻色哟哟久久| 国产欧美日韩综合在线一区二区 | 十八禁网站网址无遮挡 | 伦精品一区二区三区| 一二三四中文在线观看免费高清| 精品熟女少妇av免费看| 久久久久网色| 日韩欧美 国产精品| 青春草亚洲视频在线观看| 男女国产视频网站| 老女人水多毛片| 亚洲av.av天堂| 丝瓜视频免费看黄片| 日韩中字成人| 丰满饥渴人妻一区二区三| 日韩av不卡免费在线播放| 久久久久精品性色| 精品国产一区二区三区久久久樱花| 国产综合精华液| 亚洲国产精品999| 国产成人精品福利久久| 精品午夜福利在线看| 国产91av在线免费观看| 国产黄色免费在线视频| 国产亚洲精品久久久com| 成人免费观看视频高清| 美女xxoo啪啪120秒动态图| 国产av一区二区精品久久| 亚洲精品色激情综合| tube8黄色片| 国产精品国产三级国产av玫瑰| 伊人久久国产一区二区| 亚洲国产精品专区欧美| 99热这里只有精品一区| av福利片在线| 久久久久久久大尺度免费视频| 桃花免费在线播放| 国产淫语在线视频| av在线播放精品| 日韩在线高清观看一区二区三区| 国产亚洲最大av| 蜜桃在线观看..| 一级毛片 在线播放| 国产日韩欧美亚洲二区| 亚洲av成人精品一二三区| 欧美 日韩 精品 国产| 最黄视频免费看| 久久久久久人妻| 亚洲av成人精品一二三区| 亚洲美女搞黄在线观看| 国产伦精品一区二区三区视频9| 老熟女久久久| 亚洲av男天堂| 十八禁高潮呻吟视频 | 黑人巨大精品欧美一区二区蜜桃 | 一本色道久久久久久精品综合| 亚洲精品一二三| videossex国产| 男人舔奶头视频| 99久久人妻综合| 国产精品欧美亚洲77777| 国产欧美亚洲国产| 丝袜脚勾引网站| 国产一区二区在线观看日韩| 男人添女人高潮全过程视频| 亚洲欧美日韩卡通动漫| 91久久精品电影网| 蜜桃久久精品国产亚洲av| 夫妻午夜视频| 国产男女超爽视频在线观看| 韩国av在线不卡| 精品久久久久久电影网| 亚洲精品一二三| 人妻人人澡人人爽人人| 免费人妻精品一区二区三区视频| 日本91视频免费播放| 99九九线精品视频在线观看视频| 大陆偷拍与自拍| 建设人人有责人人尽责人人享有的| 99久久中文字幕三级久久日本| 亚洲无线观看免费| 99久国产av精品国产电影| 国产黄频视频在线观看| 成人美女网站在线观看视频| 国产高清有码在线观看视频| 久久久久久久久久久久大奶| 99久久人妻综合| 国产精品一区二区在线不卡| 欧美日韩一区二区视频在线观看视频在线| 国产无遮挡羞羞视频在线观看| 大码成人一级视频| 大香蕉久久网| 国产精品99久久99久久久不卡 | 婷婷色综合www| 少妇人妻一区二区三区视频| 日韩电影二区| 大片免费播放器 马上看| 2022亚洲国产成人精品| av在线老鸭窝| 色视频在线一区二区三区| 人妻夜夜爽99麻豆av| 国产91av在线免费观看| 日韩强制内射视频| 人体艺术视频欧美日本| 成人二区视频| 一级a做视频免费观看| 夫妻性生交免费视频一级片| 国产精品一二三区在线看| 五月玫瑰六月丁香| 伦理电影大哥的女人| 秋霞在线观看毛片| 日日撸夜夜添| 国产成人freesex在线| 精品国产国语对白av| 久久99蜜桃精品久久| 精品一区二区三卡| 嫩草影院入口| 中文欧美无线码| 久久精品熟女亚洲av麻豆精品| 中文字幕av电影在线播放| 亚洲精品亚洲一区二区| 丝袜喷水一区| 日本av手机在线免费观看| 中文天堂在线官网| 又爽又黄a免费视频| 丰满迷人的少妇在线观看| 国产成人精品久久久久久| 日本黄色片子视频| 亚洲天堂av无毛| 热re99久久精品国产66热6| 另类精品久久| 亚洲av日韩在线播放| 婷婷色综合www| 欧美三级亚洲精品| 国产片特级美女逼逼视频| 精品人妻偷拍中文字幕| 国产无遮挡羞羞视频在线观看| 色吧在线观看| 在线观看免费视频网站a站| 有码 亚洲区| 99re6热这里在线精品视频| 人人妻人人添人人爽欧美一区卜| 女性生殖器流出的白浆| 婷婷色综合www| 久久久久久久久久久久大奶| 久久99热6这里只有精品| 视频区图区小说| 夜夜爽夜夜爽视频| 久久人人爽av亚洲精品天堂| 久久女婷五月综合色啪小说| 免费观看的影片在线观看| 国内少妇人妻偷人精品xxx网站|