2012美国大学生数学建模竞赛 MCM A题.doc

Team #14749 page 30 of 30SummaryMany scholars conclude that leaf shape is highly related with the veins. Based on this theory, we assume the leaf growth in each direction satisfies a function F. For the leaves in the same tree, the parameters are different; for those of separate trees, the function mode is different. Thus the shape of leaf differs from that of another. In the end of section 3, we simulate one growing period and depict the leaf shape.Through thousands of years of evolution, the leaves find various ways to make a full use of natural resources, including minimizing overlapping individual shadows. In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches. As to the tree structure and the leaf shape, first we consider one species. Different tree shapes have different space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown. These are the factors which affect the leaf shape according to the model in section 1. Here we analyze three typical tree shapes: Small canopy shape, Open center shape and Freedom spindle shape, which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of that affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile/branching structure is proved.Finally we develop a model to calculate the leaf mass from the basic formula of m=V. By adjusting the crown of a tree to a half ellipsoid, we first define the function of related factors, such as the leaf density and the effective ratio of leaf area. Then we develop the model using calculus. With this model, we approximately evaluate the leaf mass of a middle-sized tree is 141kg.Dear editor,How much the leaves on a tree weigh is the focus of discussion all the time. Our team study on the theme following the current trend and we find something interesting in the process.The tree itself is component by many major elements. In our findings, we analyze the leaf mass with complicated ones, like leaf shape, tree structure and branch characteristics, which interlace with each other.With the theory that leaf shape is highly related with the veins, we assume the leaf growth in each direction satisfies a function F. For the leaves in the same tree, the parameters are different; for those of separate trees, the function mode is different. Thats why no leaf shares the same shape. Also, we simulate one growing period and depict the leaf shape. In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches.As to the tree structure and the leaf shape, different tree shapes have different space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown that affect leaf shapes. Here we analyze three typical tree shapes which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of that affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile or branching structure is proved.Finally we develop the significant model to calculate the leaf mass from the basic formula of m=V. By adjusting the crown of a tree to a half ellipsoid, we first define the function of related factors and then we develop the model using calculus. With this model, we approximately evaluate the leaf mass of a middle-sized tree is 141kg.We are greatly appreciated that if you can take our findings into consideration. Thank you very much for your precious time for reading our letter.Yours sincerely,Team #14749Contents1. Introduction42. Parameters43. Leaves have their own shapes53.1 Photosynthesis is important to plants53.2 How leaves grow?63.3 Build our model73.4 A simulation of the model94. Do the shapes maximize exposure?134.1 The optimum solution of reducing overlapping shadows134.1.1 The distribution of adjacent leaves134.1.2 Equilibrium point of photosynthesis and respiration144.2 The influence of the “volume” of a tree and its branches164.2.1 The coronary hierarchical model164.2.2 Spatial distribution model of canopy leaf area174.2.3 Transmission model of the solar radiation175. Is leaf shape related to tree structure?195.1 The experiment for one species195.2 Different tree shapes affect the leaf shapes215.2.1 The light distribution in different shapes215.2.2 Wind speed and humidity in the canopy225.2.3 The concentration of carbon dioxide235.3 Conclusion and promotion236. Calculus model for leaf mass246.1 How to estimate the leaf mass?246.2 A simulation of the model267. Strengths and Weakness277.1 Strengths277.2 Weaknesses278. Reference281. IntroductionHow much do the leaves on a tree weigh? Why do leaves have the various shapes that they have? How might one estimate the actual weight of the leaves? How might one classify leaves?We human-beings have never stopped our steps on exploring the natural world. But, as a matter of fact, the answer to those questions is still unresolved. Many scientists continue to study on this area. Recently, Dr. Benjamin Blonder (2010) achieved a new breakthrough on the venation networks and the origin of the leaf economics spectrum. They defined a standardized set of traits density, distance and loopiness and developed a novel quantitative model that uses these venation traits to model leaf-level physiology.Now, it is commonly thought that there are four key leaf functional traits related to leaf economics: net carbon assimilation rate, life span, leaf mass per area ratio and nitrogen content.2. ParametersSpthe area a leaf grows decided by photosynthesisSthe additional leaf area in one growing periodthe leaf growing obliquitythe total photosynthetic ratethe dark respiration rate of leavesthe net photosynthetic ratethe height of the canopythe distance between two branchesthe illumination intensity of scattered light from a given directionthe solar zenith anglethe truck highthe crown high3. Leaves have their own shapes3.1 Photosynthesis is important to plantsIt is widely accepted that two leaves are different, no matter where they are chosen from; even they are from the very tree. To understand how leaves grow is helpful to answer why leaves have the various shapes that they have.The canopy photosynthesis and respiration are the central parts of most biophysical crop and pasture simulation models. In most models, the acclamatory responses of protein and the environmental conditions, such as light, temperature and CO2 concentration, are concerned Ian R. Johnson, John H. M. Thornley,et al. A model of canopy photosynthesis incorporating protein distribution through the canopy and its acclimation to light, temperature and CO2. Annals of Botany, 2010, 106: 735749.In 1980, Farquhar et al developed a model named FvCB model to describe photosynthesis Farquhar GD, von Caemmerer S, Berry JA. A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species.Planta, 1980, 149: 7890.:The FvCB model predicts the net assimilation rate by choosing the minimum between the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate.Assume An, Ac, Aj mol CO2m-2s-1 are the symbols for net assimilation rate, the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate respectively, and the function can be described as:An=minAc,Aj (1)while Ac=VcmaxCi-*Ci+Kmc1+OKmo-Rd Aj=JCi-*4Ci+8*-Rd (2)where Cjbar and Ombar are the intercellular partial pressures of CO2 and O2, respectively, Kmcbar and Kmombar are the MichaelisMenten coefficients of Rubisco for CO2 and O2, respectively, *bar is the CO2 compensation point in the absence of Rd (day respiration in mol CO2m-2s-1 and Jmol e-m-2s-1 is the photosystem II electron transport rate that is used for CO2 fixation and photorespiration Farquhar GD, von Caemmerer S. Modelling of photosynthetic response to environmental conditions. In: Lange OL, Nobel PS,Osmond CB, Ziegler H, eds. Physiological plant ecology II. Water relations and carbon assimilation. Encyclopaedia of plant physiology, New Series, 1982,Vol. 12 B. Berlin: Springer Verlag: 549588. Yin X, van Oijen M, Schapendonk AHCM. Extension of a biochemical model for the generalized stoichiometry of electron transport limited C3 photosynthesis. Plant, Cell and Environment 2004,27: 12111222.We apply the results of this model to build the relationship between the photosynthesis and the area a leaf grows during a period of time. It can be released as:Sp=AnSTKk1,k2,k3,kn (3)Sm2 and Tsare the area of the target leaf and the period of time it grows. Kk1,k2,k3,kn m2mol CO2 is a function which can transfer the amount of CO2 into the area the leaf grows and the k1,k2,k3,kn are parameters which affect Sp. Sp can be used as a constraint condition in our model.3.2 How leaves grow?As the collocation of computer hardware and software develops, people can refer to bridging biology, morphogenesis, applied mathematics and computer graphics to simulate living organisms Prusinkiewicz, P. Visual models of morphogenesis. Artificial Life, 1994, 1, 1/2:6174., thus how to model leaves is of great challenge. In 2001, Dengler and Kang Dengler N,Kang J.Vascular patterning and leaf shapeJ.Current Opinion in Plant Biology,2001,4:50-56. brought up the thought that leaf shape is highly related to venation patterns. Recently, Runions Runions A,Fuhrer M,Lane B,et al.Modeling and visualization of leaf venation patternsJ.ACM Transactions on Graphics, 2005, 24(3):702-711. brought up a method to portray the leaf shape by analyzing venation patterns. Together with the Lindenmayer system (L-system), an advanced venation model can adjust the growth better that it solved the problem occurred in the previous model that the secondary veins are retarded.We know leaves have various shapes. For example, leaves can be classified in to simple leaves which have an undivided blade and compound leaves whose blade is divided into two or more distinct leaflets such as the Fabaceae. As to the shape of a leaf, it may have marginal dentations of the leaf blades or not, and like a palm with various fingers or an elliptical cake. Judd et al defined a set of terms which describe the shape of leaves as follows Judd W W,Campbell C S,Kellogg E A,et al.Plant systematic : A phylogenetic approachM.Sunderland,MA:Sinauer Associates,1999.:Figure 3.1 Terms pertinent to the description of leaf shapes.We chose entire leaves to produce this model as a simplification. Whats more, they confirmed again that the growth of venations relates with that of the leaf.To disclose this relationship, Relative Elementary Rate of Growth (RERG) can be introduced to depict leaves growth Hejnowicz, Z., Romberger, J. Growth tensor of plant organs. Journal of Theoretical Biology, 1984, 110: 93114. RERG is defined as the growth rate per distance, in the definitive direction l at a point p of the growing object, yieldingRERGlp=1sdsdt (4)Figure 3.2 A sample leaf (a) and the results of its: (b) marginal growth, (c) uniform isotropic (isogonic) growth, (d) uniform anisotropic growth, and (e) non-uniform anisotropic growth.Considered RERG, the growth patterns of leaves are also different. Roth-Nebelsick et al brought up four styles in their paper Roth-nebelsick, A., Uhl, D., Mosbugger, V., Kerp, H. Evolution and function of leaf venation architecture: a review. Annals of Botany, 2001, 87: 553566.:3.3 Build our modelFigure 3.3 The half of a leaf is settled in x-y plane like this with primary vein overlapping x-axis. The leaf grows in the direction of .We chose marginal growth to build our model. Amid all above-mentioned studies, we assume that the leaf produce materials it needs to grow by photosynthesis to expand its leaf area from its border and this process is only affected by what we have discussed in the previous section about photosynthesis. The border can be infinitesimally divided into points. Set as the angle between the x-axis and the straight line connecting the grid origin and one point on the curve, and s as the growth distance in the direction of . To simplify the model, we assume that the leaf grows symmetrical. We put half of the leaf into the x-y plane and make the primary vein overlap x-axis.This is how we assume the leaf grows.In one circle of leaf growth, anything that photosynthesis provided transfers into the additional leaf area, which can be described as:Sn=Sp (5)Figure 3.4 The curves of the adjacent growing period and their relationship.while 0xnynxdx-0xn+1yn+1xdx=Sn in the figure.In this case, we can simulate leaf growth thus define the leaf shape by using iterative operations the times N a leaf grow in its entire circle For instance, if the vegetative circle of a leaf is 20 weeks on average, the times of iterative operations N can be set as 20 when we calculate on a weekly basis.First, we pre-establish the border shape of a leaf in the x-y plane, yielding y0x.where x=x0,n The first number in the subscript expresses which period the leaf is in, and the second number represents a typical location on x-axis. Refer to the Figure 3.5. In this case, we assume the beginning state is period 0. Other subscripts occurred in the following context in the same structure are analogical., the relevant 0,1 satisfies:tan0,n=y0x0,nx0,n (6)0,n=tan-1y0x0,nx0,n (7)In the first growing period, assume s0,n, the growth distance in the direction of 0,n, satisfies:s0,n=F0,n The function is defined precisely by the nature of the tress. For instance, species A fits well in linear function, species B in power function while that of another kind of tree is exponential. (8)s0,nx=s0,ncos0,ns0,ny=s0,nsin0,n (9)It releases the relationship of the coordinates in the adjacent growing period. In this case, we can use eq.(9) to predict the new border of the leaf after one period of growth That means, in the end of period 1.:x1,n=x0,n+s0,nx y1,n=y1,n+s0,ny (10)And the average simple recursions are As we both change the x coordinate and the y coordinate, in the new period, these two figures relate through those in the last period in the functions.:Figure 3.5 s,sx,sy and the recurrence relations.xn+1=xn+sx yn+1=yn+sy (11)After simulate the leaf borders of the interactive periods, use definite integral The relationship must meet eq.(5). to settle parameters in the eq.(8) then s can be calculated in each direction of , thus the exact shape of a leaf in the next period is visible.When the number of times N the leaf grows in its life circle applies above-mentioned recursions to iterate N times and the final leaf shape can be settled.By this model, we can draw conclusions about why leaves have different shapes. For the leaves on the same tree, they share the same method of expansion which can be described as the same type of function as Eq.(8). The reason why they are different, not only in a sense of big or small, is that in each growing period they acquire different amount of materials used to expand its own area. In a word, the parameters in the fixedly formed eq.(8) are different for any individual leaf on the same tree. For the leaves of different tree species, the corresponding forms of eq.(8) are dissimilar. Some are linear, some are logarithmic, some are exponential or mixtures of that, which settle the totally different expansion way of leaf, are related with the veins. On that condition, the characters can be divided by a more general concept such as entire or toothed.3.4 A simulation of the modelWe set the related parameters by ourselves to simulate the shape of a leaf and to express the model better.First we initialize the leaf shape by simulating the function of a leaf border at the beginning of growing period 1 in the x-y plane. By observation, we assume that the movement of the initial leaf border satisfied:y0x=1x+0.4-0.5 (12)Suppose the curve goes across the origin of coordinates, then the constraint conditions can be:y0x=1x+0.4-0.5y00=0 (13)Thus the solution to eq.(13) is:y0x=0xy0xdx+y00=lnx+0.4-0.5x-ln0.4 (14)By using Mathematica we calculate where x5.3208(cm), y0x=0(cm) and the area of the half leaf is:Figure 3.6 y0x=lnx+0.4-0.5x-ln0.4S=0xy0xdx=0xlnx+0.4-0.5x-ln0.4dx2.82108cm2 (15)We settle An=33 mol CO2m-2s-1 according to the research by S. V. Archontoulisin et al S. V. Archontoulis, X. Yin,et