2013美賽特等獎(jiǎng)原版論文集15065_W

1、Team Control Number15065Problem ChosenAFor office use onlyT1 T2 T3 T4 For office use onlyF1 F2 F3 F4 2012 Mathematical Contest in Modeling (MCM) Summary SheetIn our paper, we construct a mathematical model for estimating total leaf mass of a tree and investigating the relationship between leaf shape

2、 and tree profile.Our approach consists of two main models. The vector tree model uses vector and linear transformation to simulate the geometrical structures of a tree, based on empirical and theoretical research on tree structures. A key assumption for this model is that the branching structure of

3、 a tree is paracladial. The sunlight model simulates light irradiance inside the tree crown. It starts by simulating the spiral motion of the sun in the course of a year using the brightness function for the sky. Then, the sunlight irradiance inside the tree crown is evaluated using a model based on

4、 the Monsi-Saeki equation.To investigate the relationship between leaf shape and tree profile. We use thevector tree model to generate different tree profiles. We change the leaf shape under the same branching structure and see whether the real world leaf shape maximizes sunlight exposure among diff

5、erent shapes. Projection of shadow is used to measure sunlight exposure. Spherical integration is used to calculate the weighted sunlight exposure rate based on the brightness function for the sky. For each tree branching structure, leaves of three different leaf shapes are tested for the total amou

6、nt of exposure to the sunlight. Comparing the simulated data to real-life data, we found that leaf shapes generally maximize the total exposure. Furthermore, our algorithm is good for medium and small sized leafs, but tends to be unreliable and generate answers with large variance for large sized le

7、aves.To estimate the total leaf mass of a tree, we incorporate the tree profile with thephotosynthesis model, and use indexes like leaf mass per area (LMA) and leaf area index (LAI) to estimate the total leaf mass of a tree. Photosynthesis rates are assumed to be affected only by sunlight irradiance

8、. We compute the leaf mass of Cinnamomum camphora using this proposed method. Comparing to the real-life data, our method is accurate enough with the limited amount of data.更多數(shù)學(xué)建模資料請(qǐng)關(guān)注微店店鋪“數(shù)學(xué)建模學(xué)習(xí)交流”/RHO6PSpAGeometrical TreeLeaf mass & leaf-tree relationshipTeam # 15065Content1.In

9、troduction31.1Outline of Our Approach31.2General Assumptions42Vector Tree Model42.1Leaf classification42.2Branching Structure Model63Sunlight Model103.1Orbit of the sun & Brightness function for the sky103.2Modeling the light irradiation at a certain position inside the tree crown114Leaf shape & tre

10、e profile/branching structure124.1Sunlight exposure rate124.2Matching leaf shape with tree profile/branching structure145Leaf Mass of a Tree155.1Computation of Leaf Mass per Area Ratio (LMA)165.2Total leaf Area185.3Case study: a real tree Cinnamomum camphora195.4Correlation between the leaf mass and

11、 the size characteristics of the tree216Improving the model236.1Leaf classification236.2Exposure area of leaves236.3Determinants of photosynthesis rate237Conclusion238Letter to a scientific journal editor24Reference24Appendix: Matlab code for implementing our model261. IntroductionIn this paper we p

12、resent a mathematical model for investigating the relationshipbetween tree profiles and leaf shapes and estimating the total leaf mass of a tree. Specifically, we created a tree profile using 3-dimensional coordinates and linear transformation. We simulate the sunlight irradiance in the tree crown u

13、sing the brightness function of the sky and Monsi and Saeki equation (Monsi, 1953). We then incorporate the sunlight irradiance into different tree profiles to find a leaf shape that maximizes sunlight exposure. The chosen shapes match with real leaf shapes, suggesting a close relationship between l

14、eaf shape and tree profile: leaf shape maximizes the overall sunlight exposure under a given tree profile. We then incorporate the tree profile with the photosynthesis model (Tsukaya, 2006), and use indexes like leaf mass per area (LMA) and leaf area index (LAI) to estimate the total leaf mass of a

15、tree.1.1 Outline of Our ApproachWe first introduce two models which will be useful in the latter applications:Vector Tree Model: The first part of our paper will be devoted to presenting thetheoretical framework of this model. Our objective is to create a spatial structure of the tree crown (branchi

16、ng structure and leaf distribution). We use vectors to simulate leaf shape, distribution of leaves on branches, and the tree profile/branching structure. Linear transformation of vectors is used to simulate the relationship between daughter branch and parent branch.Sunlight Model: The second part of

17、 the paper will introduce the sunlight model.Our objective is to simulate solar irradiance across a year. Brightness function over the celestial sphere is used to describe solar irradiance from different directions, Monsi and Saeki equation is used to calculate the light attenuation in a tree crown

18、due to overlapping of leaf shadows.The latter sections present two applications of the models.Investigating the Relationship between Leaf Shape and Tree ProfileWe use the vector tree model to construct different tree profiles. Combining the tree profile with the sunlight model, we are able to calcul

19、ate the sunlight exposure rate of the leaves. We then adopt different leaf shapes for the same tree profile and find the one that maximizes the sunlight exposure. Comparing this chosen leaf shape with the real leaf shape of the tree, if there is a match, we may conclude that the leaf shape is associ

20、ated with the tree profile in that it maximizes the sunlight exposure.Estimating Total Leaf Mass of a TreeGiven a tree, we use the vector tree model to simulate its profile, and use the sunlight model to determine the light irradiance at each leaf. Then, we derive the relationship between light irra

21、diance and the leaf mass per area (LMA) according to a photosynthesis model. Thus, we can derive the LMA for each leaf and calculate the weighted average LMA for the entire tree. We then use the sunlight model again to find the shadow of the tree crown and calculate the total leaf area of the tree u

22、sing leaf area index (LAI). Finally, the total leaf mass is calculated by multiplying LMA with the total leaf area.1.2 General AssumptionsTrees are assumed to be paracladial, i.e. if any branch is cut off, it has the samestructural characteristics, apart from size, as the parent from which it is cut

23、. The majority of leaves will grow on the last generation of branches.Sun light is assumed to propagate in a straight line. Diffraction and refraction of lights are ignored.Photosynthesis rate is assumed to be affected only by the rate of light irradiance. Other factors such as CO2 concentration in

24、the air are assumed to be homogeneous at any part of the tree crown.Photosynthesis due to light sources other than the sun is neglected. (e.g. moonlight at night and artificial light)The environmental destructive effects are minimized and thus negligible, such as natural disasters and herbivory.Nutr

25、ient supplies are sufficient.2 Vector Tree Model2.1 Leaf classificationThe leaf is a major part of the plant-body plan. How to classify different leaves?Traditional plant taxonomy focuses on leaf functions and leaf shape. Recent research also suggests that venation is a strong indicator in leaf clas

26、sification.The general leaf shape is an important factor for the plant to receive sunlight.Traditional parameters in describing leaf shape include presence/absence of leaf petiole, flatness, leaf index (a ratio of leaf length to leaf width), margin type, and overall size. (Tsukaya, 2006)Since we are

27、 going to construct a computerized simulation of a tree, is it importantfor us to classify leaves quantitatively such that they are easy for representation and simulation. Hence we develop a model to classify the leaves into 4 basic shape categories.To classify a given leaf to one of the four basic

28、shape categories, we focus on 3factors of the leaf: shape convexity, leaf index (ratio of leaf length to leaf width), and the position of the longest width on a leaf.To determine the shape convexity, we first simulate a leaf using a polygon. We first fitthe leaf into a grid with each cell size 0.5cm

29、0.5cm. Next we select the outmost grids that have been covered more than half by the leaf. Then we plot and connect the centers of these grids. Hence we have a simulating polygon of the leaf.Two illustrations are shown as below.We are now able to determine theconvexity of the polygon of leaf convexp

30、olygons are such that all diagonals lie entirely inside the polygon; concave polygonsare such that some diagonals will lie outside the polygon. Hence we can determine the convexity of the leaf.If a leaf is concave, we immediately classify it as Palmate. Convex leaves are left forfurther determinatio

31、n.To classify other leaves, the second factor we look at is leaf index, which is the ratioof leaf length to leaf width. According to empirical data (Tsukaya, 2006) (Johnson,1990), we classify leaves with leaf ratio of 3.5 and above as Linear. Those with leaf ratio below 3.5 will be either Elliptic o

32、r Deltoid.To classify Elliptic leaf and Deltoid leaf, we look at the position of longest-width on aleaf. In order to have a more quantitative view, we are interested in the ratio of distance between leaf tip and longest-width-position (D) to the leaf length (L).If the ratio (D/L) lies within 1/4, 3/

33、4, we classify the leaf as Elliptic. Leaves whoseratio3/4 are Deltoid.To summarize the decision rule of classifying a leaf, we provide a decision flow chart.ConcaveConvex3.53.51/4, 3/43/42.2 Branching Structure2.2.1Biological Backgrounds and Existing ModelsThe existing geometrical models simulating

34、branching structures of trees areessentially empirical based on a rule that specifies the relative angular direction and length of a daughter branch to its parent branch (Johnson, 1990). In order to obtain a more realistic model, studies considering whorls and bifurcations as rule in branching simul

35、ation have also been carried out (Fisher, 1977) (Fisher, 1979).DeltoidEllipticRatio of distance between leaf tip and longest-width-position (D) to the leaf length (L)LinearLeaf indexPalmateLeaf polygon: convex?2.2.2Model Description(/01/002_en.html)In this model, our objecti

36、ve is to represent each branching point and the position ofeach leaf using their position vector (x,y,z) in a 3D coordinate shown above with z- axis set to be the main stem of the tree. Child Branch FormationGiven the length and location of a parent branch, we want to find the direction andle

37、ngth of the child branch. At this point, the Child branches generated will subject to the following:a) Number of child branches at the branching point N is given byN= (),where : A species-wise constant coefficient which is related to the mean of the number of child branches for the species. : Uncert

38、ainty compensator which incorporates the genetic uncertainties and environmental uncertainties. : A function which maps the species of a tree to the mean number of child branches generated at this branching point.b) At each branching point, the length and direction of the N child branches aredetermi

39、ned by multiplying the direction vector of the parent branch with transformation matrices 1, 2, , respectively. is defined as the following: (0)(, ) =where is the length change, is the rotated angle with respect to the parentbranch about and is the rotated angle with respect to the parent branch abo

40、ut local x axis with respect to the parent branch. Note that at each branching point, the coordinate system adopted for this transformation will be the local coordinate system with respect to the local parent branch, and the parent branch will lie on the z-axis.c) Now, we are able to determine the c

41、oordinate of each branch.We first number the branches according to the generating sequence. Let denotesthe vector representing the branch of the Nth generated branch. If branch of , then the equation: = is to be satisfied, where is the transformation matrix that governs the ruleis the parentbranch a

42、nd kth branch (abbreviated as branch t and branch k in laterbetween tth context).Let denote the coordinate of Nth generated branch. Then 1 = 1 and = + .In particular, in the situation where only bifurcations are concerned (Johnson, 1990),2 = 12+1 = 2and2 = 2 + 2+1 = 2+1 + Leaf FormationNow,

43、we are ready to construct leaves on the branches.Define set = | , and the growing of leaves is subject to the following:a)Basic phyllotaxis patterns:(/phyllo/About/Classification.html) Distichous Phyllotaxis, in distichous phyllotaxis, leaves or other botanical elements grow

44、one by one, each at 180 degrees from the previous one.;Whorled Phyllotaxis, In whorled phyllotaxis, two or more elements grow at the same node on the stem.;Spiral Phyllotaxis, In spiral phyllotaxis, botanical elements grow one by one, each at a constant divergence angle d from the previous one.;Mult

45、ijugate Phyllotaxis, elements in a whorl (group of elements at a node) are spread evenly around the stem and each whorl is at a constant divergence angle d from the previous one.The spacing of leaves on the branch is determined by the function d(I) which takes in the intensity of sunlight irradiance

46、 and outputs the spacing of the leaves.The probability that a leaf grows at a certain spot is determined by the function () which takes in the intensity of sunlight irradiance and outputs the probability that a leaf is likely to grow on a certain spot of the branch.b)c)In our model, for each , we gr

47、ow the leaves on branch p according to thearchived Basic phyllotaxis patterns for different species. The spacing between twodifferent potential leaf-growing spots is d(I) and the probability of a leaf growing at a certain spot is ().2.2.3Different tree profiles generated by our modelUsing the model

48、introduced above, we are able to generate a wide range of treeprofiles by varying the coefficients in the model. Three examples are shown below.Zooming in to the picture, the leaves are represented as this:3 Sunlight ModelPhotosynthesis is important for the growth of leaves, and it is driven by sola

49、r energy.Thus, in this section, we will consider the light irradiance in a tree crown. By modeling the sun radiation using brightness function, and using Monsi-Saeki equation (Monsi, 1953) to describe the light attenuation within a tree crown, we will be able to calculate light irradiance at any poi

50、nt within the tree crown.3.1 Orbit of the sun & Brightness function for the skyOrbit of the sunSince the sun moves in a spiral during the course of a year (in the view of an observer standing on the earth), we will first model its orbit using celestial sphere. At different latitude, the sun will tra

51、ce out a different trajectory in a year. Three examples are shown below.North temperate zoneEquatorthe Arctic PoleBrightness function for the skyThe brightness function (Johnson, 1990) for the sky is defined in terms of spherical polar coordinates (r, , ), although it does not involve the r coordina

52、te. When defining the position of the sun, is known as the zenith angle (when = 0, the sun is directly overhead) and is termed the azimuth angle.The brightness function of the sky is defined as B(, ), with units W m-2 srad-1. The steradian (srad) is the SI unit for solid angle: a solid angle is the

53、area of the surface of a portion of sphere divided by the square of the radius of the sphere. Thus the total solid angle of a sphere is 4 sradDeriving the brightness functionTo illustrate the method of deriving B(, ), we use trees on the equator as anexample. Brightness function for the sky at other

54、 latitude and longitude can be derived in a similar way with slight modification of the algorithm.ABOn the equator, the orbit of the sun will shift from circle A to circle B from summersolstice to winter solstice (strictly speaking it is not a circle but a spiral).We denote the rate of solar radiati

55、on that reaches the earth as E with unit of W m-2 s-1. E is known from previous research of solar energy.When the sun is at a certain position (, ) on the celestial sphere, the rate of solarenergy that is absorbed by the earth surface (Ea) depends on the angle between the ground and the light beam (

56、). E sin, ( = )2Thus, since we know the orbit of the sun in the course of a year, the brightnessfunction can be derived by integration of Ea over time (t). This should be done by computer simulation and discretization of the spiral trajectory is used for ease of implementation.3.2 Sunlight irradiation inside the tree crownIn plant models, light attenuat