2012 MCM北美数学建模A题论文.doc

Page 19Mathematical model research on the nature of tea treesFor office use onlyT1_T2_T3_T4_Team Control Number13499Problem ChosenAFor office use onlyF1_F2_F3_F4_2012 Mathematical Contest in Modeling (MCM) Summary Sheet Abstract: To better express a trees characteristic, furthermore to solve some actual problems. This paper used L system fractal model and Forest gap model to build an accurate and complete model to describe a tree position parameter and shape parameter. Our model is three steps: the intensity of the condition of the three model gradually diminish. First model, we use L system to create a tree without leaves. We then add leaf information to the first one which made it a complete tree. In model 3,we use forest gap model to add some more environmental factors, to amend the simulation to make it more real.To testify this model accuracy, we chose tea trees fudingdabai which are from Hunan province grown up from garden 13 years old canopy as our research object and testify the three models ,meanwhile calculate dispersion coefficient and the difference. From the dispersion we can know that the model is alright. After building the model, we did some research on the tree, use mathematical deformation to find the relations between the various parts of the tree, until at last get the weight of the tree.This paper specialized in layer modeling constructing , reducing the constraints , taking good advantage of matlab and spss, excel etc. to gradually to close to reality.Key words: L system fractal model, forest gap model, significance level testify, Layer modelingContents1. Introduction.2 2. Analysis of the Problem. .23. Models. 44. Conclusions.16 5. Future Work.18 6. References.19 I. IntroductionTo figure out the weight of all the leaves of a tree, we supposed that there are several factors that may influence the weight of the leaves: environmental factors (including sunlight, water, etc.) , leaf internal factors (leaf surface area), and the position of the leaves (where it is the location of the branches.) As long as we can figure out these elements will we be able to estimate the total mass of the leaves. 1 First we ignore the environmental elements, assuming that environmental factors are very ideal for the tree to live, and we only consider one kind of the tree ( tea tree : fudingdabaicha a kind of canopy) Leaves abstract into an oval shape, for the time being , ignore the influence of difference of tree leaves, only consider the tree branches and leaves, according to L fractal model of view, between the stem and branches of the tree has a certain relationship, further branches and tree branches there is also a relationship. Relationship between the branches of a tree can be estimated, while each of the branches and the number of leaves on each tree branches (assuming the leaves of each tree is the same size (ignores the influence of a number of new leaves), consider the number.) so you can get the number of leaves ,known leaf area (all the leaves of view), the total leaf area can be drawn according to the ideal situation. With total area of tree leaves: S, then find the thickness H of the leaves, measure the average density p of the leaves, then, SHp you can get the total mass of the leaves.Then we can take further consideration of leaf individual difference and environmental factors into account.II. Analysis of the ProblemCalculate the weight of the leaves of a tree, figure out the overall contour of a tree by a leaf, and do some research:2.1.Why are there so many leaf shapes ? Is it meaningful?On the shape of the leaves:There are many types of leaves, needles, deciduous broad-leaved, evergreen broad, oval leaves, needle-like leaves, heart-shaped leaves and so on.The general characteristics of these leaves in order to better adapt to the local geographical climatic characteristics: The factors of land and sea, latitude factors and vertical height of factors. This also confirms: the survival of the fittest token.Sea factors: the length of the vegetation in relatively arid regions: cacti, cactus, camel thorn, leaves have been allowed to degenerate into needle-like, in order to lock up more water to protect themselves, the photosynthesis transferred to the stem. In relatively humid areas, the water will not easily lose, all kinds of tree leaves are green, the leaves for photosynthesis, greatly increased by the illumination area in order to better carry out photosynthesis. Leaves of aquatic plants: living in the water: mesophyll cells arranged in loose, well-developed aerenchyma. Latitude factors: relatively warm regions: subtropical and tropical regions trees grow on the active accumulated temperature: evergreen broad-leaved forest, most of the tall trees, leaf area, and does not change with the seasons while the leaves. To warm temperate and temperate regions, the active accumulated temperature is relatively low: the leaves to adapt to the cold weather of autumn and winter, to keep the energy no longer carry out photosynthesis, leaves appear.Cold temperate and boreal regions, active accumulated temperature is very low: asked to adapt to this situation, the total for survival in a cold place mostly dominated by coniferous forest, coniferous forest, because of its relatively broad-leaved surface area contact with the cold air small area, so the cold more.The vertical height of the impact is because the temperature decreases with elevation caused by strange Chinese Hengduan Mountains scene can prove this point. From the foot of the mountain evergreen broadleaf forest to deciduous broad-leaved forest, coniferous forest, with the elevation and latitude distribution is very similar.These external factors, a deeper point of view, the shape of the leaves by many internal factors control theManufacturing leaves the carbon content, the life of the leaves, photosynthesis combination of the speed of the three elements, we can construct an infinite number of leaf shape and structure. The scientists believe that the vein is the foundation of the leaves: vein density, the distance between the veins, smaller veins of the human capillary the number of regions, determine the various shapes of the leaves of the same plant trees in order to better carry out photosynthesis, In order to better retain moisture, to seek protection, environmental adaptation to growth.The shape of the leaves of the fundamental genetically determined: the NAM, cuc3 they are responsible for a large group of plant proteins decoding. These genes are initiated at the base of the leaves and leaflets, if it is turned off so the leaves will become indescribable ill-defined lumps. Differentiation of all the leaves are inseparable from the the cuc gene family.2.2 Do the leaves tend not to cover each other? Leaves get close to not cover each other as far as possible, as well as branch status (weight status) has any effect on leaf conditions (leaf size).Biologically, it is right, and this is why we promote rational close planting.(To determine the intensity of photosynthesis :through the release amount of oxygen.). When leaf area is greater than a certain value, the efficiency of photosynthesis is no longer enhanced, respiration continued to increase.The excess leaves cant do photosynthesis , but still they need to do respiration, the accumulation of a substance tend to increase slowly.2.3 Leaf shape and Tree shapeLeaf shapes has to do with shapes of the tree ? Is it possible to judge the overall situation of the tree with only a leaf of the tree?.Access to relevant information, we tried to figure out that it is sure: certain tree has certain sort of leaves. We can figure out the tree by the leaf. Even one kind of tree , a certain leaf may indicate the tree state.It is certain that these things are true, how can we prove it ?As we build the model we will get futher into it. III. Models3.1 L - System fractal graphics improvement and the main branches of the model L, - system constructed axiom and production set through experiential generalizations and abstract fractal objects , then generate the character development of the sequence, then each character interpreted as the corresponding graphic elements, in order to get the target sub-fractal graphics of the model .2The majority of woody plants in its backbone structure, including:1) root, the initial node, all the branch and trunk origins from here.2) The relay node. on the way generating plant structure, we choose at least one relay node.3) the terminal nodes: the terminal of all paths to generate plant structure .4) Straight branches and lateral branches. Straight branch is the trunk of the plant, lateral branch is branch. Straight branches and lateral branches together formed the complex plant bifurcation structure. To describe the plant bifurcation structure we need 4 characters: “ ”、“ ”、“ + ”、and“ - ”。during the characters rewriting in, when it comes to these 4 charaters, copy them into result set sequence directly, to get it identical replaced. When translating the string means Mapping the location coordinates onto the stack;“ ” means Popping from the stack of mapping the location coordinates,They play the role of the temporary location to save the coordinates.“ + ”means To keep the starting coordinates of the origin ,counterclockwise direction of rotation.“ - ”means keep the starting coordinates of the origin , clockwise direction of rotation.2.The simulation of plant tree trunk and computer realization First, determine the roots and the growing direction until the first node, and placed in the diagram line thickness value,When drawing to any stage, we can use the (X, Y, and ) 3 amount to determine the graphics state.Among them, the (X, Y) coordinates of the current drawing position , describes the direction of the current line direction angle.Then, what need to consider is how to convert between state to state.Through some simple status symbols( Translation and Rotation) defined transform action will we be able to get the various states in the Drawing.By trunk (branch) fractal theorem: the length of a branch and the next branch length ratio fixes, the diameter of a branch and the next branch ratio fixes. The roots of plants are always thicker than the branches , the upper tree body is thiner and thiner. To achieve this goal we take action :L system ,Branches of plants in the string defined are distinguished from each other by and . when it comes to more iterations there are or nesting,it can be known from L-system rewriting the nature of fractal : outter and represent for tree start and tree end, the inner the smaller branches are, so we can difine a variable :width, to record the starting position and the ending position of the branch When meeting a width-, a width+, obviously the trunk width is larger, for that the smallest branch width is not less than 1, we should consider when initializing width. This way is only to describe the problem in brief , the width only +1 or -1.When it comes to actual problems, and then consider.3L system mainly contains Initial element we can assume:The initial element w = AProduction P1: A-Bf (A) P2: B-g (B)Generate state S0 = A S1 = Bf (A) S2 = g (B) of f (Bf (A) = g (B) f (S1) S3 = g (B1), f (S2) . Sn = g (Bn-2) f (Sn-1)L system fractal images with self-similarity,This feature is Consistent with plant morphology. L system is suitable for describing the plant topology.Represent the L system in the form of character and string. By translating the characters to draw the final fractal graphics.Initial thickness :initthiLength of the zoom : lenmultThickness scaling : thimultBranches bifurcation angle :anglemultThe number of separating branches: numlimb The trunk surface and the lower surface ratio: stemFacThe number of top branches: numTopLimbBranch segment model L represents for length ,R represents for diameter of the branch(or trunk), T: The angle around the Y axis,E:Around the z-axis rotation angle . F:Around the X-axis rotation angle. F，R，r，Eeach represents for Properties of the branch segment , branch and branch segment has the relationship below :4 3.2 description of tree leavesThe leaves are an important part of the trees,N denotes the number of the separation of each one branch.Here every tree leaf is set on the random height of the branch. There is angle between the leaf and the branch.N denotes number of the leaves, L denotes length , E denotes angle of rotation around the Z axis , F denotes angle of rotation around the X axis ，L，E，Feach denotes properties of the branch segment.3.3 Environmentally sensitive modelThrough research on the conceptual model of the interaction of plants and the environment, analyze and construct the environmentally sensitive model.Due to the complexity of the mechanism of plant growth, it is very hard to set up an entire growth model. Here, we made some changes and expansion to L system and environmental parameters.Tea tree as the specific object of study. We take how to get the environmental factors integrated into the research on growth of the a tree for the core of the model. 5TO qauntify the relationship between environmental factors and the growth of a tree. The morphology of the plant is determined not only by its internal growth factors, but also by many external environmental factors. according to the conceptual model put forward by Prusinkiewicz in Material9:When the plants receive sunlight, moisture, climate and other environmental factors from outside, its internal growth system will be affected to some degree which result in the changes of leaf shape.The growth of a tree is that it interacts with environment :recieve message, respond to it. It is a chain of event. we it observes stimulation from the environment ,some message can be sent to its inside and the tree respond conditionally. This reaction would also counteract on the environment making the environment change.For instance:Roots in the soil depend on the water concentration, absorb moisture or release moisture.According to this principle, plant-environment interactions can be assumed as two interactive synchronization processes.Therefore, it forms an information flow chain reaction.(graphic 2)9For plant,the main processes to deal with are:To receive environmental information through the various tissues and organsTransfer the information obtained to the internal and process; Growth (such as growth of new branches) will deal with the information and feed back to the environment.Similarly, the environment also has a similar operating mechanism.6 Perception of the feedback information of the plant; environmental internal data processing ; to form a new environment, and being perceived by the plant. Forestry has researches on the environment and plant growth ,a famous one put forward by watt:Forest gap modelForest gap model is based on the dynamic theory of the forest cycle. In this paper, we take the tea tree forest gap model for example to implement this model.It mainly includes temperature effects, light effects and other Environmental suitability functions.Generally speaking, a tree by the following characteristics:Diameter at breast height(DBH),height(H),Crown(CD),crown height(CH), with which we can describe the trees outline.The tea tree Forest gap model provide us a growth model,This growth models assume that it is in the ideal environment. Then amend it with environmental factors (G, V), finally get the actual growth, DBH optimal growth equation: 、 are DBH and tree height; Dmax和Hmax:maximum diameter at breast height and tree height of trees,is crown diameter ,t is age. G is sunshine growth parameter (we suppose that sunshine is the main factor of the external environment)which affects the evolution of community and it is the key parameter which has to do with environment in the vicinity.Environment for growth parameter V ranges from 0 to 1. If the growth environment is superior, then G is the maximum 310. If the growth environment is not pleasant, G is smaller. It can be seen from the above formula, its main function is to establish a relationship between the trees and Environment, to express that Tree growth is affected by the external environment.To describe the tree in more details , like tree diameter ,tree growth angle of the branches, so we Refer to the constraint equations 9-13, 7 through Interface parameters (fDBH (t) fH, (t), fCD (t) to realize the effective connection with the forest gap mo