2019MCM_A_M奖论文.pdf-资料库

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For office use only T1 T2 T3 T4 For office use only F1 F2 F3 F4 Team Control Number 1907723 Problem Chosen A 2019 MCM/ICM Summary Sheet Dragon in the Real World Summary In the Game of Thrones based on the epic fantasy novels A Song of Ice and Fire, ”The Mother of Dragons”, Daenerys Targaryen raised three dragons. Given that these dragons are living today, we’re going to analyze dragon’s features, behaviors, habits, diet and interaction with the environment. Our tasks mainly include the following four aspects: 1.Dragon’s characteristics; 2.Daily behaviors; 3.Interaction with ecolog- ical system; 4.Environment’s support to the growth of the dragons. We firstly build the dragon model, analyzing the dragon’s shape, behav- iors and other characteristics. Then, we build an ecosystem model to analyze the interaction between dragons and the environment. In task 1, we conducted a comprehensive analysis and modeling of the dragon in order to maximize the description of its physical characteris- tics, including morphological features such as the weight, wingspan and length of the body. In the analyzing of the weight of dragons, we used the Gompertzcurve curve to model the change of the weight of dragons over time. In task 2, we analyzed dragon’s behaviors of flight, migration and fire breathing. We also modeled the metabolism of dragons based on the famous Kleiber’s law. In task 3, we mainly analyzed a temperate deciduous broad-leaved for- est ecosystem. Based on the species coexistence model by Tilman, we established the Dragon’s invading model to analyze the interaction. We came up with the concept of Development Index (DI) to measure the development status of a group in the ecological system. In task 4, we quantitatively analyzed the environmental requirements for maintaining the growth of dragons. Keywords: Gompertzcurve curve;Kleiber’s law;Dragon’s invading model

Team # 1907723 Page 1 of 21 Contents 1 Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Restatement of the Problems . . . . . . . . . . . . . . . . . . . . . 2 Notation 3 Dragon’s Characterisics 3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Body Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wingspan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Body Length . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Physiological Characteristics-Metabolism . . . . . . . . . . . . . . . 3.4 Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Breath of Fire . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Ecological System 4.2.1 Establishment of the Model 4.2.2 4.2.3 4.1 Assumption and Justification . . . . . . . . . . . . . . . . . . . . . 4.2 Model of Dragon’s invading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of the Model . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 4.3 Support from Ecological System . . . . . . . . . . . . . . . . . . . . 4.3.1 Energy Flaw . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Energy Demand . . . . . . . . . . . . . . . . . . . . . . . . 5 Varying Environment 6 Conclusions 6.1 Strengths 6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 3 3 3 4 4 4 5 6 6 6 6 7 8 8 10 10 11 12 13 13 14 15 16 16 16

Team # 1907723 7 Extensions 8 Letter to George R.R.Martin Appendices Appendix A First appendix Appendix B Second appendix Page 2 of 21 16 18 19 19 20

Team # 1907723 1 Introduction Page 3 of 21 1.1 Background In the TV series Game of Thrones, the three dragons, Dragon, Rhaegal, and Viserion have attracted the attention of numerous fans with their tremendous power. We were astounded by the dragon’s enormous size and destructive fire. However, what about the authenticity of the dragon in the film? In this paper, we try to explore the realistic basis of dragons, including their characteristics and their interaction with the environment. 1.2 Restatement of the Problems Given that dragons are living today, we need to construct models about their body and behaviors to understand them. It’s no doubts that dragons which depend on the ecosystem will have an impact on the ecosystem. To understand the interaction between the enormous dragons and ecosystem, we need some models. Our specific tasks are the following： • Construct Model of the dragons , study the dragons’ shape, behaviors and other characteristics. • Establish ecosystem model and study the interaction between dragons and environment. • Consider other fields where the model can be used. • Write a letter to the author of A Song of Ice and Fire, George R.R. Martin, to provide guidance about how to maintain the realistic ecological underpinning of the story. 2 Notation Here we list some symbols and notations used in this paper,as shown in Tabel 1. Others will be explained when used. 3 Dragon’s Characterisics 3.1 Analysis Because of the illusory nature of the dragon being analyzed, we do not have enough effective data to construct all the characteristics of the dragon, so we only give some obvious characteristics of the dragon, such as weight, wingspan, body length,

Team # 1907723 Page 4 of 21 Table 1: Notations Symbol Description M Ws Ad v Ld PBM R The weight of the dragon The wingspan of the dragon The wingspan area of the dragon The flight speed of the dragon The length of the dragon The baisc metabolic rate of the dragon metabolic rate, flight, and fire breathing, which are enough to describe the dragon. When analyzing the characteristics of the dragon, it is assumed that the dragon can obtain sufficient external resources, and the external environment will not have obvious influence on the growth of the dragon, and the dragon is only limited by its own internal factors. 3.2 Body Characteristics 3.2.1 Weight Firstly, we need to address the problem of the dragon’s weight, assuming that exter- nal resources are sufficient and that the weight change of the dragon is only related to the internal causes (biology) of the dragon. The Gompertzcurve[7] performs well in fitting the growth process of organisms and is adopted by most biologists. We use the Gompertzcurve to simulate the relationship between weight M (Kg) and age t (years) during the growth of the dragons. The expression of the Gompertzcurve is: eabt M = ke (1) Specific analytic expression will be given below. 3.2.2 Wingspan Considering that the fictional dragons are similar with pterosaurs that existed in the history, we use the biological studies on pterosaurs for reference to estimate the physical characteristics of dragons. Based on the studies on pterosaurs by Stuart Humphries, Richard h. C Bonser, Mark P Witton, David M Martill [1] and others, we give the relationship between the wingspan and the body weight of dragons: M = 0:299W 2:215 s where M is weight, Ws is the width of the wingspan (2) The author,George R.R. Martin has said that The dragon is the size of a Boeing 747.We assume that after 10 years since hatching, the wingspan of the dragon

Team # 1907723 Page 5 of 21 reaches the wingspan of a Boeing 747, which is about 60m. Thus, we can obtain the following set of equations: eabt M = ke M = 0:299W 2:215 s M (0) = 10 M (1) = 35 Ws(10) = 60 According to the equation set above, we can get the parameters: a=1.758, b=0.2432 and k= 3309. Figure 1 shows the change of dragon’s weight with time. The weight increases rapidly when dragon is young. It still increases after the age of 20 but not obvious. At the age of 10, dragon is already in his youth. In the article of Figure 1: Dragon’s weight Stuart Humphries et al., pterosaur’s wingspan area calculation formula was given. Similarly, the wingspan area of dragon can be expressed as: Ad = 0:0886W 1:778 s Ws is the wingspan area. (3) 3.2.3 Body Length The main function of the body length for flying animals is to maintain the stability during flight. Based on the proportion of the body length and wingspan of birds in the world, we give the proportion of the body length and wingspan of dragons: Ld = 0:382Ws Ld is the body length of dragon (4)

Team # 1907723 3.2.4 summary Page 6 of 21 The morphological characteristics of dragons are summarized in the Table 2. Table 2: Morphological Feature of Dragon e1:7580:2432t s M 3309e M 0:299W 2:215 0:0886W 1:778 Ad 0:382Ws Ld s For a 10-year-old dragon, the specific features are shown in Tabel 3. Table 3: Morphological Feature of Dragon at the age of 10 M 2000 Kg Ws Ad Ld 60 m 128 m2 23 m 3.3 Physiological Characteristics-Metabolism In the biological field, studies on animal metabolism are relatively complete. Kleiber summarized various metabolic models In Body size and metabolic rate[3]. We have adopted the famous kleiber’s law[2] to simulate the metabolism level of dragons in the resting state. Stuart Humphries et al. studied the metabolism of many pterosaurs, we have refered to their experimental data[1]. We then give the basic metabolic rate(PBM R(Kcar/Day)) of the dragon: PBM R = 50M 0:72 (5) 3.4 Behaviors 3.4.1 Breath of Fire In this section, We mainly consider the maximum energy that can be released by a dragon. Assuming that the maximum energy is equivalent to Ten Percent of dragon’s weight, and that the heat required by a dragon to increase by 1Kg is H0 (H0 = 7000Kcal, the conversion ratio refers to human[4]). we use to represent

Team # 1907723 Page 7 of 21 the conversion coefficient of heat (Kcal) to energy (Kj). Then the maximum energy that can be released by a dragon is: E = 0:1M H0 (6) For a 10-year-old dragon weighing 2000Kg, the maximum energy that can be re- leased is: E10 = 0:1 2000 7000 4:18 = 5:852 106Kj (7) The energy is equivalent to 1396Kg of TNT. 3.4.2 Flight We mainly consider the energy problem faced by dragon. Considering the energy consumption in the process of animals’flying go almost similar ( related to speed, windward area and other factors),.We use pterosaurs’ flying to estimate dragon’s energy consumption. Drawing on the work of Stuart Humphries and others (ref), We give an expression for the energy consumption of a dragon in flight: Pf = 0:5SCbv3 + 2kM 2g2 Wsv + C (8) Pf is calculated as power (W), is the air density, S is the projected frontal area of the animal’s body (calculated as 8:13×10−3M 2 3 ), Cb is a coefficient related to the dragon’s body resistance (0.75), v is the speed, k is the coefficient related to the wing movement(1.2), M is the mass, W is the wingspan. C is a coefficient associated with mass M[1]. For a 10-year-old dragon, M=2000Kg, W=60m, and C is calculated as 7.7 Kw. The relation between power Pf and speed v is shown in Figure 2. When v equals 14.7385m/s, Pf gets the minimum value. Considering that the dragon may take long-distance migration, we need to esti- mate the maximum distance for dragon. For a 10-year-old dragon, assuming that there is no energy supply or fire-breathing during the flight, the maximum energy that can be consumed in one day’s flight is close to the maximum energy(E10) that can be released in an instant which we have calculated in (7): E10 = 5:852 105 Kj. Assumed that the flying speed of the dragon is 14.7m/s(the minimum power con- sumed), and the power is: Pf 0 = 1:5358 104 W. From above, it can be calculated that the maximum flight distance(represented with Sm) in a day is : E10 Pf 0 = 10:6 Hour t = Sm = vt = 560127:6m (9) (10) Consider three habitats: arid regions (25 degrees north latitude), warm temperate regions (50 degrees north latitude), and the arctic (65 degrees north latitude).When calculating the distance between the three regions in the process of migration, the

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