没有合适的资源?快使用搜索试试~ 我知道了~
2019美赛O奖论文-MCM2019A-1910246.pdf
1.该资源内容由用户上传,如若侵权请联系客服进行举报
2.虚拟产品一经售出概不退款(资源遇到问题,请及时私信上传者)
2.虚拟产品一经售出概不退款(资源遇到问题,请及时私信上传者)
版权申诉
0 下载量 25 浏览量
2024-03-17
21:35:25
上传
评论
收藏 6.45MB PDF 举报
温馨提示
试读
29页
美国大学生数学竞赛获奖论文,历届,单项文件,内容丰富,大学生数学,数学竞赛,参考资料,极具参考价值
资源推荐
资源详情
资源评论
For office use only
T1
T2
T3
T4
Team Control Number
1910246
Problem Chosen
A
For office use only
F1
F2
F3
F4
2019
MCM/ICM
Summary Sheet
Long May They Reign: How Majestic
Dragons Can Survive In The New Realm
Summary
The dragon has always been imagined by people in both East and West . If a dragon
lives in our world, it will be very meaningful and interesting to study its characteristics
and interaction with ecosystem.
First, the basic physical characteristics of the dragon are studied. According to the
growth characteristics of creatures, we establish the Dragon’s Weight Logistic Growth
Model. On the basis of this model, it is estimated that the weight of a mature dragon
is between 10 tons and 20 tons. Then, by using the mechanics knowledge and analogy
method , we analogize the trunk of the dragon to the elastic cylinder and estimate that the
length of the trunk of the mature dragon is about 4 meters to 6 meters and that the length
of the whole body of the mature dragon is about 20 meters to 35 meters. Secondly, we
analyze the daily energy intake and expenditures of the dragon. Comparing the dragon
with the creatures that have similar characteristics, we estimate the daily energy intake
and diet of the dragon, and the rate of energy consumption during its flight. In addition,
given the conditions of breathing fire, we establish the Dragon’s Fire Chemical Kinetics
Model so as to analyze the mechanism and energy consumption of breathing fire.Thirdly,
on the basis of the results above, we consider the minimum requirement of the habitat for
maintaining the dragon. In order to analyze the impact of the dragon on the ecosystem,
we establish the Dragon-Sheep-Habitat Model based on the Logistic model. More impor-
tantly, we compare the difference of the resources for maintaining a dragon in arid, warm
temperate and arctic regions, and analyze the living status of the dragon in these regions.
What’s more, in the sensitivity analysis, we focus on three factors, namely the inherent
growth rate of a dragon’s weight, environmental factors and environmental temperature,
in order to analyze their impacts on the weight of a mature dragon as they change. Finally,
we write a letter to George R.R. Martin, the author of A Song of Ice and Fire, to provide some
advice on maintaining the realistic ecological basis of the story. Although the dragon does
not exist in the real world, the models established in this paper can also be applied to solve
the problem about the growth of dinosaurs, the invasion of alien species and the design of
new flamethrowers.
Note: To avoid copyright infringement, we draw and generate all the figures on our own.
Keywords: Physical Characteristics; Energy Expenditures and Caloric In-
take; Dragon-Sheep-Habitat Model; Climate Impacts
Team # 1910246 Page 1 of 26
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1 Problem Restatement . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Problem Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Assumptions and Justifications . . . . . . . . . . . . . . . . . . . . . . . 4
3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Dragon’s Physical Characteristics and Diet . . . . . . . . . . . . . . . . . 5
4.1 Description of Dragon’s Weight and Length . . . . . . . . . . . . . . 5
4.1.1 Dragon’s Weight Logistic Growth Model . . . . . . . . . . . 6
4.1.2 Dragon’s Length Analysis Based on Analogy . . . . . . . . . 7
4.2 Dragon’s Energy Expenditure and Caloric Intake . . . . . . . . . . . 9
4.2.1 Dragon’s Diet and Energy Consumption during Flight . . . 9
4.2.2 Dragon’s Fire Chemical Kinetics Model . . . . . . . . . . . . 10
5 Dragon-Sheep-Habitat Model . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1 Analysis of Dragon’s Habitat Demand . . . . . . . . . . . . . . . . . 12
5.2 Dragon’s Impact on The Ecosystem: Balance or Collapse? . . . . . . 16
6 Impact of Three Regions on Dragon’s Survival . . . . . . . . . . . . . . . 16
6.1 Difference in Daily Energy Intake of Dragons in Three Regions . . . 16
6.2 Three Regions’ Capability about Self-Recovery and Supporting Drag-
ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.3 Minimum Resource and Habitat Demand for Dragon’s Survival in
Three Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.1 Impact of Inherent Growth Rate r on Dragon’s Weight . . . . . . . . 18
7.2 Impact of Environmental factor σ on Dragon’s Weight . . . . . . . . 19
7.3 Impact of Environmental Temperature T on Dragon’s Weight . . . 19
8 Model Evaluation and Further Discussion . . . . . . . . . . . . . . . . . 20
8.1 Evaluation of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.2 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9 A Letter to Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Team # 1910246 Page 2 of 26
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix A Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix B Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1 Introduction
1.1 Problem Restatement
Dragon, the most legendary behemoths from antiquity appeared in the phe-
nomenal TV series Game of Thrones adapted by George R.R. Martin’s best-seller
Song of Ice and Fire, which attracted eyeballs around the world. In the story, at the
very beginning of their birth, they were small and thin, on the other hand, they
were able to grow up at a rapid rate. Now, we assume that the three dragons in
the series live in the real world and their growth meets basic ecological princi-
ples. We need to build mathematical models to solve the following problems:
1. Analyze the physical characteristics of a dragon and estimate its daily en-
ergy expenditures and the minimum of calorie intake.
2. Investigate the impact of the dragon’s appearance on the ecological en-
vironment in the real world, and figure out the area that could sustain
the dragons’ life as well as the ecological conditions and the extent of the
biomes that satisfy the necessary living conditions of them.
3. Resembling other animals, the dragon would migrate to various places as
the local temperature changes. Therefore, we need evaluate the differences
in the vital requirement of them living in an arid region, a warm temperate
region, and the arctic region.
1.2 Problem Analysis
Since the dragon is a kind of beasts in fiction, for avoiding starting from
scratch, we ought to consider them by analogy to animals existing in reality.
We apply the related knowledge about ecology and biology etc. and the rule
of thumb to help us clarify the ecological characteristics of the dragon. To help
us conceive the image of the dragon better, we draw the conceptual graph, cf.
Figure 1. n.b. To avoid disputes caused by copyright law, all figures in this paper
are drawn, or generated by computer.
First of all, concerning the physical characteristics of the dragon, we focus on
the trend between their weight and length change in accordance of time. Con-
sidering the above change is continuous, we set up a logistic growth differential
equation, using the dragon’s birth weight and the weight next year as two initial
Team # 1910246 Page 3 of 26
Figure 1: The Hand Drawing Schematic Diagram of Dragon
values to obtain the weight function changing by time. Consequently, we could
estimate the approximate range of the mature dragon’s weight.
When analyzing the length change of the dragon, we assume that the dragon’s
limbs are on the ground when it is staying still, and regard it as an elastic cylin-
der. Based on mechanics, we can calculate the ratio of weight and length. Hence
according to the initial value of weight and length of the dragon’s birth, we can
finally confirm the scale coefficient and the functional relationship of length to
time. Considering that a dragon needs to maintain primary metabolism and life
activities, the more weight it holds, the more energy it consumes. We estimate
the daily energy consumption of a mature dragon according to an adult man. To
maintain its long-term life activities, the dragon’s daily energy intake must be
approximately equal to its consumption.
Based on the above analysis, we study the habitat area of the three dragons
and the conditions for the ecosystem and community size of the dragon. We
assume that the three dragons will not compete with each other so that we only
need to deal with the individual dragon entering a balanced ecosystem. Without
loss of generality, we consider the dragon as the top predator of the ecosystem,
regarding all animals as sheep and the area of the ecosystem as the habitat area
Team # 1910246 Page 4 of 26
of the dragon.
Then we let the number of sheep represent the supply of resources to the
dragon in the habitat, and assume that this value is proportional to the ecosys-
tem’s area. Considering the factors affecting the number of sheep, we establish
the Dragon-Sheep-Habitat Model based on differential equations and obtain a
curve of the number of sheep over time.In order to ensure the sustainable sup-
ply of the ecosystem to the dragon, the initial number of sheep should have a
minimum, otherwise the ecosystem will collapse due to the slow recovery.
To meet the minimum requirements for the dragon’s activities, we consider
this minimum as the community’s capability to support the dragon in the habitat
as the minimum standard for measuring the size of the ecosystem. Then, from
this minimum value, we can refer the minimum habitat area for maintaining the
dragon’s survival. Hence, based on the above results, we discuss the impact of
dragons on the ecosystem.
Considering the migration of dragons, the survival pressures of dragons in
varied climate regions are different. We focus on the three different climate re-
gions of the arid region, the warm temperate regions and the arctic region, to
analyze the effect of different regions have on the survival resources of mature
dragons. According to the above analysis of the daily energy intake of the dragon
and the Dragon-Sheep-Habitat Model, we discuss the minimum requirement of
resources for the dragon’s long-term survival based on the different climatic con-
ditions, the dragon’s consumption and the ecosystem’s supply difference, as well
as the impact of the dragon on the local ecosystem.
2 Assumptions and Justifications
By adequate analysis of the problem, to simplify our model, we make the
following well-justified assumptions.
1. The growth of dragons conforms to fundamental biological laws. If the
dragon lives in the real world, as a species on the earth, its growth and
maturation should be the same as other creatures, to meet the fundamental
biological laws.
2. The dragon is a homothermal animal whose body temperature is not af-
fected by the environment.
3. The dragon can fly and withstand huge wounds, and it will not easily die
due to physical, chemical and biological attacks. According to the mythol-
ogy and the TV series, we can consider that the dragon is with a capabil-
ity to fly, and its flight satisfies the aerodynamics. In the meantime, the
dragon is overgrowing, and its matured shape surpasses all the terrestrial
creatures, and any species is unable to bear the Dracarys from the dragon.
剩余28页未读,继续阅读
资源评论
阿拉伯梳子
- 粉丝: 1395
- 资源: 5503
上传资源 快速赚钱
- 我的内容管理 展开
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功